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From synaptogenesis to map formation - modeling visual system development exploring the contribution… Godfrey, Keith B. 2008

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FROM SYNAPTOGENESIS TO MAP FORMATION  -  MODELING VISUAL SYSTEM DEVELOPMENT EXPLORING THE CONTRIBUTION OF CELLULAR MECHANISMS TO THE EMERGENCE OF RETINOTOPIC PROJECTIONS AND EYE-SPECIFIC SEGREGATION  by KEITH B. GODFREY B.S., University of Alaska, 1992  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy in The Faculty of Graduate Studies (Neuroscience)  The University of British Columbia October 2008  © Keith B. Godfrey 2008  11  Abstract  The brain is composed of many anatomically distinct areas that control different functions. A common feature of these areas is that information is represented in a spatially organized manner. In the visual system, retinal representation is spatially mapped onto visual areas such that neighboring neurons respond to adjacent retinal locations, forming a retinotopic map. When axons from two retinas project to the same target structure, both produce similar retinotopic projections on the large scale but these segregate into eye-specific domains locally. How these spatial representations are formed is not well understood. Experimental studies have shown that many mechanisms are involved. Several modeling studies have addressed how such organization arises, with most representing different varying subsets of the mechanisms known to be present and showing how the particular representation of mechanisms can produce the emergent properties of organization. This results in models producing similar outputs yet coming to different conclusions that often cannot be rec onciled. By omitting behaviors that are present and likely to be involved in organization, such as spiking neurons and the dynamics of axon and synapse growth and retraction, the models are poorly constrained. This limits their explanatory and predictive scope regarding how organization develops, and further limits their ability to examine how the different mechanisms interact. To more accurately analyze both how such organization develops and the interactions between underlying mechanisms, a model of the developing retinocollicular pathway was produced that represented a wide range of cellular and subcellular phenomena, including spike-timing dependent plasticity (STDP), chemoaffinity, spontaneous retinal activity, trophic factors, and growth and re traction of synapses and axons. The model demonstrated retinotopic refinement and eye-specific segregation across a wide range of parameters and variations in implementation. Results indicated that the mechanisms necessary for organization were chemoaffinity, retinal waves, trophic factors and homeostatic controls. Analysis of the relative roles of activity and chemoaffinity suggested that these mechanisms play distinct and complementary roles. Among the predictions of the model are that smaller synapses produce more refined projections and, surprisingly, that STDP does not play a significant role in organization.  111  Contents Abstract  ii  Table of Contents  iii  List of Tables  viii  List of Figures  ix  Acknowledgements  xi  Dedication  xii  Chapter 1 Introduction 1.1 Neural organization 1.2 Biological mechanisms 1.2.1 Spontaneous neural activity 1.2.2 Chemoaffinity 1.2.3 Spiking neurons 1.2.4 Growth and trophic factors 1.2.5 Synaptic plasticity 1.2.6 Rapidly diffusible molecules 1.2.7 Homeostatic mechanisms 1.2.8 Axon and synapse growth and retraction 1.3 Computational modeling 1.4 Existing models 1.4.1 Chemoaffinity models 1.4.2 Correlated retinal activity 1.4.3 Synapse density, trophic factors and other approaches 1.5 Bringing it all together 1.6 Organization of the document  1 1 7 8 9 10 11 11 12 12 13 14 18 19 21 24 27 28  Chapter 2 Retinal Wave Behavior through Activity-Dependent Refractory Periods 2.1 Introduction 2.2 Methods  .  30 30 32  iv  2.3  2.4  37 39 39 42 42 48 48 49 50 50 50 54 54  2.2.1 Data analysis 2.2.2 Variable duration depolarization 2.2.3 RGC spike generation Results 2.3.1 Ferret Waves (P2—P4) 2.3.2 Reproduction of Wave Statistics in Different Species Rabbit Mouse Chick Turtle 2.3.3 Chaotic Behavior 2.3.4 Variable Duration Depolarization Discussion  Chapter 3 Large scale phenomenological modeling of retinocollicular development 3.1 Introduction 3.2 Model Overview 3.2.1 Conventions and mathematical notation 3.2.2 Extracellular diffusion 3.3 Computational methods and implementation  .  61 61 62 67 68 69  Chapter 4 Axon development 4.1 Axon model 4.1.1 Implementation: Charm and axon resources 4.1.2 Implementation: Axon growth and retraction 4.1.3 Implementation: Direction of axon growth 4.2 Results  72 72 76 77 79 82  Chapter 5 Synapse and neuron models 5.1 Synapse model 5.1.1 Implementation: Synapse formation 5.1.2 Implementation: Synapse survival and retraction 5.2 Neural model 5.2.1 Dendrite growth 5.2.2 Synapse vesicle release 5.2.3 Implementation: Neural model 5.2.4 Implementation: Firing rate estimation 5.2.5 Implementation: Vesicle release  85 85 88 89 91 92 93 93 95 96  Chapter 6 Additional component models 6.1 STDP 6.1.1 Implementation: STDP  97 97 100  V  6.2 6.3  104 106 108 109 110 111 ill 113 114 115 117 117 119  Molecular guidance cues 6.2.1 Implementation: Molecular guidance cues Growth and trophic factors 6.3.1 Growth factors 6.3.2 Trophic factors 6.3.3 Implementation: Growth factor release 6.3.4 Implementation: Trophic factor release NMDAR activation 6.4.1 Implementation: NMDAR activation Homeostatic controls 6.5.1 Implementation: Homeostatic firing rate control Rapidly diffusible molecules 6.6.1 Implementation: Nitric oxide release .  .  6.4 6.5 6.6  Chapter 7 Retinotopic mapping and eye-specific segregation 7.1 Introduction 7.2 Methods 7.2.1 Quantification of retinotopic refinement 7.3 Results 1D 7.3.1 Retinotopic organization and eye-specific segregation 7.3.2 Component analysis Spontaneous retinal activity (retinal waves) Chemoaffinity Spike-timing dependent plasticity Model neurons Nitric oxide release Synapse dynamics and trophic factor Diffusible growth factors NMDA receptor activation 7.4 Results 2D 7.4.1 Retinotopic organization and eye-specific segregation 7.4.2 Activity and chemoaffinity in retinotopic organization.  120 120 121 125 125 125 127 128 132 134 134 137 139 141 143 145 145 150  Chapter 8 Discussion 8.1 Model components 8.1.1 Retinal waves 8.1.2 Axon model 8.1.3 Molecular guidance cues 8.1.4 Synapse dynamics and trophic factors 8.1.5 Neuron model 8.1.6 STDP 8.1.7 NMDAR activation  155 156 157 159 161 163 165 167 169  -  .  -  vi  8.2  8.3  8.1.8 Growth factors 8.1.9 Nitric oxide 8.1.10 Homeostatic controls Analysis of model behaviors Justification of model design Complexity of model Eph3A knock-in experiments Comparison to other models 8.3.1 Local excitation and distal inhibition  170 171 172 173 173 174 176 179 180  Chapter 9 Summary 9.1 Accomplishments of the model 9.1.1 Retinotopic order 9.1.2 Retinotopic refinement 9.1.3 Eye-specific segregation 9.1.4 Model of retinal waves 9.1.5 Comprehensive modeling framework for studying retinocollicular devel opment 9.1.6 Examining the contribution of different physiological mechanisms to de velopmental organization 9.1.7 Examining the relative roles of retinal waves and chemoaffinity 9.1.8 Spatial representation of afferents 9.2 Experimental predictions 9.2.1 Retinal waves result from activity-dependent refractory periods 9.2.2 Developmental organization very tolerant to patterns of correlated retinal activity 9.2.3 Retinal waves and molecular guidance cues play complementary and dis tinct roles 9.2.4 Bias from molecular guidance cues is weak in visual brain areas in chicks and rodents, and larger animals 9.2.5 Retinal waves, molecular guidance cues, trophic factors, and homeostatic mechanisms are all required for organization 9.2.6 STDP not required for retinotopic refinement or eye-specific segregation 9.2.7 Small synapses produce more refined projections 9.2.8 Synapses use the equivalent of a resource-based mechanism for survival during development 9.2.9 Synapse survival is supported by other nearby synapses on the same axon 9.2.10 Cooperative synapse survival mechanism allow for alignment of connec tions to different cell types 9.2.11 Local excitation and distal inhibition not required for organization .  .  183 183 183 183 184 184 184 185 • 185 186 186 • 186 187 • 187 187 • 188 188 189 189 189 • 190 190  vii  9.3  9.2.12 Burst intensity is determining factor in spatial representation of afferents, not overall firing rate 9.2.13 The mechanism governing retinotopic organization and refinement is tol erant to perturbation Future directions  190 191 191  Bibliography  193  Appendix A Variables and parameters A. 1 Variable and parameters for equations in Chaps. 3-7 A.2 Parameter values for equations in Chaps. 3-7 A.3 Mathematical functions and symbols  214 214 216 217  viii  List of Tables 2.1 2.2 2.3  Spatiotemporal properties of retinal wave Spiking properties of RGCs Parameter sets that reproduce waves in different species  42 44 49  4.1  Parameters for axon growth  84  5.1 5.2  Parameters for synapse growth Neural model parameters  91 96  6.1 6.2 6.3 6.4 6.5  Parameters Parameters Parameters Parameters Parameters  7.1 7.2 7.3  Spatiotemporal properties of retinal wave Experimental perturbations and effect on retinotopic refinement Synapse resource sharing  used by STDP model for growth and trophic factor release and receipt for NMDAR activation for homeostatic control of firing rate for nitric oxide release  103 113 115 117 119 129 135 142  ix  List of Figures 1.1 1.2 1.3  Topographic organization Refinement of the retinocollicular projection Multiple RGC termination zones in EphA3 knock-in mice  3 6 20  2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11  Network topography Variations of simulation time step and retina size on wave properties Effects of smoothing behavior on waves RGC arrangement and excitation Examples of wave behavior History of waves passing through a single point Comparison between model and ferret wave data Distribution of wave initiation points Deterministic wave behavior Variable depolarization intervals Effects of parameter variations on wave properties  33 34 38 40 43 45 46 48 52 55 59  3.1 3.2 3.3 3.4  Developmental paradigm Axon description Schematic of 1D and 2D models Output of the function E(n,x)  63 64 66 68  4.1 4.2 4.3  Branch depth Axon growth Axon development  75 81 83  5.1  Synapse stability  87  6.1 6.2  Synaptic modification by spike interval Relative positioning in retina and colliculus  98 107  7.1 7.2 7.3 7.4 7.5  1D and 2D model schematics 1D retinotopic projection and measuring retinotopic refinement Eye-specific segregation Effects from variation of retinal activity Chemoaffinity experiments  122 126 127 130 133  .  .  x 7.6 7.7 7.8 7.9 7.10 7.11 7.12  Changes to binocular projection from changes, in synapse strength Synapse resource sharing and NO upregulation Development of retinotopic projection in group of RGC axons Accuracy of retinotopic projection Development of retinotopic projection in a group of RGC axons Eye-specific segregation 2D EphA upregulation  138 142 147 148 149 151 154  8.1 8.2  Synapse weight distribution Arbor size without correlated activity  168 177  -  xi Acknowledgments  First and foremost, I wish to thank Miriam Hebé, for her unconditional support and love, and for motivating me to achieve my goals. I would also like to thank Nick Swindale for providing me the opportunity to pursue a PhD, and for the financial support, academic freedom, encouragement to pursue my interests, and instruction in academic writing. Thanks also go out to the many present and former people around the lab, especially Tim Blanche, who gave me a crash course in physiological experimentation and data analysis, guidance, and advice on navigating the university bureaucracy; as well as Robert Douglas and Martin Spacek, for always providing a helping hand. I’d also like to thank Stephen Eglen and Evelyne Semagor for help they have provided me in understanding retinal data. Finally, I would like to thank Rob and Leslie Liston, who have been like family and have helped us immeasurably since we arrived in Vancouver.  xii  for Miriam Hebé, and for my grandparents  1  Chapter 1 Introduction “The basic concepts and laws which are not logically further reducible constitute the indispensable and not rationally reducible part of the theory. It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” Albert Einstein  1.1  Neural organization  For 200 years it has been thought that different parts of the brain were responsible for different functions. Around 1800, Franz Joseph Gall hypothesized that different sections of the brain were responsible for specific behaviors and that bumps on the skull indicated the strength of a particu lar mental function. This field of study, which came to be known as phrenology, was discounted by science. However, the idea that the brain could be partitioned into functionally distinct areas proved to be very correct. One hundred years later, Korbinian Brodmann was the first to iden tify 52 anatomically distinct areas of the human cortex. Subsequent research showed that these areas controlled specific brain functions, areas that are now known as the visual cortex, motor cortex, sensory cortex and auditory cortex, as well as brain areas important for understanding and producing speech, such as Broca’s and Wernicke’s areas.  1.1. NEURAL ORGANIZATION  2  After areas were identified as having specific functions, the spatial organization of these areas was explored. Studies in the sensory cortex showed that tactile stimulus to different parts of the body elicited responses in distinct subregions of the cortex (Adrian, 1941). Similarly, electrical stimu lation of distinct areas in the motor cortex caused muscle responses in different parts of the body (Penfield and Rasmussen, 1950). Mapping out these patterns of stimulus and response showed that both sensory and motor cortices have distorted but largely continuous mapping of the entire body (Fig. 1. 1A, B). In a similar manner, the primary visual cortex displays a continuous representation of the visual space as represented on the retina (Fig. 1. 1D). This spatial organization, where the cortex has a spatial and ordered representation of the outside world, is commonly called a topo graphical mapping. A topographical mapping in the cortex is characterized by a neighborhood of neurons in the brain mapping to continuous series of points in physical or sensory space. It is very similar to the mathematical notion of topological projections, with one space (e.g. retina) producing a largely continuous projection into another (e.g. visual cortex). Topographical mappings are a common feature of the brain and are observed in the cortex, thala mus and brainstem. In vision, these mappings are called retinotopic projections, with the retina having a continuous though distorted projection onto different brain areas. The superior colliculus (optic tectum), lateral geniculate nucleus, primary visual cortex and higher cortical areas associated with vision all have this type of retinotopic organization. An unresolved question is how such or ganization occurs. For example, how can neurons from any given portion of the retina predictably and reliably project to the same topologically correct locations in their target areas of the brain, and how can this behavior be consistent across an entire species? As a solution to this question, Sperry (1963) hypothesized that there were gradients of chemical markers on neurons that axons could read out, providing a “latitude and longitude” by which they could navigate to their correct termination zones. One experimental system for examining this organization, and the possible roles for chemical markers, has been in the retinotectal projection. Experiments in frogs and fish were performed where parts of the retina andlor tectum were removed andlor rotated and the resulting patterns of connectivity observed (reviewed in Prestige and Willshaw, 1975; Udin and Fawcett, 1988). Results from these experiments supported a role for chemical labels to guide axons to their approximate  1.1. NEURAL ORGANIZATION  3  A  B  Medial  Lateral  Medial  Lateral  Fig. 1.1. Topographic organization Cartoon representation of the body in both sensory (A) and motor cortices (B). Image adapted from Penfield and Rasmussen (1950); Amaral (2000). The entire body surface is represented in an orderly and largely continuous fashion. Representation of parts of the body is not proportional to size but to the sensory innervation from the part, where the hand has a larger representation, and more sensory innervation, than for example the legs. The motor cortex is similarly organized, with spatial representation indicating the amount of motor control of each part. C, D. Deoxyglucose autoradiograph of primary visual cortex, D, showing the projection of the stimulus shown in C (figure adapted from Tootell et al., 1988; Swindale, 1996). The retinal projection is mapped to the cortex such that neighboring points in the retina map to neighboring parts of the cortex. The black and white stripes reflect ocular dominance columns, not the black and white patterns in C. Similar to the representation of the sensory and motor cortex, the projection is distorted. The retinotopic projection closely approximates a mathematical topological log-polar mapping, projecting neighboring points in the retina map onto neighboring parts of the cortex and preserving the spatial relationship of nearby points (i.e. angles are conserved in the projection; Schwartz, 1980). The fovea receives a much larger representation than peripheral areas. The stimulus was presented monocularly and cortex was flattened before sectioning. Black and white pattern in stimulus was contrast reversed at 3 Hz. Labels indicate regions of the visual field: S = superior visual field; F=fovea; 1= inferior visual field; H=horizontal meridian. The semicircles in A (labeled 1, 2, and 3) are positioned at 10, 2.3° and 5.4° eccentric to the fovea, respectively.  1.1. NEURAL ORGANIZATION  4  termination zones in the tectum. For example, if part of the tectum was removed, rotated and reimplanted, retinal axons altered their projections and continued to project to similar groups of cells in the tectum, regardless of the degree of rotation (Yoon, 1975). Other experiments showed that, in some species, removal of half of the tectum and/or half of the retina results in the remaining portion of the retina projecting to the remaining portion of the tectum (Udin and Fawcett, 1988). This behavior indicated that, while chemical labels provided guidance, the targeting they provided was not absolute and that other forces were at play. More recently, one family of molecules responsible for these chemical labels has been identified. Supporting the molecular gradient hypothesis (Sperry, 1963), orthogonal gradients of ephrins and their Eph receptors have been found in the retina as well as both tectum and colliculus. Along the anterior-posterior axis of the tectum (colliculus), both ephrin-A and EphA are expressed in coun tering gradients, and axons from retinal ganglion cells (RGCs) exhibit a similar pair of gradients based on the position of the RGC on the nasal-temporal axis. Similarly, the lateral-medial axis of the tectum (colliculus) expresses countering gradients of ephrin-B and EphB, gradients which  are also expressed by axons from RGCs based on their location on the dorsal-ventral axis (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005). Thus, axons originating from cells along each axis of the retina have markers that correspond directly to those expressed on neurons along each axis of the colliculus, providing a clear framework for the development of retinotopic organization. Such gradients of molecules have been found in many parts of the brain, including the auditory system (Ellsworth et al., 2005; Siddiqui and Cramer, 2005), thalamocortical projections (Dufour et al., 2003), sensory cortex (Vanderhaeghen et a!., 2000), hippocampus, motor areas and spinal cord (Flanagan and Vanderhaeghen, 1998), indicating that patterned molecular gradients are a general phenomena used by the brain. Despite this strong match between theory and experimental findings, other experiments made it clear that part of the puzzle was still missing. While molecular markers were clearly involved in driving retinotopic organization, retinal activity was also shown to be required for the devel opment of connections. For example, in eye rotation experiments in Xenopus larvae, when the eye was rotated 180°, retinal axons connected to their same tectal location, forming an inverted map in the tectum (see Yoon, 1975). Intertectal connections in such animals were also shown  1.1. NEURAL ORGANIZATION  5  to reorganize. This reorganization of intertectal axons was prevented by blocking retinal activity (Keating and Feldman, 1975). A more recent study in mouse colliculus has shown that retinal axons arborize diffusely throughout the colliculus and that retinal activity is required for these arborizations to refine into focused, adult-like projections. However, it was not simply retinal ac tivity that was required, but rather correlated patterns of such activity (McLaughlin et al., 2003). During retinal development, and before the retina is responsive to light, patterns of RGC bursting  are observed, where spatially localized groups of RGCs synchronously burst and the pattern of bursting propagates across finite portions of the retina (Maffei and Galli-Resta, 1990; Meister et aL, 1991; Wong et a!., 1993). This activity, known as retinal waves, is characterized by infrequent and non-repeating patterns of activity (Feller et al., 1996, 1997; Semagor et aL, 2003; Syed et al., 2004) and is required for refinement of the retinotopic projection (Sretevan et al., 1988; Thompson and Holt, 1989; Penn et al., 1998; Wong, 1999; McLaughlin et al., 2003; Ruthazer and Cline, 2004; Chandrasekaran et a!., 2005; Torborg and Feller, 2005; Cang et al., 2008) (Fig. 1.2). Retinotopic organization and refinement is not the only form of spatial organization observed. When a colliculus or tectum is innervated by a single retina, a smooth and continuous retinotopic mapping is produced. When the colliculus or tectum is innervated by two retinas, however, the axons from each retina are still mapped retinotopically but segregate into distinct sections of the target structure. This can be observed as a result of experimental perturbations, such as the “three eyed frog” (Law and Constantine-Paton, 1981) or from unilateral tectal ablation (Ruthazer et al., 2003), resulting in the axons of each retina innervating discrete sections of the tectum. The lateral geniculate nucleus (LGN), which is normally innervated by two retinas, has very characteristic patterns of eye-specific segregation (Rakic, 1976; Linden et al., 1981; Shatz, 1983; Sretevan and Shatz, 1986; Penn et al., 1998) as does the primary visual cortex, even though it receives innerva tion only indirectly via the LGN (Hubel and Wiesel, 1963, 1968, 1977). Segregation is also seen in ferrets whose RGC axons have been re-routed to innervate the medial geniculate nucleus (MGN), which normally processes auditory information. Axons from each retina segregate into distinct and neighboring clusters in the MGN (Angelucci et al., 1997). Eye-specific segregation can also be observed naturally, such as in colliculi that are binocularly innervated, as in the hamster. In such animals, the contralateral eye produces a retinotopic projection across most of the colliculus, with axons from the ipsilateral eye projecting to discrete clumps of “puffs” in the colliculus (Graybiel,  1.1. NEURAL ORGANIZATION  6  A  B  C  Fig. 1.2. Refinement of the retinocollicular projection A. Normal development of the retinocollicular projection between postnatal day 1 (P1) and P8. All RGC axons originating from a spot in the retina are visible after focal injection of a fluorescent tracer in the temporal retina. At P1,  axons extend across the entire A-P axis and are distributed across the L-M axis. Retinotopically correct termination zone shown by circle. By P4, many axons have eliminated overshoot and have branched preferentially in the area of the retinotopically correct termination zone. By P8, a dense, focal termination zone is visible and resembles the mature form. All axon arbors extending outside of the termination zone have been eliminated. B. Early development of the retinocollicular projection in mice genetically engineered to lack a subunit of the acetylcholine receptor ($2—!—) that is necessary for retinal wave activity. The axons show a weak bias for directing to the retinotopically correct termination zone (circle) but refinement is strongly inhibited. Inset in lower right is outline of retina. Spot indicates location of focal tracer injection. Right side of retina indicates temporal side. C. Retinotopic refinement at P20. Retinal waves resume in $2—!— mice by PlO, but the projection fails to refine into a single, dense cluster, indicating the presence of a critical period. Axons, branches and arbors are found in aberrant locations (arrow). Scale bar is 250 tm. Figure adapted from McLaughlin et al. (2003).  1.2. BIOLOGICAL MECHANISMS  7  1975; Thompson and bit, 1989). It is generally believed that the source of such segregation is the independent patterns of retinal waves that occur in each retina. While there is a high degree of correlation between nearby RGCs in the same retina (Wong et al., 1993), there is thought to be no correlation in activity between eyes, as retinal waves are generated independently in each retina and RGCs at any given location in a retina burst very infrequently. As a result, neighboring RGCs, whose activity is strongly correlated, have a bias to connect to similar sets of target neurons (“cells that fire together, wire together”) while RGCs from different retinas, whose activity is uncorrelated, project to different sets of target neurons. The majority of evidence indicates that correlated retinal activity is required for retinal axons to segregate in these target structures (Shatz and Stryker, 1988; Sretevan et al., 1988; Thompson and bolt, 1989; Penn et al., 1998; Grubb et al., 2003), although this conclusion is not universally accepted (Crowley and Katz, 2002; Chalupa, 2007). It thus appears that both molecular markers and correlated patterns of retinal activity play an im portant role in organization, both for development of the retinotopic projection and for segregation of retinal axons into eye-specific domains. As discussed below, several other biological mecha nisms are also involved in development, and the role of all of these mechanisms and how they interact, is not understood.  1.2  Biological mechanisms  In the developing brain there are a wide variety of biological mechanisms active that are present during and contribute to retinotopic organization and eye-specific segregation. A brief description of many of these is provided below. This includes the aforementioned phenomena of sponta neous retinal activity (Sec. 1.2.1) and chemoaffinity (Sec. 1.2.2), as well as other physiological mechanisms known be directly involved in retinotopic organization and eye-specific segregation, including trophic feedback (Katz and Shatz, 1996; Cabelli et al., 1997; Elliott and Shadbolt, 1999; Sec 1.2.4) and nitric oxide (Cramer et al., 1996; Sec. 1.2.6). Several other physiological behaviors 1  This quote is commonly attributed to Carla Shatz but its origin could not be confirmed  1.2. BIOLOGICAL MECHANISMS  8  are known to be present but whose importance is not fully understood are also addressed, such  as spiking neurons (Sec. 1.2.3), spike-timing dependent plasticity (Bi and Poo, 1998; Zhang et al., 1998; Sec. 1.2.5), homeostatic controls (Turrigiano, 1999; Turrigiano and Nelson, 2004; Sec. 1.2.7), growth factors (Sec. 1.2.4), and the growth and retraction of both synapses and axons (Sec. 1.2.8). In addition to contributing to organization in the visual system, many of these mechanisms are found throughout the brain, indicating that their importance is widespread. The wide range of physiological behaviors involved shows that neural development is a complex processes involving many different phenomena.  1.2.1  Spontaneous neural activity  Neural activity plays an active role in organization of the developing nervous system (Shatz, 1996; Torborg and Feller, 2005; Cang et al., 2005; Pfeiffenberger et al., 2006). Early in devel opment, when initial sets of connections between neurons are being formed and before the onset of stimulus-driven activity, neural circuits are spontaneously active in many parts of the developing nervous system, including the auditory system (Gummer and Mark, 1994; Lippe, 1994), neocortex (Yuste et al., 1992), hippocampus (Garashuk et al., 1998), spinal networks (O’Donovan and Chub, 1997; Spitzer and Gu, 1997), brainstem nuclei (Ho and Waite, 1999) and retina (Maffei and Galli Resta, 1990; Meister et al., 1991). In the retina, this activity is called “retinal waves” as it takes the form of waves of activity in retinal ganglion cells (RGCs) that slowly propagate over portions of the retina. These locally correlated patterns of spontaneous activity are thought to drive activ ity dependent synaptic remodeling, refining circuits from the brain’s initial “best guess” of neural connectivity (Katz and Shatz, 1996). Manipulations of retinal wave behavior affect the refinement of retinotopic projections in the superior colliculus (McLaughlin et al., 2003) (Fig. 1.2) and LGN (Rossi et al., 2001; Sretevan et a!., 1988), alter the size of the layers in the LGN that represent each eye (Penn et al., 1998; Stellwagen and Shatz, 2002) and affect eye-specific segregation in both LGN (Huberman et al., 2002, 2005) and Vi (Stryker and Harris, 1986). Manipulations of visu ally driven activity also affect the form of eye-specific segregation, where monocular deprivation causes a reduced representation by the deprived eye in Vi (Hubel et al., 1977), strabismus sharp ens eye-specific segregation in Vi (Adams and Horton, 2006), and synchronous activity between  1.2. BIOLOGICAL MECHANISMS  9  eyes interferes with eye-specific segregation (Crair, 1999). Under experimental conditions in frog tectum, when RGC axons from two eyes are forced to innervate a single tectal lobe, the axons segregate into eye-specific bands (Ide et al., 1983; Law and Constantine-Paton, 1981; Ruthazer et al., 2003) showing that activity is sufficient to drive eye-specific segregation and molecular labels are not required. Spontaneous patterned activity in the retina is a common developmental behavior and has been observed in many species, including rat (Maffei and Galli-Resta, 1990), ferret (Meister et al., 1991; Wong et al., 1993; Feller et al., 1996), mouse (Singer et al., 2001; Demas et a!., 2003), cat (Meister et al., 1991), turtle (Sernagor et al., 2003), rabbit (Zhou, 1998; Syed et al., 2004) and chick (Catsicas et a!., 1998; Wong et al., 1998; Sernagor et al., 2000).  1.2.2  Chemoaffinity  Though correlated activity is required for axon arbor refinement and eye-specific segregation, a coarse retinotopic projection is achieved in the absence of correlated activity (McLaughlin et al., 2003). The mechanism behind this organization is summarized by Sperry’s chemoaffinity hypoth esis (Sperry, 1963) which proposes that brain areas have chemical markers that allow a neuron to determine its “latitude and longitude” and guide its connectivity accordingly. The Eph family tyrosine kinase receptors and their associated ligands, the ephrins, are one group of chemical mark ers that fit this description. Gradients of these markers are found throughout the brain and body, including the auditory system (Ellsworth et al., 2005; Siddiqui and Cramer, 2005), thalamocorti cal projections (Dufour et al., 2003), sensory cortex (Vanderhaeghen et al., 2000), visual system, hippocampus, motor areas, spinal cord (Flanagan and Vanderhaeghen, 1998) and even the gluteus maximus (Lampa et al., 2004). In the visual system, these receptors and ligands are expressed in RGC axons and in the bodies and dendrites of cells in the SC/OT (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005), LGN (Huberman et al., 2005) and Vl (Cang et a!., 2005). These studies as well as others (Feidheim et al., 1998; Pfeiffenberger et al., 2006) show that disruption of this expression impairs topographic organization. Results by Crowley and Katz (1999; 2000), who observed ocular dominance segregation in ferret visual cortex despite removal of both retina at an early age, suggests that molecular cues are responsible for forming ocular dominance columns  1.2. BIOLOGICAL MECHANISMS  10  in VI, although their search for eye-specific markers that would mediate such segregation in the LGN has not so far been successful (Kawasaki et al., 2004). Given the clear ability of activity to accomplish this role (Law and Constantine-Paton, 1981; Ruthazer and Cline, 2004), plus obser vations that blocking retinal activity blocks OD formation (Stryker and Harris, 1986), what drives eye-specific segregation remains an open question, but evidence suggests that it is activity-based and not the result of different molecular cues.  1.2.3  Spiking neurons  Neurons communicate through discrete signals called action potentials, or “spikes”. When a neu ron spikes, a wave of depolarization is produced that procedes down the axon. When this depo larization reaches the presynaptic terminal, it causes the release of neurotransmitter vesicles (Süd hof, 2004). This behavior is common to most known neuron types, and to all neurons that form synapses with cells outside of their immediate vicinity. From an information theoretic perspective, the timing of individual spikes can contain significantly more information than the average firing rate as there exist many distinct spike trains for any given firing rate, whether from spike trains differing by the phase of individual spikes, the temporal pattern of the spikes, or their bursting characteristics. It seems unlikely that the brain would not take advantage of such a mechanism and evidence suggests that neurons do use the timing of individual spikes. For example, while the re sponse in the neuromuscular junction may be proportional to the average firing rate of innervating axons, motor neurons require independent spiking patterns to achieve synaptic segregation onto individual muscle fibers during synapse development (Busetto et al., 2000). Other evidence of the importance of the timing of individual spikes include their role in synaptic plasticity (Bi and Poo, 1998; Zhang et al., 1998), perceptual grouping (Singer, 1999a) and olfactory sensation (Stopfer et al., 1997). Also, spiking with millisecond level precision is observed in the visual (Dan et al., 1998) and auditory systems (see Singer, 1999b), suggesting a strong importance for such behavior (but see Shadlen and Newsome, 1994).  1.2. BIOLOGICAL MECHANISMS  1.2.4  11  Growth and trophic factors  One important contributor to neural development is a group of molecules called neurotrophins, which includes nerve growth factor (NGF) and brain derived neurotrophic factor (BDNF) (see McAllister et a!., 1999; Vicario-Abejón et al., 2002; Cohen-Cory and Lom, 2004, for reviews). Neurotrophins modulate neural excitability (Du and Poo, 2004), growth in axons (Cohen-Cory and Fraser, 1995) and dendrites (McAllister et al., 1995), synaptic connectivity (Cohen-Cory and Lom, 2004), stabilization of axon arbors and synapses (Hu et al., 2005), and possibly influence the production of new synapses (Alsina et al., 2001; Poo, 2001). Exogenous application of NGF and BDNF to the visual cortex is able to prevent the effects of monocular deprivation (Lodovichi et al., 2000), and both excessive (Cabelli et al., 1995) and blocked (Cabelli et al., 1997) BDNF prevents OD formation. Several behaviors of BDNF are activity dependent, including synapse potentiation (Kovalchuk et al., 2002) and modulation of dendrite growth (McAllister et al., 1996). It is quite possible that the patterns of synaptic connectivity result from activity dependent responses to NT presence (Katz and Shatz, 1996; Poo, 2001). Additional groups of molecules also operate as growth factors that are able to attract and repel axons (McFarlane and Holt, 1997), indicating that such mechanisms are important physiologically.  1.2.5  Synaptic plasticity  Since the time when Donald Hebb theorized that the timing of activity of two neurons relative to each other altered the efficacy of connections between them (Hebb, 1949), researchers have been investigating synaptic plasticity. Modem research has shown that plasticity is dependent on the timing of individual spikes between pre- and postsynaptic cells (Markram et al., 1997; Zhang et al., 1998; Froemke and Dan, 2002). This phenomena, known as spike-timing dependent plasticity (STDP), produces long term potentiation/depression (LTPILTD) when a presynaptic spike imme diately preceeds/succeeds a postsynaptic spike (Dan and Poo, 2006) and is thought to contribute to synaptic stabilization and retraction and mediate circuit refinement (Torborg and Feller, 2005; Dan and Poo, 2006). STDP has been shown to occur through natural visual stimuli (Dan and Poo, 2006). LTP and some forms of LTD require activation of the NMDA receptor (Torborg and  1.2. BIOLOGICAL MECHANISMS  12  Feller, 2005) which is thought to organize topography by acting as a coincidence detector (Debski and Clime, 2002), and stabilize synapses, possibly through the action of BDNF or other retrograde messengers (Schmidt, 2004). BDNF is required for induction of LTP (Korte et al., 1996; Poo, 2001), indicating a linkage between synaptic plasticity and trophic feedback. Another feature of STDP is that it is saturating, meaning that a finite amount of plasticity is realized for a spike pairing with a given interval (Zhang et al., 1998; Froemke et al., 2006).  1.2.6  Rapidly diffusible molecules  Another activity dependent mechanism present in development is the release of rapidly diffusible molecules such as nitric oxide (NO), a diffusible membrane permeable molecule released from the postsynaptic terminal (Fitzsimonds and Poo, 1998) that modulates axon arbor remodeling through increased branching (Cogen and Cohen-Cory, 2000). It partially mediates organization in the retinogeniculate and retinocollicular pathways (Mize and Lo, 2000) and it induces RGC growth cone collapse, a phenomenon prevented by the presence of BDNF (Ernst et al., 2000). NO is released by NMDA receptor activation and it triggers various presynaptic second messenger path ways, including cyclic GMP. NO is required for some types of LTP/LTD and its absence disrupts some neural segregation and can delay retinotopic organization and eye-specific segregation (Mize and Lo, 2000; Schuman and Madison, l994a). Blocking NO prevents spatial ON/OFF segregation in ferret LGN (Cramer et al., 1996). Another such molecule is arachidonic acid (AA), which acts as a retrograde messenger and mediates the growth promoting pathway in RGC axons (Schmidt, 2004).  1.2.7  Homeostatic mechanisms  There are several mechanisms used by neurons to regulate their behaviors. Some of these are truly homeostatic, such as where a neuron adjusts its excitability or level of innervation to maintain a set level of activity, while others are the result of compensatory responses to maintain a behavior despite variations in underlying mechanisms (see Davis, 2006), or compensatory responses that seek to restore a set behavior or level of activity, even if that behavior or level of activity is not  1.2. BIOLOGICAL MECHANISMS  13  fully reached. For simplicity, “homeostatic” as used here will refer to all of these mechanisms, as they each offset disruptive changes to maintain a stable neural behavior. Neurons homeostatically control their firing rate. In response to perturbations in firing rate, neurons respond by altering their intrinsic firing properties (MacLean et a!., 2003; Zhang and Linden, 2003) or scaling the strength of synaptic input (Turrigiano and Nelson, 2004; Davis, 2006) to compensate for such perturbations. Increases in synaptic strength are observed in the neuromuscular junction within 10 minutes of blockade of glutamate receptors (Frank et al., 2006) and changes to synapse strength are observed in central synapses, as a result of transcription changes, within hours (Ibata et al., 2008). Heterosynaptic LTD is observed in synapses on neurons following the induction of LTP (see Dan and Poo, 2006), helping balance total excitation to the cell. Homeostatic mechanisms also occur on a larger scale. In mice where early retinal wave behavior is disrupted, preventing the proper refinement of RGC projections, each collicular neuron conserves its total input by forming connections with a larger number of RGCs than normal (Chandrasekaran et al., 2007). While there are relatively limited examples of homeostatic mechanisms measured experimentally, from a practical perspective, the complexity of each neuron, and the connectivity found in the brain, strongly argues for homeostatic controls or similar regulatory mechanisms to be widespread. Physiological development is prone to variations from several sources, including genetics, the environment or mutations. Homeostatic controls are much better able to compensate or adapt to disrupting changes than fixed or threshold based mechanisms.  1.2.8  Axon and synapse growth and retraction  The retinocollicular projection is composed of RGC axons projecting to the colliculus, and what defines the functional and spatial behavior of this projection is the distribution of synapses on each axon. Each synapse must form based on information available to a particular section of axon and must retract based on information available to the synapse. An understanding of how the retinocollicular projection develops and refines must take into account the perspective of individual axon segments and synapses.  1.3. COMPUTATIONAL MODELING  14  Axon and synapse growth are influenced by several factors. For example, neurotrophins modulate growth and stabilization of axons (Cohen-Cory and Fraser, 1995; Hu et al., 2005) and growth and connectivity in synapses (Alsina et al., 2001; Poo, 2001; Cohen-Cory, 2002; Goda and Davis, 2003; Cohen-Cory and Lom, 2004). The diffusible factors nitric oxide and arachidonic acid influence synapse and axon growth and stabilization (Fitzsimonds and Poo, 1998; Cogen and Cohen-Cory, 2000; Schmidt, 2004). Axon branching is also associated with the presence of synapses (Alsina et a!., 2001).  1.3  Computational modeling  The several physiological mechanisms present during development, and the numerous ways they interact, makes it difficult to develop an intuitive understanding of how all of these behaviors fit together. A technique that is commonly used to investigate complex or incompletely understood phenomena, such as developmental organization in the visual system, is the use of computational models. Computational models are simplified representations of observed phenomena that are used to gain understanding and to predict characteristics about the phenomena. As discussed in Sec. 1.4, the last several decades have seen the production of many such models that investigate organization in the visual system. These models vary in detail and in the behaviors and mechanisms they address. Before discussing these, however, it might help to look at computational modeling in more general terms, and categorizing different models based on the level of detail they incorporate. As defined here, models with a narrow descriptive scope attempt to accurately describe the behav ior of a an experimentally observed phenomenon using as simple of a representation as possible. Narrow-scope models are often used to reproduce the phenomenological behaviors of experimental observations and, while they may be accurate at describing observed phenomena, they are generally not able to provide significant insight into the mechanisms underlying the phenomena. Newton’s law of gravity is a good example of such a model. Using a mathematical formulation very re moved from many of the physical details of matter, it is able to describe the gravitational influence of two or more objects to high degrees of accuracy and it makes no underlying assumptions (or predictions) about how the characteristics of matter, such as chemical composition or subatomic  1.3. COMPUTATIONAL MODELING  15  properties, result in this behavior. There are several narrow-scope models of the organizational  properties of the visual system. At the opposite end of the spectrum that is considered here are wide-scope models, whose descrip tive scope includes not just an experimentally observed phenomenon but also the mechanisms that are known (or believed) to underlie the phenomenon. This type of model seeks not only to de scribe a phenomenon but also to describe, predict andlor analyze the role of causative elements of the modeled phenomenon. Large scale climate modeling is a good example of a wide-scope model. Such models include many of the physical phenomena that are thought to influence weather, such as ocean currents, cloud cover, air temperature, ocean heat storage, sea ice, solar radiation intensity or atmospheric CO 2 (see Washington and Parkinson, 2005, for an example of climate modeling). This modeling approach does not live up to the concept of Occam’s razor of using minimal com plexity to describe a behavior as it seeks complexity in order to produce a more realistic, and hopefully more accurate, modeling environment. Or maybe it does use the razor. Climate is a complex phenomenon that is influenced by many underlying mechanisms that contribute to its be havior. Until we understand which of these mechanisms are truly important, and the roles of the many mechanisms, both in relation to influencing overall climate and in relation to each other, it will be difficult to produce a simpler model that adequately represents climate, if indeed such a simpler model is even possible. Incorporating phenomenological approximations of the behaviors of ocean currents, cloud cover, air temperature, etc., in a climate model helps to examine the im portance and interplay between each of these behaviors. If only a wide-scope model is able to adequately represent climate, this wide-scope model uses minimal complexity to describe the be havior, as a more minimal representation has not (yet) been found, thus observing Occam’ s razor. The developing retinocollicular pathway is arguably another complex system that can be addressed using wide-scope models. Unfortunately, this type of wide-scope model has not yet been used to study retinotopic organization and eye-specific segregation and the mechanisms underlying these phenomena. Most models of visual system development have an intermediate scope and seek to reproduce ex perimentally observed phenomena, such as retinotopic organization and eye-specific segregation, and at the same time represent some of the underlying mechanisms of this phenomenon. The ad-  1.3. COMPUTATIONAL MODELING  16  ditional constraints provided by incorporation of some underlying mechanisms can improve the ability of the model to describe the phenomenon in question. As discussed in Sec. 1.4, many models of developmental organization in the visual system are of this category. These models can accurately reproduce observed patterns of organization as observed in the superior colliculus, LGN and visual cortex, and can form predictions about the characteristics of such organization. Incorporation of some underlying mechanisms, such as those discussed in Sec. 1.2, could poten tially improve the accuracy of these models and their predictions. However, care must be taken when considering the results of such a model because of the implicit, and sometimes explicit, pre dictions about the mechanisms underlying the phenomenon being modeled. Unlike wide-scope models, intermediate scope models seek to explain phenomenological observations using only a subset of the mechanisms known to be present. This is arguably analogous to modeling climate by representing ocean currents, cloud cover and sea ice but omitting solar radiation, atmospheric CO 2 and air temperature. If such a model is still able to reproduce phenomenological observations, the omissions mean that either the unrepresented mechanisms do not contribute to the phenomenon in question, or that the mechanisms present in the model were not represented in an accurate way and are thus able to compensate for those that were omitted. Any resulting predictions about these mechanisms, or the ones not represented, must therefore be scrutinized, based on the implicit in accuracies of representing only a subset of known behaviors. The same conclusions hold true for modeling in the retinocollicular pathway. While many existing intermediate-scope models are able to generate retinotopic organization and eye-specific segrega tion, these models fail to represent many of the behaviors that are known to be present, such as axon growth, spiking neurons and synapse formation and retraction. It follows that the represen tation of the underlying mechanisms in these models is likely to be wrong and thus little weight can be given to the predictions about these underlying mechanisms. This analysis is not meant to suggest that intermediate-scope models do not provide interesting and insightful ideas about how the mechanisms underlying a modeled phenomenon might contribute, only that the inherent inac curacies in such models draw into question their ability to make predictions about how represented (and unrepresented) mechanisms contribute. Only by incorporating all of these mechanisms in a wide-scope model, where the role of each is represented and the interactions between the mecha nisms is also considered, can more reliable predictions be made about the behaviors and roles of  1.3. COMPUTATIONAL MODELING  17  the underlying mechanisms. On the surface, the task of producing a wide-scope model of the developing retinocollicular path way seems unrealistic and unconstrainable, as each of the underlying phenomena are governed by a myriad of factors, including varying chemical processes and gene expressions, and many functions vary between species as well as between different brain areas in the same species. Fortunately, reproduction of the exact details of each of these phenomena does not appear to be necessary. While certain biological mechanisms and/or components are highly conserved, such as synapse vesicle proteins (Ferro-Novick and Jahn, 1994), many phenomena are achieved through multiple biochemical pathways that vary between species and even in the same species at different stages of development. For example, retinal waves are ubiquitous across the mammals, birds and turtles studied, and the patterns of activity are very similar. However, the underlying chemical processes that govern these waves vary, both between species and across different development stages in the same animal, with waves being mediated by different neurotransmitters as development progresses (Wong, 1999; Syed et al., 2004). It is possible that exact wave properties (i.e. frequency, size, ve locity, duration) as seen in a given species/age are required for development in that species, but this seems unlikely given the high degree of variability between different animals of similar age in the same species (Feller et al., 1997). It thus appears that it is the general phenomenological behavior of waves that is important, not the mechanisms underlying their generation. Similarly, Eph receptors are either repulsive or attractive and different combinations of receptors/ligands are expressed in the equivalent brain areas of different species (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005). Further, the particular expression of receptor/ligand pairs changes between different brain areas within the same animal (Feidheim et al., 1998), and retinotopic organiza tion occurs despite perturbed expression through genetic introduction of additional molecules (e.g. Brown et al., 2000; Reber et al., 2004), suggesting that it is the functional property of axon guid ance that ephrins provide that is important, not the particular sets of molecules used. For another example, LTPILTD are seemingly ubiquitous phenomena in the brain but they use a wide range of different mechanisms in different brain areas (Malenka and Bear, 2004). Again, it is possible that the particular mechanism underlying LTP/LTD in each brain area is important or necessary for proper function, but it is also possible that it is the general synaptic plasticity mechanism that is important for functioning and not the particular implementation.  1.4. EXISTING MODELS  18  This variation of mechanism as seen with retinal waves, molecular guidance cues and LTP/LTD, yet the conservation of function across species and even within individual animals, implies that it is the functional mechanisms that are important when seeking to understand neural system devel opment at a general, species-independent level, not specific molecular implementations. As such, a model of retinotopic organization and eye-specific segregation with a wide descriptive scope is not constrained to incorporate exact replicas of physiological functions but can instead use phe nomenological approximations of such behaviors.  1.4  Existing models  Many computational models have been created to explore and explain neural behavior and orga nization in the brain, with many examining retinotopic organization and eye-specific segregation. Owing largely to limits both in computational resources and experimental knowledge at the time of their creation, these models typically represent only a subset of the biological behaviors de scribed in Sec. 1.2. Most of these derive from two categories of early studies, focusing either on activity-dependent mechanisms being responsible for driving organization (e.g. Willshaw and von der Malsburg, 1976) or on chemoaffinity generating observed patterns of organization (e.g. Prestige and Wilishaw, 1975; Willshaw and von der Malsburg, 1979). The models vary in physio logical detail, but as a general rule they are of intermediate scope, focusing on a particular subset of physiological mechanisms underlying organization and making simplifying assumptions about the remainder. Several of these models are addressed below, organized by the physiological mech anisms conceptually underlying each. Given the large number of existing models, only a few can be discussed here. There are many good reviews available that describe and discuss a variety of the available models (Erwin et al., 1995; Swindale, 1996; Obermayer and Sejnowski, 2001; van Ooyen, 2001; Swindale, 2003; Goodhill and Xu, 2005; Goodhill, 2007).  1.4. EXISTING MODELS  1.4.1  19  Chemoaffinity models  Several models are based on the hypothesis that molecular guidance cues are responsible for the observed patterns of visual system organization and a good review of such models can be found in Goodhill and Xu (2005). One of the original models in this category (Wilishaw and von der Malsburg, 1979) is still a basis for modeling research today (Wilishaw, 2006). The underlying hypothesis of this model is that of “marker induction”, which states that molecular markers on tectal neurons are not fixed and that their level of expression is induced by markers present on RGC axons. According to this theory, synapses inject molecules into the postsynaptic neurons, and these molecules diffuse among nearby postsynaptic neurons. The set of molecules injected by each axon are unique, representing a chemical code that indicates the relative location of the presynaptic neuron. Receipt of these molecules induces the expression of markers in the target neurons and the strength of a neural connection between presynaptic neurons and postsynaptic neurons is a function of the similarity between the molecules of the presynaptic axon and the markers on the postsynaptic cell. This model is able to reproduce several experimental results, including where part of the retina and/or part of the tectum were removed, as well as tectal rotation experiments, where part of the simulated tectum was rotated 180°. A notable characteristic of this model is that it implements synapse formation and retraction as well as a simple form of axon representation, elements that are uncommon in models even today. A recent version of this model has been used to address the findings of genetic manipulation ex periments (Wilishaw, 2006). These experiments performed a “knock-in” of EphA3 into a spatially distributed subset of RGCs, resulting in these RGCs having a set of molecular markers that corre sponded to a more rostral location than normal and that are different than those of many neighbor ing RGCs. This results in nearby RGCs projecting to two different collicular locations (Brown et al., 2000; Reber et al., 2004) (Fig. 1.3). Willshaw concluded that the marker induction model was able to account for the experimental observations of EphA3 knock-in mice and that activity-based mechanisms were not required. The same Eph3A knock-in experiment was modeled in a different study, using the hypothesis that RGC targeting was based on the relative, not absolute, levels of expression of molecular guidance cues (Reber et al., 2004). In that study, RGCs effectively per formed a comparison of their total EphA expression relative to all other RGCs and targeting was  1.4. EXISTING MODELS  20  Fig. 1.3. Multiple RGC termination zones in EphA3 knock-in mice Termination zones from RGCs originating in a single spot in the retina. A flattened P8 mouse retina is shown (A) with a single injection of tracer (Dii, marked with an asterisk). The RGCs from this spot form two termination zones in the colliculus (B, marked with asterisks). A. Image from Reber et al. (2004). Scale bar is 1 mm.  based on this ratio. While the Reber et al. study also concluded that molecular markers were nec essary for retinotopic organization, they instead found that activity-based mechanisms were also required to refine the retinotopic projection. A resolution to the contradiction in the results of these two studies is not clear, as both models address the ability of molecular guidance cues to produce the emergent properties of retinotopic organization and refinement. These models do not represent other physiological properties that are also likely to be involved in organization, such as axon growth and branching, trophic factors and spiking neurons, and are intermediate-scope models. As discussed previously, predictions about the behaviors underlying modeled phenomena are not reliable in such models, likely explaining the source of this contradiction. A separate theoretical study, based on axon branching that occurs as a function countering ephrin-A and EphA gradients also concluded that molecular guidance could explain the patterns of organization but that activity was required to refine the retinal projection (Yates et al., 2004). This study was able to link specific molecular interactions to the higher level behavior of retinotopic refinement. However, the model is only 1D and it is not apparent if this could scale to 2D. Further, while it represents activity-based mechanisms, the representation is abstract and based on the concept of ‘branch density’, where there is an increased axon branching probability in locations where an axon has already branched. The authors note that the same branch  1.4. EXISTING MODELS  21  density mechanism could represent a number of other underlying biological mechanisms besides activity. This model is also of intermediate scope and it is thus difficult for it to resolve the conflict in results between Willshaw (2006) and Reber et al. (2004). A wide-scope modeling approach that represents the set of physiological mechanisms involved appears to be required to better address this issue.  1.4.2  Correlated retinal activity  Many more theoretical models have addressed retinotopic organization based on correlated retinal activity than based on chemoaffinity. Many of these models also address eye-specific segregation, a phenomenon that chemoaffinity-only models cannot accomplish without the use of distinct eyespecific molecular labels (see Crowley and Katz, 1999) or through subsequent neural spike activity (discussed in Willshaw, 2006). One of the early and prominent models that operated on the hy pothesis that retinal activity was responsible for retinotopic organization was by Willshaw and von der Malsburg (1976). They hypothesized that the geometric proximity of presynaptic cells (i.e. RGCs) was coded in the form of correlations in electrical activity induced by visual experience. While it was later found that organization in the visual system occurs before the retina is sensitive to light (Wong, 1999), electrode recordings show spontaneous discharges in neighboring ganglion cells in the developing retina (Maffei and Galli-Resta, 1990), providing the required source of cor related activity in the absence of vision. The model of Willshaw and von der Malsburg, like most activity based models, has an implicit representation of chemoaffinity guidance, or an equivalent mechanism, that provides coarse guidance to axons. These mechanisms vary between models and include “polarity markers” (Willshaw and von der Malsburg, 1976), retinotopic bias based on the timing of arrival of retinal axons (Eglen, 1999), restricting axon arbors to specific domains in the target structure (Elliott and Shadbolt, 1998) or a bias in initial connectivity (Goodhill, 1993). As such, most activity-based models are technically hybrid models that combine chemoaffinity bias (or its equivalent) with activity dependent mechanisms. Activity based models are based on Hebbian learning. As theorized by Donald Hebb, “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency,  1.4. EXISTING MODELS  22  as one of the cells firing B, is increased.” (Hebb, 1949). A second feature common to most of these models is a local interaction function among target neurons that provides local excitation and distal inhibition, such that when a target neuron is activated, its immediate neighbors are also excited while its more distant neighbors are inhibited. See Swindale (1996) for further discussion  and comparison. One family of Hebbian model that has been successful at producing realistic patterns of organiza tion was originally described by Kohonen (1982). This model operates as a dimension reduction mechanism, spatially mapping connections from a presynaptic sheet of cells onto a postsynap tic sheet in such a way that presynaptic neurons having similar characteristics (such as location and ocularity) project to the same or neighboring postsynaptic neurons. The model works in a winner-take-all fashion: the representation of activity in the presynaptic sheet excites cells in the postsynaptic sheet, and the postsynaptic cell with the highest response “wins”. The winner and its neighbors then adjust their preferred responses to the representation of activity in the presynaptic sheet. Derivations of this model have been used to demonstrate retinotopy and eye-specific segre gation (Goodhill, 1993) and other organizational features that are observed in primary visual cortex (Obermayer et al., 1991, 1992). An algorithmically different approach for dimension reduction, called the Elastic Net, can reproduce similar patterns of organization (Durbin and Mitchison, 1990; Goodhill and Wilishaw, 1990). The Elastic Net was inspired by and is similar to the “tea-trade” model (Durbin and Wilishaw, 1987), which is a precursor to the marker-induction model discussed in the previous section on chemoaffinity models (see Swindale, 1996). Its was originally de signed to solve a mathematical optimization problem known as the traveling salesman problem and functioned by minimizing an energy function that represented the length of a path connecting an arbitrarily distributed group of points (Durbin and Willshaw, 1987). It was realized that this problem could be reformulated to address eye-specific segregation (Goodhill and Wilishaw, 1990). These dimension reduction models reproduce experimentally observed patterns of organization very well, but it is not clear whether or not they function in a similar manner as the brain or if they simply generate similar results. Their mathematical formulations are sufficiently abstracted from the biological mechanism that it is difficult to draw conclusions about underlying behaviors, which is fully appropriate as these models are models of narrow descriptive scope. They seek to explain the fundamental principles governing visual system organization and to phenomenologi  1.4. EXISTING MODELS  23  cally describe it, not explain how the underlying mechanisms produce it. What is remarkable about the dimension reduction models, especially given their relative simplicity, is that the brain exhibits similar high-level patterns of organization. Another category of activity-based model for segregation derives from the work of Miller et al. (1989) and is based on the statistical structure of correlations between independent groups of input neurons. This model has been shown to generate eye-specific segregation (Miller et a!., 1989) as well as segregation of ON and OFF RGCs (Miller, 1994; Lee et al., 2002). This form of segrega tion is possible because the firing patterns of ON and OFF RGCs are strongly correlated with other ON and OFF cells, respectively (Lee et al., 2002), while the firing patterns of ON and OFF RGCs are temporally distinct (Kerschensteiner and Wong, 2008), reducing the problem of ON/OFF seg regation to that of eye-specific segregation, as hypothesized by Miller (1994). This class of model is algorithmically distinct from models performing dimension reduction, though both reproduce similar experimental observations and are two different ways of minimizing the same cost func tion (Dayan, 1993). It is difficult to determine whether correlation based or dimension reduction approaches are more accurate in regards to the description of cortical maps. The approach of Miller et al. (1989) is of intermediate scope as it addresses an observed phenomenon and a subset of its underlying mechanisms. Its ability to reproduce physiological observations, particularly those re lated to cortical organization, is reduced versus that of dimension reduction models (see Swindale, 1996), and thus misses the benefit of incorporating underlying mechanisms to better reproduce an observed phenomenon. That is not to diminish the value of this or other intermediate-scope mod els, however, as even if their mechanisms differ functionally from the physiological mechanism, these models do address important questions and propose solutions to them, such as to the issue of ON/OFF segregation. However, a model of wide descriptive scope, which represents both the observed phenomena and its underlying mechanisms, in particular spiking neurons, is arguably better suited to analyze the ideas proposed by intermediate-scope models such as those derived from Miller et al. (1989).  1.4. EXISTING MODELS  1.4.3  24  Synapse density, trophic factors and other approaches  Many theoretical approaches are based on other physiological mechanisms and have examined how these different mechanisms might produce experimentally observed patterns of organization. One interesting and elegant approach to eye-specific segregation was through the concept of synapse density, and how the presence of synapses from one eye, at a location in the postsynaptic sheet of cells, would attract additional synapses from that eye and would induce the retraction of synapses from the opposite eye (Swindale, 1980). While this model has similarities to intermediate-scope models, its scope is actually quite narrow, as the underlying mechanism it represents, the distri bution of synapses, does not attempt to reproduce a specific behavior but to describe a general phenomenon which can be physiologically realized in many ways. Further, it’s predictions are restricted to eye-specific segregation, including characteristics of the shape of the different eye do mains and about the critical periQd for development. Like the dimension reduction models, this is a good example of a narrow-scope model. A conceptually similar version of Swindale’s model has been used to describe eye-specific segre gation in the tectum, based on observations of developing retinotectal axons, with axon retraction (but not extension) being influenced by the number of opposite-eye axons present at a given lo cation in the tectum. Though the simulation was smaller, it generated similar results (Ruthazer et al., 2003). Other models address how different mechanisms can contribute to observed patterns of organization, such as volume learning, where it is hypothesized that potentiation realized by one synapse can be spread, through release of rapidly diffusible molecules such as nitric oxide, to other nearby and simultaneously active synapses (Gaily et al., 1990; Montague et al., 1991). Some mod els use spatially varying patterns of retinal activity (i.e. retinal waves), and the spatio-temporal patterns of this activity produces segregation (Eglen, 1999; Elliott and Shadbolt, 1999; Butts et al., 2007). These models analyze different sets of components underlying the emergent properties of organization, and like many other intermediate-scope models, provide insight and ideas about how these mechanisms might work. However, when models representing different mechanisms can generate similar patterns of output, especially when the subsets of mechanisms represented are non-overlapping, it becomes unclear what mechanisms are truly important in producing observed patterns of organization and what models are more insightful.  1.4. EXISTING MODELS  25  A general principle behind most activity-based neural models is that neural activity can be repre sented by the average firing rate of the neuron, and that the timing of individual spikes is not impor . Given the success of models which demonstrate retinotopic refinement and eye-specific seg 2 tant regation, there is clearly evidence to support that argument. However, synaptic plasticity is known to depend on the timing of individual spikes (Bi and Poo, 1998; Zhang et al., 1998; Froemke and Dan, 2002) and it has never been demonstrated that the timing of individual spikes doesn’t matter. Some researchers have more recently postulated that it is the timing of spike bursts that is impor tant (Butts et al., 2007), but such an approach doesn’t take into account the number of spikes in a burst nor the duration of spiking, nor the fact that different types of bursts would require differ ent learning rules (Dan and Poo, 2006). There are relatively few computational models based on spiking neurons. Some of these explore the dynamics and formation of features observed in visual cortex, such as orientation selectivity (McLaughlin et al., 2000; Wielaard et al., 2001; Choe and Miikkulainen, 2004), but few address how retinotopic organization might be produced by using spiking as the mechanism of communication between neurons (e.g. Song and Abbott, 2001). The relatively few spiking models explaining how spiking behavior can be represented in a model of visual system organization suggests that this might be a difficult problem. However, if spiking activity is indeed a relevant and necessary behavior that directly contributes to retinotopic refine ment and eye-specific segregation, it is further evidence that intermediate-scope models that do not represent it have functionally incorrect implementations of one or more of the mechanisms they do represent. Again, that is not to suggest that the results of such models be disregarded. They should continue to be considered and analyzed, not only because they provoke thought and analysis about the mechanisms driving development, but it may be that spiking activity, in this example, is not required by the brain to produce the observed features of organization. The benefit of a model of wide descriptive scope is that it is able to address the question about what mechanisms are and are not required to generate the observed patterns of retinotopic organization and eye-specific segregation. One other physiological mechanism that has received a fair amount of attention from the modeling 2  Models which address correlations of activity (e.g. Miller et al., 1989, and derivations) can be interpreted to address the timing of individual spikes. However, these can alternatively represent the timing of bursts or of simple correla Lions of activity in general and so are not included here. Their level of representation cannot be reduced to the firing of individual spikes, although the spiking between different neurons can be represented using correlations.  1.4. EXISTING MODELS  26  community involves trophic factors. Trophic factors play a role in the growth and survival in axons and synapses (Cohen-Cory and Fraser, 1995; Alsina et al., 2001; Poo, 2001; Goda and Davis, 2003; Cohen-Cory and Lom, 2004; Hu et al., 2005). Competition for trophic factors, and other forms of axonal and synaptic competition, have been addressed in several modeling studies and often address the segregation of afferents on target cells or musde fibers (reviewed in van Ooyen, 2001). Unlike many other models that address segregation, many of these trophic-based models have components that represent the behavior of individual receptor molecules or other molecular pathways. Having a more direct association with the physiological mechanism makes these models more experimentally testable. However, these models tend to be similar to other models of segregation in that they focus on specific biological phenomena and see how these phenomena might underlie experimental observations and neglect the rest, often with significant simplifying assumptions. For example, most trophic models address segregation in the visual system or neuromuscular junction but only represent a single postsynaptic cell (see van Ooyen, 2001, for review). This limited spatial representation makes it difficult to compare the results of such models to others that represent segregation (e.g. Swindale, 1980; Goodhill and Willshaw, 1990). All of these models are able to represent aspects of the emergent patterns of organization, and make predictions about such organization, but like other intermediate-scope models, they offer limited insights into the underlying mechanisms. Some models do address eye-specific segregation using not only trophic factors but also using retinal waves to represent activity (e.g. Elliott and Shadbolt, 1999). Incorporating larger amounts of details into a model increases its scope and, one could argue, the accuracy of the mechanisms it implements. While the model of Elliott and Shadbolt (1999) provides insights not possible using a simpler model, and is quite detailed for its time and for the amount of computational resources available, its scope is still intermediate as it does not account for several behaviors known to be present, in particular spiking neurons and the growth dynamics of individual axons and synapses. How these phenomena would influence the model and its results, if they were represented, remains an unanswered question.  1.5. BRINGING IT ALL TOGETHER  1.5  27  Bringing it all together  The present study describes a model of wide descriptive scope and applies it to investigating retino topic map formation and eye-specific segregation, including the contribution of each of the mod eled behaviors in producing such organization. The model was also used to explore the contribution of molecular guidance cues and different forms of retinal activity to retinotopic organization, and what aspects of activity are important for spatial representation. Due to its wide scope and level of detail, future investigations using this model can address many other experimentally observed behaviors, including development in the retinogeniculate and retino-geniculo-cortical pathways, the role of interneurons and cortical feedback, and the segregation of geniculate cells into parvo and magnocellular pathways, as well as the segregation of the ON/OFF pathways. When the model is able to reproduce experimental observations, insight can be gained into the physiological mech anisms responsible and their interaction. If the model is unable to reproduce an observation, the components of the model can be analyzed to see if they misrepresent biological behaviors, phys iological results can be compared to model behaviors to see if experimental evidence might be misleading or misinterpreted, or the model can be used to help determine if there exists an as yet undiscovered (or under-appreciated) behavior that is required to explain experimental observations. Despite its degree of detail and complexity, the model is sufficiently fast to allow a modem personal computer (circa 2008) to simulate —7,000 neurons in real time, enabling system level behavior and organization, such as retinotopic projections and eye-specific segregation, to be explored in a large number of neurons exhibiting a wide range of interrelated physiological behaviors. The design of the model is also generic and as it is not designed to address a specific experimental observation, it can be used to investigate a variety of topics and behaviors. The results of this study indicate that the known physiological mechanisms are sufficient to ex plain retinotopic organization and refinement as well as eye-specific segregation, and that some of these mechanisms play a much greater role than others. For example, molecular guidance cues, correlated retinal activity and trophic factors were all found to be necessary for development. In contrast, spike-timing dependent plasticity was found to play a very minor role. Other factors, such as rapidly diffusible molecules that influence synapse growth and retraction, were found to be redundant with other model mechanisms. While the model assumed that molecular guidance  1.6. ORGANIZATION OF THE DOCUMENT  28  cues were active throughout development and that activity contributed to molecular guidance in later development stages, it found that each of these mechanisms played a largely independent role, where molecular cues would mediate a rough retinotopic organization and correlated retinal activity was able to refine the projection without further contribution of molecular guidance.  1.6  Organization of the document  This thesis consists of a model of retinocollicular development that incorporates phenomenological reproductions of the biological behaviors described in Sec. 1.2. The model begins at the point in development when RGC axons first arrive at the colliculus and it runs until a refined retinotopic projection is produced. Analysis of the model includes how each of these biological behaviors contributes to retinotopic organization and refinement, and special attention is given to the relative roles of molecular guidance and activity-dependent refinement, given the historical debate between these two schools of thought. Eye-specific segregation is also examined, including the role of the different biological behaviors in producing it. The thesis is composed of 9 chapters. Chaps. 2-6 describe the implementation of the model, including biological descriptions of the behaviors represented. Chap. 2 describes a model for retinal waves that was developed and used in this study to generate phenomenological patterns of retinal activity as are observed in several species and at different development stages. It has been . How spatial patterns of retinal activity, 3 published previously (Godfrey and Swindale, 2007 a) as produced by the retinal wave model, are converted to spiking behavior in individual RGCs is described in an appendix to this chapter. Chap. 3 provides an overview for the model, describing the environment of the model and the components it is comprised of. The model is implemented in both 1-dimensional (1D) and 2dimensional (2D) forms, identical except that axon growth does not occur in the 1D model, the focus there being the spatial pattern of synapse generation on a static axon. Experiments were conceived, designed and performed by KBG. Data analysis was performed by KEG. Manuscript was written by KEG and NVS.  1.6. ORGANIZATION OF THE DOCUMENT  29  Chapters 4-6 describe the individual components of the model and their implementation. Chap. 4 describes the axon model, including results of axon growth occurring as a result of molecular guidance cues only. Chap. 5 describes the implementation of the models for synapse and neuron behaviors. Chap. 6 describes the component models of synapse plasticity, molecular guidance cues, growth and trophic factors, NMDA receptor activation, homeostatic controls, and rapidly diffusible molecules. The variables and parameters for the component models described in Chaps. 4-6 are in App. A. The results of the model are presented in Chap. 7, with the results split into sections for 1D and 2D simulations. The 1D model is shown to produce retinotopic refinement and eye-specific segregation. The different biological components are analyzed and their contribution to retinotopic refinement and eye-specific segregation is addressed. The 2D model was used to demonstrate over all development and refinement of the retinotopic projection and the behavior of individual axons in this process. Eye-specific segregation is also demonstrated using the 2D model. The model is then used to analyze the different contributions of molecular guidance and activity-dependent development both by deactivating correlated activity or molecular guidance at different points in development and by reproducing a recent experiment in which normal guidance molecule expres sion was perturbed. The last two chapters cover the discussion and summary of the model. Each of the model compo nents is discussed in Chap. 8, including analyses of the components and/or their role in organiza tion. The behavior of the model is then addressed, followed by a brief comparison of the present model to that of previous theoretical approaches. The closing chapter, Chap. 9, provides a list of the accomplishments of the model, a list of experimental predictions and directions for future research.  30  Chapter 2 Retinal Wave Behavior through Activity-Dependent Refractory Periods  2.1  Introduction  In the early stages of neural development, when initial sets of connections between neurons are being formed, neural activity helps organize and refine developing circuits. Before the onset of stimulus-driven activity, which helps refine neural organization in later developmental stages, neu ral circuits generate spontaneous patterns of activity which guide early development (Katz and Shatz, 1996). This spontaneous activity has been observed in many areas of the developing ner vous system, including the auditory system (Lippe, 1994; Gummer and Mark, 1994), neocortex (Yuste et al., 1992), hippocampus (Garashuk et aT., 1998), spinal cord networks (O’Donovan and Chub, 1997; Spitzer and Gu, 1997), brainstem nuclei (Ho and Waite, 1999), and retina (Maffei and Galli-Resta, 1990; Meister et al., 1991). In the retina, spontaneous activity takes the form of coordinated bursts of spikes in neighboring retinal ganglion cells (RGCs) that slowly spread across the retina (Meister et at, 1991; Wong et al., 1993). Retinal waves occur in a variety of species before visual experience, including cat, turtle, chick, mouse, and ferret (Wong, 1999). They have non-repeating boundaries (Feller et al., 1996, 1997), propagate with no directional bias, and can initiate at any retinal location (Meister et al., 1991; Wong et al., 1993; Feller et al., 1996; Sernagor  2.1. INTRODUCTION  31  et a!., 2003). The entire retina is covered in minutes (Feller et al., 1996, 1997; Sernagor et a!., 2001). Retinal waves drive activity-dependent organization in the visual system (Katz and Shatz, 1996; Wong et al., 1993; Wong, 1999; Torborg and Feller, 2005). They have been shown to refine retino topy in the LGN, superior colliculus, and cortex (Shatz and Stryker, 1988; Sretevan et al., 1988; Thompson and Holt, 1989; Simon et al., 1992; Penn et al., 1998; McLaughlin et aL, 2003; Ruthazer et al., 2003; Cang et al., 2005), to drive segregation of the LGN into eye-specific layers (Torborg and Feller, 2005; Sretevan et al., 1988; Penn et al., 1998) and to drive responses in Vi neurons (Hanganu et al., 2006). While the physiological mechanisms underlying retinal waves have been studied extensively (Wong, 1999; Torborg and Feller, 2005; Firth et al., 2005), there have been few attempts at modeling them. The first model was based on extracellular diffusion of potas sium driving RGC activity (Burgi and Grzywacz, 1994). Experimental evidence contradicted this premise (Feller et a!., 1996) and another model was put forward, based on random amacrine cell activity and long refractory periods where amacrine cells are non-responsive (Feller et al., 1997). Subsequent physiological evidence has shown these assumptions to be invalid, as amacrine cells regularly depolarize during waves and release excitatory transmitter when doing so (Zhou, 1998; Zheng et al., 2006). Other limitations are that the model produces non-uniform net coverage of the retina (Elliott and Shadbolt, 1999), that it has only been demonstrated to produce waves similar to postnatal day 2 (P2) to P4 ferret, and that the properties of the generated waves, including wave size, frequency, and velocity, can be very sensitive to small changes in network state or parameters (Butts et al., 1999). In this study we describe a retinal wave model that is robust to parameter variation and generalizes across species. We make use of the findings that amacrine cells receive input and depolarize during local wave activity (Feller et al., 1996; Zhou, 1998; Zheng et al., 2006), that they have variable periods between spontaneous depolarizations (Zhou, 1998; Zheng et al., 2006), and that the period between depolarizations appears to be a function of recent local excitation (Zheng et al., 2006). These observations lead to the central principle behind the model, that the refractory period, or the period between spontaneously occurring bursts in cells, is a function of the amount of excitatory input recently received by the cell. The resulting model produces waves with randomly  2.2. METHODS  32  distributed initiation points and non-repeating borders across a wide range of parameter settings. The velocity, domain size, and interwave interval (IWI) of the generated waves can be configured to match those seen in ferret, rabbit, mouse, turtle, and chick retinas. We show that a single homogenous group of cells can produce these wave behaviors, in contrast to claims that such behavior requires two independently functioning cell types (Feller et a!., 1997). We also show that the model exhibits chaotic behavior, producing seemingly random patterns of waves in the absence of stochastic input. The uniformity of retinal coverage provided by the model and the realistic spatio—temporal patterns of activity should also make it useful as an input to developmental models of the retino—geniculo—cortical pathway (Elliott and Shadbolt, 1999; Eglen, 1999).  2.2  Methods  The model focuses primarily on waves mediated by ACh (Wong, 1999; Torborg and Feller, 2005). However, the principles behind the model, namely that wave behavior results from spontaneous ac tivity and that refractory periods are a function of recent input, are intentionally general so as to be unconstrained by specific biophysical implementations. Hence, the model is adaptable to the de scription of waves in several developmental stages and species even though different physiological and pharmacological mechanisms might be responsible for their generation. The model relies on a single cell type, cholinergic amacrine cells, which are responsible for me diating early (Ach-mediated) retinal waves (Firth et a!., 2005). Amacrine cells in the model are spontaneously active and form excitatory connections with other nearby amacrine cells. Here, “ac tive” means depolarized and/or actively exciting its neighbors. These cells have a varying threshold for activation which is high immediately after depolarization and gradually decays, similar to, but slower than, the change in spike threshold following a spike in a normal neuron. The magnitude of the threshold change is a function of recent input to the cell: a cell receiving more input has a higher threshold immediately after depolarization. When the cell’s threshold decays to zero, or when its level of excitation exceeds its current threshold, it depolarizes and becomes active, ex citing neighboring amacrine cells. When a sufficient density of amacrine cells becomes active, local excitation brings other nearby amacrine cells to threshold, producing a wave of excitation.  33  2.2. METHODS Amacrine cells  •  •  . • •/•  .  •  •  •  •__i  F  .  •  •• • • • • • • • • . • ‘\ ‘\• • 34 pm I  •  •  .  II.•  ./. •  /85pm  •  Dendrite overlap  Fig. 2.1. Network topography. Amacrine cells are arranged in a triangular lattice with a distance between cells of 34 sum. The dendrite of each amacrine cell extends 85 um from the soma, and the excitatory coupling between two cells is proportional to the area of overlap of their dendrite arbors.  Amacrine cell activity propagates as long as there is an adequate density of nearby amacrine cells close enough to threshold to depolarize from the excitation of the advancing wave front. The model retina is circular and its amacrine cells are arranged in a triangular lattice (Figure 2.1). 2 (Feller et al., 1996), Based on the measured cell density of 1,000 starburst amacrine cells per w the distance between adjacent amacrine cells was estimated to be 34 jim. A dendrite radius of 85 jim was used, midway between observations reported from different studies in newborn rabbit (Butts et al., 1999; Zheng et al., 2006). The excitatory strength between amacrine cells was pro portional to the area of overlap of dendritic arbors. On initialization, the location of amacrine cells was precomputed, and all nearby cells having an overlapping dendritic arbor were stored in a list  ). 1 (set Z  To determine a suitable size for our model retina, we explored the spatio—temporal properties 2 and (velocity, size, and IWI) of simulated P2—P4 ferret waves for model retinas between 0.65 mm 8.11 mm 2 in size. Wave behavior varied across this range but was stable, with wave size showing ) produced waves which 2 the most variation (Figure 2.2). The smallest retina tested (0.65 mm ). For computational speed, we 2 were 27% smaller than those seen on the largest retina (8.11 mm , which produced waves with measured properties within 5% 2 selected a retinal size of 3.65 mm of those produced on the largest retina. Similarly, we explored different time steps for the model  34  2.2. METHODS  111  vetodt{{{fJfJ  too  If  Area (n,&) .93 RedO,, (mm) .54  1.64  2.53  .72  .90  3.65 4.00  4.93 1.25  6.42 1.43  0.41 1.64  9  40  25  50  100  200  Time step (milliseconds)  Retina Size  Fig. 2.2. Variations of simulation time step and retina size on wave properties. 2 (left) and simulation time Variations in wave velocity, size, and frequency for retinal sizes between 0.65 and 8.1 mm steps between 5 ms and 200 ms (right; horizontal axis on log scale). Except as otherwise noted, simulations were based 2 and used a time step of 25 ms (indicated by vertical gray lines). Mean (closed circles) on a retinal size of 3.65 mm and standard deviation (vertical bars) are shown. The model is stable across a wide range of simulation time steps and retinal sizes.  (Figure 2.2). For the simulations, we used a time step (iT) of 25 ms except as otherwise noted, which produced waves with measured properties (velocity, size, and IWI) within 5% of those produced with the smallest time step tested (5 ms). The input, N, to each amacrine cell was the weighted sum of all nearby and active amacrine cells:  (2.1)  N= wA(j) 1 jEZ  where wj is the excitatory strength between amacrine cells i and amacrine cell  j.  A(j)  =  j,  and A (j) is the output of  1 for an amacrine cell that was active in the previous time step (i.e.,  depolarized) and 0 otherwise. The value  equals the relative area of dendrite overlap between  the two cells (i.e., the area of overlap divided by the total area of a dendrite arbor).  35  2.2. METHODS  The excitation level of each cell, X, approached the current level of input to that cell at a rate:  Ni_XiT K  (2.2)  1 exceeded the firing threshold, the cell was said to where K is the time constant. Whenever X become active. Each cell maintained an independent threshold, R, which was time-varying. When 1 was 1 > R, the amacrine cell became active for a duration D. After this interval had passed, X X reset to zero. Threshold, as used here, is not to be taken literally. It is meant to reflect the increase in excitatory input required for a cell to become active (i.e., depolarize) and includes factors such as the AHP (Zheng et al., 2006) and activity-dependent changes to chloride concentrations (Marchetti et al., 2005). 1 reached Each cell’s threshold slowly decreased and the cell spontaneously depolarized when R  zero. The parameter P represented the length of time between spontaneous depolarizations for a cell receiving no external input. When a cell was depolarized (i.e., A (i)  =  1), its threshold also  increased by a constant amount plus a function of input received. The threshold changed according to:  =  1 +A(iH1 +NiH2)T (HM  (2.3)  2 is the incremental change in threshold, whose 1 is a fixed rate of threshold increase and H where H  contribution varied based on how much input a cell had. D is the depolarization duration of the cell and M is a normalizing factor equal to the maximum excitatory input to a cell divided by the 1 maximum excitatory input of a cell in the center of the retina. M  =  1.0 for amacrine cells through  out the entire retina except the border regions, where they have dendrites that extend beyond the retinal boundary, thus having fewer innervating cells. This reduced input results in border cells 1 after activation and so requires slower rates of decrease in order having smaller increases in R  to help normalize the frequency of spontaneous activations between central and border regions. ) produces a larger change of threshold for amacrine 2 The incremental change in threshold (NH  cells that are in the central region of a wave compared to those near the edge or that depolarize  2.2. METHODS  36  in isolation. It is this differential in threshold behavior that is most critical for generation of finite non-repeating waves. 1 a uniform random value selected The model was initialized by assigning each threshold value R  from the interval (0.5, 5.0). Because this is unlikely to represent anything achieved during normal activity, the model retina was allowed to run for a time in order to achieve a stable operating state. This was defined as a state where, for a given set of parameters, mean 1WI and domain size changed 10% or less compared to a run of 5 h simulated time. In all cases measured, 30 simulated mm was sufficient to reach this state. A 1 h initialization period was then used as a “warmup” period for all simulations to further minimize any possible influence of starting conditions. Unless otherwise indicated, all simulations were run for 180 mm following the warmup period. Various methods of model initialization were explored (all involving assignment of R ). All methods tested reached 1 a stable state within several hours of simulated time with the exception of the trivial symmetric case where all R, were equal. Initializing all R to be equal except for one cell, used to break symmetry, was sufficient to produce stable waves after several hours, even when the model was run deterministically (i.e., no “noise”, see below). The initialization method described above was  selected as it provides minimal initial bias and approaches a stable state reasonably quickly. Except where explicitly noted, references to time in this study refer to simulated time, not model execution time. To introduce “noise” into the model, P was varied among cells and with each depolarization, pro ducing variability in the interval between spontaneous depolarizations. To do this, P was multiplied by a normal random variable with mean of 1.0 and standard deviation of 0.2, to give a cell-specific . This value for P was used by the cell until its next acti 1 value, P, used for the calculation of z\.R  vation, when a new value was chosen. The model was also run as a strict cellular automaton where no randomness was introduced to the model beyond the initial starting conditions. The free parameters of the model are P, which regulates the interval between spontaneous amacrine , which control the increase in threshold after an amacrine cell de 2 1 and H cell depolarizations; H  polarizes; the depolarization interval D, during which an amacrine cell actively excites its neigh bors, and the time constant K, which regulates how fast cells react to excitation from neighboring cells. Table 1 lists parameter sets used to produce waves similar to those seen in different species.  2.2. METHODS  37  Parameters P, D, and K are in units of seconds. Most simulations were performed on a 2.4 GHz Core 2 Duo desktop running linux (Ubuntu 7.04). The model was implemented in C++ and most simulation times varied between 1 and 15 real-time mm when running the model for 180 simulated mm, the variable time depending on the retina  size, the time step used, and the level of amacrine cell activity. For analysis, amacrine activity was saved to file and Java-based tools were used to analyze wave properties. Parameter exploration on some alternate implementations of the model was performed on a Beowulf cluster supercomputer. Source code (C++ and Java), including applets for viewing waves, is available upon request.  2.2.1  Data analysis  In order to better compare amacrine cell behavior to the experimental results of calcium imaging studies, a rough approximation of a calcium response was produced and the spatial patterns of active (i.e., depolarized) amacrine cells were measured. The retina was partitioned into pixels, one for each amacrine cell, with each pixel assigned a luminance value based on the activation level of all cells with dendrites passing through that point in the retina. Each pixel operated as a leaky integrator and had an intensity calculated according to  =  0.005A(j))  +.O1A(i)+ 1 —.15L  (2.4)  fEZ  where L is the pixel intensity at pixel i (bounded on [0,1]), j is a set of all amacrine cells with den drite overlapping i, and dT = 100 ms. All dendrites passing through a point on the retina contribute to the calcium response, with the soma generating a stronger response than the dendrite. Because of the short dendritic spread of RGCs (Penn et al., 1994), the addition to the calcium signal due to RGCs should have minimal effect on the spatial dynamics of the signal. This transformation is a coarse approximation and was not required for the model to produce wave behavior. It was primar ily used to smooth wave progression and wave boundaries, assisting in automated wave tracking, and also to make model output resemble the experimental results more closely (Figure 2.3). The  38  2.2. METHODS  A  B  -  Fig. 2.3. Effects of smoothing behavior on waves. Simulated calcium imaging smoothes wave boundaries but does not change the overall shape of waves. A. Recent (15 s) wave activity as measured through simulated calcium imaging. Different colors show the advancement of a wave over 500 ms intervals. B. Amacrine cell activity measured over the same interval and using an identical color scheme. The timing of amacrine cell depolarizations differs slightly from the observed wave seen through calcium imaging but wave boundaries are very similar. The small islands of activity in B are amacrine cells undergoing spontaneous depolarizations that are not part of wave activity (comparable to “intrinsic bursts” as reported in Zheng et a!. (2006)).  wave propagation images in Figures 2.5, 2.6, 2.9, and 2.10 are based on simulated calcium imag ing. A wave was detected when a) the luminance of a pixel exceeded a threshold (L  0.30); and b)  the pixel was not adjacent to any pixels that were assigned to a pre-existing wave. The pixel was considered to be part of a wave until its value fell below a lower threshold (L < 0.25). This threshold range was used to minimize pixels near threshold from repeatedly joining a wave when oscillating near threshold. The initialization point of this wave was the centroid of all connected pixels which exceeded the lower detection threshold (0.25) on the first frame threshold was crossed. Wave velocity was calculated by measuring the distance between the centroid and the farthest point reached by the wave and dividing by the time required to reach that point. The velocity of each wave was stored, and the average of these values calculated. When two waves collided, both were omitted from the calculations as there was no longer an unambiguous starting or most distal point. Analysis did not demonstrate any significant difference between these joined waves and waves which remained independent, so their exclusion should not significantly bias the measurements. Wave size was calculated using the number of connected pixels that crossed threshold during the  2.2. METHODS  39  lifespan of the wave (each pixel was counted only once). To reduce border effects in 1W! and retinal coverage calculations, pixels associated with amacrine cells within one dendritic radius of the retinal boundary were omitted from the analysis. The 1W! distributions were calculated by measuring the inter-wave interval of all analyzed pixels and storing these values in a histogram.  2.2.2  Variable duration depolarization.  The model is framed in the simplest form we found that produced robust wave behavior. One simplification involved the duration of amacrine cell depolarizations, which was constant in the model, although studies show that it varies, depending on whether the cell depolarizes in isolation or contributes to a wave (Zheng et al., 2006). To explore if our simplified amacrine cell behavior affected wave production, we modified amacrine cells by allowing them to depolarize for brief fixed intervals and remain depolarized as long as sufficient excitation was present. This was done by (a) slowing the onset of threshold increase, where the factors governing the refractory period, such as the AHP, take seconds to be fully realized, thus allowing prolonged depolarizations to ), allowing it to always reflect current input to 1 occur, and (b) not resetting the excitation level (X the cell. As long as excitation was greater than the threshold, the amacrine cell was depolarized and excitatory to its neighbors. A 3 s refractory period was imposed after each fixed depolarization interval to prevent multiple triggerings during a single wave. In these simulations, D was set to 0.45 s and the maximum change of threshold (zIRj) per second was limited to 4.0. Larger threshold changes took more than 1 s to be fully realized. Other values for D and threshold onset rates were explored and produced similar results.  2.2.3  RGC spike generation  To generate spiking activity for use in modeling retinocollicular development, a layer of RGCs was 2 simulated added to the simulated retina. RGCs were arranged in a triangular lattice on a 3.65 mm  40  2.2. METHODS No 9 oooooooo o o o o o o  o  a  a  a  a  °Amacrine c cells  /  ooo.oooo  OOOOOOOG G-RGC5 0 0 0 0 0 0 o 0000000000  Fig. 2.4. RGC arrangement and excitation Both amacrine cells and RGCs were arranged in an overlapping triangular lattice. As described in Chap. 2, amacrine cells had a 34 Lm inter-cell spacing. RGCs had an inter-cell spacing of 17 For computational convenience, each amacrine cell (e.g. red circle with X) was associated with 4 RGCs (e.g. small green circles). Each RGC received excitation from 7 amacrine cells; the amacrine cell that it was associated with plus that cell’s 6 neighbors. For example, each of the RGCs in green received excitation from the 7 shaded amacrine cells. Each RGC was also gap connected to its 6 neighbors, indicated by gray lines for the green RGC with an X.  , producing an 1 retina. Spiking activity was not generated for RGCs at the border of the retina . Except as otherwise noted, the parameters for producing P2-P4 2 effective retinal size of 3.4 mm ferret waves were used. RGC cell density was four times the density of amacrine cells, producing an inter-RGC spacing of 17 pm, similar to previous models (Feller et aL, 1997) and comparable to RGC/amacrine cell ratios in P6 ferret, after RGC levels become stable (Henderson et al., 1988). As implemented, each amacrine cell was associated with 4 neighboring RGCs, and each RGC received excitation from the 7 nearest amacrine cells (Fig. 2.4). Anatomically, each RGC likely receives input from many more amacrine cells, but a focal set was used so RGC activity could better reproduce the measured spatiotemporal properties as described in Sec. 2.3. The algorithm to produce RGC spikes was focused on producing spike bursts with an average duration of 1 second and an average firing rate of 0.2 Hz. Physiological reports of burst length vary considerably. In ferret, the burst duration is reported as 1-4 seconds in P0 ferret and 1-3 seconds in P5 ferret (Wong et al., 1993). The same study, however, shows a cross-correlogram between nearby RGCs (P0 ferret) having —1 second width at half height. This half-height width served as the target Specifically, spikes were not generated in RGCs that were positioned at the border such that they could receive input from amacrine cells outside of the simulated retina.  41  2.2. METHODS  burst duration for simulated RGCs. The target firing rate was based on a reported range of 0.15 Hz to 0.4 Hz in P0-P15 ferret, with a range of -0.15 Hz in P4-P5 ferret (Wong et al., 1993). RGC firing rates between 0.1 and 0.3 Hz were used in the present study and no qualitative differences were noted. The stage of retinal waves that drives retinocollicular organization is mediated by ACh (McLaughlin et al., 2003) which corresponds to P0-P7 in ferret (Wong, 1999). In addition to target firing rates and burst durations, RGC excitation was designed to produce higher levels millisecond-level synchrony between adjacent RGCs than would occur by chance. This is based on reports of millisecond level synchrony between nearby RGCs (Brivanlou et al., 1998) and personal analysis of raw MEA retinal data of P0-P5 mouse, kindly provided by Evelyne Semagor. The spike waveforms in this data were strongly suggestive of such behavior. RGC spikes were also produced without these correlations to evaluate the contribution of millisecond level synchrony. RGCs were excited by nearby amacrine cells that were depolarized. Each RGC also received input from neighboring RGCs. Amacrine-RGC excitation influenced the overall excitation level of the RGC. Inter-RGC excitation was restricted to inducing simultaneous spikes in neighboring RGCs. Specifically:  Av  =  O.O2+Hi1Cexc(pama) +I (Prgc  5h)  y  Zi  (2.5)  thresh  where t is the excitation level of RGC i, z is the set of depolarized amacrine cells innervating i, iç is the excitation strength of each amacrine cell connection,  ((z)  =  (z) is a probability function  1 if a uniform random variable selected on [0, 1) is less than X’ and I()  =  0 otherwise),  Pama is the probability that an amacrine cell will excite an RGC on any given millisecond that it is  depolarized, Prgc is the base probability that an RGC will induce a spike in its neighbor, and 1 qhresh 1 is a is the excitation threshold an RGC must be above in order to have a spike induced in it. H  comparable to the same variable in Eq. 6.14 providing a homeostatic control of firing rate and was calculated the same way, using Farget  =  0.2Hz and  req 5 t  =  20 mm. To achieve —1 second burst  duration, amacrine cell depolarization, as used to define set zj, included only the first 0.7 seconds of the amacrine cell’s actual depolarization time. Eq. 2.5 was recalculated every millisecond and 1 was decremented by 1 RGC i produced an action potential when u> 1.0, after which V reset  2.3. RESULTS Label  42 Description  IWI (Sec.)  Velocity .tmIsec  Size mm 2  P  1 H  ff2  D  K  A  Control (P2-P4 ferret)  117  176  .156  43  4.0  0.75  1.3  0.25  B  High velocity  115  435  .154  36  4.0  0.6  0.5  0.1  C  Low velocity  116  111  .155  43  4.0  0.85  2.3  0.35  D  Large waves  114  171  .31  50  4.0  0.45  1.3  0.25  E  Small waves  118  177  .097  28  3.0  1.25  1.4  0.25  F  Short IWI  39  182  .159  15  4.5  0.85  1.3  0.25  G  Long IWI  204  176  .149  75  4.0  035  1.3  0.25  Table 2.1. Spatiotemporal properties of retinal wave Spatiotemporal statistics of the different types of retinal waves used in this study (see Sec. and the parameters used to generate them. Values in bold indicate wave properties that were significantly changed.  A hard refractory period of 5 ms was imposed on all RGCs. The spike generating algorithm described by Eq. 2.5 was designed to produce spike bursts with a specific characteristic and was not an accurate representation of the physiological pathways involved in RGC excitation. Waves with different spatiotemporal properties were used in this study. Table 7.1 shows the prop erties of these waves as well as the parameters used to produce them. The parameters used to produce RGC spiking behavior of with different characteristics are shown in Table 2.2, along with the statistics of such behavior.  2.3 2.3.1  Results Ferret Waves (P2—P4)  The waves produced by the model are qualitatively comparable to published images of physio logical waves (Feller et al., 1996, 1997; Singer et al., 2001; Syed et al., 2004). Figure 2.5 shows two examples of wave activity. Waves begin when several nearby amacrine cells are at or near the point of spontaneous depolarization and the excitation produced by the depolarization of some cause premature depolarization in others. If a sufficiently high density of depolarized amacrine cells is present, a wave develops. The wave continues to propagate in all directions where there is a high enough density of amacrine cells capable of depolarizing as a function of the excitatory input from their neighbors.  43  2.3. RESULTS  Amacrine cell activity  Simulated Ca response  Wave progression  C R  x  Time (minutes)  Fig. 2.5. Examples of wave behavior. A, B Left circles show the instantaneous states of amacrine cells at two different times in the same retina. Resting cells are white, and depolarized cells are green. Center circles show simulated responses seen with calcium imaging. Circles on the right show retinal activity over the proceeding 15s. Different colors show the progression of waves during 500 ms intervals. Colors fade to white with time. Widths of the different color bands show variations in the velocity of the advancing wave front. C. The behavior of threshold and excitation in a randomly selected amacrine cell over 11 mm simulated time. The threshold plot (R: black lines) shows slow, linear decay with periodic increases coinciding with the depolarization of the cell (vertical blue lines). The magnitude of threshold increase varies as a function of input received by the cell when it is depolarized. Large threshold increases generally indicate that the cell is in a central and fast-moving portion of a wave. Smaller threshold increases occur when a cell is in a slow-moving part of a wave or near a wave boundary, or when it depolarizes in the absence of a wave. The excitation plot (X) shows input to the cell due to nearby wave activity. This input is sometimes enough to overcome threshold and cause the cell to depolarize. The third graph (X-R) shows the difference between excitation and threshold levels. The cell fires when excitation exceeds threshold (red line segments). This plot resembles voltage recordings in amacrine cells Zheng et al. (2006), consistent with an AHP playing a large but not exclusive role in the threshold change. Peak levels of excitation are small compared to threshold levels. Most amacrine cell depolarizations are the result of nearby wave activity with approximately 10% occurring “spontaneously” (*).  44  2.3. RESULTS RGC activity  Hz  Label  IBI  Burst  Independent  (Sec.)  len (Sec.)  spikes (ratio)  85.7  1.08  0.52  0.20  A  -1.0  0.5  0.4  Unsynchronized  83.7  1.09  1.0  0.19  A  -1.0  0.5  High frequency  68.4  1.16  0.53  0.30  A  -1.0  0.5  Low frequency  100.5  0.92  0.53  0.10  A  -1.0  High velocity  71.0  0.59  0.50  0.18  B  -1.0  Low velocity  77.1  1.38  0.55  0.19  C  Large waves  94.4  1.13  0.51  0.20  D  Small waves  72.4  1.02  0.54  0.20  Control  IC,eset  Klhresh  Kexc  Prgc  Puma  Fargt  (Hz) 1.0  0.075  0.2  0.4  0  0.075  0.2  0.4  1.0  0.075  0.375  0.5  0.4  1.0  0.075  0.075  0.5  0.4  1.0  0.075  0.2  -1.0  0.5  0.4  1.0  0.075  0.2  -1.0  0.5  0.4  1.0  0.075  0.2  E  -1.0  0.5  0.4  1.0  0.075  0.2  Short IWI  33.7  0.92  0.53  0.29  F  -1.0  0.5  0.4  1.0  0.075  0.2  Long IWI  144.4  1.05  0.52  0.11  G  -1.0  0.5  0.4  1.0  0.075  0.2  Intense burst  118.9  0.77  0.44  0.16  A  -0.33  0.75  0.075  1.0  0.33  0.125  Weakburst  63.1  1.14  0.56  0.11  A  -1.0  0.5  0.55  1.0  0.055  0.35  Table 2.2. Spiking properties of RGCs Spiking and burst statistics of RGCs used in the different simulations and the parameters used to produce them. The column “Label” refers to the waves listed in Table 2.2. The independent spike ratio was the ratio of spikes that occurred without millisecond-level correlation with spikes in adjacent neurons. A value of 0.5 means that half the time one neuron spiked, one or more of its neighbors also spiked. The intense and weak bursts used the same wave files but activity of intense bursts were delayed so to not overlap with weak burst activity. Bold values indicate configuration and parameter changes.  The non-repeating wave behavior occurs because amacrine cells receive differing amounts of input during wave activity, resulting in some cells becoming more refractory than others. Figure 1C shows the threshold and excitation of a cell over time. Amacrine cells near the central regions of a wave receive more input during a wave, and hence become more refractory, than amacrine cells near the wave boundaries. This provides a deterministic and destabilizing force that inhibits the production of repeating wave domains as subsequent waves will more readily “invade” the border areas of a previous domain than central areas. Observations have shown that both the form of the original domain and the timing of these “invasions” determine the extent of the invasion and the subsequent increase in refractoriness for the amacrine cells involved. This turns the largely coherent refractoriness of amacrine cells in the original domain into several incoherent subgroups, inhibiting generation of a subsequent wave capable of following the boundaries of a predecessor. Figure 2.6 shows 40 sequential waves passing through a randomly selected spot on the retina, giving an example of the variability and non-repeating quality of the waves. The domain size and IWI distributions of simulated and physiologically recorded waves are shown  45  2.3. RESULTS  ‘I—  Fig. 2.6. History of waves passing through a single point. Snapshots of 40 successive waves that passed through a randomly selected point on the retina (identified by gray cross-hairs). Images are ordered left to right, top to bottom and show the entirety of this wave as well as all concurrent activity. Different colors show the progression of waves during 500 ms intervals and colors fade to white with time (15 s). These images were produced when the simulation was run in deterministic mode (i.e., using no “noise”), and they show no indication of cyclic or repeating activity over the observation interval. Instead, waves through a given point, and activity across the entire retina, appear to be random.  46  2.3. RESULTS  A Simulated data  B  Domain size distribution  0.10  0.10  0.00  0.0  oZ2  0.6 0.4 Area (mm 2  0:8  Interwave interval distribution  yr  10 200 Seconds  300  200 Seconds  300  Physiological 0.10 data  [Lr  0.00 0.0  0.2  0.6 0.4 Area (mm ) 2  —r 0.8  Fig. 2.7. Comparison between model and ferret wave data A. The distribution of wave sizes (also called domains Feller et al. (1996)) of the model (top) and physiological data (bottom); bin size is 0.025 mm2; physiological data are the average of five P2—P4 ferrets, data adapted from Feller et al. (1997); model data are based on 3 hours of simulated time. All distributions were normalized to 1.0. B. The IWI distribution of the model (top) and physiological data (bottom), data adapted from Feller et al. (1997). Bin size is 20 s. Table lA gives the simulation parameters used to produce this output.  in Figure 2.7. Figure 2.7A shows the distribution of domain sizes and the averaged response of five P2—P4 ferret retinas (data from (Feller et al., 1997)). The two distributions are very similar. ; the average size of physio 2 2 (median 0.119 nim Simulated domains average 0.156 ± 0.141 mm logical domains was not reported). The 1W! is defined as the period of time between successive waves passing a given location on the retina. Figure 2.7B shows plots of IWI distributions mea sured in the model retina and that are observed experimentally (Feller et al., 1997). The model’s 1W! averaged 117 ± 47 s (median 116 s) which is similar to the experimentally measured value of 115 ± 48 s (Feller et al., 1996). As with domain sizes, the model and physiological IWI statistics, and the shapes of the IWI distribution, were very similar. Experimentally reported velocities aver s (Feller et al., 1997). Corresponding figures for waves/mm / aged 177 urn/s with a frequency of 3 2 s. waves/mm / the model were 176 I rn/s and 3.0 2 The 1W! distribution measured by (Feller et al., 1996) was based on calcium imaging while other studies of similarly aged ferrets (Wong et al., 1993) used electrodes and reported notably shorter iWis (90 versus 115 s). If RGC spiking, and hence wave activity, were to occur when amacrine  2.3. RESULTS  47  cells were active but below the threshold of detection through calcium imaging, that would explain this discrepancy. Indeed, it has been estimated that RGCs which spike less than 7 times in a burst may not be detected in calcium imaging studies (Butts and Rokhsar, 2001). To test this, we halved the wave detection threshold in the model. This reduced the average 1W! to 86±43 s, producing waves with an average velocity of 162 IlrnJs. This is comparable to electrode recordings of waves that showed an 1W! of 90s and a velocity of 100—300 jim/s (Wong et al., 1993). These results suggest that wave activity and RGC spiking occurs in areas of the retina not detected by largescale calcium imaging. Physiological studies have reported that waves have a random distribution of initiation points (Feller et al., 1997; Sernagor et al., 2003, 2001). Figure 2.8A shows the distribution of wave initiation points from the model over a 60 mm period. With the exception of the edges, initiation points are distributed uniformly across the retina, consistent with physiological studies (Feller et al., 1997; Sernagor et al., 2003). Figure 2.8C shows a density plot of initiation points produced by the model after 120 h simulated time. Border effects are apparent near the retinal boundary, with each point on the boundary having two to three times the rate of wave initiation as a point in the central area. The model does not represent changing densities of amacrine cells in the peripheral and border areas of the retina, a simplification that might create altered patterns of initiation points compared to the physical retina. Physiological retinal wave studies have not yet addressed possible border effects on wave generation. One application of retinal wave models is for use in modeling development of the retino—geniculate pathway, such as done by Elliott and Shadbolt (Elliott and Shadbolt, 1999). Their model required wave activity to be relatively uniform across the retina, as areas of high relative activity would achieve disproportionately large representation in the LGN. In their study they used the most ac curate existing retinal wave model existing at the time (Feller et al., 1997) and found that it had significant non-uniformities, including areas of high relative activity they termed “hot spots”. The duration of activity of RGCs in the model averaged 96.8 ± 25.4 s after a period of simulation. While individual cells have been reported to have different firing rates (Wong et al., 1993), exper imental studies have not reported observations of hot spots or other clear non-uniformities in the spatial variability of retinal activity. A test for the current model was to determine how evenly wave  48  2.3. RESULTS  Fig. 2.8. Distribution of wave initiation points A. Initiation points for 653 waves (60 mm simulated time). Each dot shows the initiation point of exactly one wave, with waves sharing the same location being represented with adjacent dots. Points are uniformly distributed, consistent with physiological studies Feller et al. (1997); Sernagor et al. (2003). B. Density plot of wave initiation points after 79,216 waves (120 hours simulated time). Shades of gray indicate number of initiation points at each location on the  retina with darker colors representing higher activity. Shades are linearly scaled. There is a clear bias for waves to initiate near the retinal boundary.  activity was distributed across the retina. We found that each location on the retina was active for an average of 95.8 + 3.9 s over a 110 mm simulation, with no spatial groupings of above or below average activity. There were no suggested acceptable limits for variability by (Elliott and Shadbolt, 1999), but the reduction in the standard deviation from 26.2% of the mean to 4.1% over a nearly identical duration of activity should greatly improve the uniformity of retinotopic organization in such models and may be more representative of actual retinal behavior.  2.3.2  Reproduction of Wave Statistics in Different Species  Rabbit  In E24-P1 rabbit, the nicotinic acetyicholine (ACh) system is the primary driving force for retinal waves (Zhou and Zhao, 2000). During this period, calcium imaging studies have shown waves to have an IWI of 113 ±25 s and a velocity of 200 JIm/s (Syed et al., 2004). Using the parameters of Table 2.3B, the model produced an IWI distribution of 112 ± 42s and a velocity of 199 jIm/s. Physiological wave sizes for rabbits have not been published. When not constrained by wave size, several parameter sets can produce waves of this approximate distribution (this particular ). As in ferret, 2 parameter set produced waves with average domain sizes of 0.19 ± 0.17 mm  49  2.3. RESULTS A) Ferret  B) Rabbit  C) Mouse  D) Chick  E) Chick  F) Turtle  G) Ferret  P2-P4  E24-P1  P0-P13  E14-E15  E16  E23-E24  P2-P4  P  43  44  32  30  38  23  45  1 H  4.0  4.0  4.0  3.1  4.0  4.0  5.0  0.1  0.4  0.7  0.85  Parameters  2 H  0.75  0.6  0.75  D  1.3  1.05  2.3  0.8  1.05  1.0  1.3  K  0.25  0.25  0.35  0.02  0.025  0.2  0.25  Table 2.3. Parameter sets that reproduce waves in different species. A. Parameters that reproduce results of Feller et al. (1996, 1997) in calcium imaging studies. Halving the wave detection threshold reproduces the mean TV/I measured in P4 ferrets using electrodes Wong et al. (1993). B. Parameters that reproduce calcium imaging observations by Syed et al. (2004). Reducing the wave detection threshold to 1/3 reproduces the mean TV/I measured using electrodes Zheng et al. (2006) in E29-P1 rabbit. C—E. Parameters reproducing data from Singer et al. (2001); Semagor et al. (2000); Wong et al. (1998), respectively. F. Parameters that reproduce turtle data as reported in Wong (1999); Semagor et al. (2003). G. Parameters reproducing data from Feller et al. (1997), but in deterministic (i.e., non-stochastic) mode. See Results for details. Parameters P. D, and K are in units of seconds, Chick simulations (D and E) used time steps of 10 ms. Parameter values were arrived at through trial and error, with assistance of plots similar to Figure 2.11. It was often possible to find several parameter sets that would produce nearly identical outputs.  electrode recordings in rabbit have shown shorter IWIs compared to measurements done with calcium imaging. As recorded by electrodes, RGCs in E29-P1 rabbits have IWIs of 70±26 s with a median of 64 s (Zheng et a!., 2006). Lowering the calcium detection threshold to 1.3 while keeping all other parameters constant produced an IWI distribution of 74±39s with a median of 68 s, again consistent with the idea that calcium imaging is not able to detect all RGC activity during waves.  Mouse  Retinal wave velocities in P1—P 13 mice averaged 110 jim/s (Singer et al., 2001), considerably slower than in other species Wong (1999). Using the parameters shown in Table 2.3C, waves aver aging 108 jim/s were produced, with 1W! distributions of 82.2 ± 34.8 s and domain sizes averaging . This compares with physiological 1W! measurements of 83.6 + 32.3 s and sizes 2 0.19 ± 0.19 mm of 0.19 ± 0.21 mm . Other studies (Bansal et al., 2000; Demas et al., 2003) have reported wave 2 velocities in mice that are notably higher. We elected to target the lower velocities in order to  2.3. RESULTS  50  analyze the flexibility of the model. The model is also able to reproduce the higher velocity waves (unpublished data).  Chick  E14—E15 chicks are reported to have waves with velocities of 0.5 16 ± 0.118 mm/s and an IWI of 95.7 s (Sernagor et al., 2000). These velocities are considerably faster than waves observed during Ach-mediated waves in mammals. The parameters in Table 2.3D produce waves with velocity 0.525 ± 0.160 mm/s with an IWI of 99 ± 35s. Another study reported that waves in E16 chicks have average velocities ranging from 0.5 to 1.5 mm/s and that they are very large and originate at different points on the retina (Wong et al., 1998). The parameters in Table 1E reproduce these findings. The average wave velocity was 0.856 mm/s. Initiation points were distributed across the simulated retina, similar to Figure 3B, but with more clustering near the borders (unpublished 2 (median 0.83 mm ). The physiological IWI at 2 data). Average domain size was 0.91 ± 0.55 mm this age is reported to be less than 2 mm; the IWI of the model was 82±23 s.  Turtle  S23—S24 turtles have waves averaging 226 jim/s (Sernagor et al., 2003) with an IWI of 35—90 s (Wong, 1999), faster and more frequent than observed in the mammalian species considered here. Table 2.3F lists the parameters to produce waves with a velocity of 223 ±47 jim/s and an IWI of 63.5 ± 24.4 s. While the waves in turtles are rich and well-studied (Sernagor et al., 2003; Sernagor and Grzywacz, 1999), we did not attempt to investigate or represent individual neurotransmitter pathways in this or any other species. Modeling waves in turtle, chick, and mouse, which use glutamate during some or all of the ages modeled, was done to show the flexibility of the model.  2.3.3  Chaotic Behavior  Chaotic behavior is generally defined as behavior that is sensitive to small variations in initial conditions and that generates apparent randomness whose origins are entirely deterministic. To  2.3. RESULTS  51  investigate this, we initialized the retina as described in Methods and then allowed the state of the model to evolve as a deterministic cellular automaton, where the state at any given time is a strict function of the immediately preceding state, using no added simulated noise. Specifically, P (Equation 2.3) was constant. Analyses showed no significant differences between waves produced in simulations run this way compared to waves generated with ongoing noise. Table 2.3G shows the parameters used to reproduce P2—P4 ferret waves statistically similar to those measured by (Feller et al., 1996, 1997) (Figure 2.9). The IWI was 115 ± 46s, average wave velocity was 183 jimls, and . This compares with the physiological IWI of 115 ±48s 2 wave frequency was 3.0 waves/s/mm 2 (Feller et al., 1996), an average velocity of 177 JIm/s. and wave frequency of 3.0 waves/s/mm (Feller et al., 1997). The distribution of wave sizes was similar (compare Figures 2.7A and 2.9A), 2 in size. with the model producing waves 0.16 + .16 mm To investigate sensitivity to changes in initial conditions, we initialized the retina as above but changed the threshold value of a single amacrine cell by 0.1, which is 4% of the average ini tial value. We then allowed the model to evolve for 90 mm. We repeated this with 10 different amacrine cells selected from around the simulated retina and then compared the behavior of a single amacrine (the center cell) across each of these simulations (Figure 2.9E). The timing and intensity of bursting activity for the observed amacrine cell was different in all tested cases. More notable, however, was the collective change in behavior of all amacrine cells, which produced greatly altered wave patterns when viewing the entire simulated retina. Similar behavior occurs when the model is perturbed while it is running. To analyze sequential waves and determine if there were any cyclic patterns, we took snapshots of the retina when a wave passed an arbitrarily chosen point and compared sequences of successive waves (Figure 2.6). No patterns were apparent on the short time scales observed (up to 3 h). To explore the possibility of very long cycle times, we ran several very long simulations (2 or 3y simulated time) and analyzed model output. To do this, we chose 19 amacrine cells uniformly distributed across the retina, and stored the activity pattern of these cells in a list. For each time step that three or more amacrine cells were active, we created a 19 bit integer, one bit for each cell, with a bit set to “1” if the cell was depolarized and “0” otherwise, and stored this value. After the simulation, we took the final 30 patterns and searched for this sequence in the list. For  52  2.3. RESULTS  A 0.10  0.00 0.0  0Z2  B  0:8  0:6 0.4 Area 2 (mm )  0.20  0.10  IT 100  0  200 Seconds  300  E  I  II  I  I  I  I  I  I  I  I  II  I  I  I  II  I  I  I  0  1  II  I  I I  I I  I  2  3  I  I  I  4  5  6  I  I  7  8  I  I  9  I I  I  10  Minutes  Fig. 2.9. Deterministic wave behavior. A, B Domain size and tWI distributions when the model is run in deterministic mode. C, D Wave activity in the amacrine cell layer (instantaneous) and recent wave activity across the retina (15 s; each color represents the advance ment of a wave over 500 ms, colors fade to white with time). Results are comparable to Figure 1, suggesting that intrinsic noise is not necessary for the generation of random-looking wave activity. E. Depolarization times of a single aniacrine cell after variation of initial conditions. The last 10mm of a 100 mm simulation run are shown. The top line is reference behavior. The following 10 lines show the depolarization times of the same amacrine cell after a different amacrine cell has had its initial threshold value changed by a small amount (—4%). This change in behavior is representative of all amacrine cells, demonstrating a pronounced collective change in wave behavior as a result of small changes in initial conditions.  53  2.3. RESULTS simulations run this way (n  =  3), no match was found, indicating that any cycle behavior is longer  , 2 than biologically relevant time scales. We used a small model retina for this search (0.65 mm dT  =  1 OOms) on the presumption that a small retina would show shorter cycle times than a large  retina. Wave behavior was similar at the beginning and end of this interval both qualitatively and quantitatively (spatiotemporal properties of waves varied by <5%). The activity-dependent refractory period is critical for the production of non-repeating and appar ently random behavior. Analysis showed that the model produced non-repeating wave behavior so long as there was a differential in input received by cells during a wave, and hence a variable increase in their refractory periods. Extending the period where this input was summed (i.e., when the cells refractoriness increased regardless of whether or not it was active) or by reducing the parameter J2 to very low values produced stable waves which covered the entire retina and elimi nated realistic wave behavior. Such changes also eliminate realistic wave behavior when the model is run with stochastic input (i.e., randomly varying F). The observations that the model shows sensitivity to small changes in initial conditions and pro duces apparently random waves that are non-periodic and with randomly distributed initiation points suggest that it has a deterministically chaotic regime. However, we did not carry out strict mathematical tests for the presence of chaos (calculation of Lyapunov exponents or a complexity analysis of cellular automata; Wolfram, 1984), and therefore we do not claim to have demonstrated chaos in a mathematical sense. The model’s behavior is consistent with “chaotic aperiodic behav ior” as described for cellular automata (Wolfram, 1984) and is present for all time steps tested (ranging from 5 ms to 200 ms). The biological relevance of this behavior is not that the model is chaotic in a mathematical sense, which would be interesting, but that it is chaotic in a practical sense. It uses a simple mechanism which does not rely on underlying stochasticity to reproduce the non-repeating and random waves that are observed in many species and are mediated by different chemical pathways (Wong et al., 1993).  54  2.4. DISCUSSION  Variable Duration Depolarization  2.3.4  The model is framed in a simple form in order to focus on general principles behind wave pro duction. One particular simplification regards the duration an amacrine cell depolarizes, which is treated as constant but actually varies, with amacrine cells that depolarize in isolation doing so for relatively brief periods while depolarizations which are coincident with a wave are much longer (Zheng et a!., 2006). To test whether this simplification affected wave production, we modified amacrine cells to have short fixed depolarizations and allowed them to remain depolarized, and hence excitatory, for as long as they had sufficient input from neighboring cells to do so. With this modification, wave behavior still appeared normal, with the model again being able to match the spatiotemporal properties of waves in different species. We targeted reproduction of P2—P4 ferret waves, producing waves with size 113 ± 52 s, and frequency and K  =  =  =  0.16 ± 0.12 mm , velocity 2  1 3.0 waves/mm2/s (model parameters: H  =  5.0, H 2  180 tmIs, IWI =  0.25, P  =  =  36,  0.3). While waves appeared normal, amacrine cell behavior was notably different after  this modification. Cells firing in isolation produced very brief bursts while amacrine cells that contributed to a passing wave depolarized with a duration and magnitude that varied by the cell’s position in the wave (Figure 2.10).  2.4  Discussion  The main strength of this model is its ability to reproduce the statistical properties of retinal waves seen in several species using only a small set of basic principles consistent with known physiol ogy. Amacrine cells have been observed to spontaneously depolarize (Zheng et al., 2006), they regularly depolarize during wave activity (Wong et al., 1995; Zhou, 1998; Zheng et al., 2006), and they release transmitter when depolarized (Borges et al., 1996; Steliwagen et al., 1999), even immediately after passage of a wave (Zheng et al., 2006), thus contributing to wave activity. The model is based on these observations and makes two additional assumptions: that depolarization is increasingly easy to achieve the more time that has elapsed since the previous depolarization,  and that cells which receive more input have longer intervals between spontaneous depolarizations.  55  2.4. DISCUSSION  B X-R  x  4  5  7  6  t  \  L  k  Fig. 2.10. Variable depolarization intervals. A. Difference between excitation and threshold of a single amacrine cell shown over an 11 mm interval. The cell “de polarizes” and becomes active for a fixed duration (0.45 s) when excitation exceeds threshold (i.e., X > R; indicated by the vertical spike). The cell remains depoLarized and excitatory to its neighbors while X > R. After each depolar ization, the threshold increases as a function of how much excitation the cell received and then slowly decays linearly. B Wave activity for each labeled depolarization in A is shown along with excitation patterns to the cell. Position of cell in the waves is shown by arrows. The top plot (X R) is an expansion of that shown in A (gray scale bars = 2 s). The lower excitation plot (X) shows the excitation received by the cell during this same period, with the vertical spike indicating the fixed depolarization. Depolarizations 4 and 5 had no corresponding wave activity and were “intrinsic bursts” Zheng et al. (2006). While some excitation plots show clear wave activity (1, 3, 6, 7), it is not always possible to distinguish between intrinsic bursts and depolarizing amacrine cells on the outer edges of a wave (compare 2 and 5). Using variable depolarization intervals, approximately 25% of depolarizations were intrinsic bursts. —  2.4. DISCUSSION  56  These assumptions are supported by data in a recent study (Zheng et al., 2006) which showed that amacrine cells have slowly decaying afterhyperpolarizing potentials (AHPs), and they have very short refractory periods when pharmacologically isolated, longer refractory periods when they are spontaneously active while connected to neighbors, and even longer refractory periods when they depolarize during a passing wave. The model presented here differs significantly from previous models of retinal waves (Burgi and Grzywacz, 1994; Feller et al., 1997). The most recent and related model (Feller et al., 1997) was based on the assumption of random depolarization of amacrine cells and it required a second layer of RGCs to filter sparse amacrine cell activity to produce wave-like behavior. The present model is based on deterministic activity-dependent refractory periods and produces spatially dense patterns of depolarized amacrine cells. The principle of activity dependent refractory periods is very general and is not constrained by the properties of any particular neurotransmitter pathway or cell type. It produces waves with a large range of spatiotemporal properties and could underlie the production of waves at many different stages of development, in different species, and even in different brain areas. A further difference is that only a single cell layer is required to produce waves, something that was previously not thought to be possible (Feller et al., 1997). It should be stressed however, that while the principles we describe are very general, our model only addresses basic wave behavior that occurs in early development and ignores the emerging complexity of the retina as it matures. There are several ways to implement the basic principles of the model, and we explored some of the possibilities. As described above, amacrine cells were allowed to have both fixed and variable depolarization durations, and the model was run with and without stochastic input. Other strategies that we tested included: producing excitations at random points in the network, as might occur if amacrine cells, or other cells present later in development, were to depolarize spontaneously or to spontaneously release vesicles; using a layer of RGCs to filter amacrine cell activity, similar to (Feller et al., 1997); varying the connectivity radius and the connectivity strength; and using continuous (periodic) boundary conditions. None of these variations resulted in significantly dif ferent behavior, suggesting that the underlying principles of the model are more important for wave generation than their particular implementation.  2.4. DISCUSSION  57  In the model, the magnitude of the threshold regulates the refractory period, and this magnitude depends on recent input to the cell, with cells receiving more input during their periods of activity having higher thresholds. The rate of threshold decay was largely constant, resulting in longer refractory periods in cells that contributed to a wave compared to those that depolarized in iso lation. Biologically, this refractory period results from a calcium-dependent potassium current (Zheng et al., 2006) and possibly other factors, such as an activity-dependent variation in intracel lular chloride, which has been proposed to drive spontaneous activity in the developing spinal cord (Marchetti et al., 2005). The model does not differentiate between mechanisms contributing to the refractory period and only predicts characteristics of the resulting behavior. More physiologically detailed and species-specific models of the retina will be necessary for understanding the finer dy namics of retinal waves, and more experimental data will be required to adequately constrain such models. Given that calcium imaging appears to not detect all wave activity, and the spatial extent of electrode studies is limited, one experiment that would be very helpful would be simultaneous electrode and calcium imaging recordings over retinal areas sufficiently large to discern waves and their boundaries, as this would determine the frequency of very small patches of activity, how much activity is required before calcium signal detection is possible, and how focused or extensive actual wave activity is in relation to the calcium imaging responses. The model produces output that should be useful in computational studies of the developing retino—geniculate pathway (Eglen, 1999; Elliott and Shadbolt, 1999), since the parameters can be adjusted to produce retinal waves with a wide range of size, velocity, and IWI. The output pro vides a relatively uniform net retinal coverage and it is simple to convert it to RGC spike trains by using amacrine cell activity as input to integrate and fire neurons. However, there are at least two significant ways in which this model does not conform to experimental observation. First, the duration of RGC bursting is weakly explained by the present model, as during a wave amacrine cell activity at a given location in the retina typically lasts 1—3 s. We have made no attempts to reproduce or account for the burst variability seen between species (Wong, 1999), including the seconds-long oscillations of excitation observed in turtle retina (Sernagor et al., 2003) or the longer burst times observed in older ferret (Wong et al., 1993). The behavior of the model suggests that additional factors are behind the long duration bursts seen physiologically, possibly involving in put from additional cell types (e.g., bipolar cells) and the use of metabotropic ion channels andlor  2.4. DISCUSSION  58  additional neurotransmitters (e.g., GABA). Second, simple spiking patterns, as would be produced by integrate and fire neurons, are only seen during early development (but see Izhikevich 2004, for integrate and fire neurons which produce various burst patterns). As development proceeds, alpha, beta, and gamma RGCs develop distinct firing patterns (Wong et al., 1993) and ON and OFF RGCs begin to fire at different rates (Wong and Oakley, 1996). Computational studies using the output of this model as input to higher levels of the visual pathway will need to address these factors, as appropriate, according to the particular species and age being modeled. Wave behavior is stable across a wide range of parameter settings (Figure 2.11). Using Figure 2.11 as a guide, model parameters can be manipulated to produce waves quite different from those de scribed here, including waves that slowly progress across all cells, or that produce small groups of excitation that propagate very little. Analysis of the effects of changing different parameters show that the duration an amacrine cell is excitatory (parameter D) is the most important factor in reg ulating the velocity of waves, particularly at slower velocities. When simulating mouse waves, it was necessary to increase D above 2 s to achieve velocities near the 110 urn/s reported physiologi cally (Singer et al., 2001). This suggests that the excitatory mechanism used in these mice involves either a slower excitation, such as would be produced by extracellular diffusion of transmitter, or a reduced rate of transmitter degradation, compared to what occurs in other species. Alternatively, mice may have extended durations of amacrine cell depolarization and/or periods of vesicle re lease. Wave velocity was similarly affected in simulations where amacrine cells were allowed to depolarize for variable durations. Slowing the onset of the AHP thus prolonging depolarization, reduced average wave velocity. Increasing the speed of AHP onset increased wave velocity. A different situation exists in E14—16 chicks, where extraordinarily fast waves are observed (.5  -  1.5 mm/s) (Wong et al., 1998; Sernagor et al., 2003). Extremely short duration excitations can pro duce waves this fast, but the calcium imaging response from such brief depolarizations is greatly attenuated, often below the threshold of detectability. A more natural explanation for the increase in wave velocity is that the excitation time constant (K) approaches zero, causing the excitatory influence from one cell to be quickly realized in others. Physiologically, the adenosine/cAMP pathway may be related to the time constant. Adenosine has been shown to enhance transmitter release and to modulate neuronal excitability (Sebastião and Ribeiro, 1996), and both adenosine  2.4. DISCUSSION  59  150% 100% 50% I  0.3  150%  2 H  1.2 17  K  0.4  I  69  -  100% 50%1  0.5  D  I  2.1 0.1  I  Fig. 2.11. Effects of parameter variations on wave properties. TM/I (I), wave size (S), and velocity (V) for different parameters. The vertical gray lines represent the values of parameters from Table 1A (P2—P4 ferret) and serve as a baseline. Parameter values were varied ±60% from baseline and are scaled linearly. Tick marks on the vertical axis represent 50%, 100%, and 150% response versus baseline.  and cAMP have strong influences on wave velocity (Steliwagen et al., 1999). These results sug gest that adenosine/cAMP might play a decreasing regulatory role in retinas with higher wave velocities, such as chick. The model makes several experimentally testable predictions. One is that wave behavior is the re suit of activity-dependent refractory periods in spontaneously active amacrine cells—normalizing threshold changes, such as by making AHP responses nearly uniform across cells, shouid elim inate non-repeating waves. Related to this, induction of an activity-dependent refractory period in cells which are recurrently connected and spontaneously active should produce non-repeating wave behavior. A second prediction is that wave velocity should be a function of the duration of excitatory influence of an amacrine cell. Manipulating this interval through genetic or pharmaco logical means should influence wave velocity. Third, wave behavior, as measured by RGC activity, should spatially extend beyond waves detected through large-scale calcium imaging, meaning that waves as defined by spike activity should be bigger than those seen by calcium imaging. The commonalities of retinal wave behavior across species and different anatomical and neurophar  2.4. DISCUSSION  60  macological pathways suggest an underlying mechanism that is robust and capable of being im plemented in many ways. Our model displays this flexibility and has been framed to focus on mechanisms likely to be common among species and not to be constrained by specific physiolog ical implementations or specific neural cell types. Hence the principles that we describe may be applicable to the description of activity in other parts of the brain such as auditory system, spinal cord, neocortex, and hippocampus which, like the retina, also exhibit patterns of coordinated spon taneous activity during development (Lippe, 1994; Gummer and Mark, 1994; Yuste et al., 1992; Garashuk et al., 1998; O’Donovan and Chub, 1997; Spitzer and Gu, 1997; Ho and Waite, 1999; Maffei and Galli-Resta, 1990; Meister et al., 1991; Wong et al., 1993; Wong, 1999; Feller, 1999).  61  Chapter 3 Large scale phenomenological modeling of retinocollicular development  3.1  Introduction  Retinal waves, as addressed in the previous chapter, are but one of the many phenomena known to be present and active during retinocollicular development. Here is described a modeling approach that combines the phenomenological behaviors of many physiological mechanisms, including reti nal waves, together into a single, cohesive whole. These physiological behaviors are organized into component models that are described, both conceptually and mathematically, in the next sev eral chapters. Some of these component models, such as that of STDP, are derived from existing *  published models and were adapted to operate in an integrated modeling environment. Others, in particular those of axon and synapse growth, were designed to phenomenologically approximate observed behaviors, and can be expanded to serve as independent models. Each of the component models are described separately, first in the context of physiological studies about their behavior and then by a description of the mathematical implementation. Discussion about each component model is deferred until Chap. 8. Due to the complexity of representing and describing the many behaviors included in this study, each component model was implemented in as simple a way as possible while still approximating the physiological behavior it was meant to represent.  3.2. MODEL OVERVIEW  62  An overview of the model is provided below which includes how it was implemented in relation to the developing retinocollicular pathway. However, the core of the model is generic in the sense that it can be adapted to represent many developing pathways where axons innervate an existing body of neurons. A behavior model that is not currently represented, but that can be added without significant modification, is axons innervating an area of the brain where neurons are constantly being produced and their developmental state exists on a continuum, such as the tecta of fish and frogs, which better enable recreation of retinotectal experiments.  3.2  Model Overview  This model is focused on retinotopic development at the level of individual axons and synapses, starting at the point when RGC axons first arrive at the colliculus and proceeding until a stable and refined retinotopic projection is produced. Axons in the model extend, branch and retract and were designed to reproduce axon growth dynamics as observed in the developing retinocollicular pathway (Fig. 3.1). The model is based on the assumption that molecular guidance cues provide a coarse mechanism for the initial guidance of axons, enabling them to produce diffuse arborizations in their retinotopically correct termination zones, with the cues also providing a bias to synapse generation, and that coordinated retinal activity is then required to refine the axonal projections (Willshaw and von der Malsburg, 1976; Katz and Shatz, 1996; McLaughlin et al., 2003; Cohen Cory and Lom, 2004; Reber et al., 2004; Yates et al., 2004; McLaughlin and O’Leary, 2005). Development in the model closely follows the developmental paradigm as described for mouse superior colliculus (McLaughlin et a!., 2003) and that is observed more generally in chicks and rodents (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005). The model consisted of a simulated retina and colliculus, with each RGC projecting an axon into the colliculus. Model neurons were composed of a multi-segmented axon, a soma with its dendritic tree, and individual synapses (Fig. 3.2). Each axon segment was 15 urn in length and could grow, branch and retract based on the presence of growth and trophic factors, its relative chemoaffinity with local collicular neurons, and its position within the arbor. When axon growth occurred at an axon tip, the axon was considered to be extending and a new axon segment was formed growing  63  3.2. MODEL OVERVIEW  A  Axon extension and overshoot p  B  Topographic branching  ephrin-As  C  Branch guidance and arborization EphBs..  D  Map refinement  ephrin-B1  I  L  termination zone retina ePhrinAsEhA EphAs/ephrin-As RGM?  EphAs/ephrin-As TrkB/BDNF?  EphBs EphBs/ephrin-Bs TrkB/BDNF N pn s/sem a ph on n s  ACh retinal waves  Fig. 3.1. Developmental paradigm Mechanisms and molecules controlling retinotopic mapping in chicks and rodents (Figure adapted from McLaughlin and O’Leary, 2005). Axons originating from a single retinal location are shown in red. A. Axons initially extend through the superior colliculus (optic tectum) along the A-P axis, overshooting their retinotopically correct termination zone. Axons from RGCs at a given location of retina are distributed across the L-M axis. B. Topographic branching then occurs, with interstitial branches on the original axon trunk forming preferentially near the correct location on the A-P axis. C. Branches extend along the L-M axis toward the retinotopically correct termination zone. D. Correlated spontaneous retinal activity, corresponding to the ACh driven stage of retinal waves, then refines the projection. Studies of the ferret retinogeniculate pathway also appear to follow a similar developmental sequence (Huberman et al., 2005).  64  3.2. MODEL OVERVIEW  A  wth 7 j  B  Retinal ganglion cell  Synapses  l5pm  s-  I V  Growth  “Jr,Mi  Axon segments Collicular neuron  Branch  Fig. 3.2. Axon description  A. Each axon was composed of a series of axon segments, each 15 ,im in length, and each could grow by extending or branching. Axon segments formed synapses with adjacent dendrites. B. Axon segments could be of any orientation. When an axon segment extended, it had a tendency to continue in the same direction but the presence of external gradients could alter its course over a range €. Growth and retraction occurred at axon tips. Branching occurred in axon segments that had one child segment. Examples of growth capacity of different segments are shown (striped segments). Axon segments with 3 connected segments (e.g. segment with “X”) do not have further growth until one of its child segments retracts and it again has 1-2 connected segments.  in the same general direction as the existing axon. When axon growth occurred at locations other than an axon tip (i.e. on an axon “trunk”), the axon was considered to be branching and a new axon segment was formed that was largely orthogonal to the orientation of the segment it sprouted from. Axon retraction occurred only at axon tips. The following design constraints were used when cre ating the model: (1) axons had self-limiting growth; (2) growth, branching and retraction decisions were made locally by individual axon segments; (3) the same rules were used by all axons, includ ing those whose termination zones were in either anterior, posterior, lateral or medial colliculus; (4) it was required that the majority of axons should extend branches to near their retinotopically correct termination zones under influence of molecular guidance only, and (5) growth occurring only under molecular guidance cues had to allow neighboring groups of RGCs to produce initially diffuse projections near their retinotopically correct termination zones and not refined projections. Synapse creation in the model was based on similar mechanisms to those controlling axon growth, both in order to keep the model less complex and because physiological observations show that synapse density is higher near axon branches (Alsina et al., 2001), suggesting a possible connection  3.2. MODEL OVERVIEW  65  between the mechanisms responsible for synapse creation and axon growth. Synapses were formed by cooperative activity between RGC axons and collicular dendrites. Synapse survival was based on the concept of synapse resources, with each synapse having a finite amount of resource that was expended by activity and replenished through trophic feedback, providing a framework that allowed each synapse to decide if it should retract or survive using local information. Synapse resources were produced in the soma and were distributed via the axon. The soma managed production of synapse resources, which were used to regulate the size of the axonal synapse population, and it also homeostatically scaled the strength of innervating synapses to help control its average firing rate. Synapses adjusted their potentiation levels based on the timing of pre- and postsynaptic spikes and determined when to retract based both on their level of contribution to inducing a spike in the postsynaptic neuron and the availability of synaptic resources. The extracellular space was partitioned into a grid, with individual sections called grid units. Diffusion of extracellular signaling molecules occurred between grid units. Many component models were used to reproduce phenomenological behaviors observed physio logically. These include models for axons (Sec. 4.1), synapses (Sec. 5.1) and neurons (Sec. 5.2) as well as retinal waves (Chap. 2), spike-timing dependent plasticity (Sec. 6.1), growth bias from molecular guidance cues (Sec. 6.2), growth and trophic factors (Sec. 6.3), NMDAR activation (Sec. 6.4), homeostatic controls (Sec. 6.5) and rapidly diffusible molecules (Sec. 6.6). The fo cus of this study was to determine if approximations of the phenomenological behaviors of each of these components could explain the patterns of retinotopic organization and eye-specific seg regation that are observed experimentally and what the contribution of each of these was in such organization. The study addressed both 1-dimensional (1 D) and 2-dimensional (2D) patterns of growth. In the 1D model, a 1D strip of RGCs projected axons across a 1D strip of colliculus. The axon of each RGC extended across the length of the colliculus and the focus of the model was on patterns of synapse generation and retraction. Each axon was spatially static (i.e. underwent no growth, branching or retraction) but all other aspects of the model were normal. A schematic of the 1D model is shown in Fig. 3.3A and B. 1D simulations ran for 24 hours simulated time. In the 2D model, axons grew, branched and retracted following the patterns described in Fig. 3.1.  3.2. MODEL OVERVIEW  66  1D model A  Retinal ganglion cells • 5.-S  S  2D model  .b  -..e  C Superior Coil icu I us  • •  .  A Collicular neurons  CoHIcLar •:•  B Axon  Retinal ganglion cells Dendrite radius  Axon segments  •  Retina Grid/”” units Dendrite neurons ra di us  •  •  Grid units •  S  •  •  S  S  S  ••.•.•.•.•.•.•.•.•.• Retinal • • • ganglion •.•.•.•.•.•.•.•.•.•. cells •  •  S  •  S  •  S  .  • S  S  •  S S  S  S  S  S S  S S  S  S S  S  •  S  Fig. 3.3. Schematic of 1D and 2D models Retinal ganglion cells project axons into the superior colliculus, where they synapse with local dendrites. A. Cartoon diagram of 1D model. An array of RGCs project their axons across an array of collicular neurons. B. Schematic of 1D model, showing collicular neurons and their placement in grid units. The axon is composed of axon segments, each segment being 15 jim in length. The diameter of a collicular neuron’s dendritic arbor was 50 jim (radius=25 jim) and synapses could be formed with any axon segment that was within a neurons dendrite radius. Diffusion of extracellular molecules was computed in the extracellular grid. The concentration was considered uniform throughout each grid unit. Grid units were square and were 20 jim on a side. C. Schematic of 2D model. ID and 2D models were identical except for the 2D model having axons that grew, branched and retracted. Each RGC extended an axon into the colliculus and produced a diffuse axon arbor near the retinotopically correct termination zone. RGCs and collicular neurons were organized in a triangular lattice as shown. Collicular neurons were assumed to have an inter-cellular spacing of 10 jim.  3.2. MODEL OVERVIEW  67  2D simulations occurred in two stages in order to better recreate observed development patterns. The first stage of the simulation was for 24 hours of simulated time and was restricted to axon growth that was guided by molecular guidance cues. The second stage of growth, also for 24 hours of simulated time, included synapse generation and axon growth, and these were influenced both by molecular guidance cues and by activity-dependent feedback mechanisms. A schematic of the 2D model is shown in Fig. 3.3C. The model used spiking neurons with a simulation time step of 1 millisecond and generated spikes to a resolution of 1 ms. For computational efficiency, the state of each model component was updated at different intervals, depending on the time criticality and computational requirements of each process. Synapse transmission and synaptic plasticity changes occurred at the timing of pre- and post-synaptic spikes. Somatic voltage levels, and the generation of action potentials, were calculated every millisecond. The concentration and diffusion of rapidly diffusing molecules, such as nitric oxide (NO) and arachidonic acid (AA) released from synaptic activity, was calculated every 50 ms (“rapid diffusion update”). Calculation of synaptic and somatic states, including calculation of resource levels, firing frequency, diffusion of growth factors and the decision of whether a synapse retracts, occurred every 500 ms (“structural update”). Updating the state of axon segments, including generation of new synapses, occurred every 5 seconds (“axon update”).  3.2.1  Conventions and mathematical notation  A complete list of the variables used to describe model parameters in this chapter are provided in App. A. 1. Free parameters in the model are listed at the end of each section and in App. A.2, including the range over which each was analyzed. Some equations have terms that require bounds. When an element in an equation has a lower and/or upper bound, this is indicated by square brackets with a trailing superscript/subscript to indicates the bound. For example, [F (x)]gis bounded on [a, bj, meaning that the value of this term in an equation cannot fall below b or rise above a. Similarly, [F(x)]ais bounded on (-inf, a], and  [F(X)]bis  bounded on [b, ml).  The diffusion of rapidly diffusible molecules such as NO and AA was coarsely calculated to the spatial resolution of an extracellular grid, with concentration constant throughout each grid square. This coarse spatial resolution allowed for relatively slow updates.  68  3.2. MODEL OVERVIEW  1.2  1.0  0.8  0.6  0.4  0.2  0.0 0.0  0.5  1.0  3,5  2,0  2.5  3.0  3,5  Fig. 3.4. Output of the function E(n,x). Several formulas in the model utilize a sigmoid-like function that has a stable, near-unity value for small x and that decays to zero with time. The following family of functions was used for these cases:  E(n,x)  These functions have the value E(n, 0)  =  =  (3.1)  2 ’ 1 e  1.0 and E(n, 1)  0.5 for all positive n. The flatness  of E(n,x) for low x, and the steepness of its decay, varies with n. E(l,x) is standard exponential decay with a half-life of x (Fig. 3.4).  3.2.2  Extracellular diffusion  Diffusion of extracellular molecules was managed by partitioning the extracellular space into dis crete units, creating a grid, and calculating the transfer of molecules between grid units. Grid units were 20 um square. The concentration of each molecule was constant in each grid unit and diffusion was dependent on the relative difference in concentration between neighboring units. Specifically: dL  —  —  (La_i —La) t.Jjf  +  (L+i —La)  L  tjjf —  +  1 IU  (3.2) JEXu  3.3. COMPUTATIONAL METHODS AND IMPLEMENTATION  69  where L is the concentration of diffusible compound in grid unit u, x is the set of all neurons in u,  x is the number of neurons in x, and zS.L is the amount of diffusible compound released into  the grid square. The last term is normalized by the number of neurons in the grid unit to prevent artifacts resulting from variable numbers of neurons per grid unit. When L is at the edge of the grid, the term of the equation referencing the L+i or La_i unit that is outside of the grid goes to zero. When diffusing across both X and Y axes (2D model), diffusion was first calculated for the X axis and then the Y. The parameters  dif  and  tdec  represent the time constants governing the rates  of diffusion and decay and vary according to the diffusible compound.  3.3  Computational methods and implementation  The model as described here was implemented in C++ using a hybrid design between event and clock based modeling strategies (Mafia and Giudice, 2000). Its source code, and all relevant scripts, is available upon request. Model neurons were updated on every simulation clock cycle (1 ms) while synapses were updated only on the occurrence of a pre- or postsynaptic spike. When a neuron fired, its would cycle through all axonal synapses, “pushing” excitation onto the target cell of each, and updating synaptic potentiation based on STDP learning rules for a presynaptic spike. The neuron then cycled through its dendritic synapses, updating potentiation for each based on the occurrence of a postsynaptic spike. The model is made up of neurons, axons and synapses as well as a grid representing the extracel lular space and was implemented in an object-oriented framework. The mathematical description of the model components maintains this object oriented style. Modeled collicular neurons were distributed on a triangular lattice with an inter-cellular spacing of 10 jim. Model neurons had a list of all axon segments that comprised its axon as well as lists of axonal and dendritic synapses. Each axon segment stored its location plus a link to its parent and child segments, and its orien tation was computed from the vector between it and the location of its parent segment. Synapses were formed between the dendrites of collicular neurons and overlapping axon segments (den dritic radius = 25 jim). Each synapse had a reference to its target (i.e. postsynaptic) neuron and the axon segment that it was a part of. The extracellular grid was represented as a 2D array of grid  3.3. COMPUTATIONAL METHODS AND IMPLEMENTATION  70  units, each 20 ,im square, and each axon segment and neuron had a link to the grid unit it resided in. Simulations were performed in blocks of 24 hours. The 1D model used one such block, with synapse generation active the entire time. The 2D model used two blocks, the first for chemo affinity driven axon growth and the second with synapse generation and other activity dependent mechanisms active. The simulator used several threads of execution  to spread the computational load across a number  of CPU cores, enabling it to take advantage of both multi-core personal workstations and shared memory supercomputers for larger simulations. Each thread was assigned an approximately equal number of neurons and these were distributed in an arbitrary order. Each simulation step consisted of three stages. The first stage involved performing a “domestic update” on all neurons, where each neuron updated its internal state (e.g. excitation level) and plasticity updates were performed on dendritic synapses if the neuron spiked. The second stage was the “foreign update” where values were exchanged between neurons, such as excitation being pushed onto the dendrite by neurons undergoing an action potential. Plasticity resulting from a presynaptic spike was updated during the foreign update. The third stage occurred every 50, 500 and 5000 ms, when diffusion and structural updates were performed. A barrier method was used to synchronize thread transitions between stages (i.e. all threads had to finish the domestic update before any could begin foreign updates). Diffusion occurred synchronously and all threads blocked waiting for diffusion calculations to complete. Random number generation was done through a modified arcfour cryptographic stream cypher to minimize the possibility of high-order correlations in the random number stream, with each thread of execution having an independent generator so the simulation could produce deterministic 2 to be detected. Data output, allowing individual simulations to be repeated and race conditions from each simulation was stored in an embedded SQL database (sqlite3). Simulation “snapshots” were periodically taken during each simulation and written to the database, allowing analysis of the retinotopic projection at different times during development. Database storage allowed interactive and programmatic analysis of these snapshots using standard SQL. The simulator could resume execution using any of these snapshots as a starting point. The source code for each simulation was 2  A race condition is a situation that arises when the output of an otherwise deterministic process is dependent on the sequence or timing of events, such as the order and relative timing of commands executed by the different threads.  3.3. COMPUTATIONAL METHODS AND IMPLEMENTATION  71  archived to allow any previous simulation to be reproduced and to prevent and correct unanticipated source code ‘drift’. A scripting framework was written in perl to more easily initialize, define and/or customize each simulation. Analysis tools were written in C, C++, per! and Java. Graphical results were produced using perl scripts and Java applications.  72  Chapter 4 Axon development  4.1  Axon model  Model axons were designed to approximate the growth behavior as observed in mouse superior col liculus (McLaughlin et al., 2003), which is similar to the patterns observed in chick and rat (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005), and was modeled as growth under only molec ular guidance cues producing a coarse retinotopic projection and correlated retinal activity refining the connections. Model axons were composed of a connected series of axon segments. Each axon segment was 15 jim in length and could have 0, 1 or 2 child segments. Each axon segment had an “affinity” for growth based on growth factors, trophic factors (Sec. 6.3) and chemoaffinity (Sec. 6.2). The “charm” of each axon segment was a score indicating the compatibility of the axon seg + influx or, possibly 2 ment with its surrounding area, and could be thought to be mediated by Ca more likely, the presence of second messengers or phosphorylated proteins. Axons in the model were assumed to have growth mediating machinery. This machinery, referred to here as axon re sources, was delivered to axon segments that had above average charm, with the amount delivered proportional to the amount each segment was above average. Axon resources slowly accumulated and decayed and were allowed to diffuse between connected axon segments. The amount of axon resources in a segment directly influenced synapse creation, axon growth and axon retraction. The contribution of trophic factor to the charm score, and thus to axon resource accumulation and axon  4.1. AXON MODEL  73  and synapse growth is consistent with the hypothesis that neural activity provides cues that drive the generation of new synapses and axon branches and stabilize existing ones (Katz and Shatz, 1996). Axon growth in the model occurred in multiple stages (Fig. 3.1). In the first stage, axons extended across the length of the target structure (Fig. 3. 1A) and had no preference for their lateral position. In the second stage, axons would branch interstitially and extend across the width of the target structure towards the retinotopically correct termination zone (Fig. 3. 1B,C). Growth was allowed to proceed until it reached a relatively stable state (i.e. minimal continued axon growth and re traction). In the third stage, synapses began to form, allowing activity-dependent mechanisms to contribute to axon guidance and providing additional information to the axon segments and allow ing the axon arbor to refine (Fig. 3.1D). The 1D model was assumed to begin at the end of the first axon development stage (i.e. Fig. 3. 1A), with synapse formation beginning to occur at this point. The 1D model had no subsequent axon growth or retraction, but the development and refinement of synaptic projections followed the same rules as in the 2D model. When an axon grew, a new child segment was formed. The orientation and location of the new segment was based on 3 factors: the orientation of the parent segment, chemoaffinity gradients and growth factor gradients. An extending axon tended to grow in a similar direction to its ex isting trajectory while a branching axon tended to sprout orthogonally from the direction of the parent segment (Fig. 3.2B). The direction of the branching was influenced by external gradients. Gradients were exclusively driven by chemoaffinity for the first 24 hours of development. During the next 24 hours, gradient sensitivity slowly transitioned to be driven primarily by growth factor gradients. To recreate the developmental paradigm shown in Fig. 3.1, axons were assumed to have specific developmental stages. Physiologically, axons initially extend along the entire length of the A-P axis, a direction coinciding with the ephrin-A and EphA gradients, and their position along the L-M axis is highly variable. After axons reach the posterior colliculus, interstitial branches form near the retinotopically correct location along the A-P axis, and these branches extend along the L-M axis, a direction coinciding with the eprin-B and EphB gradients. Subsequent branching and axon growth contribute to refinement of the axonal projection (McLaughlin et al., 2003; Lemke  4.1. AXON MODEL  74  and Reber, 2005; McLaughlin and O’Leary, 2005). To approximate this behavior in the model, axon segments had different behaviors based on the concept of “branch depth” (Fig. 4.1). Each axon that initially extended through the colliculus had a branch depth of zero and its direction of growth was determined only by ephrin-AIEphA gradients. The depth of zero indicated that none of the segments were formed by branching and the growth of depth zero segments corresponds to the development as shown in Fig. 3.2A. After reaching the posterior colliculus, interstitial branches formed on this axon trunk and extended along the L-M axis (Fig. 3.2B). Axons extending from these interstitial branches had a branch depth of one and their growth was determined only by ephrin-BIEphB gradients. Axon segments of branch depth one branched in the vicinity of the retinotopically correct termination zone. Axons extending from these branches had a branch depth of two. The direction of growth in segments with branch depth of two and greater was based on both chemoaffinity gradients (ephrin-AJB and EphAIB) and growth factor gradients (Fig. 3.2C, D). This mechanism was used as it was able to phenomenologically reproduce observed patterns of axon growth. As discussed later (Sec. 8.1.2), it is possible that other implementations might be more effective or biologically accurate than as described here. The most important aspect of the axon model, in relation to producing retinotopic refinement, was that similar mechanisms drive both axon growth and synapse creation and that feedback from successful synaptic connections positively influenced these mechanisms. For example, trophic feedback from a synapse that was successful at inducing a spike in its target cell increased the likelihood of both synapse generation and axon growth (within the synapse’s axon) in the vicinity of that synapse. Reproducing the spatial patterns of axon growth, including the order of growth and the ability of nearby axons to collectively produce diffuse arborizations, was a complex task and the model described below was the most simple version found that could reproduce these growth patterns. Different axon models were used at various stages of model development and the model was not dependent on the use of a particular implementation. Due to the long simulation times of the 2D model (3-5 days per simulation), a systematic exploration of axon growth parameters could not be performed.  75  4.1. AXON MODEL  Superior collicutus Branch Branch Branch Branch  depth depth depth depth  0 1 2 3  RGC  Fig. 4.1. Branch depth Demonstration of branch depth (zjh). Each RGC sends an axon to the superior colliculus and the axon extends and branches through the colliculus. Axons consist of a connected series of axon segments and a continuous series of axon segments is referred to as a trunk. In the image above, every continuous section of axon having the same color is a trunk. Every segment of a trunk is at the same branch depth. Branches off the trunk are of one depth lower than the trunk. These branches extend and form new trunks. The branch depth of an axon segment determines its growth properties, with branches of depth zero being sensitive to ephrin-AIEphA, branch depth one being sensitive to ephrin-BIEphB, and all other branch depths sensitive to both molecular guidance and gradients of growth factors.  76  4.1. AXON MODEL  4.1.1  Implementation: Charm and axon resources  The charm a 112 of axon segment h of neuron i was based on chemoaffinity, growth factors and trophic factors:  ah  =  che,noih 3 I  + 13activity  (cogrowth log (1  (4.1)  + G) + Ptrophic log (1 + Nih))  where Yh is chemoaffinity (Eq. 6.8), G is the amount of growth factor present in the grid unit, u, in which the axon segment resided (Sec. 6.3), Nih is the amount of trophic factor present in the axon segment (Eq. 6.11), and c°growth and ç°trophic are the scaling factors for growth and trophic factor, respectively. /3 is a time dependent scaling factor to control changes to the sensitivity of different factors during development. In the 1 D model, f activity was 0 at the start of the simulation and 3 increased linearly to reach 1.0 at the simulation end. In the 2D model,  activity 3 I  =  0 for initial axon  development and before the onset of synapse generation i.e. the first 24 hour simulation period. -  In the second 24 hour stage of development, f activity started at zero and increased linearly to 1.0 3 by the end of the simulation. Results of 1 D simulations showed that a gradual onset of activity dependent mechanisms improved refinement compared to a constant value of improvement was minor, however, and simulations with a constant normal.  /3chemo  =  actiWty 3 I  =  13activity  =  1. The  1 were qualitatively  1 for all simulations except those involving experimental perturbations.  Eq. 4.1 was recalculated every 5 seconds. Logarithmic values for growth and trophic factors were used to enforce a soft upper bound on these values. Appropriately scaled linear values can be substituted for log values, and were sometimes used during model development, but logarithmiclike responses were found to make the model more fault tolerant and stable when model changes were made or perturbations were introduced, as the effect of changes to growth and trophic factors was restricted to a much narrower range. Axon resources were delivered to segments which had above-average charm, with one unit of axon resource distributed throughout the axon arbor every axon update step (5 sec). Delivery was proportional to how much each segment was above average, and axon resources diffused between adjacent axon segments. Specifically:  77  4.1. AXON MODEL  dsh  (s,  =  WEZh  —  Sjh)  —  Sih  1 wez [a  dec  dif  —aj]  +  (4.2)  —  where Sih is the amount of axon resources present in axon segment h of neuron i, z is the set of all axon segments connected to segment h, segments,  ‘ec  ‘Cjf  is the diffusion constant for resource transfer between  is the decay constant, z is the set of all axon segments for neuron i and  average charm level of all axon segments of neuron i (i.e. ã  =  with  ä  was the  Iz I  being the size of  Sih,  were greater than  set Zj).  4.1.2  Implementation: Axon growth and retraction  In general terms, axon growth occurred in segments whose resource levels,  75% of the maximum resource level in the arbor. Specifically, the threshold fluctuated randomly about this point and increased with increasing axon size and branch depth. Specifically, the thresh old for axon extension,  rowth  rowth =  for axon segment h in neuron i was:  h) Yoo5(l.O+O.O 5 O.7 l  [] Ksegs  max(sh) 1.0  (4.3)  h  with 0.05 being a normal (Gaussian) random variable with mean 1.0 and standard deviation 0.05, Th 9 1  was the branch depth of the segment (see Fig. 4.1),  in neuron i that had above average charm and  icsegs  was the number of axon segments  is a parameter indicating a reference num  ber of axon segments. The actual number of segments in an axon varied with several factors. If the number of segments exceeded  ‘segs  growth began to be inhibited in order to provide a  self-limiting growth mechanism within the axon. Visual inspection of simulations showed that a value of and  lCsegs  lCsegs =  =  75 segments worked well for simulations with 9,000-14,000 collicular neurons  65 for simulations with 6,500-9,000 collicular neurons, based on collicular inter-cell  spacing of 10 tm and axon segment lengths of 15 tim. Smaller values were used for smaller colli culi. Use of higher and lower values (e.g. 45-100) produced functional arbors, with higher values tending to produce heavy arbors while smaller values resulted in fewer axons reaching sufficiently  4.1. AXON MODEL  78  close to their retinotopically correct termination zone to correctly refine. The values listed here provide a balance between restrained arbor size and good retinal coverage. The model assumed that an axon segment would grow (i.e. extend) before it would branch (see Fig. 3.2). The thresh old for axon branching was 20% higher than for axon extension, and only axon segments that had exactly one child segment (i.e. that had undergone extension but did not yet support any branches) were able to branch. Each axon segment had a probability of growth,  ”, ° 1 pf,  that was calculated on each axon update  step (5 sec):  pOW =  [Pgrow  —  (sih  rowth)  (l.O+Nolog(l.O+mjh))]’  (4.4)  where Pgrow was the base probability of axon growth and çoNo was the scaling factor for the sen sitivity to nitric oxide (çoNo  =  1). The upper bound was included here for mathematical accuracy,  but in practice it was never approached.  m  was a modifying factor based on the coincident expo  sure of the axon segment to nitric oxide when the axon segment was recently depolarized (i.e. by a somatic action potential) and it was calculated using:  diiij,,  dt  l.0———  M  (4.5)  where a was the time since the most recent presynaptic spike, coNo was the time window of nitric oxide sensitivity and Mwas the nitric oxide present in grid unit u, the grid unit where axon segment h resided. The value of mth accumulated during each rapid diffusion update (i.e. 50 ms) and was  reset to zero after each calculation of Eq. 4.4, which occurred every axon update (i.e. 5 sec). The threshold for retraction of an axon segment,  Yretract,  was calculated similarly to the threshold  for growth, with the retraction threshold being 90% of the core growth threshold (compare to Eq. 4.3). Specifically:  rac ye t  =  (0.9) (0.75)  [-14] lCsegs  max(s) 1.0  (4.6)  79  4.1. AXON MODEL The probability of retraction was:  retract Pih  —  —  Pretract  1  ‘‘  Sjh  ,,retract Ij  0  where Pretract was the base probability of retraction. Only axon segments that had no child seg ments (i.e. that were axon branch tips) were subject to retraction.  Implementation: Direction of axon growth  4.1.3  A single vector,  =  (q,, qy),  was used to describe the net force from external cues on axon  growth. The lateral-medial (L-M) component of the gradient, qj, was calculated using:  qjj  =  2 gx,yTO.02) + ZjhB + (To. J3activity (gx,yTo02 x+ 1 ,yTO.02) + f O.02 T activity (gx— 1 ,y 3 —  —  —  1.0) (4.8)  where I activity was a time-varying constant used to regulate the onset of growth factor mediated 3 guidance, Z B was the chemoaffinity gradient (Sec. 6.2) and T was a normal random variable 1 of Gaussian distribution with mean of 1.0 and standard deviation of z. Each instance of T was calculated independently. Differences were only calculated for values of x, y in gx,y that resided inside the extracellular grid. For calculations where gx±i  ,  resided outside of the grid, the section  of the equation containing that term was omitted. The heavy use of random variables here was found to be necessary to prevent axons from projecting directly to their retinotopically correct termination zone under the guidance of molecular cues only (discussed in Sec. 8.1.2).  tjwtY 3 J  =  0  for the initial 24 hour stage of axon development and linearly increased from 0 to 1 during the second stage. While growth factor was likely present during the entire period of development modeled, early concentrations were assumed to be largely uniform and so calculating them was not necessary. The effective strength of growth factor, gx,y. in the grid unit x, y where the axon segment resided, was:  80  4.1. AXON MODEL  gx,y  =  (4.9)  log(l.0+G)  where G is the growth factor present in grid unit u, with u corresponding to a particularly location x,y.  A logarithm was used to produce an effective upper bound and eliminate the need for a scaling  factor. The anterior-posterior (A-P) gradient,  qihy.  was calculated similarly to Eq. 4.8 but using  ZjA. Eq. 4.8 was recalculated every axon update step (5 sec).  Eq. 4.8 is conceptually related to Eq. 4.1, where the axon’s charm for a particular section of the colliculus changes with time, as activity dependent mechanisms combine with charm from molecular guidance cues. As with Eq. 4.1, the model was relatively insensitive to changes in the rate of onset of activity dependent mechanisms. Simulations were performed with growth factor gradients always influencing axon guidance and no qualitative differences were observed. The growth behavior of a new axon segment depended on its branch depth (Fig. 4.2). Segments of depth 0 grew along the anterior-posterior (A-P) axis towards the posterior end of the colliculus, and were assumed to be under the guidance of ephrin-AJEphA gradients. The axon entered the colliculus at a random location in the central 70% of the L-M axis. Segments always grew toward the posterior end of the colliculus but were allowed to “wander” a small amount. Specifically, each axon segment of branch depth 0 extended 15 jim along the A-P axis and varied ±3jim along the L M axis, with the variability along the L-M axis determined by a uniform random variable selected on the interval (-3 jim,3 jim). This stage of growth was performed in an initialization script and was complete by the beginning of the simulation (i.e. T  =  0 hours).  Axon segments of branch depth 1 grew along the L-M axis towards the area of the retinotopically correct termination zone and were assumed to be under the guidance of ephrin-B/EphB gradients. This behavior of different growth sensitivities based on branch depth was used in order to ap proximate growth as observed in mouse colliculus (McLaughlin et al., 2003). As discussed later (Sec. 8.1.2), other mechanisms could be used to achieve this same pattern of growth. Segment growth was mechanistically similar to growth in depth 0 segments, with growth instead occurring along the L-M axis, with each segment extending 15 jim along the L-M axis having a variability of ±3 jim along the A-P axis. Each segment of depth 1 that was the result of branching in a depth  81  4.1. AXON MODEL  ephrin-A/ EphA  ephrin-B/EphB 4  ephrin-A/B, ephA/B growth factors  jni 1 ±3  Branch depth 0  Branch depth 1  Branch depth 2+  Fig. 4.2. Axon growth Graphical description of axon growth. Growth of axon segments of various depths are shown originating from existing striped segments. Axon segments of branch depth 6 = 0 grew along the ephrin-AIEphA gradients, which coincides with the A-P axis (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005). Growth largely was along the axis but was allowed to wander small amounts laterally. Segments of branch depth 6 = 1 grew along the ephrin-BIEphB gradients, which coincided with the L-M axis. When growth occurred as a result of axon extension, the direction remained similar to the existing direction of growth. When a segment of depth 1 was produced by branching, the direction was determined by the detected chemoaffinity gradient by the root axon segment (i.e. Zja from Sec. 6.2). The direction of axon segments of branch depth 6 2 was determined by orientation of the root axon segment and the combined gradients of chemoaffinity and growth factors. An example vector of these combined gradients is shown um but its actual size could vary slightly due to implementation of by the red arrow. Segment size was defined as 15 4 the mechanisms described here and by rounding each axon segment position to the nearest micron.  82  4.2. RESULTS  0 segment needed to determine if it was to grow in a lateral or medial direction. This decision was determined by the polarity of the L-M vector,  qj.  A positive value meant that the axon segment  would branch towards the medial side of the colliculus, and a negative qh meant the new axon segment would branch towards the lateral side of the colliculus. Axon segments of branch depth 2 and greater grew based on the orientation of the root axon segment and of combined chemoaffinity and growth factor gradients. The unit growth vector, that described the direction of axon extension was:  -+  -+  20+Q  -÷ =  (4.10)  2O+Q  -+  where 0 is the unit orientation vector of the root axon segment and  Q  is the unit vector of the  combined chemoaffinity and growth factor gradients. When an axon segment was produced as a result of branching of the root segment, its direction of growth was:  —*  20+Q  (4.11)  20+QI J_  -÷  O which was closest to the orientation of  J_  —*--*  where 0 was a vector orthogonal to 0 such that 0 •  Q).  Q  >  0 (i.e. 0 was the vector orthogonal to  On creation, each new axon segment was initialized  to the same level of axon resources, sj, as its root segment.  4.2  Results  The development of individual axon branches occurring over 24 hours is shown in Fig. 4.3. In all 2D simulations in this study, axons were allowed to grow for 24 hours under the guidance of chemoaffinity alone. After this initial development stage, activity dependent mechanisms con tributed to development.  83  4.2. RESULTS  8 hours  16 hours  24 hours  A  B /  I’  Fig. 4.3. Axon development C. The development of the axons from 3 nearby RGCs (representative examples; same simulation) is shown over 24 hours simulated time. Images show axons at 8, 16 and 24 hours. Retinotopically correct termination zone is shown in red. In most cases, axons passed very close to their retinotopically correct termination zone, but rarely were there significant numbers of collaterals there. D. Arbors of 10 adjacent RGCs after 24 hours. Retinotopically correct termination zone shown in red. Simulated colliculus was composed of 13,327 neurons arranged in a triangular with inter-cellular spacing of 10 ,um. Results shown here were from simulations performed in Chap. 7, and a more complete description of the simulation environment can be found there.  A  —  4.2. RESULTS Parameter  iCsegs Pgrow  84 Description  Base value  Reference number of axon segments  65-75 segments  Base probability  of axon growth  Range explored  45-100  30minutes  (sute  1  1 5Thtes  Pretract  Base probability for axon retraction  45 minutes  (Pgrowth  Scaling factor for growth factors  1.0  0  -  conitric  Scaling factor for nitric oxide  1.0  0  -  1.0  0  -  ec  Scaling factor for trophic factors Decay constant of axon resources  if  Diffusion constant of axon resources  2 minutes  Time window of nitric oxide sensitivity  2 sec  Ptrophic  nit Tic 0 (  1 30 minutes  (  (  12O,,rnutej  —  1 l2Omjnutes)  —  1 l0minutes  ( O.2minutes  10.0* 10.0* 10.0* —  —  \ 1 lOOminutes)  20,ninutes)  (0.2-20 sec)  Table 4.1. Parameters for axon growth The value for lCsegs varied according to collicular size (see Sec. 4.1.2). Values in parentheses indicates that parameters in this range were not systematically explored. Asterisk indicates parameter exploration was only performed in the 1D model.  While many of the free parameters governing other model components were analyzed over a range of values, the free parameters of the axon model were not systematically explored, owing mostly to the long runtime of 2D simulations (—4 days for a 13,300 RGC retina projecting to a 13,300 neuron colliculus). Axon model parameters were selected based on subjective analysis of simulations on smaller retinas and colliculi.  85  Chapter 5 Synapse and neuron models  5.1  Synapse model  Synapses in the CNS are formed by cooperative activity of both axon and dendrite (Cohen-Cory, 2002; Goda and Davis, 2003). This was achieved in the model by each axon segment, and each dendrite adjacent to an axon segment, having a small probability of combining to producing a new synapse every axon update step (5 sec). On the axonal side, this probability was influenced by growth and trophic factors (Cohen-Cory, 2002; Goda and Davis, 2003), chemoaffinity compati bility (Dalva et al., 2000) and rapidly diffusible molecules such as nitric oxide (Sec. 6.6). When an axon segment attempted to create a synapse, an adjacent dendrite was randomly selected and queried to see if it would accept a synapse from this axon. Factors affecting whether the synapse was accepted included the number of synapses already existing between this pair of pre- and post synaptic cells and the firing rate of the postsynaptic cell. Synapse survival and retraction is a competitive process that is linked to retrograde signaling (McCann et al., 2007) and transgenic mice overexpressing trophic factor have reduced rates of synapse retraction (Nguyen et al., 1998) consistent with the hypothesis that trophic factors stabilize synapses through retrograde signaling. In the model, synapse survival was based on trophic factor being released by the postsynaptic terminal in an activity dependent manner, and being received by the presynaptic terminal where it was exchanged for resources to promote synapse survival  5.1. S YNAPSE MODEL  86  (Fig. 5.1). Each presynaptic terminal required resources from the soma in order to survive. Each synapse started with an initial resource allocation and retracted when its supply of resources was exhausted. In this manner, trophic factors acted as synaptotrophins (Snider and Lichtman, 1996; Sanes and Lichtman, 1999), where the synapses on each axon compete for a limited amount of trophic agent, or, in this case, they competed for an agent produced through receipt of trophic factor. Trophic factor was released by the postsynaptic terminal following every spike in the post synaptic cell when a vesicle was recently released by the presynaptic terminal (Sec. 6.3). Trophic factor received by the presynaptic terminal was relayed to the axon, where it was converted to synaptic resources and redistributed to local synapses. Synaptic resources were expended with each vesicle released. Two homeostatic behaviors (Sec. 6.5) were mediated through the release and effect of trophic factors. To provide a soft limit on the number of axonal synapses, the amount of trophic factor that a synapse was required to receive in order to survive was controlled by the presynaptic cell, with neurons having increasing numbers of axonal synapses requiring increasing amounts of trophic factor to be received by each synapse. Thus, the more synapses on an axon, the more trophic factor required to be received by each synapse for it to survive. Postsynaptic cells regulated trophic factor release, reducing the amount released per spike when they were over-excited (i.e. above their target firing rate). The effect of this reduction was to make the innervating synapses less competitive within their axons, encouraging the lesser effective of these synapses (or the synapses on more populated axons) to retract, thus reducing total input to the postsynaptic cell. Variations of the model for synapse growth and retraction were explored. First, different paradigms of synapse resource delivery were tested, including the “cooperative” model, as described here, where trophic factor was exchanged for resources within the axon, and then distributed to all nearby  synapses, and an “individualist” approach was explored, where trophic factor was exchanged for resources on a synapse-specific basis. Individualist synapses were found to produce more discon tinuous maps, as there was no bias within the axon for synapses to be spatially organized, although this could be partially rescued by upregulating nitric oxide sensitivity. Second, trophic delivery to synapses was either blocked or uncoupled from the synapse’s ability to induce a spike in the tar get cell. When trophic feedback was blocked or uncoupled, retinotopic refinement was prevented,  87  5.1. S YNAPSE MODEL  A Vesicle release Resources  Trophic feedback Resources +  -  B  ç  Li’  c  1. I  I  II  7 III  III  III  II  Unstable  Stable  Fig. 5.1. Synapse stability  A. Each synapse was assumed to have a finite supply of resources, possibly metabolic or structural in na ture, and these resources were diminished with each vesicle released. Resources were replenished by trophic feedback. When the resources of a synapse were depleted, the synapse retracted. Homeostatic mechanisms within the axon, regulating the number of axonal synapses, regulated the rate at which synaptic resources were replenished. Homeostatic mechanisms in the dendrite, regulating the firing rate of the postsynaptic neuron, regulated the amount of trophic factor released. B. The stability of synapses as a function of the timing of pre- and postsynaptic spikes. The relative times of pre- and postsynaptic spikes are shown as vertical lines, with presynaptic spikes on top. Trophic factor was assumed to be released when the postsy naptic cell fired immediately after the presynaptic cell, a parallel mechanism to STDP induced potentiation. Trophic release is indicated by green dashes. Synapses receiving large amounts of trophic factor were stable and did not retract. The less trophic factor received, the more likely the synapse was to retract.  88  5.1. SYNAPSE MODEL  indicating the need for an activity-dependent feedback mechanism for regulating synapse survival.  5.1.1  Implementation: Synapse formation  The probability of synapse formation, the amount of axon resource pngen =  (sjh)  pehi,  in axon segment h of neuron i, was proportional to  present:  [psyngensih (çogrowth  (5.1)  (1+ log (1+ Ga)) c°No (1+ log (1 +M)))]  where Psyngen was the base probability for synapse generation, ç°growth was the scaling factor for the sensitivity to growth factor, G was the amount of growth factor present in the grid unit, u, where axon segment h resided (Sec. 6.3), çoNo was the scaling factor for the sensitivity to nitric oxide, and M was the nitric oxide concentration. The logarithmic functions provided soft limits on the influence of growth factors and nitric oxide, similar to Eq. 4.1. Psyngen was time-varying, linearly decreasing to 33% of its initial value by simulation end. This behavior was not important to retinotopic organization but was used as it helped reduce the number of outlying synapses in the 1 D model, making quantitative measurements of retinotopic refinement less variable. On every axon update (5 see), each axon segment had a probability,  pehi,  of attempting to gen  erate a synapse with a local dendrite. When this happened, a dendrite was selected at random from the set of dendrites overlapping the axon segment and this dendrite was queried to see if it would accept a synapse from this particular target cell. The probability,  paccePt  that the dendrite of post-  synaptic neuron j would accept a new synapse from presynaptic neuron i was based on three main , of collicular neuron j, the total number of innervating synapses on 1 components: the firing rate, F  j, and the number of existing synapses from i. Specifically:  pcePt  =  2 (3 E  Farget ‘‘  )  E (4  where l arget is the target firing rate (Farget 3 neuron j and  Iy.i  =  iYj lCsyns asoma  )  1.0—  Yj  (5.2)  Kmax_ratio 0  0.2), Yj is the set of synapses on the dendrite of  55 is the size of this set, lCsy! is a reference number of synapses (K  =  40),  cY.,jma  89  5.1. SYNAPSE MODEL is the size (as measured through relative conductance) of the postsynaptic neuron (Eq. 5.4), the ratio of dendritic synapses on neuron  j that  are from neuron i and  Kmax_ratio  is  is the maximum  ratio of dendntic synapses that can come from a given presynaptic neuron. E ()is a sigmoid-like function (Eq. 3.1)1. The higher the value of lCmaratjo  =  iC_ratjo,  the more refined the projection. A value of  0.1 was used in all simulations here to force each collicular neuron to be innervated  by several RGCs. However, larger values for projections, with  K,ratio  =  lCm_ratio  produced increasingly refined retinotopic  1.0 producing much more refined retinotopic projections than that  used in the simulation. The smaller value was used to reflect the tendency of geniculate neurons to accept input from a large number of RGCs during development, before the number of inputs is pruned with maturation (Chen and Regehr, 2000). In summary, this function was designed to reduce the probability of a dendrite accepting a synapse if it was at or above its target firing rate, if there were too many synapses on the dendrite, or if there were too many synapses on the dendrite from the same presynaptic neuron. Different forms of this equation were used and no negative changes to organization were noted, but a systematic analysis of different forms of the equation was not performed.  5.1.2  Implementation: Synapse survival and retraction  Synapse resources in synapse k between presynaptic neuron i and postsynaptic neuron  j are rep  resented by Fjjk. Upon formation, each synapse started with an initial level of resources (.Finiriai 25). The maximum resources allowed in a synapse was  =  =  50. On each postsynaptic spike,  trophic factor was released to the presynaptic terminal, where it was converted, indirectly, to synapse resources (see Sec. 6.3). On each vesicle release, synapse resources were decremented by an amount determined by the total number of axonal synapses:  —  IFijk  =  lCexpectIYiI  (5.3)  The power of 2 on the first sigmoid function was an unintended implementation artifact, providing stronger emphasis to frequency sensitivity than desired. This element of the equation is not important to functioning in the model and retinotopic refinement is improved with its removal. It remains here as the artifact was not discovered until after all simulation data from this study was afready collected. The model was stable across changes to this equation as it was to changes in other mechanisms and to parameter changes.  5.1. SYNAPSE MODEL  90  where yj is the number of axonal synapses and lcxpect is the expected amount of trophic factor to be received by each synapse, per vesicle released, on an axon with 100 synapses (icxpect =0.25). Synapses on less populated axons thus required less trophic factor to be received per spike than did synapses on more populated neurons. Resources were delivered to synapses from the axon every structural update (500ms; see Eq. 6.12). Strictly speaking, decrementing synapse resources by the number of axon synapses (Eq. 5.3) requires unrealistic information to be had by the synapse, which has no practical way of knowing the size of the synapse population. However, this mechanism is roughly equivalent to decrementing resources by a finite amount and altering the rate of replenishment based on the number of axonal synapses, in essence creating an exchange rate, regulated by the soma, of trophic factor for synapse resources. This equivalent mechanism is within the realm of available information as the soma can realistically be assumed to have information on the approximate number of axonal synapses. The method described here was used for computational convenience. The free parameters for synapse survival and retraction were the initial and maximum values for synapse resources, Fjnjtjai 100 synapses, iCexpect  =  25 and F,  =  50, and the expected trophic factor to be received per  0.25. Values explored for Finitiat ranged from 5 to 50, and values explored  for Fm ranged from 20 to 200. Smaller values tended to allow the retinotopic projection to mature more quickly, as analyzed in the 1D model, but there were no qualitative differences observed between different values. The value selected for ICexpect had a strong effect on synapse stability. Because this value was tied directly to the relative amount of synapse potentiation resulting from a postsynaptic spike, it was necessarily bounded by the amount of trophic factor that could be received on each spike (i.e. between 0 and 1). Too low values (e.g. <0.1) resulted in the majority of synapses being stable and not retracting, while values that were too high (e.g. >0.5) resulted in extremely high rates of synapse turnover. The range ICexpect work best.  =  0.20 to  iCexpect  =  0.45 was found to  91  5.2. NEURAL MODEL Parameter  Description  Base value  Amount of trophic factor expected to be received per vesicle released  0.25  Maximum ratio of synapses on a neuron from a single presynaptic source  0.1  Reference number of dendrite synapses  40  ’initial 1  Initial resource allocation to new synapse  25  rmw,  Maximum number of resources in a synapse  50  Psyngen  Base probability for synapse generation  1  .  360 ,ninu(e  Values explored 0 0.6 -  0.05  -  1.0  -  10-50 (5 50) -  25-100(10 200) -  ‘  1 —  I lOSOmini#e)  Table 5.1. Parameters for synapse growth Parameter exploration only performed in 1D model. Values in parentheses indicates that parameters in this range were not systematically explored.  5.2  Neural model  The model used integrate and fire neurons similar to those used in other computational studies (e.g. Troyer and Miller, 1997) but that were modified to support more physiological behaviors and for computational efficiency. Synaptic input was summed at the soma of collicular neurons and when accumulated excitation exceeded firing threshold, the neuron would emit a spike and its excitation was reset. Some of the modifications to the basic integrate and fire model include homeostatic regulation of firing rate, unreliable vesicle release, approximating the effects of dendritic growth, and a more simple form of synaptic current decay. Other neural modeling approaches were ex plored, including model neurons with complex dynamics (Izhikevich, 2003) and two-compartment neurons capable of bursting behavior (see Pinsky and Rinzel, 1994; Kopecs and Lisman, 2003). A more limited implementation of the model described here using Izhikevich neurons did demon strate retinotopic organization and eye-specific segregation (Godfrey and Swindale, 2007b), as did a version using two-compartment neurons. However, the simpler integrate and fire model was used instead because of limitations and unnecessary complexity in the other models. Examination of Izhikevich neurons showed they had a strong after-hyperpolarization that constrained short term spiking behavior by a strong and non-linear inhibition of spikes 10-30 ms after a neuron spiked, limiting its generality. The two-compartment model was not considered necessary as, at least in the developing LGN, immature neurons are more depolarized than mature neurons and only ex hibit tonic firing mode (Liu and Chen, 2008). Further, the slow time course of excitation decay partially mimics the effect of the dual compartment model in producing rapid spiking after strong  5.2. NEURAL MODEL  92  input, thus the simpler 1-compartment model was used to reduce model complexity. The particular neural model used was not found to be of significant importance in the development of retino topic projections. The strength of synaptic EPSPs, however, had a notable effect on the degree of refinement achieved, with weaker synapses achieving the best refinement.  5.2.1  Dendrite growth  Biological dendrite growth is modulated in an activity dependent manner, with innervation from afferent fibers inducing changes in the dendrites to cause them to grow from simple projections into complex shapes that are characteristic of a particular type of neuron (Cline, 2001; Wong and Ghosh, 2002). While dendritic arbors can be highly dynamic, their spatial extent is small com pared to the range over which axons extend, which can be across the entire length of the target structure (Huberman et al., 2005; McLaughlin and O’Leary, 2005). This limited spatial range im plies that dendrite dynamics regulate local functions, such as synapse creation and maintenance, while axon dynamics regulate more regional functions, including how projections are spatially or ganized. Model dendrites started small and grew with time, with growth measured by the electrical size of the dendrite based on observations that total dendrite branch length, and correspondingly volume, increases as dendrites mature (Cline, 2001). For computational convenience, dendrites were assumed to have a constant arborization radius and physical growth was considered to re sult from an increase in arbor complexity. The effect of model dendrite growth was to reduce the somatic EPSP produced by each innervating synapse, which in turn made room for additional synapses as new innervating synapses would be sought by the cell in order to maintain its target firing rate (see Sec. 6.5). Physiologically, dendrite growth is influenced by several activity dependent mechanisms, includ ing trophic factors (McAllister et al., 1996; Yacoubian and Lo, 2000), extracellular signals (e.g. CPG15; Nedivi et al., 1998), by NMDA and AMPA receptor activation (Rajan and Cline, 1998; Haas et al., 2006) and by the activation of intracellular molecules (e.g. CaMKII; Wu and Cline, 1998). In the model, the net results of these mechanisms was approximated by the firing frequency of the postsynaptic cell. The dendrite grew when the postsynaptic cell was firing near or above its  93  5.2. NEURAL MODEL  target firing rate, as this was considered to imply sufficient synaptic input to activate the requisite growth pathways. Dendrite growth, in the manner described here, was found to be important for increasing the degree of refinement in retinotopic projections. The larger the electronic size of the mature dendrite, the better the degree of retinotopic refinement.  5.2.2  Synapse vesicle release  An action potential in a real neuron initiates in the soma and propagates down the axon. When it + influx in the synapse induces synaptic vesicles to release 2 reaches the presynaptic terminal, Ca their contents into the synaptic cleft, which in turn activates postsynaptic receptors and creates a postsynaptic response (Südhof, 2004). Synaptic vesicle release, however, is believed to be an unreliable process in central synapses, and most of the time a synapse does not release a vesicle after an action potential (Goda and SUdhof, 1997). Synapses in the model had a parameter that governed the probability of vesicle release after each action potential. Vesicle release probabilities had a minor effect on retinotopic refinement, with non-reliable release slightly improving refinement, but only to a point lowering the rate of release too far negatively -  impacted refinement.  5.2.3  Implementation: Neural model  As described above, the model neurons used in this study were integrate and fire neurons modi fied for homeostatic scaling of input and dendritic growth. For both flexibility of design and for computational convenience, the non-temporal variables described here were not scaled in a physio logically realistic way. For example, the excitation level of a neuron,  U,  corresponds to the voltage  of the cell with the intracellular resting potential serving as point of reference (i.e. resting potential =  0 mV) but actual values for v were not scaled to the mV scale and so are presented without units.  Similarly, a represents the total electrical conductance of the cell but uses the computationally  94  5.2. NEURAL MODEL  convenient value of o  =  1.0 to represent the conductance of an immature neuron, instead of a  physiologically relevant value, and so it also is presented without units. The conductance of the neuron,  Ysoma,  started at a base value of cYsoma  total somatic and dendntic conductance of an immature neuron.  =  agrowth  1.0, corresponding to the represents the change in  conductance realized by the neuron as a result of dendritic (and somatic) growth. An immature neuron had  growth 0  =  0. The total somatic and dendritic conductance was thus  The maximum growth realizable by a neuron was cYm  =  3.0  soma 5  (0’growth < 0max).  =  1+0 rowth•  When the neuron  received sufficient input to bring it near its target firing rate, its physical size, and thus its total con ductance, would grow. Growth continued with the neuron asymptotically approaching its mature size:  dYgrowth  growth  mtu  —  where  Ftarget  F) is the average firing rate of the collicular neuron (Eq.  the target firing rate (Farget  =  (  —  ut  0.2 Hz).  agrowth  5.7 in Sec. 5.2.4) and  .  )  Ftarget 5  was recalculated at every structural update step  (500 ms). ‘r represents the growth time constant (‘ru  =  12 hours). The total conductance,  which approximates the relative size of the neuron, was also used when calculating the likelihood of a dendrite accepting/creating a new synapse (Sec. 5.1.1). The excitation level, vj, for postsynaptic neuron j was calculated using:  dV dt  where  H”” +  —  —  is the soma decay constant  =  6.14), n 3 represents the summed synaptic input,  (v  (Vexc  —  v)  30 ms), H is a homeostatic scaling factor (Eq.  Dexc  is the reversal potential of excitatory synapses  50) and r is the conductance of an individual synapse. When  tential occurred and v 1 was reset to zero.  (.)  > 1 V  10, an action po  was calculated to produce a specific peak rate of  depolarization in the soma of the target neuron by activation of a single synapse. A non-potentiated synapse would produce an EPSP, 1.0.  iç,,  in the soma of an immature dendrite (i.e.  agrowth  =  0) of  95  5.2. NEURAL MODEL  Excitation to neuron  j from synaptic input, nj, was calculated using: dn  n  —  —  — -  +  exc  where  t  neuron  J’  iTi  ,  rijk Vvjjk  i,key  is the time constant for excitatory input, and Yj is the set of all dendritic synapses on  ‘Pk  is the state of vesicle release (‘I’ijk  =  1 if a vesicle was released and 0 otherwise,  see below) for the k-th synapse between presynaptic neuron i and postsynaptic neuron  j, and Wk  is the strength of synapse k (Eq. 6.5). Eqs. 5.5 and 5.6 were recalculated every millisecond.  5.2.4  Implementation: Firing rate estimation  The average firing rate Fj of collicular neuron with a time constant of ‘r5req  =  j, in Hertz, was calculated using a leaky integrator  10 minutes:  (5.7)  Fj= tfreq  where  4 denotes an accumulator of spiking activity that slowly decays with time. Specifically:  =  —c (i  _e_1/12Yre)  +S  (5.8)  where 3 is the number of action potentials occurring in neuron j over the integration time (500 ms). The constants 120 and 60 were used to align time units: time constant in minutes, frequency in Hz, and the integration time step at 0.5 sec intervals. Eqs. 5.7 and 5.8 were recalculated every 500 ms. The average firing rate was used throughout the model to control several mechanisms and was al ways used in reference to the cell’s target firing rate (Farget  =  0.2 Hz). These uses include dendritic  growth (Sec. 5.2.3), growth and trophic factor release (Sec. 6.3), synapse formation (Sec. 5.1.1) and homeostatic control of firing rate (Sec. 6.5).  5.2. NEURAL MODEL [_Parameter ICEPSP  96 Description Peak synapse EPSP Somatic excitation decay constant Synaptic input decay constant  ‘r tq  Dendritic growth constant Tme constant of firing rate estimator Maximum increase in conductance due growth  Base value 1.0 30 ms  Values explored 0.5  -  3.0  20 50 ms -  2 ms  -  12 hours  -  10 mm  5-20 mm  3.0  0- 5.0  t  ‘Reversal potential’ of synaptic input  50.0  Fa.get  Target firing rate of collidular neuron  0.2 Hz  Pv.,c  Vesicle release probability  0.5  -  -  0.25  -  1.0  Table 5.2. Neural model parameters  5.2.5  Implementation: Vesicle release  On every action potential, synapse k between presynaptic cell I and postsynaptic cell probability Pvesc  =  0.5 of releasing a vesicle. ijk represents if a vesicle was released on a given  action potential, with Wijk 0 otherwise.  j had a  =  1 if the synapse released a vesicle after an action potential and ijk  97  Chapter 6 Additional component models  6.1  STDP  Excitatory synapses are bidirectionally modifiable, with the magnitude and direction of plasticity depending on the timing of action potentials in pre- and postsynaptic neurons (e.g. Bi and Poo, 1998; Zhang et al., 1998). This behavior is reproduced here using a phenomenological model of STDP derived from the STDP triplet model of Froemke and Dan (2002). If a postsynaptic neuron spikes shortly after the presynaptic neuron spikes, the synapse is potentiated, with the amount of potentiation being a function of the time between spikes. If the postsynaptic neuron fires first, the synapse is similarly depressed (Fig 6.1). Froemke and Dan (2002) extended this mechanism to handle triplets of spikes (i.e. 2 pre and 1 post, or 1 pre and 2 post), by assigning pre- and postsynaptic spikes an “efficacy” which was based on the time since the most recent spike in that neuron. The effect of this mechanism is that the second in a series of two spikes in the same neuron has less influence on plasticity than the first. The original model of Froemke and Dan (2002) phenomenologically reproduced the results of con trolled spike triplet experiments. It was modified here to accommodate unreliable synapse vesicle release (Sec. 5.2.2), STDP saturation (Zhang et al., 1998; Froemke et al., 2006) and continuous spike trains. An additional modification was made for computational efficiency where only tern-  6.1. STDP  98  150%  -30 ms  -10 ms +lOms  +30ms  50%  I Fig. 6.1. Synaptic modification by spike interval A plot of synaptic potentiation based on repeated spikes in pre- and postsynaptic cells. Potentiation is shown as a function of the time of the postsynaptic spike relative to presynaptic spike. Vertical axis shows the saturating level of potentiation (depression) that occurs after 60-100 spike pairings (Zhang et al., 1998; Froemke et al., 2006). This is a stereotypical STDP plot (e.g. Zhang et al., 1998; Froemke and Dan, 2002) and was generated from the STDP behavior of the model.  6.1. STDP  99  porally adjacent spike pairs were considered in plasticity calculations. These modifications did not change the qualitative behavior of the original model. All non-potentiated synapses in the model were considered to have a unitary base strength and potentiation and depression was relative to this unitary strength. Experiments show that synapses have characteristic sizes at various points in development (Chen and Regehr, 2000), consistent with this assumption. The magnitude of potentiation(depression) observed experimentally from repeated pairings of pre and postsynaptic spikes is approximated in Fig. 6.1. This observed level of potentiation (de pression) is an asymptotic value and is only reached after 60-80 spike pairs (Zhang et al., 1998; Froemke et al., 2006). The implementation of STDP in this study included saturating behavior to prevent the summation of a large number weakly potentiating spike pairings (e.g. n=400 spike pairs with 30 ms separation) from unrealistically achieving the strong potentiation observed between a much smaller number of closely spaced spikes (e.g. n=80 spike pairs with 7 ms separation), as typ ically occurs in many STDP modeling implementations (e.g. Song et al., 2000; Song and Abbott, 2001; Lubenov and Siapas, 2008). The saturating behavior was also observed in order to pre vent synapses from achieving bimodal distributions of potentiation (Song et al., 2000; Song and Abbott, 2001) instead of the unimodal distribution observed physiologically (van Rossum et al., 2000). Analysis of synaptic weights from simulations in Chap. 7 shows a unimodal distribution with a mean between 0.9 and 1.0 (see Sec. 8.1.6). A complication of using a nonlinear (e.g. saturating) approach for calculating plasticity changes arises when considering continuous spike trains that have highly variable spike pairings, as it be comes unclear what the cunent saturation level for a given spike pair should be. For example, what is the effect of alternating spike pairs that, by themselves, would produce weak and strong potentiation respectively? Would the synapse stabilize at a moderate potentiation level, or would strong potentiation “win” over weak potentiation? Saturation was addressed in one STDP model by Froemke et al. (2006), but that implementation was based on a finite number of spike pairs and did not scale well to continuous spike trains. Another issue not addressed by STDP studies is the combined effect of alternately occurring po tentiating and depressing spike pairs. Several assumptions were made to address these issues.  6.1. STDP  100  First, the present model assumes that a weakly potentiating spike pairing will potentiate a synapse only to the appropriate saturation level for that spike pair and it will exponentially approach this saturation level. Second, a weakly potentiating spike pairing will have no depressing effect on a strongly potentiated synapse. In other words, a spike pairing that saturates at 10% potentiation will cause no net depression (or potentiation) to a synapse afready potentiated by 20%. Third, an individual spike pairing that produces potentiation will, at maximum, potentiate a synapse to the degree observed if the synapse were unpotentiated. LTP and LTD operate through different mecha nisms (Sjöström et al., 2007) and, as such, a spike pair that could only weakly potentiate in normal (non-potentiated) circumstances was considered to always be wealdy potentiating. In other words, a spike pairing that saturates at 10% potentiation, when occurring in a strongly (-40%) depressed synapse, only produced weak potentiation. The absolute potentiation amount was based on the 10% difference between a non-potentiated synapse and the asymptotic level. It was not based on the 50% difference (i.e. 110%  -  60%) of the current synapse strength and the asymptotic level.  This third assumption may appear arbitrary but it is necessary to prevent extremely weak potenti ating (depressing) changes from minimally associated spike pairs (e.g. separated by 100 ms) from driving already depressed (potentiated) synapses quickly to zero. As discussed in Sec., this permissive approach, which is designed to maintain plasticity changed, achieved very minor amounts of plasticity change from baseline in refined retinotopic connections (most synapses stabilize between 90% and 110% of baseline). The implicit repre sentation of NMDAR activation as represented by the STDP model used was found to be very important to achieve retinotopic organization, particularly for regulating trophic factor release. However, the synapse potentiation and depression behaviors of STDP were found to be unneces sary for refinement of the retinotopic projection, making STDP a largely redundant component of the model.  6.1.1  Implementation: STDP  The model described here is derived from the STDP triplet model of Froemke and Dan (2002). To aid intuitive description, the implementation is described in 3 separate steps: the core STDP, as exampled in Fig. 6.1; a compensation to weight changes to account for triplets and quadruplets of  6.1. STDP  101  spikes (Froemke and Dan, 2002); and the saturating behavior of STDP changes. The core STDP behavior, and the use of efficacy to account for triplets and quadruplets of spikes, is from Froemice and Dan (2002). Implementation of saturating behaviors, and the restrictions in the application of plasticity in already potentiated or depressed synapses, are behaviors added in this implementation. The baseline level of synaptic weight change, CjJk, for synapse k between presynaptic neuron i and postsynaptic neuron  j was: —a  (  cijk where ‘r  =  34.5 ms and  =  —  ) ‘  1.0—q÷e postsynaptic spike —a 1.0— pe pre presynaptic spike  ‘6 1  13.3 ms, which were the time constants governing the time win  dow for STDP sensitivity, and q  =  1.03 and q  =  —0.51, which were the peak magnitudes of  potentiation and depression, respectively, a was the time between temporally adjacent spikes in pre- and postsynaptic neurons and was always considered to be positive. More specifically, be cause this implementation of STDP here incorporates unreliable vesicle release, a as measured upon a postsynaptic spike was the time since the most recent presynaptic vesicle release and a as measured upon a presynaptic spike was the time since the most recent postsynaptic spike. Cik represents the potentiation change that produces the stereotypical STDP curve Fig. 6.1. Values for ‘r and q, are from Froemke and Dan (2002). Studies of triplets of spikes (Froemke and Dan, 2002) showed that the direct implementation of the STDP curve of Fig. 6.1 was not adequate to explain potentiation resulting from several spikes occurring over a short time frame and an additional element was required to compensate for multi ple pre- and/or postsynaptic spikes. This element was called “efficacy” (Froemke and Dan, 2002) and acted to reduce the influence of plasticity changes from multiple closely spaced spikes in the same neuron. Each neuron’s efficacy, s, was set to zero immediately after a spike in a postsynaptic neuron, and vesicle release in a presynaptic neuron, and exponentially recovered to 1.0:  E=1.0—e  (6.2)  102  6.1. STDP  The efficacy time constant r 8 was different for pre and postsynaptic neurons: =  tire  =  34 ms and  75 ms (Froemke and Dan, 2002). For postsynaptic efficacy, j, a was the interval between  spikes in the postsynaptic neuron. For presynaptic efficacy,  the interval a was the time since  the last vesicle release by the presynaptic terminal of synapse k. The saturating level of potentiation, SjJk, for a given spike pair was thus:  SJk  =  1  ejC posts)aptic spike £jCi presynaptic spike  (6.3)  Synaptic potentiation (depression) approached its saturation level asymptotically. The strength of an individual synapse, Wik, was unitary for an unpotentiated synapse. The weight change realized by a synapse, IXWjJk, after a pre- or postsynaptic spike was:  1 ‘  where  t  (  [(i +siJk) ’ijk 1 ‘  —  {wJk] 110 postsynaptic spike  [(1 +Sjfk)  jO  —  [4fk]  ]  (6.4)  presynaptic spike  was the time constant regulating how quickly the synapse approached its saturation  plasticity level and  Wijk  =  {0, 1 } indicates vesicle release by the presynaptic cell (Sec. 5.2.5).  The inner brackets []in Eq. 6.4 limit the magnitude of synaptic change in order to limit a weakly depressing(potentiating) spike pair in a previously potentiated(depressed) synapse to the magnitude as would occur in a non-potentiated synapse (i.e. a weakly depressing spike pair was limited to small magnitude changes). The outer brackets prevent spike pairings that saturate at lower magnitude potentiation(depression) from weakening an already potentiated(depressed) synapse. =  0 for synapse k if it did not release a vesicle as a result of the current presynaptic spike  based on the assumption that a synapse not releasing a vesicle would not influence the postsynaptic terminal and would not be subject to depressing effects. The value  t  =  Pvesc 4 ‘  spikes was a  constant governing how quickly synaptic weights would approach their saturation level and was approximated using the observed rate of STDP saturation (Zhang et al., 1998). The value  Pvesc  is  the probability of synapse vesicle release, defined in Table 5.2. Weight changes to each synapse were additive in nature, such that the potentiation level of synapse k between neurons i and .j, WJk. is the sum of all weight changes:  103  6.1. STDP Parameter  Base vaIu  Description  c’--  Peak potentiation realized through STDP  1.03  q,.  Peak depression realized through STDP  -0.51  r  STDP saturation time constant (re is vesicle release probability)  re  Time constant for presynaptic efficacy recovery  25vesc spikes 4 l 34 ms  Time constant for postsynaptic efficacy recovery  75 ms  re  Standard STDP time constant governing synapse potentiation  13.3 ms  roa  Standard STDP time constant governing synapse depression  34.5 ms  ,,  Table 6.1. Parameters used by STDP model Except for ;,all parameters used here are from the base STDP model of Froemke and Dan (2002). r was estimated based on the rate of plasticity saturation reported in experimental studies Zhang et al. (1998); Froemke et al. (2006).  (6.5)  Wijk =W0+ZM4’fk  0 with W  =  1.0 being the starting level of potentiation in a synapse.  The magnitude of trophic factor (see 6.3) and nitric oxide (see 6.6) release was based on the amount of potentiation occurring after a postsynaptic spike. Because these mechanisms seem unlikely to be bounded by behaviors such as saturation and efficacy, the unscaled baseline potentiation level, was used by these model components. Specifically:  k 1 C  -  CLJk  (6.6)  =  0  The parameters of the STDP implementation were not varied as they were derived from the ex isting STDP model of Froemke and Dan (2002), which were calculated to give the best match to experimental data, or were otherwise constrained by experimental measurements. Manipulations of STDP involved disabling changes to synaptic weights (i.e. setting AWJk  =  0 for all spikes).  104  6;2. MOLECULAR GUIDANCE CUES  6.2  Molecular guidance cues  Molecular guidance cues (also referred to here as chemoaffinity or molecular markers, e.g. ephrins and Eph receptors) exist in both RGC axons and neurons in the superior colliculus (and optic tec turn). These cues take the form of orthogonal pairs of opposing ephrin and Eph receptor gradients  (Lemke and Reber, 2005; McLaughlin and  O’Leary, 2005) which provide each RGC axon with a  target “latitude” and “longitude” (Sperry, 1963) in the target structure it projects to. Ephrin gradi ents are found through many brain areas, including visual, auditory and sornatosensory (Feldheim et al., 1998, 2000; Vanderhaeghen et al., 2000; Lampa et al., 2004; Ellsworth et al., 2005; Sid diqui and Cramer, 2005) and appear to be a general organizational mechanism used by the nervous system. While it is becoming accepted that molecular guidance cues and correlated retinal activity play complementary roles in retinotopic refinement (e.g. Luo and Flanagan, 2007, but see Chalupa, 2007), the precise roles of molecular guidance cues and correlated retinal activity in the refinement of retinotopic projections remains controversial (Eglen et al., 2003). On one hand, recent studies by Gosse et al. (2008) show that patterned retinal activity was not required to produce accurate retinal projections in zebrafish tectum (but see Gnuegge et al., 2001, who report that retinal activ ity is required). On the other hand, a large body of evidence indicates that patterned activity plays an instructive role in the refinement of axonal projections and that molecular guidance cues alone are not sufficient to accomplish this task. For example, retinal waves play an instructive role for retinotopic organization (Chandrasekaran et al., 2005) and appear to be required for refinement of retinal projections (Reber et al., 2004; Yates et al., 2004). Further, retinotopic refinement is pre vented by decorrelating RGC activity (Grubb et al., 2003; McLaughlin et al., 2003) or by blocking it (Gnuegge et al., 2001). For review, see Cohen-Cory and Lorn (2004); McLaughlin and O’Leary  (2005). The expression of molecular gradients are well defined on large scales, their representation appears noisy at the local level (Feldheim et al., 1998; Reber et al., 2004), supporting the hypothesis that activity is required to supplement molecular guidance to achieve refined connections. Finally, while activity could be responsible for the regulated distribution of molecular markers (Willshaw and von der Malsburg, 1979; Crowley and Katz, 2002; Willshaw, 2006), thus allowing correlated retinal activity to play a permissive role in organization, Eph and ephrin expression is observed  6.2. MOLECULAR GUIDANCE CUES  105  to be normal under experimental conditions that disrupt both correlated retinal activity and the re finement of retinal projections (Huberman et al., 2005; Pfeiffenberger et al., 2005) and the pattern of disruption to retinotopic refinement is different when either activity or molecular expression is altered (Huberman et al., 2005). A possible resolution to this dilemma is that molecular guidance is itself sufficient to produce refined retinotopic projections in the tecta of small animals, such as the zebrafish (Gosse et al., 2008) or frog, whose tectum is —2% the length of the tectum in chick (McLaughlin and O’Leary, 2005; Lemke and Reber, 2005), as the molecular gradients occur over a limited spatial extent and hence are relatively steep (but see Gnuegge et al., 2001, which suggests that retinal activity is required for refinement even in zebrafish). In larger neural areas, such as the colliculus (tectum) in mouse, chick, monkey or human, the gradients are much more shallow, esp. given their exponential expression pattern (Reber et al., 2004; Lemke and Reber, 2005), and an additional mechanism is required to achieve refinement that occurs only through chemoaffinity in zebrafish and other small animals. Correlated retinal activity is a good candidate this role. From an evolutionary perspective this description is reasonable, as molecular guidance cues would be sufficient to guide refined retinotopic development in small target brain structures but such refinement would be restricted to animals with such small structures. Later, after activity dependent mechanisms had evolved, the size constraint was removed and brain structures could become larger and yet maintain similar degrees of topographic refinement. The model is therefore based on the scenario where molecular guidance cues provide coarse guid ance information to guide the development of synapses and axons (see MartInez and Soriano, 2005; McLaughlin and O’Leary, 2005; Luo and Flanagan, 2007; McAllister, 2007; Segura et al., 2007, for reviews), but these cues are themselves insufficient to produce a refined retinal projection. The strength of the chemoaffinity bias in the model is relatively weak and subject to probabilistic events and ‘noise’ to prevent high degrees of refinement from occurring as a result of chemoaffinity alone. The bias is based on the distance of an axon segment from its target latitude and longitude. Only the bias itself was modeled, not the interactions between the different molecules that produced it, which are many and which vary between species, nor were the interactions between countering gradients that generated the resulting bias (McLaughlin and O’Leary, 2005) modeled. How ephrin  106  6.2. MOLECULAR GUIDANCE CUES  and Eph interactions can produce observed patterns of growth are addressed in other modeling studies (e.g. Yates et al., 2004). Molecular guidance cues were a critical component of the model, providing axons necessary infor mation to direct themselves to appropriate parts of the colliculus, and in a larger context, finding the colliculus in the first place. After axon arbors reached the area of their retinotopically correct termination zones, molecular guidance cues were of greatly reduced or even negligible importance.  6.2.1  Implementation: Molecular guidance cues  Each simulated retina was circular and the position of each RGC was measured in terms of its relative location in the retina, with JA representing the relative location on the NT axis and JB representing the relative location on the D-V axis (Fig. 6.2). Similarly, the colliculus was circular and had a similar set of relative coordinates, with KA being the relative location in the colliculus along the AP axis and KB being the relative location along the L-M axis. KA and KB were calculated to the resolution of the extracellular grid. For convenience, all figures in this study display a retina that was was flipped and rotated so that the retinal point JB, JA was in the same relative visual location as the collicular point KB, KA. The retinotopically correct termination zone of an RGC at JB,JA  was thus KB,KA (Fig. 6.2).  As the model only represents the chemoaffinity bias and not the different molecular gradients, the relative position of an RGC indicated its chemoaffinity in the colliculus, and the affinity of an axon segment for any particular location in the colliculus was a function of the distance to its retinotopically correct termination zone. The distance, djh, of axon segment h of neuron i from its retinotopically correct location was:  dh  2 / K M ) + (JiB V (JiA 1 —  =  —  KjhB)  2  (6.7)  where J and JB represented the position of RGC i. KIhA and KjhB represented the position of the grid unit in which axon segment h of RGC i was located. The 1D model was a special case of the 2D model, using a vertical stripe of neurons in both retina and colliculus, with JB  =  KjhB.  6.2. MOLECULAR GUIDANCE CUES  Retina  107  Superior colliculus  0.0  Fig. 6.2. Relative positioning in retina and colliculus Each position on the retina is measured by its relative location, with a value of 0 indicating the most temporal location along the ephrin-AIEphA gradient (JA) or the most dorsal location along the ephrin-BIEphB gradient (JB). Positional values increase to 1.0 on the opposite side of the retina. A similar coordinate system is imposed on the colliculus resulting in RGCs from retinal location (J , JA) having a retinotopically correct termination zone at (KB, KA), with 8 gradients of axon segments from each RGC were then calculable and chemoaffinity and affinity The KA. K JA JB based on the position of the axon segment within the colliculus compared to the soma position in the retina. For clarity, the retina has been flipped and rotated to align its ephrin-AIEphA and ephrin-BIEphB axes with the colliculus.  108  6.3. GROWTH AND TROPHIC FACTORS  The affinity of an axon segment for a particular location due to molecular guidance cues is based on its distance from its retinotopically correct termination zone, with segments near the correct location having maximal affinity. Specifically:  (6.8)  Yh=E(2,djh)  where Yh is the chemoaffinity of the axon segment h of neuron land EQ is a sigmoid-like function (Eq. 3.1). The choice of function used to convert distance into an affinity score was not found to be particularly important. A sigmoid-like function provides better results as it produces near maximal affinity in a region about KB, KA, allowing (forcing) axons to disperse into a loose pattern near the retinotopically correct termination zone, consistent with physiological observations (Katz and Shatz, 1996; McLaughlin et al., 2003). Use of a simple linear relationship (e.g.  Yih  =  h) is 1 d  also viable but this type of function tends to produce rather sharp refinement even in the absence of correlated retinal activity, and shows only limited improved refinement with activity, inconsistent with physiological observations. Model axon growth followed external gradients of molecular guidance cues (see Sec. 4.1). Be cause chemoaffinity was equated with position, calculation of chemoaffinity gradients only re quired calculating the vector between was ZjhB  =  JB  —  JB,JA  and KB, KA. Thus the horizontal gradient magnitude  hB and the vertical gradient was Z 1 K  =  —  M. 1 K  The chemoaffinity magnitude value Yh is used for axon segment affinity in Sec. 4.1.1. Chemoaffin ity gradients Z and ZhB were used to calculate the direction of axon growth in Sec. 4.1.3.  6.3  Growth and trophic factors  An important contributor to neural development is a group of molecules called neurotrophins, which includes Nerve Growth Factor (NGF), Brain Derived Trophic Factor (BDNF) and Neu rotrophins 3, 4/5 (NT3, NT4/5), which bind to both the p75 receptor and a family of tyrosine  kinase (Trk) receptors, TrkA, TrkB and TrkC (see McAllister et al., 1999; Vicario-Abejón et a!.,  6.3. GROWTH AND TROPHIC FACTORS  109  2002; Cohen-Cory, 2002; Cohen-Cory and Lom, 2004, for reviews). Neurotrophins modulate growth in axons (Cohen-Cory and Fraser, 1995) and dendrites (McAllister et al., 1995, 1996), synaptic connectivity (Cohen-Cory, 2002; Cohen-Cory and Lom, 2004), stabilization of axon ar bors and synapses (Hu et al., 2005), and they modulate, influence and promote the production of synapses (Alsina et al., 2001; Poo, 2001; Goda and Davis, 2003). Infusion of neurotrophins, or introduction of neurotrophin antagonists, prevents eye-specific segregation in the cortex (Cabelli et al., 1995, 1997). Evidence indicates that these compounds can have widespread effects, such as release into the extracellular space to influence populations of neurons, or highly localized effects, such as synapse stabilization (Bonhoeffer, 1996). In the model, the functional behaviors of growth and trophic factors were based on the behavior of these compounds (e.g. NGF, BDNF, NT3, NT4/5) and were split into two distinct categories based on their mechanism of release. Growth factor was assumed to be released into the extracellular space by collicular neurons seeking additional synaptic input, and it was allowed to diffuse from its point of release. Trophic factor release was assumed to be activity dependent and was restricted to the immediate vicinity of the synapse. The function of both growth and trophic factors was to increase synapse and axon growth in the area of the axon where it was received. It was also assumed that growth factors guided growing axons to areas of higher release.  6.3.1  Growth factors  Growth factors in the model were assumed to be released by neurons that were firing below their target firing rate to attract axons and synapses. This assumption is consistent with the hypothesis that neurons have a homeostatic mechanism to control their firing rate that involves the growth of new synapses (Turrigiano, 1999). Released growth factor diffused through extracellular space, increasing axon and synaptic growth in the area of release and providing a gradient for axons to follow. In addition to the neurotrophins listed above, extracellular concentration gradients of sev eral families of molecules act as chemoattractants and chemorepellants for axon growth cones and would be able to play a role in attracting axons (and synapses) to under-innervated cells (McFarlane and Holt, 1997; Song and Poo, 1999). While many molecules can act as attractants and produce functional axon guidance gradients, only a single resultant gradient was used in the model.  6.3. GROWTH AND TROPHIC FACTORS  110  Growth factors were found to accelerate retinotopic organization but were not critical. Simula tions of experimental conditions where growth factors would likely play a significant role, such as attracting axons to newly formed or denervated areas, were not explored however.  6.3.2  Trophic factors  Trophic factors were used locally to signal to the presynaptic terminal how successful and “de sirable” each particular synapse was at producing a response in the postsynaptic neuron and their receipt regulated synapse survival. Neurotrophins have been postulated to be retrograde messen gers that stabilize synapses (Bonhoeffer, 1996). BDNF has been shown to rescue synapses from destabilization and elimination resulting from NMDAR antagonists (Cohen-Cory and Lom, 2004), consistent with this behavior, and implicating NMDA receptors in possibly contributing to its re lease. Trophic factor receipt by presynaptic neurons was proportional to STDP potentiation, an assump tion based on several observations, including that BDNF is needed for LTP and its production is upregulated with activity patterns capable of producing LTP (Poo, 2001), blocking Trk receptors prevents LTP (O’Dell et al., 1991), neurotrophic factors regulate synaptic plasticity (Korte et al., 1996; Schinder and Poo, 2000), NMDAR activation is thought to drive synapse potentiation in STDP (Karmarkar and Buonomano, 2002) and trophic factor is released through NMDAR acti vation (Schmidt, 2004). Algorithmically, this is an appropriate mechanism as synapses that are successful at eliciting a postsynaptic response are the appropriate ones to be reinforced through trophic feedback. Release was restricted to the synapse terminal (Poo, 2001; Cohen-Cory, 2002). Trophic factor was released by the postsynaptic terminal after every spike in the postsynaptic cell in an amount related to the time since the most recent pre-synaptic spike, with amounts propor tional to potentiation realized through STDP rules, and trophic factor received by the presynaptic terminal was relayed to the axon. The role of trophic factor in regulating synaptic plasticity was implicitly represented in the model of STDP and was not represented here. In addition to stabilizing synapses, trophic factors also enhance axon growth and synapse gen eration (Cohen-Cory and Fraser, 1995; Alsina et al., 2001; Poo, 2001; Goda and Davis, 2003;  111  6.3. GROWTHAND TROPHIC FACTORS  Cohen-Cory and Lom, 2004). In the model this occurs both at the point in the axon that received the trophic factor and in neighboring axon segments. The tendency for an axon to grow or to sprout new synapses was related to chemoaffinity cues, local amounts of growth factor, and the recent amount of trophic factor received by an axon segment or its neighbors. Trophic factor was also involved as a homeostatic control for firing rate that regulated synaptic connectivity (Turn giano, 1999), with the target neuron up- and downregulating trophic factor release to manipulate axonal synapse competition and thus affect the number of innervating synapses. Trophic factor release was found to be a critical component of the model, providing feedback to individual synapses by stabilizing successful and appropriately targeted synapses while inducing less successful and mistargetted synapses to retract. It also provided a mechanism to mediate homeostatic balancing of axonal synapse numbers.  6.3.3  Implementation: Growth factor release  Release of growth factor from collicular neuron  j  in grid unit u increased the concentration of  extracellular growth factor, G, by an amount: zG=  where  [i.o_  F) was the average firing rate of the neuron  ‘‘ Farget  ]  (6.9) o  (Eq. 5.7 in Sec. 5.2.4) and  Farget  is the target  (reference) firing rate for collicular neurons. Diffusion occurred according to Eq. 3.2, using the values  =  1 mm and  =  5 sec. Updates to Eq. 3.2 for growth factor diffusion were calculated  every 500 ms.  6.3.4  Implementation: Trophic factor release  Trophic factor was released by the postsynaptic neuron after every spike in the postsynaptic cell in synapses that had recent pre-synaptic activity. Release was proportional to potentiation as deter mined by STDP rules for a given spike, CJk (Eq. 6.6), meaning that spike pairings that produced  6.3. GROWTH AND TROPHIC FACTORS  112  greater amounts of synaptic potentiation also caused the release of greater amounts of trophic fac tor. The amount of trophic factor,  rjfk,  presynaptic cell i and postsynaptic cell  rJk  released to the presynaptic terminal of synapse k between  j is:  CJkDifkE (2, \  ‘  (6.10)  “‘arget  /  where rk is the trophic factor received by the presynaptic terminal after a postsynaptic spike, Jk 1 D  was an activity dependent boost to postsynaptic release (Eq. 6.13), Fj is the firing rate of  postsynaptic neuron  j (Eq.  5.7), Farget is the target firing rate of collicular neurons and EQ is a  sigmoid-like function (Eq. 3.1). The function E(2, z) produces a near unitary response for small values of  z’  equals 0.5 when X  1, and decays in a largely exponential manner for larger  =  thereby gradually reducing trophic release with increasing firing rates in  z,  j.  Trophic factor received by the synapse was relayed to the axon. Within the axon, trophic factor diffused between adjacent axon segments and contributed to an axons affinity for growth (Sec. 4.1.1). Trophic factor with the axon was also converted to synapse resources and delivered to local synapses (Sec. 5.1.2). The amount of trophic factor Nh in axon segment h of neuron i was calculated using:’  dIVh  1 —Nih”\ (Ni  =  IEZh  where  zh  I  \  N tdif  I  J  NIh —  dec  Nih  + r tconvert  is the set of all axon segments connected to segment h,  diffusion between connected axon segments,  ihk  (6.11)  kyh  t  was the constant regulating  was the decay constant, tonvert was the time  constant regulating conversion to synapse resources and Yih was the set of all synapses residing on  axon segment h. Eq. 6.11 was updated every 500 ms. 1  Different labels are used here to indicate individual synapses, ijk and ihk, and both are equivalent. Lety be the set of all retinocollicular synapses. The synapses in y can be labeled ijk, representing all synapses k between presynaptic neurons i and postsynaptic neurons j. The synapses in y can also be labeled ihk, representing all synapses k on axon segments h on all neurons i, as synapses from i only connect to j. Thus, there is a one-to-one mapping between all synapses ijk and synapses ihk, and variables using one label can unambiguously be interchanged with variables using the other label. Such exchange is done here for rhk and rlJk.  6.4. NMDAR ACTWATION Parameter Lf  113  Description  Base value  Values explored  Axon NT diffusion constant  20mm  (2— 200mm)  Axon NT decay constant  20mm  (2— 200mm)  Time constant for synapse resource production  5 mm  -  rlif  Growth factor diffusion constant  5 sec  0.5-50 sec  ?e  Growth factor decay constant  1 mm  0.1-10 sec  Table 6.2. Parameters for growth and trophic factor release and receipt Values in parentheses indicates that parameters in this range were not systematically explored.  Synapse resources were produced from trophic factor in the axon and delivered to all synapses on the axon segment every 500 ms. The amount of synapse resources, Fihk, delivered to synapse k of axon segment h in neuron i was: IFihk  =  r  tconvert IYih  (6.12)  I  where IYih is the number of synapses residing on segment h. In other words, a set amount of trophic factor in each axon segment was converted to synaptic resources and distributed among all synapses on the segment. If no synapses were present (i.e. Ymh I  =  0), the synapse resources went  unused.  6.4  NMDAR activation  NMDA receptors are involved in many aspects of neural behavior, including synaptic plasticity (Shouval et al., 2002), trophic factor release and nitric oxide release (Schmidt, 2004). Some STDP models explicitly represent NMDARs and associate NMDAR activation with synapse potentiation (e.g. Karmarkar and Buonomano, 2002; Shouval et al., 2002) so synapse potentiation in the model was assumed to also be associated with NMDAR activity. The release of trophic factors and nitric oxide in the model was made proportional to STDP potentiation, thus also being implicitly linked to NMDAR activity. NIvIDA receptors are ligand and voltage regulated, requiring not only the binding of glutamate to allow ions to pass, but also the intracellular space to be depolarized. Further, they take much longer  114  6.4. NMDAR ACTIVATION  to be activated than other glutamatergic receptors, such as AMPA receptors. These characteristics result in NMDARs being principally activated during temporal or spatial summation of excitatory input (Hickmott and Constantine-Paton, 1993). Input resulting in a limited postsynaptic response, or a single spike in the postsynaptic neuron, is thus unlikely to elicit as strong of an NMDAR acti vation as input that is sufficient to drive many postsynaptic spikes. It follows that an STDP model that addresses only individual pairs or triplets of spikes cannot completely account for NMDAR activity. To partially compensate for this limited NMDAR response, a simple mechanism was im plemented to alter nitric oxide and trophic factor release such that maximal release only occurred when the postsynaptic cell fired several spikes over a relatively short window (i.e. hundreds of milliseconds). The goal of this mechanism was to provide a coarse approximation of the effects Ca influx and second messenger activation. As + of NMDA receptor response and the resulting 2 discussed in, while alterations of this mechanism did alter retinotopic organization of the model, all changes could be explained by non-independent changes in nitric oxide and trophic factor release. This mechanism is thus a redundant component of the model.  6.4.1  Implementation: NMDAR activation  The modulation of postsynaptic response by NMDAR activation, DjJk, in synapse k between presy naptic neuron i and postsynaptic neuron j, was based on the timing of the previous four postsynap tic spikes, based on the assumption that if the previous several spikes occurred recently, maximal NMDAR activation would occur, with reduced response with only one or two recent spikes, and further reduced response if there was no recent postsynaptic spiking. This was approximated by: 4 l+cONMDARlE  ( 2,  0 [o,Mlj?—aJ] NMD  (6.13)  where ç°NMDAR is a scaling factor, the time of the  tth  NMDAR 0  most recent spike in  is the length of the time integration window and  is  j. It was assumed that the timing of the previous 4 spikes  were sufficient to approximate postsynaptic activity and NMDAR activation, including secondary Ca influx and other second messenger activation. DIJk + effects such as 2  =  0.2 when there were  6.5. HOMEOSTATIC CONTROLS Parameter  115  Description  Base value  ‘PNMDAR  Scaling factor for NMDAR activity  NMDAR 10  Length of integration window for coincident postsynaptic activity  1 1 sec  Values explored 0- 10 0.3  -  3.0  Table 6.3. Parameters for NMDAR activation no recent postsynaptic spikes and Dfk approached unity when the previous 4 spikes were very recent. Different forms of this equation were tried, having low values when there was no recent activity and increasing with postsynaptic activity, and none were observed to improve retinotopic organization. The free parameters of NMDAR activation were  =  1, which was varied between 0 and  1.0 sec, which was varied between 0.3 sec and 3.0 sec.  10, and  NMDAR  6.5  Homeostatic controls  =  (PNMDAR  Neurons have several mechanisms for balancing their input. One includes up and down-regulating the strength of innervating synapses in response to changes in firing rate (Turrigiano and Nelson, 2004; Gonzalez-Islas and Wenner, 2006; Ibata et al., 2008), a process which involves multiplicative scaling of innervating synapses (Turrigiano and Nelson, 2004). The ionic conductances of neurons are also regulated in such a way that inward conductance is lowered when activity is high, thus reducing excitability, and inward conductance is increased when activity is low (van Ooyen, 1994; Turrigiano, 1999). Neurons also conserve total input in the face of changed connectivity patterns (Chandrasekaran et al., 2007). How this is achieved is not yet known but it has been hypothe sized to involve the growth of synapses, axons and dendrites being enhanced/inhibited in response to changed levels of activity (Turrigiano, 1999). Consistent with this theory are observations that axons preferentially extend into sparsely innervated territory (Ruthazer et al., 2003) and that block ing activity or NMDA receptors results in increased axon and dendrite arbor branching (Schmidt, 2004) as neurons are deprived of input and activate mechanisms to obtain more. The increased number of synapses and the increased size axons and dendrites after increased exposure to trophic factors (McAllister et al., 1995; Alsina et al., 2001) implies that homeostatic mechanisms also exist which balance the number and size of these neural components.  6.5. HOMEOSTATIC CONTROLS  116  In the model, multiplicative scaling was used to adjust the EPSP magnitude of all input synapses of a neuron to compensate for it being above or below its target firing rate, approximating a combined effect of regulation of synaptic strength and regulating the ionic conductance, and hence excitabil ity, of the neuron. Previous modeling studies have implemented this behavior before using an “integral controller” (van Rossum et al., 2000) to minimize the error between desired and actual firing rates. The current model uses a simpler implementation that is less exact but that is very flex ible and computationally efficient. Additionally, while multiplicative scaling does alter synaptic weights to compensate for changes in cell activity, it has not been demonstrated to alter synaptic input sufficiently to maintain a particular firing rate and use of a less powerful mechanism allows for additional mechanisms to operate, such as increased synapse formation (Turrigiano, 1999), that also help conserve total synaptic input (Chandrasekaran et al., 2007). Each model neuron has multiple mechanisms to conserve total synaptic input. The first is attractive and involves the release of growth factor to enhance local axon sprouting and synaptogenesis, and to attract new axon branches. The second is regulatory and involves dendritic acceptance of new synapses. The probability that a dendrite accepts a new synapse varies with firing rate. Neurons that are below their target firing rate are much more receptive to synaptic input than neurons at or above their target rate. The third is repulsive and involves reducing trophic support for innervating synapses. Axonal synapses receiving reduced trophic feedback will be seen by the axon machinery as being less “successful” than their peers, resulting in a higher likelihood of removal, which in turn results in less dendritic innervation. The number of synapses and axon segments for each model neuron was also subjected to homeo static controls. The number of dendritic synapses was controlled through manipulation of trophic factor distribution, as described above. The number of axonal synapses was regulated through al tering the expectation of trophic factor receipt for each synapse as a function of the total number of synapses on the axon: axons with more synapses required more trophic factor to be received by each synapse, per vesicle released, than less populated axons, thus raising the bar for synapse survival on more populated axons (Sec. 5.1). The number of axon segments was regulated by in creasing the threshold for axon growth and branching with increasing numbers of segments (Sec. 4.1).  6.6. RAPIDLY DIFFUSIBLE MOLECULES Parameter Ftaret ICH  117  Description  Base value  Target firing rate of collicular neuron  Values explored  0.2 Hz  Upper bound for scaling of synaptic strength  3  -  -  Table 6.4. Parameters for homeostatic control of firing rate Different firing rates were used during model development (typically .1-.3 Hz) and the model showed no dependency on the particular rate used, but the target firing rate for neurons was not systematically altered.  6.5.1  Implementation: Homeostatic firing rate control  The homeostatic scaling factor for synaptic strength, H, is defined as:  H=  [Ftaretf’’  (6.14)  where Farget represents the preferred or target firing rate of a type of neuron (i.e. RGC or collicular neuron) and Fj is the actual firing rate of neuron  j (see Eq. 5.7 in Sec.  scaling factor was given an arbitrary upper bound of  ICH =  5.2.4). The homeostatic  3 to prevent runaway synapse strength  in certain perturbation experiments (i.e. if F fell to 0). The scaling factor for synapse strength was used to calculate neural excitability (Sec. 5.2.3). Some homeostatic mechanisms mentioned here are described in other sections: release of growth factor to attract new input is described in Sec. 6.3.3; postsynaptic acceptance of new synaptic connections is described in Sec 5.1.1; regulation of trophic factor release to alter the number of innervating synapses is described in Sec. 6.3.4; and axonal regulation of synapse count is described in Sec. 5.1.2.  6.6  Rapidly diffusible molecules  Activation of postsynaptic NMDARs by presynaptic vesicle release causes the production and re lease of membrane permeable compounds such as nitric oxide (NO). NO activates the presynaptic guanylyl cyclase / cyclic-GMP pathway (Schmidt, 2004), it is present during the time of retinotopic  6.6. RAPIDLY DIFFUSIBLE MOLECULES  118  map development and refinement, it modulates axon and synapse growth and stabilization (Mize and Lo, 2000; Schmidt, 2004) and it diffuses rapidly and has a short half life. Blocking NO release disrupts normal axon arborization and prevents the spatial segregation of ON/OFF cells in the fer ret LGN (Cramer et at., 1996) and retinotopic refinement is delayed in NO-synthase knock-out mice (Wu et al., 2000). There is some question about the role of NO in developmental organiza tion, however, as blocking it fails to prevent eye-specific segregation in the “3-eyed frog” (RenterIa and Constantine-Paton, 1999) or in the ferret LGN (Mize and Lo, 2000), and in vitro studies show that NO causes the collapse of growth cones (RenterIa and Constantine-Paton, 1996). However, in vivo optical recordings have shown that NO donors enhance axon growth and sprouting (Cogen  and Cohen-Cory, 2000) and growth cone collapse through other pathways, also mediated by cyclic GMP, induce axon branching (Campbell et al., 2001), supporting the evidence that NO plays an active role during growth (see Mize and Lo, 2000, for review). Arachidonic acid is another postsynaptically released membrane permeable molecule that influences synapse and axon growth and stabilization (Fitzsimonds and Poo, 1998; Schmidt, 2004). In the model, synapses release a diffusible molecule, here referred to as NO for simplicity but that could alternatively be arachidonic acid or any combination of molecules with similar properties. NO diffused locally after release and the amount of release was proportional to potentiation from synaptic plasticity. Release was additionally modulated by a mechanism to more accurately re flect NMDAR activation than that implied by the STDP model. The presence of NO increased the probability of synapse generation and axon growth in axon segments that had correlated activity. This implementation is conceptually related to that hypothesized by Gally et al. (1990) and Mon tague et al. (1991). These previous models assumed a form of polysynaptic plasticity mediated by NO. While experiments have shown that diffusible factors are unlikely to be involved in such a role and that plasticity is synapse specific in retinotectal synapses (Tao et al., 2001), NO has been shown to contribute to synaptogenesis (Nikonenko et al., 2003; Trimm and Rehder, 2004) which could produce a similar potentiating effect from co-active afferents. While NO is involved in LW induction (Fitzsimonds and Poo, 1998; Mize and Lo, 2000), its role was not addressed as the phenomenological model of STDP used did not go into this level of detail. Nitric oxide release, and its resulting effects on synapse and axon growth, were found to provide  6.6. RAPIDLY DIFFUSIBLE MOLECULES Parameter  119  Description  Base value  Values explored  t  Decay constant for extracellular nitric oxide concentration  2 sec  0.2-20 sec  dif  Diffusion constant for nitric oxide (between adjacent grid units)  1 sec  0.1-10 sec  Table 6.5. Parameters for nitric oxide release only minor improvements in retinotopic refinement. However, it was also found that NO release could partially compensate for the negative effects on organization resulting from disabling coop erative synaptic survival mechanisms where nearby synapses on an axon share synapse resources (see Sec. 5.1).  6.6.1  Implementation: Nitric oxide release  Synapse k between presynaptic neuron i and postsynaptic neuron  j released NO in the grid unit  u that k resided in. The amount of NO, M, present in grid unit u, was incremented after each  postsynaptic spike:  IXMu  =  (6.15)  CkDifk  where zXM represents the change to the amount of NO present,  ijk  was the potentiation for the  present postsynaptic spike based on STDP rules (Eq. 6.6), Djk is the activity-dependent regulation of release (Eq. 6.13). Diffusion occurred according to Eq. 3.2, using the values and t.jf  =  t  =  1 second. Updates to Eq. 3.2 for NO diffusion were calculated every 50 ms.  2 seconds  120  Chapter 7 Retinotopic mapping and eye-specific segregation  7.1  Introduction  The results of modeling the developing retinocollicular pathway are described in this chapter, in cluding a description of how the model described in Chaps. 3-6 was implemented. The results are split into 2 sections. The first addresses a 1-D version of the model and shows how retinotopic refinement and eye-specific segregation develop as a result of synapse growth and retraction me diated by the physiological behaviors represented in the model. Analysis of these mechanisms is then described, where each was perturbed and the resulting changes to retinotopic refinement and eye-specific segregation were observed. The second section addresses 2D development, showing that the principles of the 1D model scale to 2D with the inclusion of growing axons. Retinotopic organization and refinement is addressed as well as eye-specific segregation. Additional simu lations were performed to explore the relative roles of chemoaffinity and correlated activity in organization.  7.2. METHODS  7.2  121  Methods  2 simulated retina as described in Chap. 2 was used to drive retinal activity. Retinal A 3.65 mm waves were converted to spikes in individual RGCs as described in Sec. 2.2.3. Twenty-four hours of simulated retinal activity was calculated and stored in a data file for each of the retinal wave patterns and RGC spiking paradigms addressed (see Sec. 2.2.3). These files were used to drive RGC activity. For simulations extending beyond 24 hours, retinal activity was repeated. Except where otherwise specified, all simulations used the same retinal activity patterns. For simulations requiring 2 retinas, a second file, based on independent but statistically similar patterns of retinal activity, was used to drive the second retina. A simulation environment was created as described in Chaps. 3-6. All simulations were run under  this environment and used parameters as described there and as summarized in App. A.2. The model was implemented in two forms, a two-dimensional (2D) model, which used all of the model behaviors described in Chaps. 3-6 and was used to explore 2D retinotopic organization and eye secific segregation, and a one-dimensional (1 D) model, which was identical to the 2D version except that axons in the 1D model were spatially static (i.e. no axon growth, branching or retrac tion). The ID model was used to explore retinotopic organization and eye-specific segregation and also to evaluate the contribution of the model’s components to produce such organization. The 1D model consisted of a lD array of RGCs projecting to a 1D array of collicular neurons as shown in Fig 7. 1A,B. The axon of each RGC was composed of a series of individual axon segments  15 urn in length that extended across all target cells. Each axon segment was able to produce synapses and exchange resources and trophic factor with adjacent segments but was unable to grow or retract. The 2D model supported axon extension, branching and retraction but was otherwise identical to the 1D model. A schematic of the 2D model is shown in Fig. 7.1C. The dendrite of each collicular neuron had a radius of 25 jim and an inter-cell spacing between collicular neurons of 10 jim was assumed. For computational convenience, the soma was considered to reside at the center of the dendritic arbor. The extracellular grid on which neurons and axons resided was used to track the concentration and diffusion of extracellular molecules. The grid was composed of connected grid units that were square and were 20 jim on a side. Some 2D simulations used grid units that were 13x13 jim or 21x21 jim and no differences were noted.  7.2. METHODS  122  2D model  1D model A  C  Retinal ganglion cells  ••..  Superior Colliculus  :.— Collicular neurons  B  ColHcular  Retinal ganglion ceHs  Axon  iT)  1 Grid units Dendrite radius  Dendrite radius  Axon segments  Collicular neurons  Grid units  Retina .,...,..  Retinal gangiion cells  Fig. 7.1. 1D and 2D model schematics  A. Cartoon diagram of 1D model architecture. 99 RGCs each extend an axon across an array of 150 SC neu rons. Each axon was spatially static (i.e. no branching, growth or retraction) but all other model behaviors were represented. B. Schematic of 1D model. Each RGC axon is composed of a series of segments, each 15 jim long. Axon segments can form synapses with any neuron with an overlapping dendrite. Dendrite radius was 25 jim. Collicular neurons were organized in a grid. Diffusion of extracellular molecules (growth factor, nitric oxide) as calculated to the resolution of grid units, each was 20 jim long. C. Schematic of 2D model. RGCs from a circular retina project to a circular colliculus. Collicular neurons were also in a grid, with each grid unit 20 jim square. RGC and collicular neurons were organized into a triangular lattice, as shown. Binocular projections were simulated in both 10 and 2D models. In each case, 2 retinas, each with independent spiking behavior, projected to the same colliculus. All other model behaviors were unchanged.  7.2. METHODS  123  For 1D simulations, a vertical and centered 1D slice of the retina consisting of 99 simulated RGCs projected to a collicular array that consisted of 150 neurons. The number of RGCs was arbitrarily selected. The size of the colliculus was selected to be larger than a single retina but smaller than two, to enable both monocular and binocular projections to use the same sized colliculus and be more easily analyzed. Different sized retinas were used (49-199 RGCs) as were different size colli culi (75-300 collicular neurons), with simulated colliculi having fewer, the same, and more neurons than simulated retinas. Results of all simulations were qualitatively similar. For 2D simulations, both retina and superior colliculus were circular and consisted of 6,505-13,327 cells (1.7-3.4 mm ) 2 and 7,915-13,327 cells (0.7-1.2 mm ), respectively, the size depending on the simulation. All sim 2 ulations used the same retinal activity data files as described above, with smaller simulated retinas using only a portion of the simulated retinas. No difference in behavior was noted between small and large simulations, and the size of each simulation reported in Sec. 7.4 was as large as was prac tical using available computational resources. 1D monocular simulations required approximately 30 minutes real time to simulate 24 hours, while full-size 2D monocular simulations (26,654 cells; 3•4 mfl 2 retina to 1.2 mm 2 colliculus) required -4 days real time to simulate 24 hours. The simulation time step was 1 ms. Unless otherwise specified, all time references made here are  in units of simulated time (1 hour  =  3 .6x10 6 iterations). Every 50 ms, 500 ms and 5 sec, the  simulation was paused and the model underwent various additional updates. Every 50 ms, the diffusion of rapidly diffusing molecules (e.g. nitric oxide) was computed (NO diffusion was only calculated to the resolution of the extracellular grid). Every 500 ms, the diffusion of growth factor was calculated, and the neurons in the model underwent a structural update step, which included updating the estimated firing rate, calculating synapse retraction and calculating diffusion within in the axon. Every 5 sec, the axonal resources were distributed and axon segments individually decided if they would extend, branch or retract, and if they would attempt to create new synapses. All simulations were 24 hours long (simulated time) except as otherwise noted. In the 1D model, the simulation began with RGC axons extending across the length of the colliculus, before any synapses were present. Synapse creation began at the start of the simulation and was based on chemoaffinity. Activity dependent influences on synapse generation began slowly, starting with no influence and then linearly increasing to maximal influence at 24 hours. This slow onset of  7.2. METHODS  124  activity dependent contribution to synapse generation produced a minor (<5%; see Sec. 7.2.1) but consistent improvement in retinotopic refinement. However, this behavior was not required as sim ulations having maximal activity dependent feedback from the beginning of the simulation showed qualitatively similar organization. Simulations of the 2D model consisted of two 24-hour stages. The first stage addressed chemoaffinity guided axonal arborizations in the colliculus. During this stage, axons from neighboring RGCs produced a diffuse arborization pattern in the vicinity of the retinotopically correct termination zone. All growth patterns were the result of molecular guidance cues as no synapses were present during this stage. Visual inspection showed —90% of axons had reached a stable state of development and did not exhibit significant additional growth after 24 hours. The second 24-hour stage was a continuation of the first and included synapse generation and retraction. Axons continued to grow, branch and retract during this period. The results of the first 24-hour simulation were saved and used as a starting point for subsequent simulations of the second 24-hour period. Retinotopic refinement continued in simulations lasting longer than 24 hours but the majority of refinement, as measured visually, had been achieved by this point. The initial conditions of the 1D model consisted of RGC axons projecting through the length of the simulated colliculus, with no synapses present. As described in Sec. 3.2, the 2D model was similarly initialized, with each RGC axon projecting across the entirety of the collicular anteriorposterior axis and having no synapses. Simulation variables, corresponding to those listed in App. A. 1, were typically initialized to zero or values appropriate for the parameters listed in App. A.2. To minimize potential artifacts resulting from variable initialization, the simulation was allowed to run for 1% of its runtime before synapse generation began, in order to allow all system variables to reach or approach stable operating values. While most variables were initialized to 0 or 1, no qualitative or quantitative difference were noted, such as perturbations to retinotopic refinement, receptive field sizes or eye-specific segregation, through use of different initial values for variables so long as realistic values were used (i.e. 0 or 100). Unrealistic values (e.g. 1 x 10100) could impact simulation results. A systematic exploration of different initial values was not performed, however. The model as implemented here is of the developing mouse retinocollicular pathway but it can also be considered to apply to the retinotectal pathway of chicks, or more generally, of the retinocollic  7.3. RESULTS-iD  125  ular/tectal pathway where target neurons are in, more or less, uniform developmental states. The current implementation does not apply to the retinotectal pathway in frogs or fish, or anywhere where the developmental state of tectal neurons lies on a continuum and where retinal wave ac tivity is likely absent. Modifications of the model can readily be made to support this form of development.  7.2.1  Quantification of retinotopic refinement  In order to compare results between different simulations, the size of the retinotopic projection in the lD model was quantitatively measured. Two measures were made, the first being the average “width” of the receptive field of a collicular neuron, and the second being the average width of an RGC projection (Fig. 7.2A). To determine each value, the center (mean) of the receptive field and of the RGC projection was measured. The width of the projection was defined as the standard deviation of the distance of all synapses from this mean. This measurement was based on the average of 5 simulation runs. In 2D simulations, the deviation of an axonal projection from its optimal termination zone, as indicated by chemoaffinity cues, was measured.  7.3 7.3.1  Results 1D -  Retinotopic organization and eye-specific segregation  The combination of physiological mechanisms used in the model produced organization and re finement of the retinotopic projection. Fig 7 .2B shows a typical projection produced by a 1dimensional (1D) strip of retina projecting to a 1D strip of collicular (tectal) cells. The projection was initially diffuse, governed primarily by molecular guidance cues. As time progressed, the activity-dependent influences became dominant, with correlated retinal activity driving refinement and trophic factor release mediating it. When two independent retinas projected to the same collicular structure, the projections from each refined, similar to the monocular case, and they also segregated into eye-specific domains.  126  7.3. RESULTS-iD  A Position of RGC  Position of collicuiar neuron  B  C  2 hours  4 hours  zz 22 hours  24 hours  16 hours  8 hours  D  Fig. 7.2. 1 D retinotopic projection and measuring retinotopic refinement  A. Schematic of how data is represented and measured. Results from 1D simulations are shown using synapse density plots. In all density plots, the vertical axis represents the position of the RGC and the horizontal axis represents the position of the collicular neuron. The number of synapses between any two cells shown by intensity of the red dot (plots are scaled so that the maximum number of synapses between a cell pair is bright red). Numerical measurements report the radius of the projection, which is defined as the standard deviation of the projection or receptive field for each neuron. B. Density plots of synapses between RGCs (vertical axis) and collicular cells (horizontal axis) overtime. No synapses exist at the simulation start (i.e. T=0 hours). The initial projection is diffuse and is governed by molecular guidance cues. With time, the projection becomes increasingly refined. All simulations ran for 24 hours and the collicular RF width and RGC projection width were measured at this point. C. Synapse generation was turned off during the last hour to reduce the number of “wayward” synapses and allow for less variation in the measured projection width. No other qualitative differences were observed as a result of turning off synapse generation near the end of the simulation. D. In approximately 25-33% of simulations, border effects caused irregularities at the edges of the projection. Their occurrence showed no clear pattern and their presence was not observed in 2D simulations. To accommodate for this irregularity when measuring RF and projection field widths, only the central 50% of RGCs and collicular neurons were used to produce measurements, as this group was generally stable.  127  7.3. RESULTS-iD 4 hours  2 hours  8 hours  16 hours  A  B  Fig. 7.3. Eye-specific segregation A. Development of a binocular projection is shown over time. Red indicates synapses from one retina and green from the other. No synapses existed at the start of the simulation (i.e. T=0). The projection from each eye is initially diffuse and intermixed with the other, but these quickly segregate into eye-specific domains in the colliculus. B. A similar pattern of organization even occurs when two groups of RGCs exhibit very different patterns of retinal activity. In this case, RGCs from the retina indicated in green bursted twice as frequently and had nearly twice the average firing rate compared to RGCs from the retina shown in red. However, when the RGCs from the red retina did burst, their burst frequency was nearly twice that of those from the green retina. In both A and B, each retina achieves full representation in the colliculus.  Synapses from each retina initially formed a diffuse projection and were intermixed. With time, the projection from each retina refined, as in the monocular case, but with the projections from each retina alternating in the colliculus, and each retina innervating discrete sections of the colliculus (Fig. 7.3A). This is similar behavior as observed in “3-eyed frog” and dually innervated tectum experiments, when two independent retinas project to the same tectum and axons from each eye innervate distinct portions of the tectum (Law and Cons tantine-Paton, 1981; Ruthazer et al., 2003). The independent spatiotemporal patterns of activity in each retina were necessary for segregation, as when both simulated retinas were driven by the same spatiotemporal patterns, segregation failed to develop.  7.3.2  Component analysis  To analyze the contribution of the different model components to retinotopic organization and eye-specific segregation, perturbations were made to the components and the resultant changes to  7.3. RESULTS-iD  128  organization were analyzed. These analyses are described below. The reader is encouraged to revisit the mathematical description of each of these model components in Chaps. 3-6 to better understand how the mechanisms were altered and tested.  Spontaneous retinal activity (retinal waves)  During the period of development when retinotopic organization and refinement occurs, the retina generates spontaneous retinal waves, in which the activity of neighboring RGCs is highly corre lated. The correlation reduces with distance (see Wong, 1999, for review). In this study, a phe nomenological model of retinal waves was used to produce spontaneous retinal activity (Godfrey and Swindale, 2007a). This model is described in Chap. 2. To measure the sensitivity of retinotopic organization to the statistical characteristics of retinal waves, the velocity, size and frequency of retinal waves were increased and decreased compared to control levels and RGC spiking patterns were altered. These alterations included increasing and decreasing RGC firing rates and removing millisecond-level synchronization between adjacent RGCs. Table 7.1 shows the statistical characteristics of the different wave styles analyzed and the results of such perturbations. The results of these simulations showed that retinotopic refinement was very tolerant to variations, showing similar organizational behavior for all tested forms of patterned spontaneous retinal activity. While the effect was minor, retinotopic organization occurring under waves with either long IWI (i.e. low burst frequency) or high velocity was less refined (E and J in Fig. 7.4B, respectively) than control values. Closer examination of development in both situations did not show any significant deficit other than the projection was slower to refine. The model demonstrated a similar degree of organization in 24 hours under control conditions as it had after 36 hours under long IWI and fast waves. Eliminating retinal waves but keeping the average RGC firing rate constant (0.2 Hz) reduced or abolished retinotopic organization (Fig 7.4B). The pattern of initial synapse generation followed the bias provided by molecular guidance cues but the lack of spatiotemporal patterns in RGC spiking prevented refinement of the connection. As the simulation progressed, the strength of  129  7.3. RESULTS-iD  Description  1j iwi  Vel  Size  RGC freq  IBI  RF rad.  Projection  (Sec.)  smIsec  2 mm  Hz  (Sec.)  % change  rad. % change  A  Control (P2-P4 ferret)  117  176  .156  .20  86  (2.86 cells)  (4.14 cells)  B  Unsynchronized RGCs  117  176  .156  .19  84  -2.5%  -2.2%  C  High freq RGCs  117  176  .156  .30  68  -1.2%  -3.1%  D  Low freq RGCs  117  176  .156  .10  100  5.8%  2.6%*  B  High velocity  115  435  .154  .18  71  13%*  12.6%*  F  Low velocity  116  111  .155  .19  77  35%*  G  Large waves  114  171  .31  .20  94  0  H  Small waves  118  177  .097  .20  73  -3.4%  0.9%  I  Short IWI  39  182  .159  .29  34  79%*  -1.8%  J  Long IWI  204  176  .149  .11  144  23.l%*  15.4%*  6.1%*  Table 7.1. Spatiotemporal properties of retinal wave The retinal wave model described in Chap. 2 was configured to produce waves of RGC spiking having the specified inter-wave intervals (IWI), wave velocities, sizes, and firing rates listed above. Bold values indicate changes to particular wave property compared to controL waves. Changes to retinotopic refinement resulting from these changes are shown at right. In summary, there was very limited change to retinotopic organization resulting from changes to wave characteristics. Some changes, particularly for long and short IWI, had secondary changes to overall firing rates. The measured average inter-burst interval (fBI) for all RGCs is also listed. The IBI and IWI values differ due to the different methods used to calculate each (see 2.3.1). Values with asterisk (*) have p<O.Ol (Student T-test).  7.3. RESULTS-iD  130  A 4.0  6.0  3.O  4.0  g -‘  -  a)  3.0 g  2.0  C  In  ci,  2.0 1.0  11.0 ABCDEFGH IJ Type of retinal activity  B 2 hours  4 hours  8 hours  16 hours  Decorrelated retinal activity (no waves)  Fig. 7.4. Effects from variation of retinal activity  A. The radius of the RGC projection (blue) and the collicular cell RF (green) are compared as a function of different patterns of retinal activity. Letters (A-J) correspond to the rows in table 7.1. The reported radius is one standard deviation from the projection (RF) center, in units of number of cells. Waves with different spatiotemporal properties had little effect on retinotopic refinement. B. The effect of eliminating correlated retinal activity while keeping the RGC firing rate constant (0.2 Hz; Poisson distribution). The retinal projection does not refine. No synapses were present at the beginning of the simulation (i.e. T=0 hours).  7.3. RESULTS-iD  131  activity dependent growth mechanisms increased, thereby reducing the relative role of molecular guidance cues, resulting in further deterioration of organization. Physiological experiments in which retinal wave activity is blocked (e.g. McLaughlin et al., 2003; Pfister and Gerstner, 2006) similarly result in lack of refinement, although they do not show such a nearly complete lack of retinotopy in the final projection. Disorder is more apparent in the ID model because the simulated axon extends across the entire colliculus while in real life the axon is restricted to topographically correct region of the colliculus, explaining this discrepancy. Results from the 2D model show more realistic effects of decorrelated retinal activity (Sec. 7.4.1). In all simulations involving two retinas projecting to one colliculus, with both retinas having rel atively equal net levels of retinal activity, qualitatively similar patterns of eye-specific segregation were produced. In physiological experiments in the LGN, when the activity in one retina is upreg ulated compared to the other, the retina with higher activity achieves a larger spatial representation (Steliwagen and Shatz, 2002). To examine the properties of retinal activity that might result in increased spatial representation, spike trains were produced for two retinas that varied in burst (i.e. wave) frequency and average firing rate. The first retina, A, had RGCs with an average firing rate of 0.16 Hz, an inter-burst interval of 119 seconds and a burst firing rate of 24 Hz.. The second retina, B, had an average firing rate of 0.29 Hz, an inter-burst interval of 63 seconds and a burst firing rate of 13 Hz. Assuming that eye-specific segregation could occur between retinas with such differences in firing activity, if a higher burst frequency or higher average firing rate would allow one retina to achieve a larger spatial representation in the colliculus, then more collicular neurons should respond to B than to A. If, on the other hand, the higher burst firing rate was more impor tant, which should correspond with the ability to elicit postsynaptic responses, then more collicular neurons should respond to A than to B. The results from this simulation are shown in Fig. 7.3B. Eye-specific segregation occurred between A and B, and interestingly, 61% of collicular neurons (average, n=5) responded preferentially to retina A (red in Fig. 7.3B) compared to 39% responding to B (green in Fig. 7.3B), despite the higher firing rate of RGCs and the more frequent bursts in B. It thus appears that it is not the firing rate of presynaptic neurons that determines their spatial representation in a target group of neurons, but the relative ability of presynaptic neurons to elicit a response.  7.3. RESULTS  -  JD  132  Chemoaffinity  Countering gradients of molecular markers are found in both the retina and the superior collicu lus (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005), providing RGC axons informa tion about their retinotopically correct “latitude” and “longitude” (Sperry, 1963) and guiding their growth towards their correct retinotopic termination zones. This was implemented as the axon from each RGC having a retinotopically correct termination zone in the same relative position in the colliculus as the RGC had in the retina. Each axon had a bias to grow towards its retinotopi cally correct termination zone as well as having a higher affinity for producing synapses in that area compared to other retinal locations. The model’s implementation of molecular guidance cues is described in Sec. 6.2. This mechanism was very important for the model’s ability to produce retinotopic organization (Fig. 7.5A). While disabling molecular guidance cues prevented global retinotopic organization, it did not prevent organization at the local level, resulting in many discontinuous patches of re fined projections. Studies which partially block molecular guidance cues show a similar pattern of behavior, where the axons that originate from one location in the retina will develop refined pro jections to multiple collicular or geniculate locations (Feldheim et al., 2000, 2004; Pfeiffenberger et al., 2006). Experiments have shown that retinal axons often produce a complete, ordered mapping on the tectum even when portions of the retina or portions of the tectum are removed (discussed in Prestige and Willshaw, 1975). Further, experiments show that retinal axons will continue to project to a similar location in the tectum even if the tectum is rotated, meaning that if part of the tectum was reversed, the axons projecting to that section of tectum would also be reversed (e.g. Yoon, 1975). Fig. 7.5B,C shows the results of simulations where either the retina or the colliculus was truncated. Similar to experimental observation, the retinal projection continues to map over what remains of the colliculus. Fig. 7.5D shows the results of a simulation where half of the colliculus was reversed. Retinal axons still produced refined connections to their retinotopically correct collicular neurons despite the reversal. The absence of chemoaffinity markers had no apparent influence on the development of eye-  7.3. RESULTS-iD  133  2 hours  4 hours  A  16 hours  8 hours  -4  —  1#  a  4.  )C  401  $1  4#  p.  .e  B  C  D  .% wI  ‘0 0’  Fig.  7.5.  Chemoaffinity experiments  Results of different “experiments” manipulating molecular guidance cues.  A.  Disabling molecular guidance  cues (i.e. setting 3 f c herno = 0 in Eq. 4.1) disrupts normal retinotopic organization, producing several discon tinuous patches of refined connections. B and C. Half of the simulated colliculus and retina was removed, respectively. In both cases, the retina projected onto the entire colliculus despite the resulting mismatch of chemoaffinity compatibility.  D.  One half of the simulated colliculus was rotated 180 degrees (which is  equivalent to reversal in a 1D model). RGC axons matched this reversal and projected to collicular cells having the most similar chemoaffinity compatibility.  134  7.3. RESULTS-iD  specific segregation. Retinotopic order of a binocular projection was disrupted similarly to that observed in Fig. 7.5A, with RGCs from each eye continuing to segregate to innervate distinct sections of the colliculus.  Spike-timing dependent plasticity  Potentiation and depression in synapses has been found to result from the timing of individual  spikes in pre- and postsynaptic neurons: if a presynaptic neuron fires shortly before the postsynaptic neuron, the synapse is potentiated; if the postsynaptic neuron fires first, the synapse is  depressed. (Bi and Poo, 1998; Zhang et al., 1998). This plasticity is commonly known as spiketiming dependent plasticity, or STDP. STDP was incorporated into the model used in this study and the implementation was derived from the triplet STDP model of Froemke and Dan (2002) which extends the spike pairing plasticity rules of STDP to account for triplets of spikes in order to estimate plasticity as a result of more realistic spiking behavior. A spike triplet was defined as one spike in the presynaptic neuron and two spikes in the postsynaptic cell, or two presynaptic spikes and one postsynaptic. See 6.1 for a description of how STDP was implemented here.  STDP was found to not be important for retinotopic organization, as disabling synaptic plasticity (i.e. changing Eq. 6.4 to  =  0) did not have a statistically significant effect on retinotopic  refinement (see Table 7.2). When STDP was active, analysis of synaptic weights showed that synaptic weights ranged between 0.85 and 1.05. Synapses at the center of a projection tended to have higher weights than those at the projection edges but this was highly variable. As discussed in Sec. 8.1.6, including the a synapse’s distance from the soma in considering its ability to produce  somatic excitation, which generated a more realistic distribution of synapse weights, also had little effect on organization. Disabling STDP had no noticeable effect on eye-specific segregation.  Model neurons  The model neurons used in this study were integrate and fire neurons (IFNs). Three different types of IFNs were used during various phases of model construction and similar patterns of retinotopic  7.3. RESULTS ID  135  -  Model component  Description  Experiment  Control  RF rad.  Projection  value  value  % change  cad. % change  (2.86 cells)  (4.14 cells)  Control STDP  Block STDP  Model neurons  Peak EPSP  Vesicle release probability  Neural growth  IWJk  0.5  KEPSP  =  3.0  2tvesc  1.0  =  5.0  Ymo.x =  1.0  -  0  -  ço,itric  Pnitric =  tflf  Syn/axon NT sharing ratio  =  =  0  =  3  =  -  3.0  Pgrowth  10  0.1 sec  rf  -  1.0  0.33  1.0  “Individualist synapses”  0  1.0  =  0.125  ICe.epect  0.375 Synapse resource range (starting/max) NMDAR, Ca 2 influx  Scale strength  init/m  =  Fj,it/m ç°NMDAR ‘PNMDAR  Window  measurements were  refers to T-test).  the  based  =  1.0  =0.3sec =  3.Osec  oNMDAR  3.1%  11.8%  8.9%*  6.6%*  .8.1%* 16.8%*  18.1%*  62.4%*  64.2%*  9•5%*  17.5%*  48.4%*  14.2%*  -0.7%  2.8%  =  -  -1.7%  0.8%  -2.6%  -0.9% -4.5%  1.8%  245.6%*  36.1%*  36.6%*  +2.3%  -2.0%  +8.0%*  +8.3%*  20.0%*  19.0%*  7.0%*  -  (ONMDAR  58.6%*  14.4%*  25/50  0  NMDAR 0  555%*  1.Osec  0.8%  3.4%  -0.8%  -2.9%  effect on retinotopic refinement  the average of 5 simulation runs. Percent change RF and projection widths. Values with asterisk (*)  on  average change of  0.25  50/100  Table 7.2. Experimental perturbations and All  =  -  10  -5.2%  549%*  25/50  =  0.7%  -8.7%  -4.5%  10/25  =  -0.6%  .7.6%* 1.0 sec  =  10.0 sec  0.67  ICexpect  1.0  =  -  (see text)  Resource expectations  1.0  =  -  0  =  0.5  7rvesc =  =  Diffusion tau  Synapse dynamics  -  ‘ec  =  1.0  =  o,,,,  Scale strength  Scale strength  ICEPSP  0.25  =  Pgrowth  Nitric oxide  -  =  max  Growth factors  0  =  CEpSP  referenced in have p< i . 0  the text (Student  136  7.3. RESULTS JD -  organization were observed in each case. These models include Izhikevich thalamocortical neu rons (Izhikevich, 2003), two-compartment integrate and tire neurons (simplifications of Pinsky and Rinzel, 1994; Kopecs and Lisman, 2003), and single-compartment IFNs as described in Sec. 5.2. All produced qualitatively similar patterns of retinotopic organization and refinement, and eyespecific segregation. Only single-compartment IFNs were used to generate the results reported in this study. While changes to the time constants used for neural excitation (Zma 20-50 ms) and for calculating the average firing rate  (rYreq  =  30 ms; analyzed over  10 mm; analyzed over 5-20 mm)  did not have significant effects on retinotopic organization, altering model components governing the strength of synaptic connectivity had significant effects. There were three main variables that governed the effective postsynaptic potential: EPSP magnitude (Pvesc)  (KEpSp),  vesicle release probability  and neuron growth (agrowth)’. Manipulations to EPSP magnitude and neuron growth, which  directly regulate the magnitude of somatic depolarization to synaptic input, showed that the smaller the relative strength of an individual synapse, the more refined the retinotopic projection. Baseline values used by the model produced weak postsynaptic effects by individual synapses, largely as a result of these findings. As such, changes to the model which decrease the strength of individual synapses had small effects. Increasing the size of somatic EPSPs notably impacted the rate and/or ability of the model to achieve retinotopic refinement. The baseline peak EPSP creasing  KEpSp  (icEpSp)  was 1.0 and a neuron’s firing threshold was  Xhresh  =  10.0. In  to 3.0 while holding all other behaviors constant increased the average RF and  projection width by 57% (see Table 7.2). Decreasing 1 EpSp to 0.5 improved refinement by 7%. Growth of neurons in the simulation was modeled by increasing the electrical size (asoma) with time which produced a correspondingly reduced somatic EPSP from each individual synapse. Baseline rates called for a mature neuron to be 400% of its original, immature size, resulting in a synapse on a mature neuron producing a somatic EPSP —25% of that realized on an immature neuron (Eq.  5.5). Reducing the maximum size of a mature neuron to be twice that of an immature neuron (c 1  =  1.0) increased the average RF and projection width by 17%. Eliminating neuron growth  Neuron and dendrite growth affected the strength of synapse connectivity because the larger the electrical size of the cell, the smaller the somatic response due the input from a single synapse.  137  7.3. RESULTS 1D -  altogether (a,,  =  0.0), such that each synapse would produce the same EPSP on an immature  neuron as on a fully mature neuron, increased the average RF and projection width by 63% versus baseline. Increasing growth such that the mature neuron was 600% of its immature size (o,  =  5.0), improved refinement by an average of 7%. Synapses in the model had a probability of releasing vesicles on each action potential. The baseline  0.5, meaning that a synapse had a 50% likelihood of  rate of synapse vesicle release was Pvesc  releasing a vesicle on a given action potential. Increasing Pvesc to 1.0, thus increasing effective synaptic strength, increased the average RF and projection width of 8%. Reducing the probability of vesice release did not improve refinement, however. Lowering Pvesc to 0.25 increased the average RF and projection width by 7.5%, despite an effective 50% reduction in the strength of an individual synapse. Similar to how changes in effective synapse strength altered retinotopic organization, eye-specific segregation was also altered. Fig. 7.6 shows the progression of retinotopic organization and eyespecific segregation under changes to cr,,  KEPSP  and Pvesc. In all cases, both retinas segregate  to innervate distinct sections of the colliculus, but the rate at which segregation develops appears notably slowed under the same conditions that resulted in reduced retinotopic refinement, such as high values for  ICE psp  and low values for  cYmax.  Nitric oxide release  The rapidly diffusible, membrane permeable molecule nitric oxide (NO) has been implicated in some forms of LTP, it modulates axon and synapse growth (Mize and Lo, 2000; Schmidt, 2004) and is hypothesized to play a significant role in spatial learning and retinotopic organization (Mon tague et al., 1991). NO is released by NMDAR activation (Fitzsimonds and Poo, 1998) and it is well suited to provide a retrograde spatial signal to simultaneously active afferents to help induce synapse generation or axon sprouting in the locality of is release. To explore this behavior, simu lated NO was released into the extracellular grid by active synapses as a function of approximated NMDA receptor response. Its presence increased the probability of axon growth and synapse for mation in axon segments that were depolarized when NO was present. This is described in Sees. 6.6 and 6.4.  138  7.3. RESULTS-iD  6 hours  12 hours  18 hours  24 hours  control  °max =1  max 0  KEPSP  =3.0  KEPSP  =0.5  Pvesc =1.0  Pvesc =0.25  Fig. 7.6. Changes to binocular projection from changes in synapse strength Development of eye-specific segregation as a function of changes to effective synaptic strength. Changes that increase the relative strength of individual synapses slow the development of eyespecific segregation. Discontinuities and reversals in the projection, as shown in Fig. 7.2D, occur more frequently in 1 D binocular projections compared to 1 D monocular development.  7.3. RESULTS JD  139  -  Altering the sensitivity of an axon segment to the presence of NO when it was depolarized had very minor effects on organization. Eliminating the effect of NO altogether  (çoNo  insignificant changes to retinotopic refinement, while increasing it 10-fold  (coNo  =  0) resulted in  =  10) improved  refinement by an average of 7% (see Table 7.2). However, further increases to çoNo PNo  =  20 and  100 had increasingly negative effects on retinotopic refinement as measured by projec  tion and RF width. Changing the rate of diffusion 10%-1000% of baseline (‘rf =  =  =  0.1 sec to  10 sec) produced no significant change in retinotopic refinement in the 1D model (<2%  change). These results suggest that a coincidence-detecting, volume based learning mechanism plays a minor role in regulating patterns of synapse generation during retinotopic refinement. However, as discussed in Sec., the effect of NO could be masked by cooperative survival mechanisms for neighboring axonal synapses as these two mechanisms produce similar end results (i.e. spatial concentration of synapses). Because this analysis only addressed synapse generation, it remains possible that NO might play a volumetric role in axonal growth behaviors. No qualitative changes to eye-specific segregation were observed under changes to NO release using values for cYNO  e  {0, 0.3, 1.0, 3,010.0} and for  E {0.l, 0.3, 1.0, 3.0, 10.0} seconds.  Synapse dynamics and trophic factor  Synapses have to make the decision whether or not to retract based on information available to each individual synapse. In this study, synapses were assumed to have finite resources and when these resources were expended, the synapse retracted. Resources were expended on each vesicle released and were replenished based on how much trophic feedback the synapse received. A brief summary of this mechanism is provided here (see Sec. 5.1 for a more detailed description). Each synapse had an expected amount of trophic factor it needed to receive, per released vesicle, and this expectation was based on the total number of axonal synapses. The more synapses there were on the axon, the less resource was available to be delivered to each synapse, thus each synapse needed to receive more trophic factor in order to survive. Each synapse could receive a maximum of 1.0 units of trophic factor per presynaptic spike, and the baseline expectation of trophic factor to be received for each vesicle released was  =  0.25,. based on a population of 100 axonal synapses. This  expectation was scaled linearly based on the actual number of synapses. For example, an axon  7.3. RESULTS JD  140  -  with 50 synapses had an expectation of 0.125. Each synapse started with resources and had a maximal amount of l’  =  initia1  =  25 units of  50 units. On the release of each vesicle, a synapse’s  resources were decremented by the expectation level and incremented by the amount of trophic factor received. When a synapse depleted its resources, it retracted. A 50% increase in the expected synaptic resources  (1xpect  0.375) resulted in a 14% decrease  in the RF and projection width (see Table 7.2) while a 50% decrease  (‘cexpect  =  0.125) resulted in  RF and projection widths 36% larger. Raising the bar for synapse survival, where only the very fit survive, thus seems to improve retinotopic refinement, or at least the rate of refinement, both by increasing the rate of maturation and by eliminating lesser performing synapses more quickly. As discussed in Sec. 5.1, the model assumed that the level of synapse resources increased after every postsynaptic spike. Different mechanisms were explored to achieve this increase. The base line mechanism involves a scaled release of trophic factor from the postsynaptic terminal, with the amount of release based on how much a particular synapse contributed to a spike in the postsynaptic neuron. The receipt of trophic factor was relayed to the axon, resulting in a distribution of synaptic resources to all nearby presynaptic terminals on the axon, allowing synapses to act cooperatively. This cooperative action biased axonal synapses to cluster into groups, as neigh boring synapses that received more trophic factor were able to assist less successful neighbors. To examine the reliance of the model on this cooperative mechanism, trophic factor delivery was altered whereby each synapse received trophic factor and this was converted directly to synapse resources at the synapse, with no sharing of resources between synapses (i.e. an “individualist” strategy). Trophic factor was still relayed to the axon to influence behaviors such as synapse gen eration, but no resources were distributed back to the synapses. Specifically, Eq. 6.12 was changed to zil’hk  =  rJk.  Fig. 7.7B shows the results of this arrangement. Restricting trophic factor to the  individual synapse resulted in a small reduction in receptive field sizes but it significantly disrupted the spatial continuity of the target neurons that an RGC projected to, greatly increasing the pro jection radius. Sections of the axon that had the most “successful” synapses still received trophic feedback from them, and so synapse generation was more likely in parts of the axon where the more successful synapses were, but this bias was insufficient to provide spatial continuity to the synaptic projection. Intermediate levels of sharing between fully individualist and fully coopera  141  7.3. RESULTS 1D -  tive modes were also explored, with axonal synapse grouping becoming increasingly disrupted at more individualist levels. Nitric oxide release, as implemented in the model, also provided a spatial bias for synapse gener ation by increasing the rate of synapse production in areas where successful and temporally asso ciated synapses already reside. The effect of NO release on the projection radius of an RGC and its influence on the axonal projection was thus examined in the context of rescuing the projection deficits caused by individualist synapses. Increasing the sensitivity to nitric oxide by 20-fold from its baseline level  (coNo  =  20) rescued the projection, reducing both RF and projection widths to  below control levels (Fig. 7.7D). This situation is interesting as both cooperative synapse resource sharing and nitric oxide release seem to act to accomplish similar roles of focusing the axonal projection, but each works best individually. Thus when an equivalent upregulation to nitric oxide sensitivity was coupled with the normal cooperative synapse model, refinement of the retinotopic projection was degraded (Fig. 7.7C). These results suggest that NO release and cooperative re source sharing between synapses may both be active in the brain, but operating independently. This could explain conflicting experimental results which show that blocking NO interrupts some forms of spatial organization while it has no effect on others (see Mize and Lo, 2000; Debski and Cline, 2002). Another assumption of the synapse model was that trophic factor had a graded, activity-dependent release that was tied to how successful a synapse was at contributing a postsynaptic spike. To test how critical this assumption was for retinotopic organization, each postsynaptic terminal was altered to produce a constant amount of trophic feedback on every postsynaptic spike. Specifically, Eq. 6.12 was changed to LSI’jhk  =  expect for 1 Z  z e {0, 0.5, 1, 1.5, 2}.  In all cases the retinotopic  projection failed to refine. This appears to be because activity dependent release of trophic factor to the presynaptic terminal provides information about the relative contribution of the synapse, without which it is difficult for each synapse to determine when or if it should retract.  Diffusible growth factors  Growth and trophic factors modulate growth in axons, synapses and dendrites (Cohen-Cory and Fraser, 1995; McAllister et aL, 1995; Alsina et al., 2001; Poo, 2001; Cohen-Cory and Lom, 2004).  142  7.3. RESULTS-iD  Fig. 7.7. Synapse resource sharing and NO upregulation  A  Table 7.3. Synapse resource sharing Projection  Experiment  RF width  A Control  2.86  4.14  Individualist synapses  2.57  9.48  C Nitric oxide upregulation  3.2  4.75  2. 1  3.77  width -  B  -  -  C  —I  D Individualist synapses and -  nitric oxide upregulation  Results of disabling cooperative synapse resource sharing and upregulation of nitric oxide. A. Control. Normal retinotopic projection, based on cooperative synapse resource sharing, with the axon distributing synaptic resources to all local synapses based on trophic receipt of the local synapses. B. “Individualist” synapses. Organization observed when the trophic factor received by each synapse is converted to synaptic resources within each individual synapse. When synapses are not reinforced together, the continuity of the axonal projection is disrupted. C. Nitric oxide upregulation. Alteration of A where the response to nitric oxide is upregulated 20-fold (pNo 20). The projection becomes much more fuzzy, with both RF and retinal projection width increased. D. Individualist synapses and nitric oxide upregulation. Upregulating nitric oxide response (çoNo = 20) partially “saved” the retinotopic projection from the effects of the individualist behavior of synapses and it reduced the receptive field (RF) width of collicular neurons by —25%. The table at right shows the RF and retinal projection widths of A-D. Nitric oxide and cooperative synapse resource sharing play similar though seemingly competing organizational roles.  143  7.3. RESULTS JD -  As defined Sec. 6.3, growth factors diffuse throughout the extracellular space while trophic factors  are restricted to individual synapses. Diffusible growth factors were released to the extracellular grid by neurons firing below their target rate, increasing the local probability of synapse creation and attracting additional axon branches to the area. Growth factor affected axons by enhancing axonal growth and increasing the probability of synapse generation. Blocking the response to growth factors  (çogrowth  =  0) resulted in a 13.5%  decrease in retinotopic refinement (see Table 7.2). Enhancing the response of growth factor 3-fold ((Pgrowth  =  3) decreased projection and RF widths by 16.3%. Visual inspection of the developing  projection showed that refinement was accelerated when growth factor response was upregulated but that an increasing number of synapses was formed in retinotopically incorrect locations. En hancing the response to growth factor 10-fold (cogrowth  10) essentially prevented refinement as  synapse formation remained widespread, similar to Fig. 7.4C, hours 4-8. Diffusible growth factors thus appear to assist in the development of the retinotopic projection, but there can be too much of a good thing. Eye-specific segregation showed a similar sensitivity to growth factor as did retinotopic organi zation. Segregation was observed for all levels of response to growth factor that were tested (cogrowth  e  {0, 1, 3, 10}). However, the high rate of synapse creation under the maximal level  (Pgrowth  =  10) produced a weak projection that was poorly refined but still demonstrated clear  segregation. Segregation was strong for  cogroh E  {0, 1, 3}.  NMDA receptor activation  Activation of postsynaptic NMDA receptors is known to influence many synaptic behaviors. For example, NMDA receptor activation is required for LTP (see Dan and Poo, 2006, for review), for the release of NO and trophic factor (Schmidt, 2004) and NMDA receptor activation causes a large Ca entry, resulting in the activation of many second messenger pathways (Dan and + increase in 2 Poo, 2006). NMDA receptor activation in the model was assumed to be proportional to synaptic potentiation from STDP rules (see Secs. 6.4, 6.3 and 6.6) and both NO and trophic factor release were thus  144  7.3. RESULTS-iD  based upon synaptic potentiation. NMDA receptor response is more dynamic than is represented by the STDP model, however, as NMDA receptors require ligand binding to be activated and they take much longer than AMPA receptors to reach maximal activation. Further, NMDARs are voltage sensitive, allowing maximal conductance only when the intracellular space is depolarized (Cull-Candy et al., 2001). It thus seemed possible that NMDAR activation was higher in synapses where the postsynaptic neuron was recently active, as this implied recent excitation of the neuron (see Sec. 6.4). The effect of such increased response was approximated by activity-dependent upregulation of trophic factor and NO release. The activity modulated release of NO and trophic factor had a default relative strength 1.0. Disabling this mechanism  (çONMDAR  =  PNMDAR  =  0) increased the RF and projection width by 20% (see  Table 7.2) while increasing its relative strength to 10 enhanced refinement by 8%, implying that 2 enhances retinotopic organization. the activity dependent increase in NMDA receptor activation Altering the temporal window for the sensitivity of postsynaptic activity had minor effects. The baseline window for activity dependent boosting of trophic factor and NO release was 1 .Osec. Lowering  NMDAR 0  to 300 ms enhanced refinement by 5%. Increasing  owMDAR  O)NMDARtO  =  3 seconds  had negligible effects (<1%). To examine whether the activity dependent regulation of NO or trophic factor, or both, was respon sible for enhanced refinement, the model was run with only one of NO release or trophic factor release being modulated in an activity dependent manner. First, NO release was decoupled from activity dependent regulation while keeping trophic factor release normal. Specifically, Eq. 6.15 was changed to LSM  =  Cik. This change had minimal effect on retinotopic refinement (< 1 % de  crease in refinement), showing that modulation did not benefit retinotopic organization through the NO pathway. Second, trophic factor release was uncoupled from activity dependent modulation while NO release was normal. Specifically, Eq. 6.10 was rewritten as r  =  -i_-). This CikE (2, 1  change decreased retinotopic refinement by 21%. Because activity dependent regulation altered the total amount of trophic factor released, the amount released under regulation was measured (release was 60.3% of control levels) and the model rerun with trophic factor release not regulated by approximated NIvIDAR response, but where release was only 60.3% of normal. Restricting 2  Specifically, NMDA receptor activation, Ca2+ influx or other second messenger activation  145  7.4. RESULTS 2D -  release “rescued” the refinement and restored it to almost control levels (1 .3% decreased refine ment compared to control). Consequently, it appears that activity dependent regulation of either trophic factor or NO release, more so than accounted for by implicit assumptions of NMDAR activation during synapse potentiation, does not enhance refinement of the retinotopic projection. The changes that occur by altering simulated NMDAR response can be explained by activity in dependent changes to trophic factor release. Thus, activity dependent regulation of trophic factor and nitric oxide release, other than as occurs in the STDP potentiation window, do not appear to contribute to retinotopic organization. Eye-specific segregation was not affected by changes to approximated NMDAR activation lev els. Segregation was strong for all tested values of  { 0.3, 1, 3} seconds.  7.4.1  e  {0, 0.3 1, 3, 10} and  As with the monocular case, higher values of  ingly refined projection and changes to  7.4  c°NMDAR  NMDAR 0  ç°NMDAR  NMDAR 0  e  resulted in an increas  had little effect.  Results 2D -  Retinotopic organization and eye-specific segregation  The results and analysis of the previous section were focused on the 1 D model, where spatially static axons extended across the colliculus forming and retracting synapses with nearby collicular neurons. This and the following section describe simulation results from a 2D retina projecting to a 2D colliculus, using axons that extend, branch and retract. Development of the simulated retinocollicular projection followed the paradigm of mouse retinocol licular development as defined by McLaughlin et al. (2003); McLaughlin and O’Leary (2005) where axons initially project through the length of the colliculus and then branch interstitially and extend towards their retinotopically correct termination zones, with activity dependent mech anisms helping to refine the projection. This is summarized in Fig. 7.8A. The corresponding development of the simulated projection is shown in Fig 7.8C. Guided only by molecular guid ance cues, axons first extended along the A-P axis of the superior colliculus, which corresponds  7.4. RESULTS 2D -  146  to axons extending along the ephrin-AJEphA gradient. These axons had no positional preference along the L-M axis. Axons then sprouted branches along the main axon trunk near the retinotopi cally correct position on the A-P axis, the location corresponding to where ephrin-A and EphA gradients were balanced for each particular axon (Fig 7.8C; 30 mm), an outcome similar to more detailed chemoaffinity-based axon branching models (Yates et al., 2004). The branches then ex tended towards the correct location along the L-M axis, following the gradients of ephrin-BIEphB, to reach the approximately correct retinotopic location as governed by molecular guidance cues (Fig 7.8C; 6 and 16 hr). Axons then started forming synapses, governed initially by chemoaffinity and, as development proceeded, increasingly by trophic feedback (Fig 7.8C; 25-48 hr). Correlated retinal activity provided spatio-temporal information to help guide refinement. Synapses produced a much higher degree of retinotopic refinement than indicated by the axonal projection (Fig 7.8D). To analyze the continuity of the retinotopic projection, a plot was made comparing the position of each RGC to the center of mass of its synaptic projection and the distance of that projection from the retinotopically correct termination zone in the colliculus (Fig. 7.9A). There were systematic shifts in the projection, as shown by the smooth color gradients, but discontinuities and reversals, as sometimes occurred in the 1D model (e.g. Fig. 7.2D), were not observed. An unexpected finding was the occurrence of individual RGCs with unrefined axon branches and synapse projections that were discontinuous from neighboring RGCs. The axons from these RGCs failed to locate their retinotopically correct termination zone and failed to refine their projections (7.9C). By the end of the simulation, these axons had many fewer synapses. It would be expected that the RGCs corresponding to these axons would either retract the axon from the colliculus or the RGCs would die as is observed experimentally (O’Leary et al., 1986). Axon development in the model is characterized by an initial, chemoaffinity driven stage of de velopment, where axons approach their retinotopically correct termination zone, and an activity driven stage, where the axon fine tunes its projection, reaching its retinotopically correct location and concentrating its synapse projections there. The first stage of development for individual ax ons is demonstrated in Sec. 4.1. Fig. 7.10 shows development of axons during the second stage. Axons initially approach their ultimate termination zone and while they may pass through it, the information is not present in the axon for this to be known. Activity dependent feedback enhances  147  7.4. RESULTS-2D  A ephrin-As  EphBS—  16 hours  6 hours  C ephrin-B].  L  M inB5  ePnrin-As  Eph8s  48 hours  25 hours  B TL  V 25 hours  E  Decorrelated retinal activity  I  I  31 hours  ii  48 hours  Corre’ated retinal activity  Fig. 7.8. Development of retinotopic projection in group of RGC axons  A. Developmental paradigm, more fully described in Secs. 3.2 and 4.1. Figure adapted from (McLaugh un and O’Leary, 2005). In summary, RGC axons extend through the colliculus, produce branches that extend toward the retinotopically correct termination zone and, using activity dependent feedback from reti nal waves, refine their projections. As modeled here, chemoaffinity drove axon and synapse development through the entire simulation, while activity dependent mechanisms contributed only after synapse creation began, 24 hours into the simulation. B. Simulated retina consists of 13,327 cells (white dots). Axons and synapses from RGCs at randomly selected point (black spot) are shown in C. C. Time lapse development of simulated colliculus (consisting of 13,327 cells, shown as white dots). All axons from RGCs corresponding in a single patch of retina (n=14 RGCs) are displayed as they grow, branch and retract during 48 hours of development. Synapses from the axons shown in C are displayed in D. Pixel intensity indicates the number of synapses. The collicular neuron with the most synapses from the patch of RGCs in C is shown in red. The color intensity of the remaining cells is scaled linearly to white based on this number. E shows development that occurs when retinal waves are blocked (i.e. spatial activity is entirely decorrelated) but RGCs continue to fire at the same average rate. Simulated retina (left) and colliculus (right). Axons (green) and synapses (red) are shown from RGCs that originate in location of black dot on retina. Refinement of the retinotopic projection is prevented by blocking retinal wave activity. C and E had identical development patterns over the first 24 hours.  7.4. RESULTS 2D  148  -  D  Fig. 7.9. Accuracy of retinotopic projection A. Mapping of the positional error of RGC projections to the colliculus. Each RGC had an optimal termination zone in the colliculus that was at the same relative position as the RGC’s location in the retina. The position of each RGC was compared to the location of the center of mass of its synaptic projection in the colliculus, and the difference in these was plotted using color to represent the direction (hue, see F) and relative magnitude (saturation) of the difference vector. Magnitude of vector is magnified to show continuity of the projection.  RGCs had systematic errors with a tendency to be offset toward the retinal center, however no discontinuities were observed, as in the 1D model (e.g. Fig. 7.2D). Samples of refined RGC projections are shown in B, D and E. Axons are shown in green with red indicating the density of synaptic connectivity with each collicular cell. Sometimes RGCs failed to properly refine, generally through insufficiently accurate targeting during the development stage mediated by chemoaffinity. Such cases are shown as isolated dots in A, and a typical example is shown in C. F. Color chart showing direction of retinotopic projection errors in A. Correctly targeted RGCs are displayed in white and the color intensity increases with the distance of each RGC’s synapse cluster from its retinotopically correct termination zone.  149  7.4. RESULTS 2D -  0 hours  4 hours  8 hours  12 hours  24 hours  H  Fig. 7.10. Development of retinotopic projection in a group of RGC axons Development of 3 axons (representative examples) during retinotopic refinement. Green shows the axon and red indicates the density of synapse connectivity with collicular neurons. Synapses tend to form in the areas nearest the retinotopically correct termination zone, with axon growth being similarly focused as a result of trophic feedback to the synapses (see Sec. 4.1). As the axon reaches its termination zone, the arbor becomes focused and auxiliary branches retract.  axon growth in the areas nearest the retinotopically correct termination zone, primarily from the overlapping axons of neighboring RGCs in the initial diffuse projection. Growth continues until the axon reaches its ultimate termination zone, where the arbor refines and a dense synaptic field develops and axon branches that are not in the termination zone retract. Experiments have shown that mice lacking a subunit of the ACh receptor lack normal correlated wave activity and that RGCs are still active, but with dramatically reduced correlations between neighboring cells. Refinement of the retinocollicular projection in such mice is greatly reduced (McLaughlin et al., 2003). This experimental situation was modeled by eliminating retinal wave activity and having each RGC fire randomly at an average rate of 0.2Hz (Poisson distribution)  7.4. RESULTS 2D -  150  and observing the resulting organization (Fig. 7.8E). As in experiments and simulations in the 1D model, the projection failed to refine, again supporting the hypothesis that correlated retinal activity is required for refinement of the retinotopic projection. It is likely that a short exposure to retinal wave activity before onset of the apparent critical period for retinocollicular development (McLaughlin et al., 2003) would allow partial refinement to occur as is observed experimentally, but this was not explored in the model. Eye-specific segregation is observed in dual-innervated tecta (e.g. Law and Constantine-Paton, 1981; Ruthazer et al., 2003) as well as normally innervated colliculi in some species (e.g. Graybiel, 1975; Thompson and Holt, 1989). The 1D model was able to show eye-specific segregation (Fig. 7.3) as a result of two retinas projecting to the same collicular (tectal) structure. This experiment ) 2 was recreated in the 2D model by projecting 2 simulated retinas (6,505 RGCs each, 1.7 mm ). The results are shown in Fig. 7.11. 2 onto a single simulated colliculus (9,763 neurons, 0.9 mm The projections from each retina segregated into distinct spatial areas. The border regions of the colliculus showed much stronger spatial segregation than the central areas, where boundaries between segregated regions became less smooth. The reason for this is unclear. Analyzing the development over time showed that the border areas were very fast to form patterns of segregation while the central area took much longer.  7.4.2  Activity and chemoaffinity in retinotopic organization  While experimental evidence shows that both molecular guidance cues and correlated retinal ac tivity are important for organization, the relative importance of each of these mechanisms remains unclear. Physiological experiments have shown that normal patterns of retinal activity are not re quired to achieve coarse retinotopic projections (McLaughlin et al., 2003; Pfeiffenberger et al., 2006), that molecular guidance cues are important for retinotopic organization (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005; Pfeiffenberger et al., 2006) and that correlated patterns of activity are necessary to refine the retinotopic projection (see Wong, 1999; Ruthazer and Cline, 2004; Torborg and Feller, 2005; McLaughlin and O’Leary, 2005, for reviews). However, exper iments have not been able to address the role of chemoaffinity in refinement of the projection. The model was able to address this situation, however. A simulation was performed where axons  7.4. RESULTS 2D  151  -  Fig. 7.11. Eye-specific segregation 2D -  A. Pattern of connectivity from two simulated retinas projecting to a single colliculus. Each dot represents a single collicular neuron. 95% of collicular neurons received 90÷% input from a single retina. Collicular neurons responding to one eye are green and those responding to the other eye are red. Neuron responding to both eyes are shaded intermediate colors (yellow-orange).  7.4. RESULTS 2D -  152  were allowed to develop under normal chemoaffinity guidance until a coarse topology was estab lished and molecular guidance cues were thereafter blocked. Subsequently only activity dependent mechanisms were available to refine the projections. Specifically, normal axon development was allowed to occur for the first 24 hour stage of development, which was guided exclusively by chemoaffinity. In the second 24 hour stage of development, sensitivity to molecular guidance cues was disabled and only activity dependent mechanisms were available to guide refinement. The 2 collicu simulation consisted of a 2.0 mm 2 simulated retina (7,915 RGCs) projecting to a 0.7 mm lus (7,915 neurons). Despite the lack of chemoaffinity during the refinement stage, the projection . These results suggest that chemoaffinity and correlated activity play largely 3 appeared normal independent roles in development, with chemoaffinity providing guidance for coarse connectivity and correlated patterns of activity refining these connections. As mentioned previously, the relative contribution of molecular markers and retinal activity in retinotopic organization was addressed in two recent modeling studies of the development of the retinocollicular pathway in mutant mice, with both studies coming to conflicting conclusions. These studies were based on experiments where the molecular markers (EphA) was upregulated in a spatially distributed subset of RGCs, resulting in individual RGCs from a given spot on the retina having different chemoaffinity gradients and projecting to different collicular locations. The upregulated RGCs produced a separate termination zone in a more posterior location in the colliculus than did unaltered RGCs, a position more consistent with their upregulated markers (Brown et al., 2000; Reber et al., 2004). One modeling study showed how the relative expression of molecular markers was responsible for retinotopic organization but concluded that activity was necessary to refine the retinotopic projection (Reber et al., 2004). A more recent study, based on the marker induction hypothesis (Wilishaw and von der Malsburg, 1979) showed that marker induction was able to explain experimental observations in EphA3 upregulated mice and that activity was not required to refine the resulting retinotopic projection (Wilishaw, 2006). 4 (Brown et al., 2000; Reber et al., 2004) were reproduced These EphA upregulation experiments in the present model by upregulating chemoaffinity along the ephrin-AIEphA axis in a spatially One simulation, running on a reduced size retina and colliculus (4,795 neurons each) and for only 12 hours of development time, showed a 5% rotation but otherwise normal organization. These results were not replicated on larger retinas running for a full 24 hours (n=2). EphA3 knock-in experiments  7.4. RESULTS 2D  153  -  distributed subset of RGCs. Specifically, every other RGC had the value of its preferred ephrin AIEphA-mediated placement (i.e.  A 1 J  in Eq. 6.7) increased by 0.5. The results of this simulation  are shown in Fig. 7.12. During initial axon innervation of the colliculus, the RGCs which were sub ject to upregulation developed coarse axon arborization in more posterior sections of the colliculus than normal RGCs. As a result, when activity dependent mechanisms refined the connections, RGCs at a given point of the retina produced twin termination zones, one corresponding to each of the coarse initial arborizations. These results are consistent with the experimental observations of Brown et al. (2000) and Reber et al. (2004) and support predictions that activity dependent mechanisms are required to refine axonal projections in EphA upregulated mice.  7.4. RESULTS-2D  A  154  C  P  L  Call i Cu I us 24 hours M  Retina N  A V  D T  I  Colliculus 48 hours  D  E  :4  Fig. 7.12. EphA upregulation Results of simulated EphA knock-in experiment (Brown et al., 2000; Reber et al., 2004). In this simulation, 50% of RGCs had their ephrin-A/EphA mediated chemoaffinity shifted up by 50% to simulate the increased posterior bias for RGCs with EphA upregulation. A. Description of the data presentation. Simulated retina is on the left. On the right are 2 plots of the colliculus, one at 24 hours, which is the end of the chemoaffinity only stage of development, and below is at 48 hours, which is the end of the simulation. B-F. Tracings of axons through the colliculus from 5 different retinal locations. Red dot in retina shows location. Axons are are in green, and synapses with collicular neurons are shown in red, with the shade of red indicating the relative number of synapses with a particular collicular neuron. The simulated retina and colliculus each had 7915 neurons. Results show that a combination of activity induced refinement and chemoaffinity based guidance is able to explain experimental observation.  155  Chapter 8 Discussion This thesis has described the construction of a model of retinocollicular development with a wide descriptive scope, incorporating many of the physiological mechanisms known to be active during development. Many of these behaviors, such as retinal waves and axon development, required the creation of novel and independent component models. This wide-scope model was then shown to reproduce patterns of retinotopic organization and refinement as well as eye-specific segregation. The roles of the physiological mechanisms present during development were then examined to explore their role in retinotopic refinement and eye-specific segregation. Special emphasis was given to the analysis of retinal waves and chemoaffinity in these examinations, given the historical questions about their relative importance. The findings of the model indicate that retinotopic refinement and eye-specific segregation is a very robust emergent behavior and that it is tolerant to perturbations of the underlying physiolog ical components. Several of these components influence organization but are not required, such as growth factors, synaptic plasticity and rapidly diffusible molecules. Other physiological be haviors are required, such as trophic factors, molecular guidance cues, correlated retinal activity and homeostatic mechanisms, but organization still occurs across a range of perturbations to these mechanisms. Organization was only prevented through disabling of these mechanisms. As de scribed in Chap. 9, this thesis has generated several experimental predictions.  8.1. MODEL COMPONENTS  8.1  156  Model components  To produce the model used in this study, the phenomenological behavior of a number of physio— logical mechanisms was taken into account. Many of these mechanisms have been the subject of previous modeling studies. However, these previous models only focused on reproducing the phe nomenological behaviors of the particular component, or certain aspects of it, and little emphasis was given on how the modeled behavior could be integrated into a larger model with more descrip tive scope. For example, the best prior model of retinal waves generated retinal wave-like activity that closely matched the spatio-temporal properties of waves observed in P2-P4 ferret, including wave size and interwave interval (Feller et al., 1997). The model was quite simple, having only 2 free parameters. However, it was only able to produce wave activity with a temporal resolution of 100 ms. Moreover, the model was very sensitive to parameter changes and it was difficult to con figure it to produce wave activity with realistic spatiotemporal properties (see Sec. 2.1) matching other species, or matching the spatiotemporal properties of P2-P4 ferret waves on an different sized simulated retina. Because of these restrictions, a new retinal wave model was created, building on the foundations of the existing model. The new model describes a general mechanism, derived from physiological data, that is able to reproduce retinal wave-like behaviors that are statistically similar to those observed in many species. The requirements of a comprehensive modeling ap proach, integrating the phenomenological behaviors of several biological phenomena, thus pushed research to improve phenomenological reproductions of retinal waves and led to the discovery of a simple mechanism, specifically activity-dependent refractory periods in a group of recurrently connected cells, which can reproduce observed wave behaviors in a general manner. In another example of how existing phenomenological descriptions are often inadequate to in corporate into detailed models, the STDP model selected for use in this study (Froemke and Dan, 2002) was done so because of its ability, to address synapse plasticity changes resulting from triplets and quadruplets of spikes, as general STDP descriptions typically only deal with individual spike pairs. Implementation of this STDP model in the context of a comprehensive model revealed many shortcomings, including the lack of support for the saturation of plasticity and an ambiguous way to handle realistic (i.e. continuous) spike trains, where the time between successive spikes was largely random and not tightly controlled and repetitive, as in experiments. While data exists to  8.1. MODEL COMPONENTS  157  address the saturation issue, and was incorporated in this study, there remain many holes in the experimental data relating to how plasticity changes occur resulting from continuous spike trains. The need for a more accurate phenomenological model thus directs specific questions to be ad dressed by future experimental studies. The “most simple” description of these phenomena as provided in these and other studies are very useful in many ways, including providing an intuitive sense to how these mechanisms behave, but their incorporation into a comprehensive model often demonstrates omissions from the models which reflect holes in our knowledge. One advantage of a developmental model of wide descriptive scope such as used in this study is that experimentalists who create phenomenological models of the data they collect are able to incorporate them into a comprehensive model, such as described here, allowing them to better realize the missing links in their descriptions. Knowledge of these missing links can then help shape future experiments, helping to improve physiological knowledge and to produce more accurate models. The next several sections include a discussion of some of the model components used in the present study, including retinal waves (Sec. 8.1.1), the axon model (Sec. 8.1.2), molecular guidance cues (Sec. 8.1.3), the synapse dynamics and trophic factors (Sec. 8.1.4), the neuron model (Sec. 8.1.5), STDP (Sec. 8.1.6), NMDAR activation (Sec. 8.1.7), growth factors (Sec. 8.1.8), nitric oxide (Sec. 8.1.9) and homeostatic controls (Sec. 8.1.10).  8.1.1  Retinal waves  The results of the model indicate that correlated patterns of retinal activity are very important for refinement of the retinotopic projection, but the precise characteristics of this activity, as measured by the frequency, intensity, size and velocity, are not very important. Given the ubiquity of spon taneous correlated activity in development as seen across species, and throughout different parts of the nervous system in the same animal, it should almost be expected that correlated activity plays an important developmental role, as is indicated here. The results of this study are consistent with previous models and analyses that hypothesize an important developmental role for correlated activity (e.g. Wilishaw and von der Malsburg, 1976; Katz and Shatz, 1996).  8.1. MODEL COMPONENTS  158  As described in Chap. 2, realistic wave behavior results from the simple mechanism of activitydependent refractory periods. This simple mechanism combined with the apparent insensitivity of retinotopic refinement to wave characteristics, suggests that waves take the form they do out of biological convenience and that other, statistically similar patterns of activity would be equivalent in achieving retinotopic refinement and eye-specific segregation. Thus, it should be adequate for modeling studies to use patterns of activity that are computationally less complex than retinal waves, such as Gaussian blurs (e.g. Eglen, 1999), without qualitatively affecting organization. Similarly, refinement in biological systems should not be significantly impaired through the use of similar, non-biological patterns, even if the statistical properties of activity patterns vary. It is possible that spontaneous patterned activity evolved as a necessary step to produce topographic projections in larger brains. The developing zebrafish tectum is —2% of the length of the chick tec tum (Lemke and Reber, 2005; McLaughlin and O’Leary, 2005), and the smaller dimensions allow for much steeper molecular gradients, possibly explaining the accurate targeting of RGC axons in zebrafish tectum in the absence of correlated activity (Gosse et al., 2008), while correlated activity is shown to be necessary for accurate targeting in larger animals such as mouse (McLaughlin et al., 2003). However, activity seems to play an additional role to just refining topographic projections. Misdirected axons retract and their parent RGCs die (O’Leary et al., 1986) and correlated activity provides information to drive this process. In the model, most axons project near to their retino topically correct termination zones under chemoaffinity guidance and, using correlated activity, are able to refine their arbors at the retinotopically correct termination zone. Some axons, however, do not manage to reach the termination zone or refine their arbors. While these axons continue to produce synapses, they are in retinotopically incorrect areas. Thus, the activity of the axon is not correlated with other axons who project to where its synapses reside, resulting in the misdirected axon being unlikely to elicit a spike in the neurons it innervates, which in turn prevents stabiliza tion of its synapses. Axons in the model do partially retract as a result of this mistargetting and the corresponding inability to establish stable synaptic connections. This retraction happens even in the absence of an active mechanism that required trophic factor for axon (and RGC) survival. Thus, activity not only provides information to refine topographic projections, it also provides information to allow the removal of mistargetted axons.  8.1. MODEL COMPONENTS  8.1.2  159  Axon model  The mechanism to produce axon growth, branching and retraction, is arguably the most complex, and most important, part of the present study. Axons provide the scaffolding on which synapses exist and their arbors are important for defining functional projections. Many variations of the equations described in Sec. 4.1 were explored for driving axon behavior, as were altogether dif ferent algorithms. Unlike other model components used in this study which had relatively stable behaviors across different parameter settings and different implementations, some changes in the axon growth model resulted in significant changes in behavior. The greatest difficulty encountered with the axon model was having each individual axon reach an approximately correct termination zone, resulting in axons from neighboring RGCs producing a diffuse projection in the area, but not an exactly correct one, which would result in production of a refined projection in the absence of activity. In many implementations of the axon model, the smooth mathematical nature of the chemoaffinity calculation resulted in neighboring RGCs focusing their axons very close to the termination zone and producing a nearly refined retinotopic projection in the absence of retinal activity. Even when noise was introduced, it was often averaged out over the long simulation runs (24 hours simulated time for development driven exclusively by chemoaffinity), resulting in accurate and largely refined projections before the onset of activity. The difficulty encountered in making axons “miss” their targets but yet systematically come close suggests that chemoaffinity gradients in relatively large target structures, such as mouse or chick colliculus, provide only a weak directional bias to growth as experimental observations show large, diffuse projections persist in mice lacking wave activity. By using a weak bias, axons in the model were able to produce only diffuse connections in the absence of correlated retinal activity, as is observed experimentally (McLaughlin et al., 2003). If the bias were strong, such as in the much smaller zebrafish tectum, It would be expected that axons would project directly to the retino topically correct termination zone and not require activity in order to refine, as also is observed experimentally (Gosse et al., 2008). These results are consistent with molecular guidance cues providing only a weak organizational bias in animals with relatively large brain areas. The current implementation of axon growth controls the direction of growth based on the branch depth, where the axon initially innervating the colliculus is sensitive to EphA and ephrinA, and  8.1. MODEL COMPONENTS  160  branches off this axon ‘trunk’ are sensitive to EphB and ephrinB. Branches off this secondary trunk are then sensitive to both gradients. Experiments show that axons change their growth be havior through changes in their internal state (McFarlane and bit, 1997), not inconsistent with this mechanism. However, a potentially more accurate model of axon growth would be to differen tiate sensitivity based on the stage of development, not on branch depth. In this case, axons would initially be sensitive to EphA and ephrinA and extend through the colliculus. Then, the internal state of the axon would change and the axon would branch, based on EphA and ephrinA counter ing gradients (Yates et al., 2004) and then extend in the relevant direction based on the EphB and ephrinB axis. Another state change would then occur where the axon branched based on counter ing EphB and ephrinB gradients. At this point, the axon could become insensitive to molecular guidance cues and turn its attention to being sensitive to growth and trophic factors. This would be a functionally similar to the mechanism used by the model and it has some biological elegance to it, as a much simpler set of signaling pathways could be used throughout all stages of growth, with  their method of activation simply changing. Although there are questions about the implementation of the axon model used here, variations in its implementation should not significantly alter the results of this study. As demonstrated in the 1D model, axon growth is not necessary for retinotopic refinement. As long as the axon is in a location where synapses are able to grow and retract at and near the retinotopically correct termination zone, they will congregate at the termination zone. In this sense, the dynamics of axon growth are only important in that it gets close enough for synapses to form. Axon branches  are associated with the presence of synapses (Alsina et al., 2001) and this behavior is reproduced in the model. The act of synapses congregating near the retinotopically correct termination zone enhances axon growth in the area, providing an increased likelihood of further synapse growth. This feedback cycle helps to refine the projection and it is not dependent on the particulars of how the axon reaches the vicinity of the retinotopically correct termination zone, only that it does so. Different implementations of axon growth that similarly associate axon growth and branching with the presence of successful synapses should demonstrate similar refinement capabilities.  8.1. MODEL COMPONENTS  8.1.3  161  Molecular guidance cues  The results of the model indicate that chemoaffinity and correlated activity play largely indepen dent roles in development, with chemoaffinity providing coarse connectivity and activity refining these connections (e.g. see Luo and Flanagan, 2007). In the model this was shown by molecular guidance cues being necessary for axons to reach the vicinity of their retinotopically correct termi nation zones and correlated retinal activity was sufficient, in the absence of further chemoaffinity guidance, to refine the connection. However, the simulations did not suggest any deficit arising from continued use of molecular guidance cues throughout the refinement process, so it is possible that chemoaffinity has a continued influence throughout development. On the other hand, it is also possible that axons go through distinct phases of development, where growth is driven primarily by a single factor during each stage. Such a mechanism would allow the growth cone machinery to have a greatly reduced complexity, being only required to focus on a single or small number of gradients at a time and, at different stages of development, to change the gradients it is sensitive to. This hypothesis is consistent with physiological data, as discussed in Secs. 8.1.1 and 8.1.2. An interesting experiment would be to record axon development, similar to the manner of McLaugh lin et al. (2003), in collicular or tectal re-innervation experiments. The experimentally measurable ephrin gradients, and the paths taken by re-innervating axons, would provide valuable insight into the mechanisms involved. While there may be interactions which alter the expression of chemoaffinity molecules (Willshaw and von der Malsburg, 1979; Wilishaw, 2006), whose inclusion in this model could enhance the results, a simple mechanism where each RGC has a preferred target in the colliculus, as deter mined by chemoaffinity gradients, and correlated retinal activity refines this projection, is able to explain the development and refinement of retinotopic organization. Moreover, the only way to phenomenologically reproduce patterns of axon growth, as observed in mouse colliculus, was to use weak gradients, suggesting that an enhanced mechanism for chemoaffinity guidance is not required andlor was not present. One simplification in the present study was the use of a circular and uniform retina projecting to a circular and uniform colliculus, and molecular guidance cues providing a direct spatial mapping between the two circles. This is not a realistic physiological situation, owing partially to the dif  8.1. MODEL COMPONENTS  162  ferent RGC densities between central and peripheral retina. Physiologically measured mappings involve a distortion of the retinal projection. In many species, this projection takes the form of a log-polar mapping, at least as measured in adult cortex (Schwartz, 1980). The simplification in the model doesn’t change the assumption that each RGC has an optimal target location in the collicu lus as defined by molecular guidance cues. Future studies should explore more realistic interaction between guidance cues (e.g. Yates et al., 2004), RGC density patterns, and spatial representation patterns in target areas within visual system. Despite its design being driven by experimental data, it might be suggested that this model comes uncomfortably close to assuming what it concludes in regards to the relative roles of activity and chemoaffinity in development. This criticism cannot be fully discounted as the axon and synapse models, in order to best represent the experimental data that served as a template for this model (McLaughlin et al., 2003), required making several assumptions. The most influential of these was that axons produced a diffuse projection in the vicinity of their retinotopically correct termi nation zones, driven by chemoaffinity, and that this projection couldn’t be refined further without activity. This is consistent with experimental data but is not the only reasonable hypothesis. It is possible that activity alters the expression of molecular markers, and that some aspect of correlated retinal activity plays a permissive role in marker expression (Nicol et a!., 2007). However, while chemoaffinity expression is clearly graded on the large scale, studies indicate that it may be “noisy” at the local level (Feldheim et al., 1998; Reber et al., 2004), which would make exact guidance me diated by only molecular marker expression much more difficult to achieve, further reinforcing the argument that activity based mechanisms are required to refine the retinotopic projection. A relatively simple activity-based mechanism, as implemented here, was able to produce both retinotopic refinement and eye-specific segregation, and it was not dependent on the core assump tion limiting the role of chemoaffinity in achieving refined projections. Physiological observations have shown that the location of axon branches are associated with the presence of synapses (Alsina et al., 2001). The mechanism of the model increases the likelihood of axon branch creation in areas where the axon receives trophic factor, which is through synapses. Thus, axon branching is more common in areas of high synapse density, in turn producing more axon segments that are able to form synapses. The more synapses that are formed in an area, the more branching that is likely to  8.1. MODEL COMPONENTS  163  occur, producing a self-reinforcing process. The assumption that synapse formation occurs prefer entially where successful synapses are already present, and that such formation is higher in areas of higher chemoaffinity, further reinforces increases in synapse density and helps to focus it in retino topically correct areas. However, as demonstrated in the 2D model (Sec. 7.4.2), once axons have arborized in their retinotopically correct areas, the role of chemoaffinity in biasing synapse growth is not required to achieve refinement and retinotopic order. The assumptions of the model were that chemoaffinity drove initial connectivity and that both chemoaffinity and activity refined this projection. The results of the model were that both chemoaffinity and activity-based mechanisms played largely independent functional roles, with chemoaffinity providing for coarse organization and activity-based mechanisms refining it. Thus, the model results differed from its assumptions. Another point to be made is that the activity-dependent mechanisms described here that result in retinotopic refinement are also able to generate patterns of eye-specific segregation, essentially for free. Or, to put it in another way, the activity-dependent mechanisms that produce eye-specific segregation can also generate retinotopic refinement, obviating the need for regulating molecular marker expression to accomplish the same task. While it is possible that the assumptions that enforced a limitation on the accuracy of axonal growth under influence of chemoaffinity alone may have biased the resulting conclusions, there is a strong support for the use of these assumptions.  8.1.4  Synapse dynamics and trophic factors  When RGC axons first arrive at their midbrain targets, they lack synaptic connectivity. Synapses must become established on dendrites in their target areas in such a way that retinotopic order is achieved, with nearby RGCs synapsing onto identical or nearby tectal, collicular or geniculate neurons. Synapse survival is an integral part of this development, where inappropriately targeted synapses are removed while correct synapses are reinforced, or at least allowed to survive. The decision to retract must be made on the basis of information available to each individual synapse, restricting the ways in which such competition can occur. The mechanism used to achieve this behavior in the model was competition for resources, where synapses would receive trophic factor and exchange that for resources that were distributed by the soma, and synapses that depleted their resource supply retracted. The soma could thus exert control over synapse number by regulating  8.1. MODEL COMPONENTS  164  its production of resources, based on its firing rate and the sum of trophic factor returned from axonal synapses, and it would require no knowledge of synapse distribution to accomplish this. Implementation of threshold mechanisms to govern synapse survival, such that a synapse was re quired to receive a set amount of trophic factor to persist, was also considered, but implementation was fragile, seemingly arbitrary, and it did not produce competition well. An active regulatory (homeo static) mechanism proved to be much more effective. The strength and simplicity of this mechanism, with trophic feedback reinforcing the stability of the synapse, and the rate of reinforcement managed by the soma, suggests that a comparable mechanism might be responsible for physiological synapse survival. If this mechanism is indeed responsible, the axon or presynaptic terminal will be responsible for triggering synapse retraction. It is algorithmically possible for an equivalent decision to be made at the postsynaptic terminal, but this would require an additional anterograde messenger, which becomes more complicated and seems less likely. A ‘different but seemingly equivalent mechanism is the inverse, where presy naptic resources are increased on each spike but with synaptic depression causing release of a retrograde messenger that depletes presynaptic resources. This mechanism was not examined but should be a viable alternative. All of these mechanisms are functionally similar and the model does not distinguish between any particular implementation. It is possible that multiple mechanisms are at play, and that different species or brain areas use different implementations. It is the functional property of resource based survival that is predicted by this study. An experimentally testable consequence of this mechanism is that if two neurons project to the same group of target cells, and these neurons are identical except for the number of axonal synapses, the neuron with the larger number of synapses will have, on average, synapses that are more effec tive at eliciting spikes in the postsynaptic neuron. This results because neurons supporting a larger number of axonal synapses require more trophic factor to be received per synapse, compared to neurons with less synapses, and trophic receipt at the presynaptic terminal is based on the ability of a synapse to elicit a postsynaptic spike. Thus, an axon that is less successful at eliciting a postsy naptic spike receives less trophic factor and can support a lesser number of synapses. Conversely, an axon with more synapses should be more successful at eliciting postsynaptic spikes. The results of the model were heavily dependent on activity-dependent trophic factor release. Dis  8.1. MODEL COMPONENTS  165  rupting this mechanism, whether through blocking trophic factor or by interfering with its activitydependent release, prevented retinotopic refinement and eye-specific segregation. In the model, trophic factor acted as a retrograde messenger and a synaptotrophin, providing important informa tion to the presynaptic terminal and stabilizing synapses that were successful at inducing a spike in the postsynaptic cell. This helped reinforce “appropriate” synapses while causing or allowing the retraction of less appropriate ones. Blocking this mechanism or interfering with information transfer prevented both the stabilization of appropriate synapses and the preferential removal of incorrect ones, preventing organization. Trophic factor, as with correlated retinal activity and molecular guidance cues, was thus a critical component of the model. The resource sharing mechanism of the model, where the effect of trophic feedback to one synapse is shared among neighboring synapses, provided a spatial bias for synapse distribution. While not explored in the present study, this support mechanism for nearby synapses can include synapses connecting to different types of nearby cells, such as relay neurons in the LGN and their adja cent interneurons, thus aligning the retinotopic projections of RGCs onto different types of target neurons at no additional physiological cost or complexity. Future computational studies should evaluate this phenomenon and experiments might also be designed to examine if such a resource sharing mechanism exists, and if so, its possible role in development.  8.1.5  Neuron model  One surprising finding of this study was that weaker synapses produced more refined retinotopic projections. The reason for this is difficult to ascertain. One possibility is that weaker synapses have less individual influence on the spiking response of the postsynaptic neuron, thus causing a more rapid removal of synapses connecting to cells that did not receive significant input from other simultaneously active afferents, thus not producing a spike. Individually, weaker synapses were less able to elicit a postsynaptic spike and thus were less likely to receive trophic feedback as a direct result of their activity. When several weak synapses project to the same area, as is more likely to happen in the vicinity of the retinotopically correct termination zone, they are collectively more likely to elicit a postsynaptic spike and thus to receive trophic support. Synapses producing larger EPSPs, even if mistargetted, have a higher likelihood of producing a postsynaptic spike than weak  166  8.1. MODEL COMPONENTS  synapses, and thus are likely to persist longer. The longer mistargetted synapses persist, the longer it takes for the collicular neuron to become primarily responsive to retinotopically appropriate RGCs from a focused retinal location, delaying organization. The weaker the average synapse, the more quickly errant synapses retract. While altering the probability of vesicle release is functionally similar to altering synapse strength, as the probability that a presynaptic spike will induce a postsynaptic spike is related to vesicle re lease probability, from the perspective of synapse survival, vesicle release probability is not related to synapse strength. An extremely powerful synapse, even if it releases a vesicle very infrequently, is much more likely to receive trophic feedback, and thus survive, than a weaker synapse that reliably releases a vesicle. As such, it was a curious finding that vesicle release probability had an effect on the retinotopic refinement, especially as both increasing and decreasing the release probability diminished refinement. It appears that this effect can be partially explained by synapse resource usage. Lower probability of vesicle release resulted in a slower rate of resource usage, and a slower replenishment rate, which can be roughly approximated by synapses having normal vesicle release but a larger pool of resources. Doubling the starting and maximum synapse resource levels  and F,, respectively) produced a similar magnitude deficit in organization (average  increase in the RE and projection widths of 7.5% with Pvesc Finit/m  =  0.25; average increase of 8.1% with  50/100). Increased vesicle release, which should use resources more quickly, did not  have the same effect as reducing the size of the resource pool, however (average increase in the RF and projection widths of 7.7% with Pvesc  =  1.0; changes of +2% with F  =  10/25). Further  investigation into this topic is required. The ability of smaller synapses to generate finer retinotopic refinement might partially explain the developmental process in the LGN where small synapses are present during generation and refinement of the retinotopic projection, with geniculate neurons being innervated by 20+ RGCs at the end of refinement. During later development, in which the pathway matures to its more adult like state, the effective synaptic strength increases 50-fold, resulting in each LGN relay neuron being driven by 1-3 RGCs (Chen and Regehr, 2000). The ability of smaller synapses to produce better refinement is likely one reason why early synapses do not have the relative ability of inducing a spike in the target neuron as do synapses in maturity.  8.1. MODEL COMPONENTS  167  As mentioned earlier, several axon branches that failed to reach their retinotopically correct termi nation zone failed to refine. It seems likely that these axons, and their parent RGCs, correspond to physiological observations of axons failing to reach their correct termination zone, resulting in the axons retracting and the RGC dying (O’Leary et a!., 1986). Enforcing a requirement on how much trophic factor must be received in order to prevent the axon from retracting and to keep the neuron alive would more accurately reflect experimental observations. However, cell death and complete axon retraction were not behaviors considered when constructing the model so these mechanisms were not considered.  8.1.6  STDP  Rather surprisingly, the results of the model suggest that STDP does not play a significant role in retinotopic refinement, which is surprising given the historical research emphasis on both it and its more traditionally known relatives LTP and LTD. On further analysis, however, this result is reasonable. If the distribution of synapse strength was bimodal, as are the results of some STDP models (e.g. Song et al., 2000; Song and Abbott, 2001), where some synapses approached maxi mal potentiation while others were drawn to zero, it would be expected that STDP would play a significant role in organization, as this could be a sole indicator of synaptic fitness and could be used to produce retinotopic organization (e.g. Song and Abbott, 2001). However, the experimen tally observed distribution of synapse weights is unimodal (Fig. 8. 1A), arguing against such a role for STDP. Correlations in the patterns of presynaptic spikes only appear to shift this unimodal dis tribution (van Rossum et a!., 2000) also arguing against its role as a segregatory mechanism. Yet another experimental observation that suggests a reduced role of STDP is that, while STDP can be induced through experimental conditions, allowing neurons to return to their normal firing pattern quickly reverses these potentiation changes (see Dan and Poo, 2006). While the results of the model indicate that retinotopic refinement and eye-specific segregation are not dependent on STDP, it is possible that the implementation of STDP in this study masks its importance, especially as many assumptions had to be made in its implementation. Existing models of STDP are useful in that they phenomenologically describe and analyze experimental observations (e.g. Froemke and Dan, 2002; Wang et al., 2005; Froemke et al., 2006; PtIster and  168  8.1. MODEL COMPONENTS  A  Experimental quantal distribution  B  Model synapse weight distribution  C  Model synapse weight distribution variable dendrite location -  50 It)  .1-,  0.10  ci) >  0.5  LU  0.05 100 Quantal current (pA)  EPSP  EPSP  Fig. 8.1. Synapse weight distribution A. Experimental quantal amplitude distribution as observed from a single cultured cortical pyramidal neuron (adapted from van Rossum et al., 2000). B. Distribution of the weight of all synapses in a refined 1D model projection. Because all synapses have equal influence on the soma, the synapse weight is equivalent to the somatic response. The distribution is unimodal but is very narrow. C. Distribution of the weight of all synapses when the model is expanded to take into account dendritic location when calculating somatic response. The distribution of somatic responses becomes much more similar to the experimentally measured quanta! values (A). The distributions in B and C were normalized to 1.0.  Gerstner, 2006). However, these models do not readily scale and respond phenomenologically to realistic input patterns. One important example is the question of the effect a weakly potentiating spike pairing has on a strongly potentiated synapse (i.e. what occurs when a spike pairing that would saturate at 20% potentiation happens in a synapse that is already potentiated to 40%). In this study it was assumed that a synapse would only experience potentiation when a spike pairing occurred that had an absolute potentiating effect (e.g. presynaptic spike before postsynaptic), and that the amount of potentiation was limited to the amount of potentiation that would be realized were the synapse to be unpotentiated. The same mechanism was implemented for depression. This behavior was designed to preserve potentiation (and depression) in synapses as much as practical while observing the constraints of physiological studies. As shown in Fig. 8.1B, the attempt to preserve potentiation was not very successful, as most synapses hovered very close to unpotentiated strength. Two other significant questions about STDP implementation involve how the saturation of plastic ity influences potentiation and depression from individual spike pairs during natural spike trains, a behavior approximated here (see Sec. 6.1), and also how repetitive exposure to potentiation/depression  8.1. MODEL COMPONENTS  169  can shift the “neutral” potentiation level of a synapse such that random spike trains no longer revert it to its original, baseline potentiation level (Zhou et al., 2003). This behavior was not represented in the current STDP implementation as it is poorly constrained, computationally expensive and its relevance to natural stimulus patterns is not clear. Its absence could have an effect on the behavior of the STDP model, however, such that the importance of STDP is underrepresented. The strongly unimodal behavior of synapses in the model compared to experimental studies (com pare Fig. 8. 1A to B) was sufficiently different to warrant further investigation. One simplifying assumption in this study was that all synapses had an equivalent ability to depolarize the soma, while in reality, the somatic response to the input of a single synapse would likely depend on its distance from the soma. Because such a simplification would have the effect of normalizing synapse weights, the model was modified to use a simplified representation of the dendrite and the . This change to the 1 strength of an innervating synapse depended on its location in the dendrite model produced a much more realistic distribution of synapse strengths (Fig. 8. 1C) and did not significantly change retinotopic refinement (3% improvement in RF width; I % improvement in RGC projection width), indicating that the synapse weight distribution did not have a significant influence on organization and that synapse location on the dendrite, when considered, can explain the wide range of synapse weights as are observed in the soma. It is entirely possible that the model is wrong and that STDP is indeed necessary for organization. Experimental results that explore this specific question will very useful. Simply blocking NMDA receptors will not be an adequate test, however, as several other mechanisms rely on these, includ ing trophic factor and NO release, and possibly homeostatic regulation in pre- and/or postsynaptic neurons.  8.1.7  NMDAR activation  The simulation results from approximating the effect of enhanced NMDAR activation after in creased amounts of postsynaptic spiking at first appeared to enhance retinotopic refinement through 1  Each new synapse was placed in a random location in the dendrite, with the dendrite represented as a unit circle. The distance d of the synapse from the circles center was used modify the somatic response of the synapse. Specifically, b = 1.0 0.75d where b is the scaling factor. The term Wjjk in Eq. 5.6 was replaced by bWJk and the maximum somatic response, IcEpsp, was increased to 2.0 to compensate for the reduced average synapse strength. —  8.1. MODEL COMPONENTS  170  modulation of trophic factor release, but on further investigation it was found to play no significant role. Other implementations of this mechanism than those described in Sec. 6.4 were also used during attempts to more accurately track the NMDAR response at the level of individual synapses and showed similar results, reinforcing the idea that such a regulatory behavior is not present and that the level of NMDAR response as estimated by the STDP model is itself sufficient to regulate the stability of synapses, and that further regulation is not beneficial. These results are not con clusive, as it is still very possible that there is a regime in which enhanced trophic factor release (or NO release), in an activity-dependent way, might enhance organization. Given the stability of the model, and how its mechanisms tend to clearly influence organization (e.g. base mechanism of trophic factor release) or to have minimal affect (e.g. STDP), it seems that such a behavior, if it were to exist, would play a minor role.  8.1.8  Growth factors  Growth factors were not found to have a significant effect on organization other than through altering the rate of retinotopic refinement. This is quite reasonable as the effect of growth factors in the model was to attract axons and synapses to areas of lesser innervation. Growth factor presence is thus most relevant when concentrations are non-uniform, indicating that areas of neurons exist which require innervation. With possible exception of border areas in the simulated colliculus, collicular neurons received a relatively uniform pattern of innervation, so there was no section of the colliculus seeking to attract input. A similar behavior might also occur physiologically, with growth factor playing a minor role once axons have arborized in an area. Both physiologically and in the model, growth factors would likely play a significant role in attracting axons to relatively denervated areas (e.g. the optic disc representation in Vi), or areas with newly formed neurons that are seeking input. Future studies should explore these behaviors in the model, both to compare model behavior to experimental results and to help further constrain the model.  8.1. MODEL COMPONENTS  8.1.9  171  Nitric oxide  As represented in the model, NO release performs a similar action to cooperative synapse survival mechanisms. This mechanism is based on local synapses pooling their resources for survival, producing a bias for synapses to cluster near each other on the axon. Nitric oxide performs a similar functional role by increasing the likelihood of synapse creation and axon branching in axon segments that are depolarized near the time of exposure to NO, tending to make the axons that fire together wire together. Simulation results in the 1D model indicate that these mechanisms operate better individually than together, which was a surprising result. It is possible that this may be an artifact of the coarse resolution of NO concentration used by the model, as NO concentration is considered constant throughout a grid square (20 Im x 20 lIm in size). Simulations using high NO sensitivity did not have deformed projections and instead were best described as being “fuzzy”, with a strong and refined retinotopic projection being bordered by many nearby individual synapses that made the border of the projection harder to define. Despite the quantitative reduction in the strength of retinotopic projection, NO in the model should qualitatively improve refinement, as was shown to occur in simulations using “individualist” synapses (see Sec. In the model, NO increased the likelihood of axon growth and branching in axon segments that were depolarized at the time of NO exposure. While this is consistent with some experimental observations (Cogen and Cohen-Cory, 2000), observations of developing axons in dual-innervated Xenopus tecta showed that axon branches occurred equally near tectal neurons that were inner vated by either eye, but that they retracted preferentially from areas dominated by the opposite eye (Ruthazer et al., 2003). If NO release had a significant influence on axon branching, it would be expected that axon growth would be enhanced in areas of the tectum that were innervated by the same eye that the axon originated from, as NO would likely be released from these areas co incidental to axon depolarization. This suggests that NO does not play a significant role in axon growth and branching in axons that are depolarized near time of NO exposure, at least in the retino tectal pathway. However, other mechanisms, in particular axon growth mediated by trophic factor, or even associated with synapse presence (Alsina et al., 2001), would also seem to enhance axon growth, or at least branching, in same-eye territory. This was not observed either. A question arises about what visual experience these frogs experienced when axon branching was observed in the  8.1. MODEL COMPONENTS  172  study by Ruthazer et al. (2003), as this could have a significant impact on axon growth dynamics, especially as Xenopus retina appears to be light sensitive during this point in development and does not appear to experience retinal waves (J Demas, personal communications), indicating that visual activity drives activity-dependent refinement. Effective visual deprivation should reduce the effect of NO and trophic factor mediated increases to axon branching, possibly biasing their results and conclusions. This matter deserves further investigation. The mechanism representing NO used a very coarse approximations of its concentration and diffu sion properties. However, alteration of the constants governing the level and duration of sensitivity did not significantly alter model behavior, despite an order of magnitude change in parameter val ues. Future investigations in the model should nonetheless incorporate a finer temporal and spatial resolution for calculating NO and AA concentration and diffusion, and use more accurate diffusion parameters.  8.1.10  Homeostatic controls  Retinotopic organization is dependent on molecular guidance cues, correlated retinal activity and trophic factor release, as disruption of any of these mechanisms disrupts retinotopic refinement andlor organization. From a practical level, retinotopic organization is also dependent on the pres ence of homeostatic controls. As defined previously, the term homeostatic as used here refers to a neuron adjusting its behavior in order to achieve a particular level of activity or size, even if that level of activity or size is not fully reached. These controls included mechanisms to maintain the firing rate of collicular neurons and to regulate the number of synapses on axons and den drites. Using homeostatic mechanisms, parameters were used to define approximate behaviors and the model was allowed to adjust and find a balance between constraints imposed by the different model components. Attempts to implement the model without homeostatic mechanisms were un successful, because of both the huge parameter space and the fragile nature of the model when behaviors were rigid. From an evolutionary perspective, it would be expected that the nervous system, due to its complexity, should be very tolerant of variations and faults and to fail grace fully, whether the variations and/or faults were caused by the environment or by genetic mutation. Homeostatic mechanisms that are able to adjust for variations can provide this flexibility. While it  8.2. ANALYSIS OF MODEL BEHAVIORS  173  is theoretically possible to define the model in such a way that fine tuned parameters and rigid be havior can produce organization, just as evolution can theoretically produce finely tuned and rigid mechanisms of development, the behavior of the model very strongly suggests that homeostatic mechanisms play a fundamental role in the development of the nervous system. When organiza tion arises from a general set of principles, as demonstrated by the model, genetic drift can alter the mechanisms without breaking functionality, allowing new behaviors to periodically appear, driving evolution. Evolutionary forces would have a much harder time driving changes under mechanisms that were finely tuned or sensitive to small variations in implementation. The assumption of the model was that neurons had a target firing rate that controlled several home ostatic behaviors of the neuron. While there is evidence for this (see Sec. 6.5), the model assumed that all neurons had the same target rate. This is not necessarily the case and it is possible that a given group of collicular neurons has a range of target rates, whether these are set independently for each neuron and are constant, or if they vary dynamically, such as through changes to somatic excitability resulting from retrograde feedback to axonal synapses (see Dan and Poo, 2004). No efforts were made to explore the effect of neurons having different or dynamic target firing rates. Given the importance of homeostatic controls and their association with having a target firing rate, future studies should examine how variability in firing rates might affect neural behavior and pat terns of connectivity.  8.2  Analysis of model behaviors Justification of model design  The model as described in Chaps. 3-6 was iteratively developed with the results of simulations from Chap. 7. Because of the iterative nature of development, criticism can be levied that this model was fine tuned to produce a specific output and thus it has little general relevance. There are several arguments against this point. First, all components were physiologically based and many had well defined behaviors (e.g. STDP, retinal waves, axonal branching, neuron model). Others are less well defined, such as synapse generation, but the general behaviors are known. Only in more  8.2. ANALYSIS OF MODEL BEHAVIORS  174  poorly defined areas such as growth and trophic factor influences are component models largely speculative. However, while many implementations of different growthltrophic factor algorithms were tried and behaved less well than those described here, most implementations still produced refinement of retinotopic projections, only they were less effective or were slower to mature. A second argument is combinatorial. The number of parameters and equations in the model, each being constrained by physiological behaviors, makes it unlikely that a correct set of model behaviors and parameters could be discovered that made the model work in a stable regime as demonstrated in Chap. 7 if it didn’t bear similarity to the system being modeled. A third argument involves the order of development. The model was finalized after the component behaviors were able to generate the observed patterns of organization (e.g. retinotopic refinement and eye-specific segregation) and before a comprehensive analysis of the model began. Many results of the model were not demonstrated, even in proof of concept form, until after this analysis was complete. One such behavior was 2D binocular segregation. It follows that the model could not have been customized to produce all of the results described here. Finally, the model is able to reproduce a range of experimental results (Chap. 7), including the behaviors documented in this study as well as others that were omitted for various reasons, such as space and complexity. From a practical perspective, it is not realistic to be able to produce a model of such complexity that reproduces a wide range of behaviors while being constrained at the level of its individual components, and have the model be fine tuned for a particular purpose. These reasons, plus the stability of the model under a variety of perturbations, suggests that the model is a reasonable representation of the underlying physiological mechanism.  Complexity of model  As complex as the model in this study is, it should represent additional physiological behaviors than it does, or at least a different set of behaviors. During development, some components of the model that seemed to have minimal contribution were removed in order to simplify its description. However, later analysis of some of the experimental perturbations showed resulting weaknesses that were not apparent under normal operating conditions, resulting from these simplifications.  8.2. ANALYSIS OF MODEL BEHAVIORS  175  One such weakness is in an interaction between synapse generation, trophic factor and growth factors. This was observed in the 2D simulation where retinal waves were blocked but RGCs continued to fire at a low average rate. The sparse RGC firing resulted in greatly reduced firing rates in collicular neurons, causing a continued high release rate of growth factor. When growth factor presence was high, synapse generation was increased, resulting in the production of more synapses than could be supported by the available level of trophic feedback. Axons in the model responded appropriately by producing more synapses, but too many synapses formed, resulting in a continuous turnover. The arbors and synaptic projections from individual axons thus remained very large and widespread. An earlier version of this model put a soft boundary on the number of axonal synapses and the simulation was much more well behaved during perturbation experiments, including those where retinal wave activity was blocked. In that version of the model, axons had a tendency to produce relatively small arbors, with the location of arbors between nearby RGCs showing little correlation and great variability. The soft limit on the number of axonal synapses was removed to simplify the model description, as the model was already very complex. In a phenomenological model such as this, however, such simplifications to the model should only be performed if they can be experimentally justified. Future studies using this model should be made using a soft limit on axonal synapse count, and other realistic restrictions. The model is currently as complex as can be reasonably run on relatively cheap computing hard ware, such as multi-CPU personal workstations. Its scalability presently seems to be limited by the rate at which data can be transferred to each individual CPU. On a 4-core workstation (Dell Op tiplex 745, Intel Core 2 Quad, 2.4 GHz), the best simulation performance for 2D simulations was achieved when running 2 or 3 cores simultaneously. It appears that the larger memory footprint of these simulations (1-4GB), and the limited bandwidth of the main system bus, results in starved CPUs when running with all 4 cores. Alternatively, thread synchronization overhead could have resulted in the bottleneck. The simulation was also run on a shared memory supercomputer (D3M 64-CPU Power5 Model 595). On this architecture, the best performance was had when the simula tion used 12-16 CPUs, and slowed appreciably when running on 32 CPUs. Whether the bottleneck is due to the bandwidth of the system bus or the overhead involved in keeping all computing threads synchronized, the result is that the simulation is only scalable to a limited number of CPUs. Thus, to increase the complexity and size of simulations using this model, CPUs having increased speed  8.2. ANALYSIS OF MODEL BEHAVIORS  176  and processing capacity must be used. Alternatively, significant improvements must be made in how data is transferred and synchronized. The current trend by computer manufacturers of in creasing computing power by adding more CPU cores is not adequate. Until this situation can be improved, it is unlikely that individual models can reach significantly larger degrees of complexity (and size) and still run in a timely manner. However, with certain restrictions not analyzed here, it is possible to redesign and extend this type of simulation to operate in a cluster environment of multi-CPU computers, partially alleviating the problem of scalability.  Eph3A knock-in experiments  Simulations of the Eph upregulation experiments in Sec. 7.4.2 reproduce some of the key findings of experimental observations, such that RGCs from a focal point in the retina produce arborizations in 2 locations in the colliculus. However, there is one notable difference between model results and experimental observations. Experimental data shows two distinct and non-overlapping maps are created by normal and modified RGCs (Reber et al., 2004), with normal RGCs producing a map in anterior colliculus and modified RGCs producing a map in posterior colliculus. In the present model, normal RGCs produce one largely normal map throughout the SC while modified RGCs produce a partially overlapping map in the posterior SC. There are several possible reasons for this . 2 discrepancy that should be examined One possibility is that the axon model is too simple or that the chemoaffinity guidance mechanism, as implemented, is too strong. Axonal arborizations in the model that develop by chemoaffinity guidance only produce relatively small arborizations compared to what appears to be observed experimentally (Fig. 8.2). The refined termination zone of a group of RGCs in the model typi cally falls within the area covered by the original and diffuse axonal projection. The location and relatively small size of these projections prevents the formation of independent, separate maps, as normal RGCs that project to the posterior colliculus have too far to relocate to create an indepen dent map in anterior colliculus. 2  One mutant strain reported by Reber et al. (2004) does produce overlapping maps similar to those in the present model, but this mutant strain appear to be a special case, exhibiting one of many possible gene combinations, with others producing separated maps. The behavior of the present model appears to be stable, however, based on obser vation the developmental patterns of RGC axons, and so its similarity in results to this mutant strain is not addressed.  177  8.2. ANALYSIS OF MODEL BEHAVIORS  A  P  B  Fig. 8.2. Arbor size without correlated activity  A. Model axonal arbor shown after development under molecular guidance only (simulation time T=24 hours). Retinotopically correct termination zone shown in red. The arbor is diffuse in the area of the termination zone, but extends across a very limited section of the retina. B. Axon arbors shown from injection of fluorescent tracer at a single retinal location in P4 2/ mouse colliculus. Retinotopically correct termination zone shown in red (adapted from McLaughlin et al., 2003). While there is a higher concentration of axons near the retinotopically correct termination zone, axons continue to occupy most of the colliculus. Retinal waves are still absent in P4 J32/ mouse retina and the large axonal arborization is likely the result of growth mediated by molecular guidance cues. The arborization in B is much larger than occurs in the model (A) and might explain why model axons from unmodified RGCs in the EphA3 knock-in simulation fail to refine in the anterior colliculus, as there is no arborization in the anterior colliculus to support development of a termination zone there.  8.2. ANALYSIS OF MODEL BEHAVIORS  178  Another possibility is that the expression of molecular markers is not static and the location of the original axonal projections is governed either by the relative expression of markers (Reber et al., 2004) or that arriving axons induce marker presence (Wilishaw and von der Malsburg, 1979) that guides the axonal arborization. Molecular markers in the present model were assumed to be static. A third possibility is that axons in the model are not sufficiently responsive to growth factors. Neu rons in the anterior colliculus, which have less axonal coverage and presumably less innervation than neurons in the posterior colliculus, should release growth factor to attract more innervation, but it is possible that axons do not respond sufficiently to this attractive force. Consistent with this hypothesis is the slow onset to responsiveness of growth and activity dependent mechanisms in the model, which may cause axons to become sufficiently entrenched in the location of their termination zones before they become sensitive enough to respond to the growth factor. Alterna tively, the anterior neurons may have received adequate innervation and thus have reduced growth factor release by the time axons are sufficiently sensitive to it, resulting in no shifting of axonal projections. Several mechanisms can help resolve the above questions. The first option is that the model can be adapted to explore the different possibilities outlined above. A possibly more insightful path would be experimental and involve watching axonal arbors through development in Eph3A knockin mice, similar to the study of McLaughlin et al. (2003), both in mice with normal waves and those having the 2/ mutation that disrupts early wave activity. This would show if axons from Eph normal RGCs in one portion of the retina produced early arborizations that were retinotopically ‘normal’ and then refined in the more anterior sections of these arborizations, or if the centers of these arborizations were initially more anterior, both with and without correlated retinal activity. The former result would suggest that molecular guidance cues provide an effectively static and course guidance and that activity dependent mechanisms produce the observed shifting in the maps. The latter result would suggest a changed molecular bias, whether through relative signaling (Reber et al., 2004) or marker induction (Willshaw and von der Malsburg, 1979), directing axons to altered sections of the colliculus, regardless of activity-dependent mechanisms.  8.3. COMPARISON TO OTHER MODELS  8.3  179  Comparison to other models  While it was the descriptions of physiological mechanisms that drove the development of the model described in this study, and not the design or results of previous models of visual system organi zation, the present model does bear some similarities to previous modeling approaches. On a conceptual level, the model derives largely from two previous studies. The first is the model of Wilishaw and von der Malsburg (1976) which was based on visually driven correlations of activity between nearby RGCs and that used weak molecular guidance mechanisms. In the present model, simulated retinal waves provided the source of correlations between nearby RGCs and a relatively weak molecular guidance mechanism provided a bias for axons to be attracted to approximately correct areas of the colliculus. This guidance mechanism is more specific than the “polarity mark ers” suggested by Wilishaw and von der Malsburg (1976), but the model in the present study is much more physiologically detailed, necessitating a more specific description. The second model that bears conceptual resemblance to the present study was by Swindale (1980), which was based on the principle that new synapses are attracted to locations where many similar synapses already exist. The functional mechanisms in the present model closely follow this behavior. Trophic receipt from synapses induce the creation of additional synapses in the areas of trophic receipt, which is predominately in areas of the axon which have many “successful” synapses. Further, the cooperative model for synapse survival that helped stabilize nearby synapses, produced a similar behavior. The alternative mechanism of nitric oxide that induced axon growth and synapse gener ation in axon segments that were depolarized near the time of exposure also supports the synapse density hypothesis. It remains to be seen whether there is descriptive similarity in the time course of eye-specific segregation. Implementationally, the present model is most similar to the neurotrophic model of Elliott and Shadbolt (1999), as this previous model also represented many of the physiological mechanisms included here, including an implementation of trophic factors and the use of spatiotemporal pat terns of retinal activity. That model went further than the present study, in a modeling sense, as it also examined development along the entire retino-geniculo-cortical pathway. However the present study has a greatly increased level of physiological detail, including growth and retraction at the level of individual axons and synapses, and a spike-based model, which more accurately  8.3. COMPARISON TO OTHER MODELS  180  reflects experimental findings. As such, the present study is more strongly constrained and is potentially able to provide more detailed insight into the physiological mechanisms. Another comparison involves the mechanisms regulating the response to molecular guidance cues. Many models that address guidance through molecular markers can be categorized as using “Type I” or “Type II” matching between RGC markers and those expressed on target neurons. Type I matching involves each presynaptic neuron (axon) having an affinity for just a small neighborhood of postsynaptic cells. Type II matching involves all presynaptic neurons (axons) having maximum affinity for target neurons at one end of the target area, and competition and the strength of this affinity managing the distribution of the projection (Prestige and Wilishaw, 1975; Goodhill, 2007). The implementation of molecular guidance cues in the present study doesn’t fall into either cate gory, although it is more closely associated with Type I than Type II. Each RGC axon in the model has a peak affinity for a specific region in the colliculus, but can grow and arborize in any loca tion. The same behavior holds true for synapse formation and connectivity. This is most apparent in perturbations of the 1D model where portions of the retina or colliculus were truncated, and where part of the colliculus was reversed, with the resulting pattern of synaptic connectivity being shown to compensate for such perturbations. Type 1 matching, on its own, cannot account for this behavior (Prestige and Willshaw, 1975; Goodhill, 2007).  8.3.1  Local excitation and distal inhibition  Many theoretical models of retinotopic development, and the development of topological projec tions in general, rely on local excitation and distal inhibition, where activity in one target cell in duces activity in neighboring cells and inhibits activity in cells farther away (see Swindale, 1996). The explicit representation of local excitation and distal inhibition that are commonly used by computational models were not incorporated in the present study and they were not found to be necessary. In this regard, the present model differs from most existing models. However, analysis of the model shows that the functional role of each of these mechanisms was represented through indirect and slow-acting mechanisms regulating synapse and axon growth and homeostatic con trols.  8.3. COMPARISON TO OTHER MODELS  181  There is little physiological evidence to support distal inhibition in the developing retinocollicular pathway. The high reversal potential for chloride early in development (Ben-Ar 2002) suggests that such inhibition is not realistic, as synapses traditionally considered inhibitory (i.e. GABAer gic) would be excitatory during retinotopic organization and refinement, and eye-specific segrega tion. Inhibitory behavior was not represented in this study. There are several plausible mechanisms that might accomplish local excitation, but these also were not implemented in the present study. The first is through gap junctions between collicular neu rons. It is possible that gap junctions between overlapping dendrites are able to perform the role of local excitation as hypothesized by these previous models by transferring excitation between coupled cells. However, there is little physiological evidence for gap junctions in the developing colliculus or LGN, and the long time constant of immature neurons suggests a high membrane resistivity, which is also not consistent with significant gap junction presence. Gap junctions were not included in the model. The second possibility is that axon collaterals of collicular neurons send branches to innervate neighboring collicular neurons. A third possibility is through GABAergic interneurons which are likely excitatory at this stage. As discussed in Sec. 8.1.4, the coopera tive synapse model should align overlapping innervation patterns between target neurons and local interneurons, producing an indirect pathway to induce local excitation. While both of these mecha nisms, i.e. axon collaterals and excitatory interneurons, could be included the model, these behav iors were not used as they were not found to be necessary. Retinotopic refinement and eye-specific segregation occurred without their presence. An interesting possibility, at least for some areas of the brain, is that GABAergic interneurons provide local excitation during early development and their shift to an inhibitory role coincides with the closure of the critical period of organizational plasticity (Hooks and Chen, 2007). Future modeling should address this possibility. The cooperative synapse model, which involved the sharing of synapse resources between nearby synapses on the same axon, did indirectly accomplish a role similar to local excitation. While it did not increase the effective strength of any synapse, it did help stabilize all synapses that were in the same area of successful synaptic connections (successful meaning synapses that were effective at inducing spikes in target neurons). The trophic factor received by the synapses caused increased synapse generation probabilities (and axon growth) in these areas, further reinforcing the local  8.3. COMPARISON TO OTHER MODELS  182  projection. Thus, while the model did not directly or explicitly represent local excitation, it did have an indirect mechanism that accomplished a similar functional role. Homeostatic mechanisms, which acted to limit the innervation of each neuron in order that the neu ron could maintain a target firing rate, accomplished a similar role as achieved by distal inhibition. As collicular neurons became more strongly innervated by one group of RGCs, their higher rate of activity caused a reduction in trophic support for all synapses and the neuron became less respon sive to input in general. This induced the less successful synapses, which tended to be from RGCs that were more distal than those which strongly drove the collicular neuron, to retract, producing “inhibition” through the slow process of synapse retraction.  183  Chapter 9 Summary This chapter briefly highlights the accomplishments of this study and provides a list of experimen tal predictions arising from it.  9.1 9.1.1  Accomplishments of the model Retinotopic order  The model shows how RGC axons can innervate the colliculus and independently develop to pro duce a diffuse arborization in the vicinity of the retinotopically correct termination zone. Growth decisions are made by individual axon segments and growth producing this arborization was me diated only by molecular guidance cues.  9.1.2  Retinotopic refinement  The mechanism for axon development that produced retinotopic order was extended to include synapse growth and mediation through activity-dependent mechanisms. When RGC activity was  9.1. ACCOMPLISHMENTS OF THE MODEL  184  driven to approximate the spatio-temporal behavior of retinal waves, axonal arbors and their synap tic projections were shown to refine the retinotopic projection. Refinement depended on RGCs ex hibiting correlated retinal activity, but it was very tolerant to the form this activity took. Growth and retraction decisions were made at the level of individual axons and synapses. Retinotopic refine ment was also observed through synapse creation in spatially static axons (the 1 D model), showing that organization was not dependent on the specifics of axonal behavior and that both axons and synapses demonstrate refinement when driven by approximations of physiological mechanisms.  9.1.3  Eye-specific segregation  The same mechanisms responsible for production of retinotopic refinement were also shown to be able to produce eye-specific segregation in both 1D and 2D models. Or, put another way, the mechanisms able to generate eye-specific segregation were also able to produce retinotopic refinement, meaning that additional physiological mechanisms, such as dynamic molecular marker expression, are not necessary to explain retinotopic refinement.  9.1.4  Model of retinal waves  A model that reproduced the spatio-temporal properties of retinal waves was created, showing that activity-dependent refractory periods in spontaneously active amacrine cells, as implied by physiological studies, was able to generate realistic patterns of wave activity. The model was sufficiently flexible to reproduce wave behavior with the spatio-temporal patterns as observed in different species. Its underlying mechanism may be responsible for spontaneous activity as seen in many brain areas.  9.1.5  Comprehensive modeling framework for studying retinocollicular de velopment  A modeling framework was designed and implemented that incorporated the phenomenological behaviors of many cellular and subcellular phenomena in order to study the emergent system-level  9.1. ACCOMPLISHMENTS OF THE MODEL  185  properties of retinotopic refinement and eye-specific segregation. Each of the behaviors of the model, and the model output, are subject to experimental verification and each component can be analyzed and/or modified independently. The model or any of its individual components can be improved with time as more experimental data becomes available.  9.1.6  Examining the contribution of different physiological mechanisms to developmental organization  The physiological mechanisms of the model were perturbed or eliminated and the effect on retino topic refinement and eye-specific segregation were observed. It was found that retinal waves, molecular guidance cues, trophic factors and homeostatic mechanisms were all necessary for de velopment and blocking or eliminating these prevented retinotopic refinement and eye-specific segregation. Disruption of the other mechanisms included in the model generally produced minor changes to organization.  9.1.7  Examining the relative roles of retinal waves and chemoaffinity  Patterns of spontaneous retinal activity were varied and were blocked to examine the role of activity in organization. To the degree that experimental data is available, these manipulations largely matched experimental results. Experiments evaluating the relative role of chemoaffinity were also recreated in the model, including truncation of the retina or tectum, rotation of the tectum, and upregulation of molecular markers. To the degree these behaviors were modeled, they largely reproduced experimental findings. The roles of activity and chemoaffinity in retinotopic refinement and eye-specific segregation were found to be very distinct, with chemoaffinity playing a critical role in guiding a coarse retinotopic projection and correlated retinal activity necessary to refine it. Refinement was very tolerant to changes in the patterns of activity, so long as it was spatio temporally correlated.  9.2. EXPERIMENTAL PREDICTIONS  9.1.8  186  Spatial representation of afferents  When two retinas projecting to the same target structure segregate their projections into spatially distinct domains, the area represented by each eye is largely equal when the activity level of both retinas are similar. Experimental and theoretical studies show that neurons that have higher rates of activity achieve larger spatial representation. The characteristics of retinal activity responsible for achieving different spatial representations were explored, concluding that it is the relative ability to drive activity in the post-synaptic neuron that causes increased spatial representation, not the absolute levels of activity.  9.2  Experimental predictions  This study has generated several predictions about the developing retinotectal pathway, some of which are more generic and can be applied to development in other neural pathways as well. One important finding was that the model was very stable across a large range of parameter variations and perturbations. The model was based on experimentally determined behaviors, and as it is likely that the biological mechanism is very tolerant to variations and perturbations, it is not surprising that the model is also tolerant. Below are listed several predictions by the model, most of which can be investigated experimentally.  9.2.1  Retinal waves result from activity-dependent refractory periods  Forms of spontaneous activity, similar to retinal waves, are observed in many parts of the nervous system. In the retina, these patterns of activity can be produced through the simple mechanism of activity-dependent refractory periods between recurrently connected amacrine cells. Extending this, any group of recurrently connected neurons that is spontaneously active and that has activity dependent refractory periods, will produce similar wave-like activity. The velocity of such activity is predicted to be related to the duration each neuron is excitatory to its neighbors.  9.2. EXPERIMENTAL PREDICTIONS  9.2.2  187  Developmental organization very tolerant to patterns of correlated retinal activity  Retinotopic refinement and eye-specific segregation require correlated activity to exist between nearby RGCs. This is accomplished through the presence of retinal waves. The results of this study show that organization occurs across a range of spatiotemporal wave properties. These results can be extended to predict that the specific wave properties are not important to retinotopic refinement and eye-specific segregation, as the exact form of waves may be the result of biological convenience (i.e. they result of general activity-dependent refractory periods). As long as the level of RGC activity, non-repetitiveness and correlation are preserved, patterns of activity that are not seen naturally (e.g. moving Gaussian blurs) will also result in retinotopic refinement and eye-specific segregation.  9.2.3  Retinal waves and molecular guidance cues play complementary and distinct roles  It is becoming increasingly accepted in the experimentalist community that chemoaffinity pro duces coarse retinotopic organization and that correlated retinal activity refines this projection. The results of the present study are directly in line with this hypothesis. While it is likely that the mechanisms responsible for activity-dependent refinement are active at the same time as those responsible for chemoaffinity driven organization, the results of the present study indicate that it is possible for both to be temporally separated, such that refined retinotopic projections will result if activity-based mechanisms are only activated after a coarse retinotopic organization is achieved and chemoaffinity-based mechanisms are silenced.  9.2.4  Bias from molecular guidance cues is weak in visual brain areas in chicks and rodents, and larger animals  The results of the axon model used in this study suggest that axon growth patterns in mouse col liculus are consistent with a very weak growth bias being provided by molecular guidance cues.  9.2. EXPERIMENTAL PREDICTIONS  188  Or alternatively and functionally equivalent, an axon’s ability to obey the bias is weak. Initial innervation in mouse colliculus is diffuse, and axonal arbors in the developing colliculus are very large while wave behavior is absent. The only way this behavior could be replicated in the model, and still phenomenologically reproduce axonal growth, was to use a weak chemoaffinity bias. The results predict that in the visual brain areas of chicks, rodents and larger species, molecular cues provides only weak guidance. These results likely extend to similar sized and larger areas through out the brain.  9.2.5  Retinal waves, molecular guidance cues, trophic factors, and homeo static mechanisms are all required for organization  Several physiological mechanisms were represented in the model in this study. Of these, only retinal waves, molecular guidance cues, trophic factors and homeostatic mechanisms were found to be necessary for development. These mechanisms could be perturbed, resulting in relatively minor changes to organization, but blocking them prevented normal refinement. The model predicts that this set of four mechanisms comprises the minimal set of behaviors required for retinotopic refinement and eye-specific segregation.  9.2.6  STDP not required for retinotopic refinement or eye-specific segrega tion  Synaptic plasticity, as measured by activity-dependent changes to a synapses ability to depolarize the target neuron (i.e. LTP and LTD), was found to have little effect on retinotopic organization. The temporal response properties of STDP potentiation were found to be very useful at regulating trophic factor release, which was necessary to model function but the model predicts surprisingly that synaptic plasticity itself is not required for retinotopic refinement or eye-specific segregation.  9.2. EXPERIMENTAL PREDICTIONS  9.2.7  189  Small synapses produce more refined projections  The results of the model indicate that the smaller the average somatic EPSP from innervating synapses, the more refined the projection. Thus, during refinement of the retinotopic projection, and of other topographic projections in the brain, the average strength of each individual synapse, with strength measured by the ability of a synapse to elicit a spike in the postsynaptic neuron, should be smaller than it is in maturity. Likewise, increasing synapse strength during development will reduce the amount of refinement.  9.2.8  Synapses use the equivalent of a resource-based mechanism for sur vival during development  The mechanism where the presynaptic terminal uses trophic factor receipt, or the receipt of another retrograde messenger, to extend its survival, is very versatile, providing a mechanism for synapses to be removed based on locally available information, a mechanism to control total axonal synapse count, and a mechanism for the postsynaptic cell to regulate the number of innervating synapses. This mechanism may vary in implementation in different synapses and in different species, but the functional behavior is simple, biologically plausible and powerful and seems likely to exist.  9.2.9  Synapse survival is supported by other nearby synapses on the same axon  The sharing of synapse resources between nearby axonal synapses, or the existence of another mechanism within an axon which provides for nearby synapses to influence each others survival, provides a spatial bias for the positioning of synapses. This bias is able to perform the role of the often hypothesized local excitation used in many computational models (see Swindale, 1996) but that has not yet been verified. As discussed below (Sec. 9.2.10), this mechanism also appears to be able to mediate the alignment of refined synaptic projections to different groups of target neurons. The power of this mechanism, and its apparent simplicity, make it also seem likely to exist.  9.2. EXPERIMENTAL PREDICTIONS  9.2.10  190  Cooperative synapse survival mechanism allow for aligmnent of con nections to different cell types  A mechanism within an axon where nearby synapses positively influence each other’s survival (see Sec. 9.2.9) would be able to duplicate a retinotopically refined projection from one group of neurons (e.g. LGN relay neurons) onto an intermixed, separate and not necessarily interconnected group of neurons (e.g. local interneurons), without the requirement of additional mechanisms, such as diffusible substances or neurons providing local excitation or recurrent connectivity. The behav ior of the model suggests that cooperative synapse support is able to align retinotopic projections onto two (or more) groups of intermixed target neurons.  9.2.11  Local excitation and distal inhibition not required for organization  The mechanisms of local excitation, where the excitation of one neuron results in excitation spread ing to its neighbors, and distal inhibition, where the excitation of one neuron results in inhibition of neurons located further away, is commonly used in computational studies, and is hypothesized to contribute to andlor be responsible for many forms of neural organization, including retino topic refinement and eye-specific segregation. The results of the present study indicate that these mechanisms are not necessary for such organization.  9.2.12  Burst intensity is determining factor in spatial representation of af ferents, not overall firing rate  When similar sets of neurons project to the same target brain structure, and the activity of both sets is not correlated, each segregates to innervate spatially distinct domains in the target structure. The relative activity of each set of neurons influences the size of the domains formed by each set of afferents. The characteristic of the activity that determines which group of neurons receives a larger spatial representation is the intensity of activity in each group, as this corresponds with a stronger ability to induce responses in target neurons, and not the overall level of afferent activity.  9.3. FUTURE DIRECTIONS  9.2.13  191  The mechanism governing retinotopic organization and refinement is tolerant to perturbation  The results of the model indicate that the physiological components present in the developing retinocollicular pathway are able to be perturbed, sometimes significantly, while having limited impacts on retinotopic refinement and eye-specific segregation. Behaviors that are critical to or ganization can be perturbed and the effects will similarly be limited, so long as these behaviors are present. It is thus sufficient for modeling studies that explore the functional roles of biological mechanisms in organization to phenomenologically reproduce biological behaviors, and/or repro duce their functional equivalents, and that exact replications of these behaviors are not required.  9.3  Future directions  There are many directions for future work emanating from the results of this study. These fall pri marily into categories of modeling additional organizational phenomena and further refinement of the model. While the model addresses development of the retinocollicular pathway, the physiolog ical components it represents are observed throughout the nervous system. It can thus be applied, with few modifications, to exploring development in many other areas, including the retinogenic ulate and geniculocortical pathways, as well as the retinotectal pathway as observed in fish and frogs. It can also be used to model other sensory and motor systems, such as in the auditory sys tem or the neuromuscular junction. Exploration of these pathways can provide insight into the similarities and differences between them as well as functional requirements that might be unique in particular brain areas. Future efforts should also be applied to continued refinement of the model. Investigating devel opment in other brain areas may indicate the presence of other mechanisms that require repre sentation, such as critical periods for growth, and also suggest how these might best be realized. Additionally, physiological studies, both new and old, should continue to be applied to the model components to improve their phenomenological approximations of physiological behaviors. The  9.3. FUTURE DIRECTIONS  192  current model implementation should also be subject to further “stress-testing” by using it to re produce observations resulting from different experimental perturbations, to help identify possible weaknesses in order that they can be addressed. Finally, alterations should also be made to the model to explore possible deeper interactions between mechanisms, such as trophic feedback and synaptic plasticity. Developing neural pathways are very complex and are made up of a wide range of interacting mechanisms that combine to generate emergent organizational properties, such as retinotopic or ganization and eye-specific segregation. Just as in large scale climate modeling, a model of neural development should have a wide descriptive scope and represent not just patterns of organization but the behaviors of the set of mechanisms that underlie it, not an arbitrary subset of these mecha nisms. Such a model helps us to better understand and predict organization and to gain insight into how the underlying mechanisms interact. This study makes for a first step in that direction.  BIBLIOGRAPHY  193  Bibliography Adams D, Horton J (2006) Ocular dominance columns in strabismus. Vis Neurosci 23:795—805. Adrian E (1941) Afferent discharges to the cerebral cortex from peripheral sense organs. J Phys iol 100:159—191. 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J Neurosci 20:6570—6577.  214  Appendix A Variables and parameters Variable and parameters for equations in Chaps. 3-7  A.1  Variable  Subscripts  Defined  a  ih  Eq. 4.1  d  ih  Distance of axon segment from it’s retinotopicafly correct termination zone  g  x,y  Effective strength of growth factor in grid unit x,y  h i  j k I  -  -  -  -  -  -  -  -  -  Description “Affinity” of axon segment for its location in colliculus  Subscript (axon segment) Subscript (presynaptic neuron) Subscript (postsynaptic neuron) Subscript (synapse) Subscript (axon segment)  m  ih  Eq. 4.5  Nitric oxide influence  n  j  Eq. 5.6  Synaptic input to neuron  p  grow, retract, vesc  Eq. 4.4  Base probability of: axon segment growth or retraction; synapse vesicle release  q  ihx,ihy  Eq. 4.8  Components of external gradient vector  r  ijk  Eq. 6.10  Trophic factor received by the presynaptic terminal  s  ih  Eq. 4.2  Axon resources present in axon segment  -  -  -  -  -  -  -  -  t  u v w x  u  y  i, j, ij, ih  -  -  j  Subscript (time) Subscript (grid unit) Subscript (synapse) Subscript (axon segment) Set of neurons Set of synapses: y axonal, yj dendritic, Yij between i and  j, y on axon segment  h z  i, h  Eq. 4.3  Set of axon segments: zh connected neighbors of segment h, z, segments on neuron I,  segments in neuron i having above average affinity  C  ijk  Eq. 6.1  Baseline STDP potentiation, before efficacy and saturation adjustments.  D  ijk  Eq. 6.13  NMDAR activation  A. 1. VARIABLE AND PARAMETERS FOR EQUATIONS IN CHAPS. 3-7 Variable  Subscripts  Defined  F  i, target  Eq. 5.7  Firing rate of neuron i, target firing rate of neuron  G  u  Eq. 6.9  Growth factor present in grid unit u,  H  j  Eq. 6.14  Homeostatic scaling factor of excitatory input  J  iA, iB  K  ihA, ihB  215  Description  Relative position of RGC i in retina, along ephrin-AIB, EphAIB gradients Relative location of axon segment 6 in colliculus, along ephrin-AIB, EphAJB gradients  L  u  M  u  N  ih  0  Eq. 3.2 -  Eq. 6.11  Nitric oxide present in grid unit u Trophic factor in axon segment h Trajectory (orientation) of existing axon segment  -  Extemal gradients influencing axon growth  Q  -  S  ijk  V  Concentration of extracellular diffusible compound  -  Eq. 6.3  Saturating level of STDP potentation/depression  Eq. 4.10  Direction of axon growth of new axon segment Excitatory strength of synapse ijk  W  ijk  Y  ih  Eq. 6.8  Chemoaffinity score of axon segment  Z  ihA,ihB  Sec. 6.2  Chemoaffinity gradient along ephrin-AJB and EphAJB axes Time interval  a  jt  j3  activity, chemo  Eq. 4.1  Time dependent scaling of a behavior  -  y  i, ih  Eq. 4.3  Threshold for axon growth or retraction  5  i  Eq. 5.8  Number of action potentials in neuron i over previous 500 ms  e  ik,j  Eq. 6.2  -  -  6  ih  Eq. 4.3  Axon segment branch depth.  IC  segs  Eq. 4.3  Reference number of axon segments  syns  Eq. 5.2  Reference number of synapses  max_pet  Eq. 5.2  Maximum allowable ratio of synapses on  expect  Eq. 5.3  Expected trophic factor to be recieved by synapse each spike  epsp  Eq. 5.5  Peak excitation in soma of immature neuron from input of single synapse  H  Eq. 6.14  Maximum level of homeostatic upregulation of synaptic input  ih  Eqs. 4.4, 5.2  t  p  STDP efficacy Temporary variable to indicate size (e.g. of a set)  j from any given i  Probability of: axon segment growth or retraction, synapse generation; dendrite of j accepts new synapse from i  soma, growth, max,  Eq. 5.4  Conductance of neuron, dendrites and synapses  syn t  dif, dec, soma, cxc.  -  Time constant. Superscript shows variable affected  freq, convert, sat i  j, cxc  Eq. 5.5  Excitation level of neuron (j), ‘reversal potential’ of synapses (exc)  0  i  Eq. 5.8  “Accumulator” variable to calculate average firing rate of neuron i  s  growth, NO, trophic,  Eqs. 4.1, 4.4,  +,  x  -  4.1, 6.4 -  -  Scaling factor indicating strength/influence of growth factor, trophic factor, nitric oxide, STDP potentation/depression General variable placeholder, such as for the expression  f(x)  o  NO, NMDAR  Eq. 4.5  Time window indicating length of behavioral effect of component  r  ijk, initial, max  Eq. 5.3  Level of synapse resources present in synapse ijk; initial and maximum number of synapse resources  A.2. PARAMETER VALUES FOR EQUATIONS IN CHAPS. 3-7 Variable  Subscripts  T  <n>  ‘P  i  Defined  Description Normal random number (Gaussian distribution) with mean of 1.0 and standard  -  deviation <n> Sec. 5.2.5  State of vesicle release for synapse jfk: 1 if synapse released vesicle in last millisecond and 0 otherwise  A.2  Parameter values for equations in Chaps. 3-7  Variable  Base value  Pgow  30,,urnues  Pretract  45 mirn,tes  Psyage  36Orniaute  Pvesc  0.5  Pgrowth t  1.0  Values explored  Description  ( lies ( $tes ( liite  l2Ominutes)  Base probability of axon growth  l2Ominatea)  Base probability for axon retraction  1080,ninwe)  Base probability for synapse generation  1.0  Vesicle release probability  —  —  —  0.25  -  0- 10.0  Scaling factor for growth factors  PNMDAR  1  0- 10  P,ric  1.0  0- 10.0  Scaling factor for nitric oxide  Ptraphic  1.0  0- 10.0  Scaling factor for trophic factors  q’+  1.03  q  -0.51  NO 0  2 sec  (0.2-20 see)  NMDAR 0 (  1 sec  0.3 3.0  Length of integration window for coincident postsynaptic activity Peak synapse EPSP  Peak potentiation realized through STDP  -  Peak depression realized through STDP  -  -  ICEPSP  1.0  0.5- 3.0  e.cpect 1  0.25  0 0.6  K.vega  65-75 segments  45-100 segments  ic  40  -  ICH  3  0.1  -  0.05  -  1.0  -  25  10-50 (5-50)  fmaa  50  25-100 (10-200)  Ftarget  0.2 Hz  Omax  3.0  v  50.0  ‘t  7tvesc spikes 4 l  re  34 ms  tre  13.3 ms  75 ms  34.5 ms  -  0 5.0 -  -  -  -  -  -  -  Time window of nitric oxide sensitivity  Amount of trophic factor expected to be received per vesicle released Maximum ratio of synapses on a neuron from a single presynaptic source Reference number of axon segments Reference number of dendrite synapses Upper bound for scaling of synaptic strength Initial resource allocation to new synapse Maximum number of resources in a synapse Target firing rate of collicular neuron Maximum increase in conductance due growth ‘Reversal potential’ of synaptic input STDP saturation time constant (1t,esc is vesicle release probability) Time constant for presynaptic efficacy recovery Time constant for postsynaptic efficacy recovery Standard STDP time constant governing synapse potentiation Standard STDP time constant governing synapse depression  20mm  (2— 200mm)  Axon NT diffusion constant  20mm  (2— 200mm)  Axon NT decay constant  5 mm tjf  Scaling factor for NMDAR activity  -  Time constant for synapse resource production  5 sec  0.5-50 sec  Growth factor diffusion constant  1 mm  0.1-10mm  Growth factor decay constant  216  A.3.  MATHEMATICAL FUNCTIONS  Variable V  Base value  Values explored 20- 50 ms  soma  30 ms  ‘r  2 ms  tG  12 hours  r#req  10 mm  ec  I 3Ominiues  2 sec 1 sec  —  1OOmirnes)  Decay constant of axon resources  2O,irnees)  Diffusion constant of axon resources  0.2-20 sec 0.1  -  Time constant of firing rate estimator 1  —  Description  Dendritic growth constant  -  1 lOminutes  217  Synaptic input decay constant  5-20 mm /  SYMBOLS  Somatic excitation decay constant  -  ( O.2mi,,u  if  ‘rjjf  AND  10 sec  Decay constant for extracellular nitric oxide concentration Diffusion constant for nitric oxide (between adjacent grid Units)  Variation of parameters over the ranges indicated typically had minor qualitative changes to organizational properties. Parameter changes which produced significant changes are noted in Chap. 7. Parameter ranges shown in parenthesis indicates values in this range were not systematically explored.  A.3  Mathematical functions and symbols  Function or symbol E(n,x)  Description Sigmoid-like function (Eq. 3.1) Average value of x  x [x] > 0 T<  Number of elements in set x a and b are upper and lower bounds of x (see Sec. 3.2.1) Normal random number (Gaussian distribution) with mean 1.0 and standard deviation cm>  


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