L O N G - T E R M POTENTIATION AND M A T H E M A T I C A L ANALYSIS OF E L E C T R I A L MODELS OF DENDRITIC SPINES by R A J P A L SINGH T H I A R A B.Sc. (Mathematics) University of Calgary, 1994 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E D E G R E E OF MASTER OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics Institute of A p p l i e d Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA July 1998 Â© R a j p a l Singh T h i a r a , 1998 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract Dendritic spines are small evaginations of the dendrites of neurons first discovered late in the 19th century. Since their discovery, many theories have been put forth to explain the physiological role of the spine. However, only recently w i t h the advent of new laboratory technology has data been available to test the various theories put forward. The two most compelling theories today are that spines are important mediators of a form of cellular memory known as long-term potentiation ( L T P ) , and that spines may be involved i n the conduction of regenerative electrical impulses within dendrites similar to action potentials in axons. We w i l l review some of the major mathematical models put forth which attempt to explain the role spines may play i n the induction of L T P . We will address the importance of calcium signals i n L T P induction and suggest how the unique morphology of the spine may allow for transient, spatially localized increases i n calcium w i t h i n the spine head, but not elsewhere i n the dendrite. This could help account for the associativity, cooperativity, and input specificity requirements of L T P . We w i l l also review some of the major mathematical models on dendritic action potentials. These models generally assume the existence of voltage-gated ion channels w i t h H o d g k i n - H u x l e y (HH) type dynamics exist i n the spine head. We w i l l employ a cont i n u u m approach i n which spines are modeled as having a certain uniform density. We w i l l further make use of the F i t z H u g h - N a g u m o ( F H N ) equations without recovery to approximate the H H equations. We w i l l examine the new set of equations i n the traveling wave frame and seek to determine how the various parameters influence the speed, and the shape of the traveling wave front solutions. We show that there is a certain balance between local excitation of the spine heads, and freedom for the electrical current to pass from spine head to dendrite required for traveling front solutions to exist. Furthermore, strict parameter spaces in which traveling front solutions exist are determined, as is the profile of the wave i n the special case of a standing wave. ii Table of Contents Abstract ii Table of Contents iii List of Figures vi Acknowledgements vii Chapter 1. Introduction to Dendritic Spines 1 Chapter 2. Spines and Long-Term Potentiation 7 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Introduction to Long-Term Potentiation The Coincidence Detector First Generation Mathematical Models of L T P 2.3.1 The First A t t e m p t : Gamble and K o c h , 1987 2.3.2 The Second A t t e m p t : Holmes and Levy, 1990 2.3.3 The T h i r d A t t e m p t : Zador and K o c h , 1990 2.3.4 Conclusions about first generation L T P models Numerical Investigation of Second Messenger Diffusion i n Dendritic Spines Linearizing C a l c i u m Dynamics through A s y m p t o t i c Expansions The Spatial Nature of C a Transients i n L T P The temporal nature of C a transients i n L T P 2.7.1 A M o d e l of C a Dynamics i n Dendritic Spines Protein Kinases and Phosphatases in L T P 2.8.1 The role of P K C and sufficiency of C a i n L T P induction C o n c l u d i n g remarks about L T P 2 + 2 + 2 + 2.8 2 + 2.9 Chapter 3. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Electrical Modeling of Spines Introduction Spines mediate synaptic potential attenuation Spines allow for linear summation of E P S P s Spine neck resistance as controller of synaptic weight Spines attenuate transient synaptic input Spines increase the low-pass filtering capacity of neurons The effect of passive spines on the cable properties of dendrites C o m p u t a t i o n a l richness of active dendritic spines Pseudosaltatory conduction i n dendritic spines in 7 8 9 12 13 15 17 18 19 23 25 27 30 35 36 38 38 38 39 40 41 42 42 46 48 Table of Contents C h a p t e r 4. 4.1 4.2 4.3 4.4 Spatially homogeneous steady states Traveling front solutions 4.2.1 Stationary front solutions 4.2.2 Positive speed traveling wave fronts N u m e r i c a l results 4.3.1 V a r y i n g a 4.3.2 V a r y i n g K 4.3.3 V a r y i n g 7 4.3.4 V a r y i n g r N u m e r i c a l simulation of the P D E model C h a p t e r 5. 5.1 5.2 5.3 6.3 7.3 T h e Full Cubic Model 57 58 60 63 64 69 69 70 71 73 75 81 83 85 86 92 96 96 97 98 100 101 108 Spatially homogeneous steady states 109 Traveling front solutions 111 6.2.1 Using integration to determine zero speed wave solutions 114 6.2.2 C a l c u l a t i n g the profile of a zero-speed front 115 6.2.3 General method for getting solution trajectories to degenerate reactiondiffusion traveling fronts 119 N u m e r i c a l simulation of the P D E 122 C h a p t e r 7. 7.1 7.2 Piecewise Linear Continuous A p p r o x i m a t i o n Spatially homogeneous steady states and stability Traveling front solutions 5.2.1 Stationary front solutions 5.2.2 Positive speed traveling fronts Numerical Results 5.3.1 V a r y i n g a over its range 5.3.2 V a r y i n g K over its range of values 5.3.3 V a r y i n g 7 over its range of values 5.3.4 V a r y i n g r over its range of values 5.3.5 N u m e r i c a l simulation of the P D E model C h a p t e r 6. 6.1 6.2 Piecewise Linear Discontinuous M o d e l Discussion 124 Theories of spine function Discussion of spines and L T P 7.2.1 Spines and L T P 7.2.2 T h e electrical properties of spines Results for the 3 analogue models 7.3.1 T h e piecewise linear discontinuous model 7.3.2 T h e piecewise linear continuous model 7.3.3 T h e cubic model iv 124 125 125 127 128 128 131 133 Table of 7.4 7.5 Contents Future areas of investigation Conclusion 134 135 Bibliography 136 v L i s t of Figures 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5.1 5.2 5.3 5.4 5.5 5.6 Speed of the wave, c, as a function of a for 7 = 1.5, n = 5, r = 10 70 Eigenvalues of the wave as functions of a for 7 = 1.5, K â€” 5, r = 10 71 Speed of the wave, c, as a function of K for a = 0.1, 7 = 1.5, r = 10 72 Eigenvalues of the wave as functions of K for a = 0.1, 7 = 1.5, r = 10. It should be noted that the abscissa here is not zero, but is a value close the value of K which corresponds to zero-speed waves as shown in Figure 4.3 73 Speed of the wave, c, as a function of 7 for a = 0.1, K â€” 5, r = 10 74 Eigenvalues, \i\ and /i2, as functions of 7 for a = 0.1, K = 5, r = 10. The abscissa is not zero i n this graph. It corresponds to the lower value of 7 which corresponds to zero-speed waves as can be seen in Figure 4.5 75 G r a p h of /13 as a function of 7 for a = 0.1, K = 5, r = 10 76 Speed of the wave, c, as a function of r for a = 0.1, 7 = 1, K = 5 77 Graphs of \i\ and ^ s functions of r for 7 = 1, K, = 5, r = 10 78 G r a p h of / i as a function of r for 7 = 1, K = 5, r = 10 79 Simulation of the traveling front for 7 = 1, K = 5, r = 10 80 2 a 3 5.8 5.9 5.10 5.11 Speed of the wave as a function of a for 7 = 1.5, K = 5, r = 10 z\ as a function of 7 = 1.5, K = 5, r = 10 Speed of the wave as a function for K for a = 0.1, 7 = 1.5, r = 10 z\ as a function of K of a = 0.1, 7 = 1.5, r = 10 Speed of the wave as a function of 7 a = 0.1, K = 5, r = 10 u; over the range of 7 values for which there are complex conjugate roots for a = = 5, T = 10 The real eigenvalues which exist on the other side of the bifurcation for a â€” = 5, r - 10 G r a p h of Z\ as a function of 7 for a = 0.1, K = 5, r = 10 Speed of the wave as a function of r for a = 0.1, 7 = 1, /c = 5 Z\ as a function of r for 7 = 1, K = 5, r = 10 T h i s is a simulation of the traveling front for a, = 0.1, 7 = 1, K â€” 5, r = 10. . 6.1 T h i s is a simulation of the traveling front for a = 0.1, 7 = 1, K = 5, r = 10. . 123 K 5.7 K vi 97 98 99 100 101 0.1, 102 0.1, 103 104 105 106 107 Acknowledgements T h i s thesis was produced under the supervision of D r . Robert M i u r a , and was funded by a scholarship from the National Sciences and Engineering Research C o u n c i l . I would like to thank D r . M i u r a for his encouragement, support, and many suggestions on how to improve the thesis. I would also like to thank D a v i d Iron and A t h a n Spiros for their technical assistance i n the production of the thesis. vn Chapter 1 Introduction to Dendritic Spines Cajal's classic work on neurophysiology [17] conclusively proved that the cell doctrine, the idea that all organs are made up of cells, applied equally well to the brain as it did to all other organs of the body. Since then, physiologists have pondered the role of the neuron's unusual morphology in its capacity for computation. The neuron contains highly arbourized appendages known as dendrites and axons, which branch off of a central body known as the soma. It is conventionally thought that the computational complexity of the brain is a consequence of the numerous connections neurons form w i t h each other at synapses. B y synapse we mean that morphological entity in which the terminal branch of an axon of one neuron, the presynaptic neuron, is directly opposed to another neuron, the postsynaptic neuron, but separated by some small distance (50 nm) [48]. It is estimated that there are approximately 10 12 neurons each forming up to as many as 2 x 10 5 synaptic connections w i t h other neurons [48]. Since it is thought that synapses are the m a i n sites of "information transfer," this enormous connectivity makes the problem of decoding how the central nervous system (CNS) functions enormously difficult, and gross simplifications are required to elucidate important principles. One classical simplification is the theory of dynamic polarization. T h i s theory holds that dendrites act as antennae which are acted upon by neurotransmitters released at the axon terminal opposed to it in response to an action potential. These neurotransmitters cause a local change in conductance of the dendritic membrane to certain ions, resulting i n a 1 Chapter 1. Introduction to Dendritic Spines passive electrical current which propagates towards the soma. T h i s current is converted into a frequency of action potentials at the axon and is the sole output for a neuron. T h e foundations of this theory were weakened by experimental and theoretical investigations of the granule cells i n the olfactory bulb. It was shown that the dendrites of granule cells release neurotransmitters which mediate the inhibition of m i t r a l and tufted cells [89]. T h i s was the first demonstration that dendrites not only could receive inputs, but also could act as output devices. T h e key compartment is the dendritic spine, "the smallest neuronal compartment capable of performing a complete i n p u t - o u t p u t operation of a single synapse" [105]. Spines are small evaginations of dendrites first noticed by C a j a l i n 1888 [17]. They are the site of type 1 excitatory synapses, which make up over 90% of a l l excitatory synapses i n the C N S [31] w i t h glutamate usually acting as the neurotransmitter [105]. The typical morphology of a spine is a small, roughly spherical head connected v i a a long, t h i n cylindrical stem to the dendrite [35]. Spines are typically just large enough to form one, or occasionally two synapses, and thus form "a structural, biochemical, and physiological compartment that is specific for that synapse" [35]. In this regard, it should be pointed out that the physical composition of spines is discontinuous from that of its parent dendrite. The most obvious example of this discontinuity can be seen i n the structure of the cytoskeleton [35] where there is the complete absence of microtubules. Instead, the cytoskeleton is composed of a loose filamentous network of actin and actin-regulating proteins which provide the scaffolding for the basic spine structure. T h e actin-regulating proteins, calmodulin ( C a M ) , fodrin, myosin, and microtubule associated protein 2 ( M A P 2 ) , interact w i t h actin i n a calcium-dependent manner. T h i s interaction has been suggested as a mechanism underlying short-term memory [23]. One of the unique morphological 2 Chapter 1. Introduction to Dendritic Spines characteristics of the spine is the postsynaptic density ( P S D ) which is an area approximately 50 n m thick lying just beneath the spine head membrane and directly opposed to the axon terminal. Its dimensions are usually proportional to the dimensions of the spine head, but not the spine neck [36]. Furthermore, it has been shown that the area of the P S D also is proportional to the number of presynaptic vesicles i n the axon terminal to which it is opposed, and this suggests that some form of p r e - and postsynaptic dynamic coregulation is involved and points to a possible role for spines i n regulating synaptic efficacy. T h e spine also is unique i n the presence of a select group of organelles [35]. A l l spines contain smooth endoplasmic reticula ( S E R ) , which is involved i n C a and membrane synthesis. synthesis. 2 + sequestration M a n y also contain polyribosomes, which function i n protein Some complex spines also contain mitochondria and multivesicular bodies which are restricted to the base of the spine. Thus, spines appear to have the capability of "independent energy metabolism, membrane turnover, and protein synthesis and thus in theory could function as a semi-autonomous unit w i t h i n a neuron" [35]. In this light, it should be noted that von Neumann suggested that the fundamental unit of integration is not likely to be the whole neuron, but instead the single synapse [109]. T h e spine seems to be uniquely situated to mediate this role w i t h i n the synapse. T h i s would not be of much interest except for the fact that there is experimental evidence suggesting that the spine may, i n fact, actually perform such roles. Spines are degenerate or irregular i n many mental disorders: t h i n and tiny i n Down's syndrome a n d long and tortuous i n mental retardation (Fragile X syndrome) [105]. Furthermore, i n partial epilepsy, cortical pyramidal cells at a distance from the site of the seizure show simplified dendritic trees and decreased spine density, graded w i t h the duration of the seizures. In less pathological cases, it has been shown that: 1) there are changes i n spine stem length during field learning i n honeybees, 2) rats trained i n spatial learning show increased spine 3 Chapter 1. Introduction to Dendritic Spines densities on basal dendrites of C A 1 hippocampal neurons, and 3) chicks show increased spine density in the lobus parolfactorius following one learning experience [35]. However, the best example of a role for spines i n memory comes from the Bruce effect where exposure of unfamiliar male pheromones to an impregnated female causes abortion. T h i s effect is thought to be mediated by metabotropic glutamate receptor 2 ( m G l u R 2 ) i n the olfactory granule spine [47] and could be prevented by the m G l u R 2 agonist, D C V - I V . A s a result, it has been suggested that m G l u R 2 s in the olfactory granule spine acts as a storage device for the original male pheromone [105]. Thus, there is an increasing amount of evidence suggesting that spines may be quite important both i n memory and i n normal cognitive function, but its exact role and the mechanism through which it accomplishes this still is largely unknown. One of the great difficulties i n studying dendritic spines is that their sizes situate them just at the edge of optical resolution using light microscopy (typical dimensions of dendritic spines i n pyramidal cells of rodent neurons are 0.1-0.2 / i m for the spine stem length, 0.04-0.2 pm for the spine stem diameter, 0.004-0.2 / / m for total spine volume, and 3 0.1-0.7 / i m for total spine surface area). 2 Furthermore, they are virtually impossible to study i n detail using traditional electrophysiological techniques as they cannot be impaled by electrodes and generally are electrically remote from the soma. Recently, more sophisticated fluorescence microscopy techniques have been developed and have allowed some insight into spine function. However, due to the long period before these new experimental techniques became available, and the great deal of interest i n the possible roles of spines, many theories about how spine morphology could contribute to its hypothesized functions have been put forward. Perhaps the simplest claim that has been made is that spines increase the surface area of the dendrites and simply serve to connect neurons [17]. However, it has been shown that even if all spines were removed from the dendrites in pyramidal cells there would still 4 Chapter 1. Introduction to Dendritic Spines v be ample room for the formation of synapses in the dendrites. A more likely possibility in this regard is that spines "allow more synaptic connections to be compacted into a limited brain volume, and hence they can be considered the microscopic parallel to sulci and gyri i n the b r a i n " [35]. A recent claim that has begun to garner a great deal of interest is the idea that the spine acts as a compartment to allow for the biochemical amplification and spatial localization of second messengers that are produced as a result of local synaptic activity. Another area of interest has been the electrical roles that spines may have i n mediating synaptic excitation. A s a model system that has drawn a great deal of interest recently, we w i l l review some theories put forth on how features of spines may be critical i n the induction and expression of a cellular correlate of memory known as long-term potentiation ( L T P ) in Chapter 2 (for a more general review of L T P , see [12]). T h i s chapter w i l l serve as background material and is an area of much current interest amongst experimentalists and theoreticians in the neurosciences. The remainder of the thesis will focus on theories put forth on the electrical roles that spines may play. We w i l l start by giving a thorough review of the literature i n Chapter 3. We w i l l focus, in particular, on theories concerned w i t h excitable dynamics. These theories suggest that there may be dendritic action potentials mediated by voltagegated sodium and potassium channels within spine heads. O f special interest w i l l be a continuum model developed by Baer and Rinzel [5] in which spines are not treated as individual entities, but rather are averaged out over space as a uniform density of spines. T h i s allows several important simplifications to be made. A t this stage, the reader will be well acquainted w i t h the various theories of excitable dynamics in spine heads, and some of the inferences made based on numerical experimentation. However, one failing of numerical simulations is that they fail to give one a good 5 Chapter 1. Introduction to Dendritic Spines grasp of what roles various biophysical features of a neuron are important i n allowing for the existence of traveling wave pulse solutions. The reason that numerical simulations are so vitally important is that the system of equations used to approximate the excitable dynamics, the H o d g k i n - H u x l e y equations, involve transcendental P D E s . A very successful approach that has been used over the years in this situation is to replace the complicated nonlinear functions w i t h a cubic function and to replace the complicated recovery equations, w i t h one simple linear O D E . T h i s new set of equations is known as the F i t z h u g h - N a g u m o ( F H N ) equations, and they are amongst the most commonly studied set of equations i n applied mathematics. Occasionally, instead of using a cubic nonlinearity as i n the F H N equations, other functional forms are used. These equations are usually piecewise linear and attempt to m i m i c some of the general features of the cubic. The advantage of this approach is that the equations can be solved exactly. In Chapters 4, 5, and 6, we w i l l use the F H N approach to simplify the equations governing the excitable dynamics in a continuum spine model of an infinite dendrite. In Chapter 4, the functional form we will use is a piecewise linear discontinuous function of the form â€”v + H(v â€” a). In Chapter 5, the functional form we will use is a piecewise linear continuous function whose shape approximates that of a cubic polynomial. In Chapter 6, we w i l l use the standard cubic used i n the F H N equations. We w i l l show how the various parameters play a role in determining whether traveling wave solutions exist and show how the speed of the propagating wave depends on the values of the parameters. 6 Chapter 2 Spines a n d L o n g - T e r m P o t e n t i a t i o n 2.1 Introduction to Long-Term Potentiation In 1949, D o n a l d Hebb proposed a coincidence detection rule i n which the strength of association between two neurons would increase if they were simultaneously active [38]. In 1973, such a phenomena was first observed in the dentate gyrus of anesthetized rabbits and was termed long-term potentiation ( L T P ) [11, 10]. L T P may be expressed as a persistent increase i n the size of the synaptic component of the evoked response recorded from individual cells or from populations of neurons [12]. B y persistent, we mean a stable increase i n synaptic efficacy which lasts at least one hour. T h i s is to be differentiated from shorter changes such as post-tetanic potentiation ( P T P ) , which lasts on the order of seconds, and short-term potentiation ( S T P ) , which is a transient increase i n synaptic weight that decays back to baseline w i t h i n the first 30 minutes. One of the most interesting features of L T P is that it is prominent i n a l l areas of the hippocampus. T h i s brain structure has long been implicated from lesion studies as being important to memory [104]. W h a t is even more intriguing is that stimulation protocols that are capable of inducing L T P i n the laboratory are qualitatively similar to those actually seen w i t h i n the hippocampus during learning. L T P exhibits three very important features that are expected of any cellular substrate of memory: cooperativity, associativity, and input-specificity [12]. Cooperativity is used 7 Chapter 2. Spines and Long-Term Potentiation to describe the requirement that L T P has a threshold wherein weak stimuli are not encoded into an increase in synaptic efficacy. Such a mechanism is required to prevent "spontaneous memory" formation. Associativity means that a subthreshold activation at one synapse may become potentiated if there is strong activity at a nearby synapse. Thus the efficacy of a synapse is dependent upon its coactivity patterns w i t h its neighbors. Input-specificity is the requirement that inactive synapses should not undergo an increase in efficacy. T h i s once again can be viewed as some form of prevention of "spontaneous memory". 2.2 The Coincidence Detector L T P has garnered a great deal of interest because of its possible linkage w i t h memory and because it has yielded many fascinating biochemical results. One of the first problems to be dealt w i t h was the identification of a coincidence detector. B y a coincidence detec- tor, we mean a cellular process which is activated during strong p r e - and postsynaptic coactivation and serves as a signal for the induction of L T P [12]. In C A l hippocampal pyramidal neurons, the coincidence detector is most likely a ligand-gated cation channel, the n i c o t i n i c - m e t h y l - D - a s p a r t a t e receptor ( N M D A R ) , which opens i n response to glutamate and N M D A . N M D A R s are thought to be capable of functioning as the coincidence detector because they are largely blocked at potentials below -20 m V by M g 2 + [73]. Thus in order for a synapse to become potentiated, it has to be active, and the neuron has to be highly depolarized in the vicinity of the N M D A R . A s a result, the neuron may be in the firing mode (threshold is typically a potential of fÂ» â€” 5 0 m V at the axon hillock). In support of this theory, it has consistently been found that N M D A R antagonists block the induction of L T P [12, 73]. The next issue becomes what feature of N M D A R activation is crucial to L T P induction. N M D A R s are highly calcium permeable, and C a i s a potent second messenger capable of 2 + 8 Chapter 2. Spines and Long-Term Potentiation initiating many different biochemical cascades involving enzymes such as protein kinases ( P K s ) , which add P 0 4 3 ~ to proteins, and protein phosphatases ( P P s ) which remove PC>4 ~ from proteins, and a variety of other proteins such as C a M and actin. Evidence 3 for the importance of calcium i n L T P induction comes from experiments which show that calcium chelators are capable of preventing the induction of L T P when administered to the postsynaptic neuron. Thus, the idea that L T P is a calcium-dependent process depending on N M D A R activation became entrenched within the discipline. T h i s led inevitably to the question about the nature of the calcium transients that are involved i n the induction of L T P . It was generally accepted that there must be some sort of threshold relating to some feature of the calcium transient. Above this threshold, L T P would be induced, but below this threshold, only S T P or perhaps even P T P could be induced. However, due to the l i m itations of experimental techniques, there was no known method of actually observing the calcium transients within dendritic spines, the presumed locus for L T P induction. A s a result, a number of mathematical models were proposed to help gain insight into the importance of various biophysical systems thought to be important i n mediating the calcium transients. 2.3 First Generation Mathematical Models of L T P The first generation models of L T P focussed heavily on what was biophysically necessary to produce large, spatially localized C a 2 + transients under stimulation protocols similar to ones that experimentally induce L T P . It was thought that memory formation should involve some sort of a biochemical threshold. Below this threshold, there is very weak activation of a critical molecule, and above this threshold there is near m a x i m a l activation of this molecule. Thus, some recurring themes i n these models were: the l i m i t a t i o n of calcium channels to spine heads which acted as the sole source of calcium for the cell, and 9 Chapter 2. Spines and Long-Term Potentiation a heterogeneous distribution of calcium pumps and calcium buffers (these are c a l c i u m binding proteins ( C a B P s ) which generally diffuse slowly and can be seen to function, i n some sense, as a capacitance [121]). The basic physical geometry of the spine was a cylindrical spine head attached v i a a cylindrical spine stem to a cylindrical dendritic shaft. In some of the models, only a single spine stem was modeled, while in others, many were. In the electrical part of models, the spine head was modeled as an isopotential compartment which is linked through a resistor to the parent dendrite. The resistor represents the theoretical total resistance conferred by the spine stem. This represents a lumpedsum resistance approximation which gives a good approximation to the actual system. Certainly, the amount of error introduced in this approximation is far smaller than the uncertainty i n the electrical parameters used. The voltage in the dendrite was determined from the passive cable equation. The motion of calcium was based on simple linear diffusion, coupled to nonlinear effects due to the pumps and buffers. The entry of C a 2 + is voltage-dependent and occurs only i n the distal part of the spine head. Otherwise, the voltage and C a m o v e m e n t equations are 2+ decoupled. However, this may not be justified in small compartments such as dendritic spines [86]. The following parameters and variables are common i n many of the following models and w i l l be defined here: 10 Chapter 2. Parameter Spines and Long-Term or Variable Potentiation name Description v The synaptic reversal potential of a channel, c tpeak,c The time to peak conductance of a channel, c c peak conductance of a channel, c 9c C specific membrane capacitance Rjn specific membrane resistance Ri specific intracellular membrane resistance m R<x>,sp spine input resistance R<oo,d infinite dendrite input resistance D calcium diffusion coefficient RN spine neck resistance Ca r n radius of spine neck In length of spine neck sp radius of spine head 1'Sp length of spine head r<i. radius of dendrite r forward buffer rate constant kb f kb backwards buffer rate constant kpi rate constant for a first-order kinetic C a b Ca K 2 + turnover rate for a C a Ps surface density of a C a K constant of dissociation d 2 + 2 + 2 + pump pump pump F Faraday's constant (9.6 x 10 Coulombs/mol) [B] concentration of C a - b o u n d buffer [M] concentration of free buffer [Bt] total concentration of buffer D diffusion coefficient of the bound buffer B 4 2+ 11 Chapter 2. 2.3.1 Spines and Long-Term Potentiation The First Attempt: Gamble and Koch, 1987 The first realistic L T P model was put forth by Gamble and K o c h ( G K ) i n 1987 [27]. In this model, the dendritic spine head contained synaptic receptors, voltage-gated calcium channels ( V G C C s ) , voltage-gated noninactivating potassium channels, and a leak conductance. The spine head was heterogeneous and contained a specialized area under the outer circular face called the shell which roughly corresponds to the post-synaptic density. The spine neck also was divided into distal and proximal portions, which correspond to the sides furthest and nearest, respectively from the dendrite. The parent dendrite was homogeneous and contained only one leak conductance. Calcineurin and C a M were present as non-mobile, spatially inhomogeneous buffers. The concentrations of calcineurin and C a M were taken to be 10 pM and 50 pM, respectively, i n the shell, and 5 pM and 25 pM, respectively, in all other areas. The binding of C a to each of C a M ' s four C a 2 + 2 + to calcineurin and binding sites was taken to be governed by first-order kinetics. There was also a non-saturable adenosine triphosphate ( A T P ) driven c a l c i u m - p u m p w i t h first-order kinetics, an equilibrium value of 50 n M for [ C a ] , the intracellular calcium 2+ i concentration, and a time constant of 2 ms located i n a t h i n compartment apposed to the subsynaptic membrane. â€¢ Some of the m a i n discoveries from this model were: 1. H F S is much more important than the total amount of synaptic activity i n inducing large increases i n [ C a ] j in small compartments such as dendritic spines. 2+ 2. The relative change i n [Ca ]^ was much smaller than the relative change i n con2+ centration of C&1 + Ca 2 + â€” C a M . This was due to the fact that the concentration of â€” C a M depends on the fourth power of [Ca ]^ . 2+ 3. Synaptic activity at a nearby spine was simulated by the injection of depolarizing current during spike activity at the spine of interest. 12 Depolarizing current more Chapter 2. Spines and Long-Term Potentiation than doubled the increase in [ C a ] 2 + . the concentration of calcium w i t h i n the s p spine head. T h i s showed the possible importance of associativity i n L T P . 4. The high input resistance of the spine allowed for much larger E P S P s i n the spine head than i n the parent dendrite. This allowed for the activation of high-threshold VGCCs. 2.3.2 The Second Attempt: Holmes and Levy, 1990 Holmes and Levy ( H L ) developed a model in 1990 looking for insights into how N M D A R s might be important for the associative nature of L T P . They based their model on a rat hippocampal dentate granule cell and explicitly included 1-115 dendritic spines i n their model. The model also contained buffers, pumps, and diffusion for C a 2 + . The model differed from the G K model i n that it included other spines and a completely different set of receptors. The H L model included N M D A R s and n o n - N M D A R s , which were thought to conduct a mixed N a / K + + current. The model had no V G C C s since they are not important i n the induction of L T P in these cells, and there is little evidence for their existence i n significant numbers. A t that point i n time there was little quantitative information about the kinetics of the NMDARs. Thus, Holmes and Levy developed a mathematical model involving simple kinetics for receptor binding and for the transitions of receptor to different conductance states for both the N M D A R s and n o n - N M D A R s . T h e qualitative features that they were trying to build into their model of N M D A R function were that the unbinding of neurotransmitters (NTs) from N M D A R s had to be slow, the transition of the N M D A R - N T complex to an open channel state had to be even slower, the N M D A R is blocked by M g 2 + i n a voltage-dependent manner, and the average number of N M D A R s on a single spine head that were open at any given time had to be small (usually < 1). The approach used to calculate C a 13 2 + influx was simply to Chapter 2. Spines and Long-Term Potentiation determine the conductance of the N M D A R s and turn it into a C a that N M D A R s were permeable to C a of Pea : 2 + 2 + - f l u x by assuming , N a , and K with a relative permeability ratio + + PK 'â€¢ PNO, of 10.6:1.0:1.0. T h e important qualitative differences between the N M D A R s and n o n - N M D A R s were the longer mean open times for N M D A R s , The one major assumption made was the linearity of conductance addition. T h a t is, the conductance changes due to activation of receptors from temporally separated presynaptic events do not affect each other. This assumption is only valid if the number of receptors bound to N T s is small compared to the number of receptors. For the C a 2 + part of the model, the spine head was split into four compartments: two just under the outer circular surface of length 50 n m representing the P S D , and two more of length 225 n m . T h e spine neck was split into three roughly equal sized compartments. To model the C a 2 + i n the vicinity of a spine, four dendritic compartments of length 0.5 pm were used, but adding additional ones didn't affect the [ C a ] 2 + s p . Ca 2 + movement between the different compartments was modeled according to simple linear diffusion between compartments, binding to buffers (based on C a M dynamics as assumed i n the Z K model), and elimination by nonhomogeneously distributed pumps. The m a i n conclusions of this model were: 1. P r i o r theories about the N M D A R acting as the source of the nonlinearity i n L T P are probably not true. It was previously thought that the voltage-dependence of the N M D A R could result i n a positive feedback loop which would generate a large Ca 2 + influx. However, C a 2 + influx could never be increased more than four-fold over the control. 2. T h e role of fast buffers is to amplify the nonlinearity i n [Ca "]^, and buffers play a 24 more important role i n modifying the C a 2 + transient than pumps, or diffusion. 3. T h e associativity of L T P could be seen i n the voltage-dependence of the N M D A R 14 Chapter 2. Spines and Long-Term Potentiation as it required a large number of coactive synapses firing at high frequency to result in large changes i n C a 2 + influx, and [ C a ] 2+ sp 4. Dendritic spines, because of their small volume, "provide a locusforphysiologically important transient increases i n [ C a ] " [41]. 2+ 2.3.3 i The Third Attempt: Zador and Koch, 1990 Later i n 1990, Zador and K o c h ( Z K ) put forward a model [122], w i t h the same spirit as the H L model. T h e basic models for the voltage equations were the same although only one spine was explicitly modeled, and the basic C a incorporating diffusion, pumps, and buffers for C a 2 + movement model was the same, . There were, however, a couple of 2 + differences which make this model worth noting. The most fundamental difference is that the Z K model is based on C A l h i p p o c a m p a l tissue w i t h pyramidal neurons which is the most thoroughly studied tissue i n the field of L T P research. T h e morphology of the model was similar to the one used i n the previous two models w i t h only slight quantitative differences. It should be noted that the value of the input resistance was much larger than i n the H L model and result i n less stringent requirements for associativity. In the electrical model, explicit, empirical formulas for the forms of the time-dependent conductances for N M D A , and n o n - N M D A R (here it was based on A M P A R s ) were found: 9non-NMDA{t) = 9nonâ€”NMDA~ - 9NMDA exp exp ( ^ ) 9NMDA(t,V) where 9nonâ€”NMDA 9non-NMDA â€” is a n ^ + alpha function with t peak M / _1 and CJNMDA â€” J , (2.1) - exp ( ^ ) g 2 + ] ^ ^ (2.2) y = 1.5ms, and a peak conductance of -5nS. T h e parameter values for 9MMDA 0.33mM , 7 = 0.06mV _ 1 l ( 1 â€” were T X = 80ms, r 2 = .67ms, 77 = 0.2 nS. T h e corresponding currents generated 15 Chapter 2. Spines and Long-Term Potentiation in the spine head were calculated under the assumption of linear conductance addition as i n the H L model. In the C a part of the model, it was assumed that all of the C a 2 + by the N M D A R s w i t h i n the spine head. 2 + influx was mediated T h e factors involved i n C a 2 + m o b i l i t y were immobile cytosolic buffers based on C a M , as i n previous models, a heterogeneous density of C a 2 + pumps obeying first-order M i c h a e l i s - M e n t o n kinetics, and C a 2 + diffusion. T h e density of pumps i n the proximal spine neck, the part closest to the dendrite, was much higher i n the spine neck due to the possible presence of the spine apparatus which may be important i n calcium sequestration. T h e method of determining the size of the calcium flux was to convert the current mediated by the N M D A R by using Faraday's constant, the valence of calcium, and by multiplying by 0.02, into the amount of the current thought to be due to C a 2 + . The basic assumption of the modeling is that the degree of potentiation of a synapse should be a monotonic function of peak [Co^CaM]. Some of the results from these simulations were: 1. T h e high surface area to volume ratio of the pumps allowed for a great deal of separation between dendritic calcium concentration and spine calcium concentration. Thus spines are to a great extent isolated from each other w i t h respect to calcium fluxes. 2. C a 2 + dynamics i n the spine are quite sensitive to spine morphology and this could be an important mechanism i n metaplasticity, the plasticity of synaptic plasticity [1] (for a model which explicitly explored this concept see [29]). 3. T h e associativity nature of L T P arises through the voltage-dependence of the N M D A R . It was found that a weak input which alone resulted i n very little elevation of [Ca " "]^ i n the spine could result i n a much greater elevation of [ C a ] if there 2 1 2+ 16 i Chapter 2. Spines and Long-Term Potentiation was enough activity at coactive synapses along the dendrite. 2.3.4 Conclusions about first generation L T P models A t this point, it would seem that these first models had done a credible job of explaining the robust experimental properties of L T P . L T P occurs when there is a sufficient amount of presynaptic activity at a synapse coupled to strong, local depolarization i n the neighbourhood of the dendritic spine to alleviate the voltage-dependent M g 2 + block of the N M D A R â€” this explains the associativity and cooperativity properties of L T P . The spine is critical since its morphology allows large local changes i n [ C a ^ due to the 2 presence of buffers, the diffusional restriction of its t h i n neck, the presence of calcium pumps, and its small volume relative to that of its parent dendrite â€” this explains the input specificity of L T P . Lastly, the threshold nonlinear type behavior expected of L T P induction can be seen i n the activation of C a B P s . Part of the difficulty w i t h the models as presented is that they are computationally intensive and fail to give much qualitative insight into the nature of calcium dynamics in a dendritic spine. One simplification is to identify a smaller number of lumped parameters whose values would give a good understanding of the nature of the calcium dynamics. If this is not possible, then a more thorough numerical investigation into how buffering, pumping, and diffusion parameters interact with spine shape to determine the shape of calcium transients should be carried out. Work has been done i n both of these areas by W o o l f and Greer who carried out numerical simulations on diffusion of second messengers in dendritic spines [118], and by Zador and K o c h [121] who carried out analytical work (based on the approach used by Wagner and Keizer [110]). 17 Chapter 2. 2.4 Spines and Long-Term Potentiation Numerical Investigation of Second Messenger Diffusion in Dendritic Spines W o o l f and Greer numerically modeled the diffusion of second messengers in the dendritic spines of granule cells of the olfactory bulb. T h e spine morphologies were based on serial electron microscopy reconstruction [119]. These cells were chosen rather than hippocampal pyramidal cells since their spines have both input and output operations, and their necks are thinner and longer than hippocampal pyramidal cell dendritic spines so the lack of spatial localization of second messengers i n the granule cell spines implies the same of the hippocampal pyramidal cell spines. T h e numerical results of their simulations suggested the following: 1. In general, longer and thinner spine necks resulted i n larger transients of second messengers i n the spine heads, and longer times were required to reach peak concentrations i n the spine neck. 2. E x t r u s i o n mechanisms often can be inefficient i n spatial sequestration of second messenger signals at physiologically plausible density levels. 3. B i n d i n g mechanisms can profoundly affect the shape of second messenger transients and the ability of the signals to reach the base of the spine. These results have some important implications. If there is a calcium-induced calcium release ( C I C R ) mechanism i n spine heads, then the effect may simply prolong the elevation of calcium i n the spine head. However, if the release mechanisms are present in the spine neck and throughout the dendrite, then there could be active propagation of calcium signals from one spine head to another. There is also a question about the diffusion of activated C a B P s such as C a M . A n important factor to consider here is the off-rate, or the rate of unbinding. C a M has a relatively rapid off-rate, so activated C a M levels w i l l quickly decay once the calcium transient begins to die. Thus the degree of 18 Chapter 2. Spines and Long-Term Potentiation localization w i l l be strongly influenced by calcium-clearance mechanisms. The picture is a bit different for membrane-bound second messengers such as D A G and P K C . They w i l l probably be restricted to the spine head and theoretically allow a much better spatial localization of the effects of synaptic activity than cytoplasmic diffusible second messengers such as C a 2 + . T h i s suggests that modeling the effect of buffering on calcium transients by scaling the diffusion coefficient of calcium may lead to qualitatively incorrect results. 2.5 Linearizing Calcium Dynamics through A s y m p totic Expansions To obtain a better understanding of the non-linear calcium dynamics, Zador and K o c h [121] show how it is possible i n certain l i m i t i n g cases to reduce the dynamics to one linear P D E w i t h constant coefficients i n cylindrical domains. This equation is formally identical to the cable equation i n the low [ C a ] j range, and simple linear diffusion i n the 2+ high [ C a ] range. Their method used asymptotics for cases in which [ C a ] j is expected 2+ 2+ ?; to be small and large relative to certain parameters and then to determine a leading order equation. The calcium dynamics involve diffusion, saturable pumps obeying M i c h a e l i s - M e n t o n dynamics, and diffusible buffers. The governing system of equations are d[Ca] D c o dt d[B] Db^r dt [Bt] ^dx^ ~ = P { [ C a ] ) + kb [M][Ca] f ~ MCa][M] fc - kb [B], b [M](x,t) + [B](x,t), + kb [B) + *f(x, b t), (2.3) (2.4) (2.5) where P [ C a ] is the contribution of the pump term, DB is the diffusion coefficient of bound buffer, r is the radius of the cylinder, and ^/(x, t) is a source term. Here we have made the i m p l i c i t assumption that [Bt] is a constant which is tantamount to a spatially homogeneous distribution of buffers and equivalent diffusion coefficients for bound and 19 Chapter 2. Spines and Long-Term Potentiation free buffers. There are two assumptions that are critical i n the reduction. T h e first is to assume that the time scale of diffusion is much longer than the time scale for buffering (i.e. for a length /, we require ^ > r where r B = B \^ ^)- If we follow the methods of Zador kbb+k Cc and K o c h , we can obtain a single equation for the C a â€žâ€ž ^d[Ca] r d (D + Ca dynamics 2 + P([Ca]) + *f(x,t). D p([Ca})) dx d[Ca] B The second assumption has to do with looking at high and low C a 2 + (2.6) limits of the terms in equation (2.6). F i r s t , let us consider the low calcium l i m i t . If we assume that [Ca] <C K , dtB Kfi iV and [Ca] <C the leading order equation becomes n ^ where (5 = + A ] Â®~dT R C a ( = n ( Dca , â€ž ffl P '~dx* 2K P ^ ^ 2 , , r [Ca] +-f(x,t) n + max DB S{ (2.7) If we multiply both sides by r/2 we can rewrite the above equation as: r(l + P) d[Ca] 2 r(D = + PD ) d [Ca) Ca 2 B 2 dt dx 2 -P K [Ca] s n*,t) = + K^PJC^Jix, max ' . f {x t] pK K t), (2.8) (2.9) Now notice that if I(x, t) = lo is & constant and we look for steady state solutions of (2.8), we get K â€ž = m ^ tâ€”>oo in l i m 20 . ( 2 . 1 0 ) Chapter 2. Spines and Long-Term Potentiation If we make the following substitutions: R-m ( -ll) 2 Ps-Kmaxi C. = (2.1*) Roo = Koo, (2.14) V = [Ca], (2.15) into (2.8), we obtain C â„¢ 3F 2 ^ ^ " - = ^ + ( 2 - 1 6 ) which is readily recognized as the cable equation w i t h a normalized current source term. The value of doing this is that the cable equation has been very thoroughly studied (see [43]). In the framework of the cable equation, we can see the pump acting like a membrane resistance, the buffers contributing to the capacitance, and the diffusion terms acting like intracellular resistances. More quantitatively, we can immediately define quantities such as the space constant, A, the time constant, r , and the input resistance of an infinite cable, Roo. In cable theory, these quantities are determined by the following formulas A = r = (2.17, R C , m (2.18) rn *â€¢ - ^ v f â€¢ If we now substitute (2.11)â€”(2.14) into the expressions listed above, we get _ , r ( l + /?) 2P K T(D . + C s 1 I3DB) max 3 K = ( 2 r ) n ^ ( D C a 21 + ~* 8D (2 22) B )P s K m a x ' Chapter 2. where X c Spines and Long-Term Potentiation is the chemical space constant, T C is the chemical time constant, and is the chemical input resistance of an infinite cylinder. Now notice that TC is dependent on the radius of the cylinder, but r is not. Thus the time required for equilibration is going to be much larger for a larger cylinder. In particular, the time constant is expected to be small for a structure such as a dendritic spine suggesting that [ C a ] 2 + s p could rise quickly during stimulation and return to normal levels quickly at the end of stimulation. Also notice that just as i n the electrical case, the space constant scales like the square root of the radius. However, the one big difference is that for reasonable physiological values of the given parameters, A <C A. It can, i n fact, be up to three orders of magnitude shorter c than the electrical space constant. Thus, it is expected that the localization of chemical signals is going to be much more efficient than the localization of electrical signals w i t h i n neurons.] A similar reduction can be performed i n the high calcium l i m i t . If one applied this assumption as we d i d with the low calcium l i m i t , one could reduce the calcium dynamics to a simple linear diffusion equation with a source. In this case, the buffers and pumps are totally saturated, and a l l the essential nonlinearities are removed. Zador and K o c h then applied this linearization i n the low C a 2 + l i m i t to their Z K model. One of their observations was that the chemical input resistance of the spine was 10 times larger than that of the parent dendrite. Thus, the same source located on the spine w i l l produce a much larger transient than if it were located on the parent dendrite. Furthermore, the chemical space constant of the spine neck was 0.27 pm m a k i n g the spine neck length 4 A c which means that steady state [ C a ] 2 + s p would decay by a factor of ~ 55 by the time it reached the parent dendrite. Furthermore, the small radius of the spine neck and head also mean that spines have much smaller chemical time constants than the parent dendrite. This w i l l only further amplify the concentration gradient between spine head and parent dendrite during calcium transients. 22 T h i s result shows Chapter 2. Spines and Long-Term Potentiation how important the morphology of the spine is i n both allowing for the development of a large, transient increase i n [ C a ] 2 + s p while simultaneously l i m i t i n g the extent to which it w i l l spread throughout a dendrite. Another interesting facet of the linearized C a 2 + dynamics can be seen i n the morpho- electrotonic transform [123]. Not only are local C a from the dendritic shaft, but C a 2 + 2 + transients i n active spines isolated transients i n the dendritic shaft are to a large extent isolated from the spine heads. T h i s is i n contrast to the tremendous voltage attenuation from spine head to dendrite when there is an input to the spine head, but when there is an input to the dendrite then there is very little voltage attenuation from dendrite to spine head [88]. T h i s gives us a good understanding of calcium dynamics and how spines provide a m i croenvironment for large changes i n [ C a ] 2 + as a result of C a s p . However, L T P cannot be explained simply influx through N M D A R s when strong presynaptic activity is coupled 2 + to strong depolarization i n the neighbourhood of the spine, w i t h the resulting large nonlinear increase [ C a ] i n the spine, and the activation of C a B P s and C a 2+ i 2 + -dependent enzymes. Thus, the modeling cannot end here. 2.6 T h e Spatial Nature of C a 2 + Transients in L T P Due to the input-specificity requirements of L T P , the C a 2 + transient' must be severely restricted spatially i n order to prevent the induction of L T P at adjacent synapses. Thus, it has been suggested that the dendritic spine may act as a separate compartment from the parent dendrite which acts to amplify the [ C a ] f o r a given level of stimulation [81]. 2+ i T h i s idea was built into the models reviewed above i n the form of calcium pumps and buffers w i t h i n the spine. However, only in the last few years have experimental techniques to measure [ C a ] i n dendritic spines been developed. 2+ i 23 Chapter 2. Spines and Long-Term Potentiation It has been found in a number of experiments that the [ C a ] 2 + Ca 2 + i n the spine head, is significantly higher than [ C a ] 2+ rf s p , the concentration of , the concentration of C a 2 + in the parent dendrite, during synaptic stimulation [84, 75]. T h i s difference could largely be negated by application of the N M D A R antagonist, D - A P V . Furthermore, bath application of N M D A resulted i n values of [ C a ] 2+ 5p greater than [ C a ] i n a manner depending 2+ d on [ C a ] , the extracellular calcium concentration [101]. Lastly, depolarization of the 2 + 0 soma to -20 m V results i n a larger increase i n [ C a ] 2 + s p than i n [ C a ] [101, 99]. T h i s dif2+ d ference could be blocked by application of L - t y p e voltage gated calcium channel ( V G C C ) blockers, but not by either a P - t y p e V G C C blocker or an N - t y p e V G C C blocker. T h i s suggests that spines have a hyperconcentration of N M D A R s and L - t y p e V G C C s compared to the parent dendrites, or a smaller concentration of C a B P s as assumed i n the mathematical models of L T P . For example , a recent experiment [32, 33] followed the dispersal of C a 2 + i n a dendrite that was temporarily made permeable to C a found that there was a significant lag in the change in [ C a ] 2+ s?) 2 + . It was compared to [ C a ] in 2 + d o n e - t h i r d of all spines. This difference was more frequently found i n spines w i t h long, t h i n stems than i n spines with short, stubby stems. However, it should be noted that the lag in spines for spreading C a 2 + was not due to a physical diffusion barrier as there was no lag in the change i n [ C o ] between spine and 2+ parent dendrite when C o 2 + was substituted for C a 2 + in the extracellular fluid ( E C F ) . T h i s suggests the possible importance of calcium pumps i n keeping [ C a ] 2 + isolated from [ C a ] 2+ d . s p somewhat However, notice that the degree of difference is considerably smaller than that suggested theoretically. This may in large part be due to the fact that calcium fluorescence techniques do not actually measure the C a 2 + directly but rather the concentration of calcium bound to certain exogenous C a B P s which fluoresce when irradiated w i t h light of certain wavelengths. The idea that calcium dynamics are not always as intuitive as one might think is an 24 Chapter 2. Spines and Long-Term Potentiation important one when dealing with such a qualitative endeavor. Nowycky and Pinter [78] performed a rigorous n u m e r i c a l analysis of the motion of C a 2 + and C a B P s i n model cells (for a less rigorous treatment of simulations of the movement of C a B P s , but one that is specific to spines see [45]). One of their chief results was the spatio-temporal distribution of C a 2 + - b o u n d buffers and C a 2 + can be quite different. T h i s is especially true for small, mobile calcium chelators such as fura-2 which are typically used i n calcium fluorescence experiments. T h i s may partially explain the paradoxical results of the above experiment. 2.7 T h e temporal nature of C a transients in L T P 2 + The experimental data shows reasonable qualitative agreement w i t h the theoretical m o d els i n terms of l i m i t i n g the spread of C a the temporal nature of the C a 2 + . Unfortunately, the same cannot be said of 2 + transients. In a recent study [64], a photolabile C a 2 + chelator, diazo-4, which increases its affinity for calcium 1600-fold upon exposure to intense light, was used to determine some of the temporal features of the C a 2 + signal required to induce L T P . A stimulus which normally induces L T P (100 H z stimulation for 1 s) was found to induce P T P if the light was applied 1 second after the beginning of the stimulus (Is of 100 H z stimulation), S T P or weak L T P if light was applied 1.5-2.0 seconds following the beginning of the stimulus, and full L T P if the light was shone 2.5 seconds or later after the beginning of the stimulus. W h i l e these results may possibly allow for the release of C a 2 + from intracellular stores such as the S E R , they do rule out any possible influences due to long-lasting C a 2 + waves or oscillations. Furthermore, since the peak [ C a ] i s reached w i t h i n 1.5 seconds when only 2+ i S T P was induced, this indicates that incomplete activation of some quickly activated [ C a ] dependent process does not distinguish whether S T P or L T P is induced. T h i s is 2+ i where the models discussed above fail. T h e C a - d e p e n d e n t processes that were assumed 2+ to underlie L T P were quick, and the peak level of [ C a ^ i n these simulations occurred 2 - 25 Chapter 2. Spines and Long-Term Potentiation well before that noticed during experiments (this can be seen in the small value of r ). c Thus, the nature of the nonlinearity that is produced during L T P could not be of the type that would have been noted i n the previous modeling efforts. We w i l l need to consider some more results from the biology of L T P before attempting to proceed. W h i l e a fairly robust result in L T P has been the requirement of increase i n [ C a ] j and 2+ the activation of N M D A R s , there are other possible sources of C a 2 + such as the S E R . C A 1 hippocampal pyramidal neuron dendritic spines contain a high concentration of m G l u R 5 [94], a metabotropic glutamate receptor, which catalyzes the hydrolysis of phosphoinositide bis-phosphate (PIP2) to inositol trisphosphate ( I P 3 ) and diacylglycerol ( D A G ) . I P acts on I P receptors located in the S E R to cause release of C a 3 3 2 + (see [9] for a review), while D A G is a potent activator of protein kinase C ( P K C ) . T h i s is of some interest since it is known that I P receptors are found in C A 1 hippocampal pyramidal 3 neurons but ryanodine receptors (RyRs) which are involved in c a l c i u m - i n d u c e d calcium release ( C I C R ) processes are not, even though they are typically found i n high concentrations i n dendritic spines [103]. Furthermore, I P is produced during the induction of 3 L T P [37], and topical application of I P to their dendrites can result i n a C a 3 propagates throughout the dendrite [44]. 2 + wave which Moreover, thapsigargin [7, 37] and d a n t r i u m [79], which act to inhibit the release of C a 2 + from intracellular stores, can prevent the induction of L T P . A direct link between these results and a role for m G l u r 5 comes from agonist/antagonist studies. m G l u R 5 antagonists have been shown to prevent the i n - duction of L T P [6], while agonists have been shown to be capable of potentiating L T P [20, 13, 14]. The effects of m G l u R 5 agonists could be prevented by thapsigargin which strengthen the link between intracellular C a 2 + stores and the induction of L T P . In light of this evidence, I w i l l now review a model of L T P which explicitly included the presence of intracellular calcium stores. 26 Chapter 2. 2.7.1 Spines and Long-Term Potentiation A Model of C a 2 + Dynamics in Dendritic Spines Schiegg et al. developed a model [97] i n 1995 which incorporated a mechanism for the release of C a 2 + from intracellular stores as a result of the experimental results noted above. In particular, they were concerned w i t h meeting the criterion for elevated levels of [ C a ] 2 + s p for 2 â€” 2.5 s as noted i n the study using the photoactivable calcium chelator, diazo-4 [64]. The modeling approach was to use a 10 compartment electrical model of a C A 1 hippocampal p y r a m i d a l neuron split into a soma compartment, a basilar bush, a lower apical branch, and an upper apical branch which is subdivided into 6 dendritic compartments. The upper apical branch was subdivided to allow better spatial resolution in the neighbourhood of the synaptic input. A set of A?'spine identical spines were attached to the fourth dendritic compartment away from the soma on the upper apical branch. The spine morphology was as before, a cylindrical spine head attached v i a a spine neck, which was reduced to a lumped potential in the electrical model, to the parent dendrite. It further was assumed that there was synchronous activity i n each spine, and thus the voltage and current loss to the dendrite was the same for each spine. A s , a result, only one spine needed to be explicitly included in the electrical model. T h e voltage channels used i n this model were based on the N M D A and n o n - N M D A R currents used by Zador and K o c h [122]. The calcium dynamics were essentially the same as in the Z K model. The spine head was split into three compartments in the discretization, with all synaptic channels located in the subsynaptic compartment. The spine neck also was subdivided into three compartments. There are a couple of differences worth noting. F i r s t , it was assumed that 10% of the current flowing through the N M D A R is due to C a 2 + , rather t h a n the 2% used in the Z K model. Rather than model the four separate sites on C a M , which was assumed 27 Chapter 2. Spines and Long-Term Potentiation to have a total concentration of RS 30 uM, it was assumed that there was a buffer w i t h a single calcium binding site with [Bt] = 120 uM i n the spine. T h e forward and backward binding constants of the buffers were the same as i n the Z K model. T h e form of the pumps were as i n the Z K model, but the pump efficiencies were larger than the values used i n the Z K model by a factor of 10. Rather than explicitly model the [ C a ] 2+ fl! , it was assumed that [ C a ] was clamped at 50 n M . However, by far the biggest difference 2+ d between the two models is the inclusion of a C I C R mechanism i n this model. B o t h the IP3 receptor and the R y R show a bell-shaped calcium response w i t h little calcium release at low [ C a ] j and high [ C a * ^ . Rather than using a detailed model 2+ 2 of b o t h I P receptors and R Y R s , Schiegg et a l . combined the two i n the form of one 3 phenomenological equation w i t h a bell-shaped calcium dependence. that there was one common intracellular C a 2 + pool which occupied j^ th It was assumed of the spine head volume. In the discretization scheme used, this intracellular pool was placed i n the second compartment, directly below the subsynaptic membrane. T h e release of C a 2 + from this store and into the second spine compartment is modeled by d where [Ca] siore [ C < 1 ^ = X ([Ca] P store is the concentration of C a 2 + - [Ca] ) (2.23) 2 i n the store, [Ca] is the concentration of 2 calcium i n the second compartment, and X is the number fraction of open channels. X is calculated from dX [X - (RA)Re([Ca] )] 2 dt 1~store (2.24) where RA is the probability of receptor-agonist binding and Re([Ca]) models the b e l l shaped calcium response of the receptors. In these simulations, when the effects of intracellular C a r e l e a s e was thought to occur, RA was set equal to 1. T h i s is equivalent 2+ to assuming the receptors are saturated by agonist molecules. T h e following functional 28 Chapter 2. Spines and Long-Term Potentiation form was used for Re: / r , Re([Ca}) = Â»([Ca]) = f 0, [Co] < [Colo, { u{[Ca]) exp [1 - u([Ca])], [Ca] > { â„¢ [Ca] , e â€¢ - â„¢ \ (2.25) (2.26) The form of the above equation is an a - f u n c t i o n (i.e. it has the form cae^ for t > 0, and 0 otherwise) which attains a m a x i m u m of 1 at [Ca] = [Ca] . max In the simulations, [Ca]e was chosen to be 150 n M as it is thought that there is little C I C R release when ( C a ] i s less than 150 n m . Thus the "intracellular stores only w i l l act as a source so long 2+ i as [Ca] is above 150 n M . 2 If RA was set equal to zero, it was found that previous models could not sustain high calcium concentrations for longer than 100 ms after the end of the stimulus. If we refer back to the analysis done on linearizing calcium dynamics, we can see this is a result of the extremely small value of the chemical time constant. T h i s is where intracellular Ca 2 + stores come into the picture. If four pulses are applied at 100 H z w i t h Nspine = 30 and RA = 1, it was found that [ C a ] 2+ could be kept above 400 n M for longer than 1 s. Because of the steep drop-off i n Re([Ca]) for [Ca] > 250 n M , i t was found that there was m i n i m a l involvement of intracellular stores until [Ca] ~ 400 n M . T h e intracellular 2 stores are able to maintain this concentration by compensating for the loss of calcium through pumps and diffusion if there is enough calcium present i n the store, and if pX (see (2.23)) is large enough. If p or X is reduced by a factor of 10, then it is not possible to sustain elevated [ C a ] 2 + s p under any stimulation protocols. If, however, p and X are large enough, the drop-off i n [Ca] store is nearly linear i n time from the time it begins until the time it ends, resulting i n a quasi-steady-state i n which [ C a ] 2 + s p is elevated for over 1 s after the end of the stimulation at a near constant value. T h e key factor that controls whether a spine is i n the high X range is receptor-agonist binding which can be thought of as a function of IP3 concentration. Thus, if a stimulus is large enough to result 29 Chapter 2. Spines and Long-Term Potentiation in significant I P release, it is expected that [ C a ] 2 + 3 amount of time. Otherwise, [ C a ] 2 + s p s p may be elevated for a significant will decay quickly as the value of pX w i l l be too small to compensate for the loss of calcium through diffusion and pumps. T h i s model nicely builds a mechanism showing how intracellular calcium stores can explain the results of the diazo-4 experiment where the previous models fail, and it also helps explain the importance of I P i n L T P . Unfortunately, there is evidence suggesting 3 that there are other calcium stores of importance i n L T P , and that the biochemical cascade involved may be a bit more complicated than initially thought. It turns out that protein kinases ( P K s ) , and protein phosphatases (PPs) may be quite important i n L T P induction. 2.8 Protein Kinases and Phosphatases in L T P A s mentioned previously, there is ample evidence for the existence of L - t y p e V G C C s i n dendritic spines. However, typically it has been found that depolarization pulses alone could not induce L T P . Some suggested that this may be due to an inability to create the appropriate increase i n [ C a ^ , although it has been found that most of the increase i n 2 - [Ca ] 2+ ?; during synaptic activity is due to V G C C s [72]. However, depolarization pulses coupled to L F S (2 H z stimulation) which did not act through an increase i n C a 2 + influx could result i n the induction of L T P [54]. T h i s suggests that some aspect of glutamate release may be important i n inducing L T P . Three immediate candidates for this effect are N M D A R s , m G l u R 5 , and a possible priming effect that action potentials have on the presynaptic cell. T h e problem w i t h N M D A R s as the candidate is that bath application of N M D A is capable of creating large C a 2 + transients, but they only can induce S T P and not L T P [12] (although, see [108]). T h e problem w i t h the p r i m i n g hypothesis is that postsynaptic glutamate iontophoresis is capable of inducing L T P [22]. Thus the most likely stimuli required for the induction of L T P is some combination of a sufficiently 30 Chapter 2. Spines and Long-Term Potentiation large increase in [ C a ] and the activation of m G l u R 5 . 2+ i Since a local, transient increase in [ C a ] 2+ ?; itself was never seriously expected to be the ultimate end of the biochemical cascade which results in L T P , the next question becomes what is? T h a t is, there may be a number of biochemical cascades involved which all have one common endpoint â€” the activation of some critical molecule or biochemical process that once initiated results in a persistent increase in synaptic efficacy. It turns out there is such a candidate molecule, C a 2 + - C a M dependent protein kinase II ( C a M K I I ) (for a review of this remarkable enzyme see [16]). A Brief Introduction to CaMKII C a M K I I is a holoenzyme composed of 8-12 basically identical monomers. E a c h monomer possesses a catalytic site, a regulatory domain, and a site which allows it to bind to other monomers. Monomers are typically inhibited by their regulatory domain, but this inhibition is relieved when C a 2 + â€” C a M binds to the regulatory domain. In its C a / â€” C a M bound state, each monomer has a very broad substrate specificity allowing it 2 to regulate such diverse processes as carbohydrate metabolism, neurotransmitter release and resynthesis, cytoskeletal function, and ion flux [98]. most common C a - d e p e n d e n t P K , comprising Â« 2+ Interestingly, C a M K I I is the 1% of total protein i n the brain, and & 2% of total protein in the hippocampus, and 30-40% of the total protein i n the P S D . Given its high concentration i n the P S D and hippocampus, it is not surprising that it is suspected of playing a major role in governing synaptic plasticity. C a M K I I has a unique set of attributes that allow it to respond to the distinctive properties of calcium signals which usually occur over a narrow range of concentrations, and are transient and pulsatile in nature rather than smooth and continuous. C a M K I I handles such signals elegantly due to its multiple activity levels which are dependent not only on its monomers C a 2 + â€” C a M bound, but also their phosphorylation state. 31 The Chapter 2. Spines and Long-Term Potentiation C a M K I I subunits are involved in autophosphorylation reactions [71] - one subunit i n its C a ^ â€” C a M bound state can phosphorylate a neighbor in the same state [34]. T h i s + phosphorylation has the effect of trapping C a 2 + â€” C a M to a monomer through a dramatic decrease i n the off-rate of Ca " " â€” C a M binding. Since there is very little effect on the 2 1 on-rate of binding, this results in a dramatic increase i n the monomer's affinity for C a M - the K d goes from RS 45 n M to RS 60 p M , one of the highest known affinities for C a M [16]. W h i l e Ca " " â€” C a M typically dissociates from a monomer w i t h i n a half-second, i n its 2 1 phosphorylated state, the typical dissociation time is on the order of hundreds of seconds. In this C a 4 + â€” C a M bound, phosphorylated state, the monomer maintains 100% of its catalytic activity. Even after the C a 2 + â€” C a M dissociates from the monomer, so long as the monomer is phosphorylated, it maintains 20-80% of its catalytic activity. Now it can be seen how the C a M K I I molecule can decode calcium signals. If a C a 2 + signal is arriving w i t h a certain frequency, a certain number of subunits per holoenzyme are going to bind Ca^" " â€” C a M . If two proximal subunits become activated, they may become involved in 1 an autophosphorylation reaction which traps the C a 2 + â€” C a M and leads to a potentiation of the calcium signal. If the frequency is low, the Ca^" " â€” C a M would escape before the 1 next spike (this can happen because there are phosphatases which also are activated by Ca 2 + â€” C a M which dephosphorylate the subunits), but if the frequency is high, then there is the possibility of even more subunits becoming activated w i t h each new spike. "This cooperativity of calmodulin trapping may allow the kinase to be a frequency detector w i t h a threshold frequency beyond which it becomes highly active" [16]. A M o d e l of C a M K I I Dynamics U s i n g some estimates of experimental parameters [68], Michelson and Schulman [69] developed a model for the activation and transition between different activity states of a ten subunit C a M K I I molecule. The j th subunit, Sj, had five activation states: inhibited, Sj, 32 Ca 2 + â€” C a M bound, 5 ^ , Chapter 2. Ca 2 + Spines and Long-Term Potentiation â€” C a M trapped, Sj, autonomous, Sf, and capped, Sf. In the capped state, it is phosphorylated at an additional site compared to the autonomous and trapped states. Once a subunit is bound to Ca " " â€” C a M , it can traverse the trapped, 2 1 autonomous, and capped states. T h e transitions between these states were a function of the O a * â€” 2 C a M concentration and the activation state of its right-hand neighbor. If we let Sf denote the activation state of the right-hand neighbor, the following tran- +1 sition m a t r i x describes the probability of transfer between the various activation states: pn([Ca Ca.M]) p, ([Ca CaM]) 4 p i([Ca CaM},Sf ) B 4 +1 B p {[Ca CaM], BB TB 4 PAl([Ca CaM},Sf ) 4 Sf ) 4 +l p ([Ca Ca.M},Sf ) 0 +1 0 +1 0 0 4 p ([Ca CaM], BT Sf ) 4 +1 p {{Ca CaM),Sf ) TT 4 +1 PAT{[Ca CaM],Sf ) 4 0 +1 0 0 0 0 0 p ([Ca CaM], TA Sf ) 4 p {[Ca CaM],Sf ) AA 4 +1 ([Ca CaM],Sf ) PCA 4 0 +1 +1 p ([Ca CaM], AC 4 CC 4 where the nonzero entries i n each row are non-negative and sum to 1. The method used to solve the problem was Monte Carlo simulation. The transition probabilities were based on half-life estimates of the various activity states from experimental data [68]. It was assumed that the transition state probabilities could be fitted to an exponential distribution. The results of simulations suggest that the trapped state of the subunits can act as a "capacitance" that allows for significant temporal summation even at low frequencies. In fact, a single InM. C a 2 + â€” C a M spike was capable of increasing the activity of C a M K I I for minutes following the stimulation. Another observation from the application of sinusoidal Ca " " â€” C a M transients is that while the proportion of subunits i n inhibited, bound and 2 1 trapped state is sinusoidal, the proportion i n the autonomous and capped state do not and thus appear to be acting capacitively. 33 Sf ) +1 p {[Ca CaM},Sf ) +1 Chapter 2. CaMKII Spines and Long-Term Potentiation and L T P The idea of C a M K I I acting as a frequency detector of high frequency calcium signals, coupled to its high concentration i n the hippocampus and P S D immediately suggests that it may be very important i n L T P induction. T h i s has been recognized and a very prominent theory i n the field has been put forward suggesting that the activation of C a M K I I is not only necessary and sufficient for the induction of L T P but may actually be the genesis of L T P and L T D expression [57, 60, 58, 59]. T h e basic idea is that low levels of [ C a ] j which are thought to be involved i n L T D induction may preferentially 2+ activate protein phosphatases which will dephosphorylate C a M K I I subunits and result in a decrease i n synaptic efficacy. However, a large increase i n [ C a ^ ^ w i l l directly lead to the phosphorylation of C a M K I I and the resulting increase i n C a M K I I activity w i l l lead to an increase i n synaptic efficacy. In between synaptic weight change episodes, the level of phosphorylation of C a M K I I will be set by the competing reactions of dephosphorylation by protein phosphatases and autophosphorylation within C a M K I I holoenzymes. In support of this theory, it has been found that postsynaptic injection of C a 2 + â€” C a M can lead to the induction of L T P [112], while postsynaptic injection of C a M inhibitors [62] or C a M K I I inhibitors [62, 65] can block L T P induction. Furthermore, evidence for the l o n g term activation of C a M K I I following L T P induction has been found [26]. A n interesting question now becomes how might C a M K I I increase synaptic efficacy. One possibility that has been suggested is that C a M K I I directly phosphorylates synaptic glutamate receptors and results i n an increase i n conductance. In this light, it has been shown that several glutamate receptor subunits have consensus phosphorylation sites for C a M K I I [67]. Furthermore, C a M K I I can phosphorylate and increase the current through kainate receptors (this is a type of n o n - N M D A R which is quite similar to the A M P A R ) threeto f o u r - f o l d in vitro. A corollary to this C a M K I I hypothesis for L T P should be that if C a M K I I already is maximally activated or cannot be activated at a l l , then it should be 34 Chapter 2. Spines and Long-Term Potentiation impossible to induce L T P . This is, in fact, observed. Transgenic mice without genes for C a M K I I are incapable of undergoing L T P [30], and when a constitutively active C a M K I I was introduced into hippocampal slices, it resulted in an increase in synaptic efficacy but prevented further induction of L T P [85]. Now the hypothesis that increased receptor conductance leads to an increase i n synaptic efficacy seems intuitively obvious, but one has to remember that efficacy here refers to the ability of presynaptic activity to result in action potential generation at the axon hillock. In fact, it has been show that under certain circumstances this may not be true [117]. T h i s w i l l be dealt w i t h i n more detail when I review results from electrical modeling of spines later. Now there is one last question which I should delve into before finishing the L T P portion of this thesis. Is an increase in [ C a ] 2+ ?; under normal physiological conditions sufficient for the induction of L T P ? 2.8.1 The role of P K C and sufficiency of C a 2 + in LTP induction The question of the sufficiency of increases in [ C a ] j i n L T P is very contentious. The first 2+ of many positive results came from experiments w i t h a calcium chelator, n i t r - 5 , which releases C a 2 + upon exposure to light [63]. However, many different experimental results have been found, and there is no clear consensus. One interesting experiment alluded to earlier [112] found that postsynaptic injection of C a o r C a M alone could not induce L T P 2 + while the injection of a c a l c i u m / C a M mixture could i n a manner that was inhibited by C a M K I I and P K C inhibitors. The role of C a M K I I in L T P was discussed previously and w i l l not be discussed again. P K C was mentioned previously as a P K which is activated by D A G which is produced during m G l u R 5 activation (along w i t h I P ). It has been a 3 fairly consistent observation that P K C is activated during L T P induction [51], and its inhibition prevents L T P induction [3, 111, 91]. 35 Chapter 2. Spines and Long-Term Potentiation T h e link between P K C activation and C a M K I I activation is not immediate as neither has any direct effect on the other. However, it has been discovered that P K C phosphorylates a protein, R C 3 (also called neurogranin), found in high concentration i n dendritic spines [113]. R C 3 is a c a l m o d u l i n - b i n d i n g protein which decreases its affinity for C a M when it is phosphorylated [28]. It has been shown that R C 3 can modulate calmodulin availability and decrease the activity of C a 2 + â€” CaM-dependent enzymes [66]. For example, i n the presence of luM of R C 3 (a more physiological concentration is 2pM), 10 pM of C a 2 + was required to achieve h a l f - m a x i m a l velocity of nitrous oxide synthase ( N O S ) , a Ca " * â€” C a M 2 1 dependent enzyme. O n l y .55pM of C a w a s required i n the absence of R C 3 . It was found 2 + that inclusion of activated P K C could prevent the R C 3 - m e d i a t e d inhibition of N O S . T h i s effect could be negated by inclusion of alkaline phosphatase which dephosphorylates R C 3 . T h i s is of some interest since R C 3 is dephosphorylated by P P 1 and calcineurin, b o t h of which are found i n dendritic spines [102]. The specific link between R C 3 and L T P comes from an experiment i n which it was found that antibodies to R C 3 which inhibit its P K C - m e d i a t e d phosphorylation could prevent the induction of L T P [25]. One possible reason for this effect may have been due to some effect on C a M availability. Some have suggested that " R C 3 is a biochemical 'capacitor' i n that it transduces C a kinetic parameters affecting the availability of C a it is made accessible to other enzymes" [28]. 2 + 2 + fluxes into / C a M and the rapidity w i t h which Thus, R C 3 may be acting as a P K C - mediated C a M store, and this is the connection between P K C and C a M K I I activation. However, there is some evidence suggesting a role for phosphorylated R C 3 i n mobilizing of intracellular C a 2.9 2 + stores i n Xenopus oocytes [21]. C o n c l u d i n g remarks about L T P W h a t exactly L T P is at this moment is not at all clear, but there seems to be a very strong connection between it and dendritic spines. T h e connections between L T P and 36 Chapter 2. Spines and Long-Term learning/memory are less clear. Potentiation T h e only solid connection between L T P and learning appears to be that mechanisms which reduce the capacity for L T P induction seem to impair spatial memory i n rats. important i n spatial memory. This is consistent w i t h the hippocampus being very Otherwise, there seems to be little effect for the most part. One has to be a bit cautious because here L T P means L T P induced at the Schaffer collaterals i n the stratum radiatum of C A 1 hippocampal pyramidal neurons. Thus, L T P induction may be dependent on similar but slightly different mechanisms elsewhere i n the C N S . O n the other hand, the local learning memory paradigm may not be the way most memories are stored i n the C N S . This should not denigrate, however, the work that has been done i n this field. L T P is a very robust memory phenomena which may provide insights into the way i n which memories are encoded i n the C N S , and this is why it is so extensively studied. Moreover, L T P offers an excellent example of theoreticians and experimentalists working side-by-side i n deciphering the riddle of biological complexity. 37 Chapter 3 E l e c t r i c a l M o d e l i n g of Spines 3.1 Introduction W h i l e intracellular C a 2 + dynamics are difficult to quantify, the cable model for voltage, and the parameters which describe it, are known w i t h much more certainty. Thus, we stand on much firmer ground when we try to obtain insights into the electrical nature of spines than when we try to understand their C a 2 + dynamics. T h e first notion of the possible electrical implications of spine necks was made by C h a n g i n 1952 [19]: "If the end bulbs of the gemmules (spines) are the receptive apparatus for the presynaptic impulses, the process of postsynaptic excitation initiated there must be greatly attenuated during its passage through the stems of the gemmules (spine necks) which probably offer considerable ohmic resistance because of their extreme slenderness." Since then, a number of theories have been put forward to explain the electrical nature of spines. I now w i l l review a few of the major ideas. 3.2 Spines mediate synaptic potential attenuation A s mentioned above, C h a n g made this proposition in 1952 and explained its functionality by requiring that a number of synapses be coactive to bring a cell to threshold for action potential ( A P ) generation. F r o m cable theory estimates, one expects the most important 38 Chapter 3. Electrical Modeling of Spines determinant of this phenomena to be spine neck resistance since the spine membrane resistance is large enough to be assumed to be infinite owing to the extremely small surface area of spines. It turns out that because of the extremely short lengths of spine necks (on the order of 1/im), there is likely to be very little current attenuation across the spine stem (this w i l l be elaborated upon later). However, because the spine input resistance is expected to be much larger than the dendritic input resistance, the spine should be able to mediate a much larger local E P S P than the same current input into the parent dendrite. A n important point which w i l l be made here is the asymmetry i n the cable equation. W h i l e a synaptic event w i l l produce a large voltage gradient between spine head and the parent dendrite, there w i l l be little voltage difference between the dendritic shaft and neighboring, inactive spines [88]. T h e large, local E P S P generated i n spine heads has implications i n terms of the activation of N M D A R s , which may be important i n L T P , and voltage-gated channels, which may allow for a type of pseudo-saltatory conduction w i t h i n dendrites (this w i l l be discussed later). Thus, as Shepherd pointed out [105]: "... a spine is a sensitive detector of its own synaptic input, but it follows closely w i t h neighboring spines the integrated potential of its parent dendrite; from this perspective, it is both a detector and an integrator." 3.3 Spines allow for linear summation of E P S P s One idea put forward was that the attenuation across a spine neck w i l l keep spines isolated from each other and allow linear E P S P summation. T h i s is a consequence of the nature of the current inputs which are proportional to the difference between the transmembrane potential and the reversal potential of the channel which can be considered to be a constant under most normal, physiological conditions (however, see [86]). However, one aspect to be pointed out is that the time course of an E P S P i n a spine is expected to 39 Chapter 3. Electrical Modeling of Spines be much shorter than an equivalent one on the parent dendrite. T h i s narrows the time frame over which non-linear interactions between neighboring spines can occur, and this can have important implications when nonlinear spine dynamics are considered. One thing that should be mentioned is that the time course of the E P S P as observed i n the dendrite and soma is fundamentally the same whether the synapse is on the spine or the dendrite because the higher capacitance of the dendrite and soma cause them to function as low-pass filters. However, the spine is not so constrained. 3.4 Spine neck resistance as controller of synaptic weight One idea that has frequently been put forth is that a spine is ideally situated to be a compartment for controlling the strength of an individual synapse. Since changes in spine morphology and density have been correlated w i t h memory [105, 35], this idea does have some merit. W i l f r i d R a i l showed the conditions under which a mechanism controlling spine stem length could be prominent [90]. His analysis was under steady state assumptions, but the fundamental underlying idea applies to transient currents as well. He demonstrated that the amount of current that reaches the parent dendrite when injected into the spine head is most malleable when the resistance of the spine neck is approximately equal to the input resistance of the infinite dendrite. It is a fairly typical observation that spines with the longest and thinnest necks tend to be located on t h i n distal dendrites. Thus, the thinking was that there is some sort of an impedance matching between RN and Roo,d which allows features of the spine neck to control synaptic efficacy. However, the bulk of evidence suggests that typically RN < #oo,d [107]. 40 Chapter 3. Electrical Modeling of Spines 3.5 Spines attenuate transient synaptic input The fundamental ideas of Rail's work led people to delve into the possibility that synaptic events i n spines may push the spine membrane potential towards the synaptic reversal potential due to the high input resistance of spines. T h i s notion was a fundamental one in the field for many years with no clear consensus. However, i n the early 1980s, several groups came forward w i t h a formal mathematical analysis of the problem [116, 49, 50, 52]. A l l suggested that there should be little attenuation of current across the spine neck, although it was accepted that there was a range of geometrical and electrical parameters over which local spine E P S P s could push the potential towards the synaptic receptor reversal potential and result i n voltage saturation. K o c h and Zador [53] showed there is insignificant attenuation of transient synaptic input if the product of the synaptic conductance and input resistance of the spine head is much smaller than one. In this case, the spine could be said to represent a current source of amplitude where g s y n is the synaptic conductance, the spine head potential. Conversely, if' V syn g n(Vsyn sy is the synaptic reversal potential, and g Rinfty syn â€” Vsp) V sp is <^ 1> then the action of a synapse can be seen as a voltage source causing the dendritic E P S P to approach â€žÂ°Â° ,d Â°Z . n The earlier conclusions that there would be significant attenuation were based on a poor understanding of the electrophysiology of spines and were based on channel densities of squid axons. W i t h more reasonable, physiological estimates, it seems as though 9synRoo,s *C 1 for most spines. Thus spines act primarily as current sources and moving P a synaptic connection from the spine head to the dendritic shaft would not appreciably change the depolarization achieved within the dendrite or w i t h i n the soma. L e t us now look at some other theories of spines and leave this idea behind. 41 Chapter 3. 3.6 Electrical Modeling of Spines Spines increase the lowâ€”pass filtering capacity of neurons One feature of spines that has not been addressed much so far is the idea that spines w i l l increase the surface area of dendrites. Thus, spines would be expected to have a reduced membrane resistance, and an increased capacitance. T h i s w i l l have the net effect of decreasing the electrotonic length and input resistance, while having little effect on the time constant [46] (see below for an analytical justification). T h i s w i l l drastically reduce spatial summation and will amplify the low-pass filtering capacities of dendrites. T h i s effect has been noticed i n the torus semicircularis of the weakly electric fish, Eigenmannia [95, 96]. It was noticed that the amplitude envelope of a stimulus (2 â€” 20 Hz) was nicely reflected i n the fluctuations of heavily spined neurons, but not i n sparsely spined or aspiny neurons of otherwise similar morphology. This has implications i n m i n i m i z i n g the j a m m i n g of Eigenmannia's electrolocation capabilities by electrical discharges from other nearby electrical fish. 3.7 T h e effect of passive spines on the cable properties of dendrites The intuitive notions of spines not affecting membrane time constant while decreasing the electrotonic length and input resistance of dendrites should be quantifiable. T h e nature of spines is such that compartmental models and numerical simulations are required to represent their discrete nature. However, Baer and Rinzel [5] developed an idea for simplifying the situation for the case of a relatively spiny dendrite w i t h identical spines. Instead of modeling each spine individually, just view them as having a certain density, p. The spines are not connected to each other directly electrically, but are coupled indirectly via the parent dendrite. Thus we can come up with a modified cable equation for passive spines i n a passive dendritic tree as follows (see the table i n Section 2.3 for some of the 42 Chapter 3. Electrical Modeling of Spines symbols) 2,r C d dV â€” Tir]d V nr â€” â€” - â€” . 2 d = m d V = IM d i + pI , (3.2) N (3.3) K N where V sp C sp and V are the membrane potentials of the spine and the dendrite, respectively, d is the specific capacitance of the spine, R is the specific membrane resistance of mjSp the spine, IN is the resistive current flowing between the spine head and the dendrite, and p is a proportionality factor. If we multiply (3.1) by R ,sp m a n d (3.2) by and make the substitutions r = R C, (3.4) A = ^ (3.5) T = -, (3.6) X = \, (3.7) rn m r we get R'm,spCsp dV p S ~V r _ dV W ~ dJO' 2^r~ - 2 jf â„¢ , A a n pRmj d Vd+ If we multiply and divide the term containing (3.8) m)Sp dV d 2 â€” R lN, sp I N ( , q ) " lN (3 d i n (3.9) by A, recognize that 9) Roo id = d let p = Ap, which is a measure of spine density per electrotonic length, then we get dV dV dT dx d 2 d -V d 2 + pR , I . . â€¢ 0o d N (3.10) Now, we can make use of physiological estimates of the parameters to reduce the problem further. Typically, we expect Rm,sp ^ RN, 43 and Roo,d 3> RN, and V sp Â« V . Let us d Chapter 3. Electrical Modeling of Spines introduce a small parameter, e = J , and expand V â€ž as iN s f*-in,sp V " = V -reV sp d + e V , ... 2 sPtl (3.11) sp 2 If we substitute the above equation into (3.8) and (3.10), and only retain 0 ( 1 ) terms i n e, we get Rm,spC p dV s d r -V - V d dT dV _ d dV 2 dT If we solve (3.12) for 2 V d + R Z ~ p V s ^ ( 3 1 3 ) substitute it into (3.13), and rearrange the terms, we obtain V i, sp> 0 dV sp sp 8V 2 d ) r Rm, + pR^, d dX pR o,dRrn, Csp\ 1 (3.12) sPtU ( d W - d X ^ - { pRoo,d\,r l + - R - ) , o i A\ V d - ( 3 " 1 4 ) Now by analogy w i t h the cable equation, the appropriate thing to do here is to m u l t i p l y b o t h sides of the equation by ^1 + j ^Â°Â°- p d dV to get dV d ~&T 2 ' m,s p r i p ' s > = Now notice that if d dX*~ = ^ T A, 2 - d P Z J ~ i (l + l T ^ ) > 1, we have r s ( 2 ^ (- ) 3 - 16 (3-17) > 1, and the effective time constant is larger than the time constant of an equivalent nonspiny dendrite. T h e opposite is true C ft if '* in s av c Fi < 1. Under reasonable physiological assumptions, we expect m , Â°* s p ss 1 i n agreement w i t h the results of Jaslove [46]. However, notice that \ < 1, and thus there s always w i l l be a decrease i n effective electrotonic length. Furthermore, the same also is expected to be true of the effective input resistance since it is proportional to the effective electrotonic length, i.e., Roo,d ff e ~ 44 Roo,d.^s- (3.18) Chapter 3. Electrical Modeling of Spines The unfortunate thing about the above analysis is that it does not show how electrotonic length and input resistance change with Rjy. We only have done a leading-order approximation which neglects the effects of RN under the presumed physiological conditions. However, the certainty w i t h which these predictions can be made must be tentative since no direct measurements from spines have been made. T h e spine cytoplasm looks quite different from the dendritic cytoplasm and might have a different value of Ri (probably bigger). Spines contain spine apparati, S E R , and occasionally mitochondria which should act to increase R^ by an uncertain amount. To clear up this issue, we look at a steady state model. T h i s should not effect the length constant or the input resistance. T h e equations now become We can solve (3.19) for V sp 0 = 0 - I (3.19) Nl X 2 ^ - V d + pR I. 00>d (3.20) N to obtain VdRm,sp Vsp = "2 â€¢ (3.21) -H-rn,sp + J*N Substituting (3.21) into (3.20), we obtain A' dV 2 d dX 2 pR<x,d 1 + RN V = 0. (3.22) d + R-rn,sp J T h i s yields an effective space constant, X ^, and effective input resistance, R^I{,, of oo,d> e = < J = A '1+ p R +R N ' , , (3.23) m I ^ J f . (3.24) Thus, we immediately can see that increasing R^ has the effect of increasing X ^ and e Rood- T h i s can be reasoned as a result of increasing isolation between spine and shaft and decreasing the conductance load of the spines on the dendrite. Rm, sp ^> RN, Notice that for we recover the leading-order multiplicative corrector to A, A , derived s earlier. 45 Chapter 3. 3.8 Electrical Modeling of Spines Computational richness of active dendritic spines Now that we have seen the effects of passive dendritic spines on the cable equation, the next obvious question is: what is the effect of active conductances i n dendritic spines? Investigations into the implications of active conductances i n dendritic spines began i n 1985 w i t h the publication of three papers on the subject [70, 82, 106]. M i l l e r et a l . [70] looked into the effects of letting the value of the spine neck resistance depend on the amplitude of the E P S P i n the dendritic shaft. T h e y found that the peak dendritic E P S P to a synaptic input was a bell-shaped function of the spine-neck resistance ceteris paribus when the active conductances were sufficiently large. Intuitively, one expects that as neck resistance initially is increased from low values, this increases the input resistance of the spine and results i n an increase i n the spine E P S P . T h i s recruits more and more active channels to open and increases the current output to the dendrite and results i n a larger dendritic E P S P . Eventually, at some critical value of spine neck resistance, there should be an action potential i n the spine head and a nonlinear increase in dendritic E P S P . A s the spine neck resistance continues to increase, the high spine neck resistance w i l l eventually cause a very quick voltage saturation i n the spine head as the spine head potential approaches the reversal potential of the active channels. Increasing i?iv further has little effect on the voltage-time integral and only serves to decrease ijv and the resulting dendritic E P S P . This is the principal driving force for current through the spine neck which is the cause of the dendritic E P S P [87]. Thus active spines confer an even greater dependence of synaptic efficacy on spine neck resistance than was first implicated by R a i l i n 1974 [90] (for the appropriate parameter ranges). These results were confirmed by Perkel and Perkel [82], who also showed that i n order to get the bell-shaped curve of dendritic E P S P vs. spine neck resistance: 1. T h e ratio of peak active conductance to peak synaptic conductance had to be sufficiently high (larger than 1). 46 Chapter 3. Electrical Modeling of Spines 2. T h e dendritic input resistance had to sufficiently small, i.e., for high enough dendritic input resistance, the dendritic E P S P is essentially a monotonically decreasing function of spine neck resistance. A thorough investigation into this problem was made by R a i l and Segev i n the late 1980s [87]. T h e y suggested that the key to the attainment of threshold for an excitable dendritic spine is I active where I tive ac > IN - I syn when is the current through active channels, V SP I S Y N = V TH (3.25) is the synaptic current, and V TH is the threshold voltage where net inward current generated at the spine head is greater than the net outward current into the passive dendrite. Thus, the early active current must exceed the difference between the current though the neck and the synaptic current. They concluded that increasing R N decreases the voltage threshold because 1. Increasing spine neck resistance increases the spine input resistance and results i n steeper spine E P S P s . 2. 1^ is a monotonically decreasing function of RN increasing the likelihood that the inequality (3.25) is satisfied. 3. Increasing RN decreases the conductance loading of the spine head membrane. The effects of increasing active channel densities i n active spines also was investigated. In general, it was found that increasing the density of active channels eventually led to a quick saturation of the voltage i n the spine head. However, here the dendritic E P S P was a monotonic function which plateaued at some finite value since IN is inversely proportional to RN but independent of the channel density. R a i l and Segev went on to address the significance of placing active channels i n spines versus on the dendrites. They suggested that to optimize synaptic efficacy, active membrane channels should be divided into spines such that each has just the right density 17 Chapter 3. Electrical Modeling of Spines of channels to fire action potentials. T h i s is preferred to equally spreading them over the dendrite since the smaller dendritic input resistance would make it more difficult to achieve threshold. Furthermore, the precise synchronous t i m i n g of inputs is very crucial to possible nonlinear interactions when dendrites contain active spines. For example, a single synaptic event which is insufficient to result i n an active spine firing an action potential, could be sufficient if the dendrite is depolarized from activity at neighboring spines. T h i s is a result of the decreased value of I due to the larger value of V in N d inequality (3.25). The possibility of precise timing is strengthened even further for those spines which appear to receive input from both an inhibitory presynaptic terminal and an excitatory presynaptic terminal. Here, conjunction of presynaptic and postsynaptic inputs w i t h i n a narrow time window could prevent action potential generation and result in a nonlinear decrease in dendritic E P S P . 3.9 Pseudosaltatory conduction in dendritic spines W h i l e it seems as though more attention was paid to R N than i n the above simula- tions, this is because they tended to focus on the implications of a single excitable spine in isolation from its neighbours. One question that arises naturally is whether or not active spines can mediate action potentials which regeneratively travel down dendrites. Since the active channels would only be in discrete locations i n the spine head along the dendrite, the situation is somewhat analogous to saltatory conduction i n myelinated axons. Here, we expect that a high dendritic input resistance is essential to ensure that a dendritic E P S P stays above V h as suggested by R a i l . The possibility of pseudosaltatory t conduction i n excitable spines was first simulated by Shepherd et al. i n 1985 for spiny, distal dendrites in pyramidal neurons of the cerebral cortex [106]. T h e i r i n i t i a l results suggested that such dendritic action potentials were possible w i t h H o d g k i n - H u x l e y type kinetics i n the spine heads if: 48 Chapter 3. Electrical Modeling of Spines 1. active channel densities were large enough, 2. the a m constant was sufficiently large, 3. spine neck resistance was w i t h i n a certain range, 4. dendritic radius was sufficiently small (this is correlated w i t h a large space constant and dendritic input resistance, both of which aid i n action potential generation), and 5. active propagation past branch points depended on activity i n sister branches. T h e y suggested such a mechanism could allow for large increases i n the efficacy of distal dendritic inputs, and allow for much quicker propagation of information to the soma instead of the classical result that distal dendritic inputs only can have a slow, small modulatory effect on the somatic potential. Furthermore, the propagation of an impulse often required several near-synchronous, spatially localized inputs and could greatly i n crease the complexity of information processing that can take place i n dendrites. T h e action of a set of inputs now not only depends on the size of the depolarization it can elicit at the axon hillock, but also depends on its spatio-temporal coactivation patterns w i t h its neighbours. R a i l and Segev showed a similar result for excitable spine clusters [88], that is, for dendrites i n which there are passive and active spines, but w i t h the active spines located i n close proximity with each other. M o t i v a t i o n of t h e spine c o n t i n u u m model Once the notions of significant synaptic attenuation across spine necks h a d been put to rest, more interest began to develop about the possible implications of excitable spine membranes. F r o m some of the major papers that had been written i n this field over the past few years, the issue of the significance of dendritic spines is not yet clear. However, over the past few years, the paradigm has shifted to spines being mediators of biochemical 49 Chapter 3. Electrical Modeling of Spines specificity of synapses (i.e. their role in L T P ) rather than electrical modulators of synaptic activity. T h e possibility of active membranes i n spines is a fascinating one, both i n terms of the possible generation of mathematical richness i n the model equations, and its capacity for fundamentally altering the way i n which the effects of distal dendritic inputs are viewed. However, the problem is plagued with analytical intractability i n its present form â€” the computational modeling of active membrane channels w i t h i n spines yields results which make intuitive sense, but we seem to lack an understanding of how the parameters work together to elicit the propagation of these dendritic action potentials. The problem has to do w i t h the discrete nature of spines. T h i s is where the ideas of averaging and switching to functional forms which retain enough of the qualitative spirit of the model, but allow for more analytical methods of investigation, become critical i n yielding new insights into the problem. One approach to the problem mathematically would be to follow i n the lines of B e l l and Cosner [8] and treat the spines as identical point sources of current which are evenly, but discretely distributed on the dendrite. Another would be to t r y the method analogous to the one previously used to derive the effects of passive spines on the electrotonic properties of passive dendrites. This continuum approach was used by Baer and Rinzel to numerically investigate the propagation of dendritic action potentials mediated by active spines i n 1991 [5]. Their approach was to have a passive, homogeneous, infinite dendrite attached to a certain density of spines as explained earlier. Here, however, the spines contained synaptic conductances and H o d g k i n - H u x l e y type dynamics. Baer and R i n z e l indicated that their simulations suggested that while the placement of active channels i n the spine head had important implications i n terms of threshold for the initiation of traveling pulses (namely, less presynaptic pulses were required for dendrites w i t h active channels i n the spines), it was relatively unimportant i n the propagation of a pre-existing pulse. 50 Chapter 3. Electrical Modeling of Spines In the next section I w i l l t r y to modify Baer and Rinzel's equations to make them more analytically tractable. D e r i v a t i o n of the Model In the paper written by Baer and Rinzel [5], they added active channels based on H o d g k i n - H u x l e y type kinetics to the spine membrane voltage equation used i n the study 51 Chapter 3. Electrical Modeling of Spines of passive spines. A s a result, their model equations were 2,r C d dV â€” d r n 7rrjd V â€” â€” 2 = d 2nr d â€” V d N{x)I , + N dV (ssp dT ryp T â€” T v s - v p IN T J-N -'act ( \ lsyn\X) t), d RN I act = lA iNa = 9Na{V p ~ IK = 9K{V -V h = dm _ dT ~ r {V ) dh _ /too ( K p ) - dT ~ dn _ {I S P + I N A S sp 9L{V V )m , Na 3 )n , 4 K L m^Vsp) - m ' sp h T {V ) h IL), -V ), S P m + K ' sp n (V )-n 00 dT sp T (V ) n sp a (V) m + p {vy o. {v) m a {V) m h a {V)+(5 {Vy h h a (V) n noo(V) <Xn(V) + p (vy n 1 T (V) m a m ( V ) + P (Vy m 1 TH(V) <* {v) + p {vy h T (V) n h I +/3 (Vy (Xn(V) n 25 - V <*m{V) 10 (e^ (3 (V) m = - l) ' V 4e"t8, 7e 20 100 1 30-V e 0. (V) n = io â€¢ +1 â€¢1 0 - V 100 ( Â§ 2 ^ _v_ Pn(V) = e so Chapter 3. I syn Electrical Modeling of Spines was modeled as an a function I yn(X,T) â€” G (X,T)(V S syn = G (X,T) syn Y^9syn{X,T If we m u l t i p l y both sides of (3.26) by R<x,d = ^ o ra (3.49) â€” Tj), T < 0 , CJsynte by A where A = y syn ,0 9syn(x,t)={ (3.48) â€” V ), sp T (3.50) T > 0. PÂ«* multiply the numerator and denominator of I N is the electrotonic length of the passive dendrite, use the identity semi-infinite dendrite, assume that N(X) = N, and let p â€” XN, we obtain r n 5T I'm = â€” JX 2 ~ d ~R ~^ N s p ~ '' ( d ^ (3.52) RmC . m The purpose of studying the system of equations stated thus far was to obtain an understanding of whether it is physiologically plausible for the H o d g k i n - H u x l e y - t y p e dynamics found i n squid axons to mediate action potentials i n dendrites if they are located w i t h i n the spine heads. T h e problem w i t h using the above set of equations is that they are intractable analytically. T h i s is where analogues of models become important. One of the more successful analogues employed for gaining an intuitive understanding of action potentials i n squid giant axons is the F i t z h u g h - N a g u m o equations (for a review see [18]). T h e F i t z h u g h - N a g u m o equations replace the I act term i n (3.27) by something more tractable analytically while still maintaining the main features of the dynamics. T h e simplification which is made is that the time scale of the activation of m, which is what controls the initiation of the pulse, is much faster than the activation of h and n which control the "relaxation" of the pulse. Using this idea, it is possible to reduce Jf 3p + I act from being controlled by three variables, m, h, and n to being controlled by two variables 53 Chapter 3. Electrical Modeling of Spines v and w. Here, v represents voltage, and it is assumed that m m ( u ) , and that h and 0O n can be combined into one recovery variable, r . T h e reduction usually takes the form +hct V s p = R'm,,sp f(v) + r, dr (3.53) Tnoiv) â€” r , where r is a constant. r In order to match these functions with experimental data, we define the quantities: I {v) p which is the transient peak current at voltage v and I which is the steady state current ss at voltage v. More formally, I (v) = I (v) = p ss l i m I (T,v), (3.55) l i m I (T,v) (3.56) m T->0+ m Tâ€”>oo where I (T, v) is the membrane current at time T and voltage v. Thus, we may conclude m that Iv(v) + ,./ x f(v) r M Since, I (v ) p = I (v ) rest ss = = P V ^ I (v) ss , 1 2 I (v) - I {v). ss (3.58) p = 0, we know that f(v ) rest (3.57) = r^v^t) rest = 0. Thus, we can simplify our i n i t i a l system of equations to the following: dV C s p -^ ^ Notice that if J , s yn = 0, then V = -f(V )-I (x,tj-I -r, = A sp sp ^ syn - (3.59) N V ^ ^ - V i ) . (3.61) = V / = 0 is a rest state of the system. Since we mainly w i l l be interested i n traveling wave phenomena not aided by synaptic currents, we can drop the I syn term from (3.59). 54 Chapter 3. Electrical Modeling of Spines In order to make the equations more tractable analytically, typically / is chosen to be a cubic p o l y n o m i a l , and hence the largest root corresponds to the sodium reversal potential, V . Na Furthermore, we usually choose r^v) = av, a > 0. Thus we can write (3.59) and (3.60) as C s p ^ = V (V acUve sp - KXWa sp - V ) sp V s p ~ V d - r, (3.62) 1 dr W g = - ( Â° V v (3.63) - r ) . If we divide (3.62) and (3.63) by V g Na and (3.61) by V active Na and make the following substitutions: r s T = = (3.64) fjactive 7*171 (3.65) (3.66) 7 = K = iP = â€ž * 9active pRo R RN , (3-67) (3.68) N dactive (3.69) t = â€”, (3.70) x = - , x_ (3.71) J'' II (3.72) v : a Na J. V Â« = 7^v Yi v (3-73) Na w = Na 55 (3.74) r (3.75) Chapter 3. Electrical Modeling of Spines we obtain dv ' â€” = (3.76) 1 , , ds m = In general, the value of r r , v{v-a){l-v)+j(w-v)-s, . < > 377 w i l l be large O(10 ) (according to [18]), and we see that for 3 t = 0 ( 1 ) , s = O(-^r). Thus, to leading order, we may assume that s = 0. T h i s simplifies the above equations to: dv â€” = dw v{v-a)(l-v)+ {v-w), (3.79) dw , 1 2 I K , . Except for the cubic term, this is the set of equations that w i l l be studied here. Having dropped the recovery term, these sets of equations w i l l not yield pulse solutions, in general, but under certain conditions I will show that traveling front solutions of constant shape and speed can be obtained. I w i l l look at three different cases. In the first case, I w i l l replace the cubic by a piecewiselinear term â€”v + H(v â€” a) where H is a Heaviside function. In the second case, I w i l l look at the case where the cubic is replaced by a piecewise linear continuous function which attains the same m a x i m a and m i n i m a at the same places as the cubic and has roots at v = 0 and v = 1. In the last case, I w i l l examine the cubic case i n (3.79) and (3.80). 56 Chapter 4 Piecewise Linear Discontinuous Model In this chapter, we begin to look at the model when the cubic polynomial is replaced w i t h a piecewise linear approximation, v â€” H(v â€” a) where H is a Heaviside function. T h i s model is quite simple, allows for explicit closed form solutions, and still has the general shape of the cubic. The purpose of these approximations is to attain the same fundamental dynamic structure for the traveling front solution of constant profile and speed, but w i t h much simpler functions. Hopefully, the approximations are close enough to show how the solution changes and how it depends on the values of the parameters. Linear functions are nice because solving O D E s w i t h constant coefficients is equivalent to solving for the roots of polynomials, which is a much simpler problem analytically and numerically. Also, it affords the possibility of using "continuation" methods to determine how speed depends on certain parameters once a single solution is found. T h i s technique was used successfully by R i n z e l to show that the modified F H N equations have a slow, unstable solution [92], which had been suspected from numerical experiments. 57 Chapter 4. Piecewise Linear Discontinuous Model T h e form of the equations that we are studying is: dv â€” = H(v dw d dt dx - a) - v+j(w - v), (4.1) 2 r 2 + -{v-w), r (4.2) where 7 > 0, K > 0, r â€¢> 0, and a > 0. Our approach w i l l initially be to look at the spatially homogeneous steady state solutions to the problem. We w i l l be especially interested i n those cases where at least two such steady states exist. We w i l l then proceed to search for constant speed solutions, and use a substitution to reduce the system of P D E s to a boundary value problem for a system of O D E s . T h e specific type of solution that we are looking for is a traveling front between the two steady state solutions. We will begin by demonstrating the existence of constant speed solutions for certain parameter sets. T h e n we w i l l use numerics to generate solutions for traveling fronts and examine the dependence of the shape and speed of the front on the different parameters. 4.1 Spatially homogeneous steady states We begin this study by examining the spatially homogeneous steady states, which are obtained by dropping the term from (4.2) and then setting the l e f t - h a n d sides of (4.1) and (4.2) equal to 0. Thus, 0 = H(v - a) - (1 + 0 = -(1 + K)W + KV. i)v + jw, (4.3) (4.4) If v < a, then H(v â€” a) = 0, and the system reduces to: 0 = - ( 1 + 7)u 0 = KV - (1 + 58 + ~/w, (4.5) K)W, (4.6) Chapter 4. Piecewise Linear Discontinuous Model which only has the trivial solution v = w = 0. If v > a, then H(v â€” a) = 1, and (4.3) and (4.4) become â€”(1 + 7 ) ^ + 71/;, -1 = 0 The solutions are v = v, and w â€” = KV â€” (1 + K)W. (4.7) (4.8) where 1+ K > a, 1+ K+ J K 1+ K+ 7 (4.9) (4.10) T h e inequality i n (4.9) can be rewritten as 7 1+ K < 1â€” a (4.11) and is a consistency condition for the existence of the nontrivial spatially homogeneous steady state: Now we shall explore the stability of these steady states. T h e stability m a t r i x corre- sponding to both steady states (they have the same stability m a t r i x because H(v â€” a) only contributes a constant) is given by: A = - ( 1 + 7) 7 -(1 K (4.12) + K) If we solve for the determinant and trace, we get det(A) = 1 + K + 7 > 0, (4.13) tr(A) = -(2 (4.14) + K + 7) < 0. (4.15) The determinant of A is positive and the trace of A is negative, which i m p l y that the 59 Chapter 4. Piecewise Linear Discontinuous Model eigenvalues of A , given by trace(A) + y/trace (A) - 4det(A) trace(A) - yj'trace (A) - 4det(A) 2 _ 2 2 (4.16) (4.17) are both negative. Thus both steady states are stable. Note that this situation is not possible for a continuous dynamical system, where stable steady states always are separated by an unstable steady state. We now ask under what conditions, i n addition to the consistency condition (4.11), do traveling front solutions between the two steady states exist. 4.2 Traveling front solutions We now seek out solutions that travel w i t h constant speed c > 0 to the right w i t h u n changing shape. It should be noted that we designate the speed to be positive arbitrarily here. Solutions w i t h negative speed do exist as well. Such solutions can be thought of as the wave dying out, whereas solutions w i t h positive speed can be thought of as the wave actively propagating forward. Introduce the following change of variables to a traveling wave frame: z t* â€” x â€” ct (4.18) = (4.19) t. Substituting this change of variables into (4.1) and (4.2), we get dv dv dt* dz dw dw dt* Â° dz H(v â€” a,) â€” v + j(w â€” v), dw w 2 60 K. (4.20) (4.21) Chapter 4. Piecewise Linear Discontinuous Model Since we are looking for solutions which go from the steady state (v = 0, w = 0) to the steady state (v â€” v , w = w ), our boundary conditions are s s lim w(z,t*) = 0, (4.22) \imv(z,f) = 0, (4.23) l i m w(z,t*) = w, s (4.24) l i m v(z,t*) = v. (4.25) zâ€”>oo zâ€”>oo 2â€”> â€” O O 2â€”> â€” O O s We are looking for continuous solutions for positive speed traveling fronts. Hence, we know that there is some value of z, say z = z , such that v(z ) = a. Notice that (4.20) 0 0 and (4.21) are autonomous i n z, and thus horizontal translation of a solution is also a solution because the boundary conditions are at infinity. Thus, we shall arbitrarily choose z = 0 for simplicity. T h a t is, our consistency condition is lim ^ +v(z,t*) 0 z 0 = a (Note, we have asked for the r i g h t - h a n d limit because solutions to the problem for c = 0 are not continuous about v = a but have a j u m p discontinuity. We can, nevertheless, satisfy this condition). For this problem, we are interested i n fronts which represent steady state solutions i n the traveling wave frame. Thus, we may drop the time-derivative terms from (4.20) and (4.21) to obtain -cv' = H(v -a)-v -cw' = w" - â€” + -(v - w), T + j(v - w), (4.26) (4.27) T where ' denotes differentiation with respect to z, subject to an overdetermined number of boundary conditions; the problem is t h i r d order, but there are four boundary conditions, 61 Chapter 4. Piecewise Linear Discontinuous Model and one consistency condition: l i m w(z) = 0, (4.28) = 0, (4.29) â€” w, (4.30) = v (4.31) zâ€”>oo l i m v(z) zâ€”Â»oo l i m w(z) s zâ€”>â€”oo l i m v(z) x 2â€”>â€”OO l i m viz (4.32) 2->0 + We can eliminate the variable v. Solving (4.27) for v yields v = W â€ž , 1+ + CW K (4.33) W and differentiation of this equation gives v ill = w , 1 + AC , il + cw (4.34) w Substituting for v and v' from (4.33) and (4.34), respectively, into (4.26), we obtain cw + [c - (1 + 7)] w" - c 2 1+ 7 + K 1+ w' + 1 + K + T w â€¢ J = -H(-z). T (4.35) Notice that if the following three boundary conditions are met, the other two boundary conditions i n (4.28)-(4.32) are automatically satisfied because they correspond to the fulfillment of steady state conditions: l i m w(z) = w , (4.36) l i m w(z) = 0, (4.37) l i m v(z) = a. (4.38) s !â€”>â€”OO 2-40+ v ' We proceed to solve for solutions of (4.35) for the special case c = 0, which is the easiest case because the third derivative term i n (4.35) drops out and reduces the order of the equation. Moreover, this case should give the boundary of the region i n the parameter space that corresponds to the existence of positive speed traveling wave solutions. 62 Chapter 4. Piecewise Linear Discontinuous Model 4.2.1 Stationary front solutions We set c = 0 i n (4.35) to obtain the second-order O D E -(1 + The general solution, w (z), g K + 1 + K w + = KH{-Z). 1 (4.39) for this O D E is w (z) g 7 = { (4.40) b2,ie-Â»* + hie"*, z > 0, where p = To match the boundary conditions, we require 6 w (z), p to (4.39) is given by 7 , M/l + K + { 1)X (441) = b 2>2 w, z < 0, 0, z > 0, s = 0. A particular solution, (4.42) To match the coefficients of z = 0, notice that from (4.26) either w or v must have a j u m p discontinuity when c = 0 at z = 0. Physically, it makes more sense for v to be discontinuous because it represents spine head voltage. Thus, we look for a solution w which is C i n z. For w to be continuous at z = 0, we require 1 h,2 + w = 6 ,i, s 2 (4.43) and for IO to be differentiable at z = 0, we require A*&i,2 = (4.44) These yield &i,2 = - & 2 , i = - y 63 (4.45) Chapter 4. Piecewise Linear Discontinuous Model Thus, the solution to (4.39) is , w(z) = { W s z -(2-e^), z<0, (4.46) K f e^ , z > 0. z Now applying (4.33) w i t h c = 0, and substituting i n for p, using (4.41), we get <;(*) = <{ S z < 0, " " 2 ( 1 + 7 ) 1 iÂ° -Â»* w z e 2(1+7) > (4.47) o ' A p p l y i n g our matching condition, (4.38), at z = 0, we get (after using (4.10) for w ) s r = 2a. r y ^ (4.48) ( l + ) ( l + 7 + Â«) 7 A t this point it is hard to say which way the inequality should go for positive speed waves, but intuitively we should expect that smaller values of a make the system more excitable. Thus, the correct condition is 7- r > 2a. r~7~~~~~ (4.49) ( l + 7 ) ( l + 7 + Â«) v ; If we rewrite the left-hand side as 7 -â€”-â€” I + 7I ' + ' K K / + , 7' x (4.50) ' v then it is easy to see that the left-hand side of the matching condition is always less than one. Thus, we see that there are no positive speed solutions for a > |, which is a classical result from the ordinary F H N equations. 4.2.2 Positive speed traveling wave fronts We look for solutions to (4.35)-(4.38) for which c > 0. If we substitute w = e yz into (4.35), we get the characteristic polynomial: cp(y) = cy + [c - (1 + 7)] y ~ c 3 2 2 1+ 1+ 7+ â€” 64 1+ K y+ K+7 -â€¢ (4.51) Chapter 4. Piecewise Linear Discontinuous Model Notice that cp(0) > 0 and cp'(O) < 0. Thus, cp always has one negative real root and may have two positive real roots, a positive root of multiplicity two, or two complex roots w i t h positive real part. Suppose there are three real roots, fj,\ < 0 < p < u-3', then, we know that using the 2 technique used above, we can find a solution for w given by ( b e^ lt2 to(z) = t + b e^ z z h3 + w, s z < 0, { b e^ , 2>1 (4.52) z>0. z Requiring continuity of w and its first two derivatives at z = 0 yields bi,2 + b - 6 ,i = -w , (4.53) ^2^1,2+^3^1,3-^1^2,1 = 0, (4.54) A , 2 + /^&i, - A , i = . 0. (4.55) l}3 2 3 s T h i s system of equations has the solution 62,1 = w h, (4.56) &1.2 = wb, (4.57) h,3 = yj b , (4.58) s s s 2 3 where (4.59) P2P3 61 b 2 (M3 - h Thus, 6 ,i > 0, 6 2 1)2 < 0, and 6 1)3 (4.60) (M3 - Ml) 0*2 - M l ) ' MlM3 (M3 - MlM2 (4.61) Ml)' (4.62) Ml)(M3 - M2)' > 0. We define p(A/,)=M + c / i - ^ , (4.63) 2 T so using (4.33), we obtain the solution 1 u(z) = â€¢( -~[hg(^)e^ - ^ ^ ( M I ^ 1 2 z + b g(p )e^ ] 2 , 2 z + v, s z < 0, z>0. 65 (4.64) Chapter 4. Piecewise Linear Discontinuous Model In order to satisfy our matching condition, (4.38), we require rbiw s g{Hi) = a, (4.65) which may be rewritten i n terms of the eigenvalues as ju /i 2 3 ( n \ + C/AI - _ q ( l + K + 7) (M3 - Mi.X/J.2 - Mi) (4.66) r If we could find a "near-by solution" for some parameter set to the above problem, then we could use continuation to get a feel for the dependence of c on the parameters. We have a solution for c â€” 0, but that solution is degenerate because p$ â€”> oo as c â€”> 0. T h e obvious method for getting an initial non-degenerate guess is to use singular perturbation techniques. Assume that K, T and a are fixed, and that c = e is a small parameter. Expand 7 in powers of e as 7 = 7o + Hi + â€¢ â€¢ â€¢ ( -67) 4 where 7oÂ« 2a. ( l + 7o)(l + 7o + Â« ) (4.68) O u r boundary condition (4.36) can be expanded as w = w + ew + 0{t ), (4.69) â€¢wi = 7 â€” ^ - â€” ' ( - Â°) tÂ«2 = s l 1 2 1 + K + 70 Will l + Â« + 7o 66 4 â€¢ . â€¢ 7 V (4.71 V ; Chapter 4. Piecewise Linear Discontinuous Model We need to obtain corrections to the eigenvalues. Here, we expect that Mi = Mi,i+eMi,2 + 0(e ), (4.72) M2 = M2,i + ^ (4.73) Mi,i = "M, (4.74) IJL ,I = p. (4.75) 2 2 2 ) 2 + 0(e ), 2 If we substitute the above equations into (4.51), and collect terms of 0 ( e ) , we get - w r h ^ r ( 4 In order to come up w i t h an expansion for p , we guess that to leading order, p 3 3 ' 7 7 ) = where p > 0. Substituting this into (4.51), we get + ^ V - (1 + 7)] f4.i - e ( l + 7 + ^ 2 ) + (4.78) We get the correct value of p if we match the leading-order quadratic and cubic terms. Thus, p = 1. We t r y to expand p as 3 M3 = â€” + M3,2 + 6/43,3 + â€¢ â€¢ â€¢ â€¢ (4-79) e Substituting the above expression into (4.51) and collecting powers of e~~ and e , re2 _ 1 spectively, we get Ms.i = l + 7o, (4-80) M3,2 = 7i- (4-81) Now we need to suggest an asymptotic expansion for the coefficients. In this case, we have an exact formula for them i n terms of the eigenvalues and the steady states (equations 67 Chapter 4. Piecewise Linear Discontinuous Model (4.56)-(4.62) for which we have expansions. One thing that should be noticed from (4.62) is that 63 is 0 ( e ) as e â€”>â€¢ 0. However, the exponential term is 0(eÂ°^) 2 so its effects are first felt i n the second derivative of w, and we can see this evolving into a j u m p condition in w as e â€”>â€¢ 0. Notice that v gets some contribution from w" i n (4.33). Thus we see that v has a j u m p discontinuity i n the l i m i t i n g case as c â€”> 0. This could be easily seen if one refers back to (4.26) and sets c = 0. Here there is a j u m p when v crosses a, a n d since it is not i n w, it must be i n v. The solution is constructed i n this manner since w is only indirectly coupled to the nonlinearity, and its effects are damped out. R e t u r n i n g to solving for the constants, we can solve for the corrections to a l l of the constants easily and can derive an answer which is only dependent upon ji which is still unknown. To get a guess for 71, we have to apply the matching condition (4.38). F r o m (4.64), this condition is equivalent to (4.82) In order to tackle this problem efficiently, we'll need some more notation. L e t us write (j(Pi) as 9oM + e#iOi) (4.83) where 1 9o(t*i) 9iM +K T = (4.84) (4.85) Mi and expand bi as 61 =Pi,i + e/?i,2 + 0 ( e ) 2 68 (4.86) Chapter 4. Piecewise Linear Discontinuous Model where = ^1,2 = ''' J 2 1 M2,l - Ml,l , / â€” M3,l(M2,l - Ml.lJ (4.87) [M3,l(M2,2 - Ml,2) + (M3,2 - Ml,l)(/i2,l - Ml,l)] ^ M2,lM3,2 + ^2,2/^3,1 (4-88) (4.89) M3,l(A*2,l We now match the coefficients of e i n (4.82) to obtain 0 = tv {ii )P 0 = '^i/?i,ibo(Mi,i)Mi,2 + 0I(A*I,I)] + ^i0i,25o(Mi,i) + '^20i,i5o(Mi,i)- ig hl hl + y, (4.90) (4-91) E q u a t i o n (4.90) is equivalent to condition (4.48) if condition (4.68) holds. If we solve (4.91) for 7 i , we can use the asymptotics to approximate a guess for a solution w i t h speed e for a particular choice of a, K, and r . Now that we have a means for making a guess, we can use numerics to t r y to determine solution sets of [p,\, fj, , /J3, c) i n terms of (a, 7, K, T) to determine the effects of the pa2 rameters on the shape (which is determined by the eigenvalues) and the speed (which is given by c) of the wave. 4.3 4.3.1 Numerical results Varying a In this section, we vary a over the acceptable ranges for 7 = 1.5, K, = 5, and r = 10. In Figure 4.1, we show the speed of the wave as a function of a. Here we get the intuitively obvious result that c is a monotonic decreasing function of a, ceteris paribus. It appears that speed may be becoming large as a â€”> 0, but strictly speaking, it is not appropriate to allow a = 0, so this result w i l l not be commented upon more here. 69 Chapter 4. Piecewise Linear Discontinuous Model 4 h 0 0.02 0.04 0.06 0.08 0, 0.1 0.12 0.14 0.16 0.18 0.2 Figure 4.1: Speed of the wave, c, as a function of o for 7 = 1.5, K = 5, r = 10. If we look at Figure 4.2, we get the expected result that p 3 â€”> 00 as we approach the value of a where speed goes to zero (a wider range for a is not shown because the scaling of the graph required to enclose p obscures P2, and p\. We also get the result that as 3 the speed goes to zero p + Mi â€”> 0. 2 4.3.2 Varying K In this section we w i l l graph the results obtained for the values of the speed of the wave and the eigenvalues as functions of K, for fixed values of a, 7 and r . In Figure 4.3 we have graphed the speed of the wave as a function of K. Here we see that there is some m i n i m u m value of K for which are no positive speed traveling front solutions. T h i s intuitively makes sense as K measures the ability of the spines to depolarize 70 Chapter 4. Piecewise Linear Discontinuous Model 1 1 1 1 1 1 - / M2 - 0 0.02 0.04 i i 0.06 0.08 a i 0.1 i 0.12 i 0.14 Figure 4.2: Eigenvalues of the wave as functions of a for 7 = 1.5, re = 5, r = 10. the dendritic shaft. Furthermore, there is some definite m a x i m u m value of c for a certain value of re. Thereafter, the speed of the wave gradually decreases asreincreases. T h i s also makes sense, since re becoming large is equivalent to the spine density going to infinity which creates a tremendous conductance load that must be overcome to bring spines at the leading edge of the front up to threshold. In Figure 4.4 we have graphed the eigenvalues of the wave as a function of re. 4.3.3 Varying 7 In this section, we graph the speed and the eigenvalues of the wave as functions of 7 for fixed a, re and r . In Figure 4.5, we have graphed the speed of the wave as a function of 7. A s was the case 71 Chapter 4. 0. 6 Piecewise Linear Discontinuous Model I 1 1 1 1 1 r 0.5 \- 0.3 - 0.2- 0.1- 0 li I I I 0 20 40 60 1 K I 80 100 I I 120 140 160 Figure 4.3: Speed of the wave, c, as a function of K for a = 0.1, 7 = 1.5, r = 10. for K, the general shape of the wave is a skewed hump. However, i n this case, there is also an upper value of 7 which corresponds to a zero-speed wave. T h e idea that there must be some m a x i m u m value of 7 above which there are no positive speed traveling waves makes intuitive sense. If 7 is too large, then the current generated w i t h i n the spine head is quickly lost to the dendritic shaft and the spine is not able to reach threshold. T h e fact that the speed of the wave should become smaller for very small values of 7 is related to the fact that the spine head saturates quickly so little current is delivered to the parent dendrite. In Figure 4.6 we have graphed p\ and p 2 as functions of 7. We again notice that as 7 approaches the two values which correspond to zero-speed waves, \i\ 4- u. â€”> 0. 2 In Figure 4.7 we have graphed ^ 3 as a function of 7. It is easier to see the tendency of /z 72 3 Chapter 4. Piecewise Linear Discontinuous Model n 1 1 1 1 r 12 M2 ,^1 30 40 K 50 . 60 80 90 Figure 4.4: Eigenvalues of the wave as functions of K for a = 0.1, 7 = 1.5, r = 10. It should be noted that the abscissa here is not zero, but is a value close the value of K, which corresponds to zero-speed waves as shown i n Figure 4.3. to become large as 7 goes to its upper limit which corresponds to zero-speed waves, but the same occurs at the lower limit as well. It is difficult to graph this region thoroughly due to numerical instability of the routine used. 4.3.4 Varying r In this section, we have graphed the speed of the wave, and the eigenvalues as functions of r for fixed values of a, 7 and K. In Figure 4.8 we have graphed the speed of the wave as a function of r . In this case, we have actually graphed c\fr as a function of r. This is due to the fact that the original scaling involved a nondimensionalization by a length scale. 73 If we plotted c against r Chapter 4. 0.45 | Piecewise Linear Discontinuous Model 1 r C 7 Figure 4.5: Speed of the wave, c, as a function of 7 for a = 0.1, K = 5, r = 10. directly, we would get the erroneous impression that c â€”>â€¢ 0 as r â€”>â€¢ 0 0 . Physically, r represents a time-scale of active dynamics i n the spine head. We intuitively expect that the speed of the wave should plateau to some m a x i m u m value as the speed of the active dynamics becomes very fast. This is, i n fact, the case when we correct for the space scaling factor introduced earlier. In Figure 4.9 we have graphed u-i and \i against r . We have performed a scaling here 2 as well to correct for the initial nondimensionalization. In this case, we see that the long-scale shape of the wave is not appreciably affected by changes i n r as one would intuitively expect. Where we would expect to see a change is i n the short-scale behaviour close to the discontinuity i n the equation. This is reflected i n /z as shown i n Figure 4.10. 3 74 Chapter 4. Piecewise Linear Discontinuous Model 0.8 1 1 1 1 1 1 i i - 0.6 0 . 4 - 0.2 0 - -0.2 â€¢ -0.4 - -0 . 6 - -0.8 -1 0 1 1 i i i i 2 4 6 8 10 12 14 16 18 7 Figure 4.6: Eigenvalues, p and p , as functions of 7 for a = 0.1, K = 5, r = 10. T h e abscissa is not zero i n this graph. It corresponds to the lower value of 7 which corresponds to zero-speed waves as can be seen i n Figure 4.5. x 4.4 2 Numerical simulation of the P D E model The analytical method we developed for the system of O D E s was extremely useful i n exploring some of the important regions of the parameter space. It allowed us to develop an intuitive understanding of some of the processes involved i n traveling wave front propagation. Unfortunately, it fails to address whether the solution we found is stable or unstable. Naturally, this would be a good thing to know since the fronts discovered w i l l not be seen experimentally if they correspond to unstable solutions. Since no proof of the stability of the system exists, we decided to examine the system of P D E s numerically using P D E c o l . 75 Chapter 4. Piecewise Linear Discontinuous Model 2 4 6 7 10 8 12 14 Figure 4.7: G r a p h of ps as a function of 7 for a = 0.1, K = 5, r = 10. Our procedure was to r u n P D E c o l using Neumann boundary conditions, a n d the initial guess was the computed solution from our analysis (slight perturbations to the initial guess were also made, and they gave the same results). 4.11. T h e result is shown i n Figure T h e traveling front appears to propagate to the right w i t h approximately the theoretically computed speed without changing shape. This acts to confirm our previous analysis, and strongly suggests that the solution we have discovered corresponds to a stable solution. 76 Chapter 4. Piecewise Linear' Discontinuous Model Figure 4.8: Speed of the wave, c, as a function of r for a = 0.1, 7 = 1,re= 5. 77 Chapter 4. Piecewise Linear Discontinuous Model 0.5 0 -0.5- -1 -1.5 - -2 - 0 100 200 300 400 500 600 T Figure 4.9: Graphs of ji\ and u, as functions of r for 7 = 1, K = 5, r = 10. 2 78 Chapter 4. Piecewise Linear Discontinuous Model Chapter 4. Piecewise Linear Discontinuous Model 0.8 0.4 - 0.3 - 0.2 - 0.1 - Figure 4.11: Simulation of the traveling front for 7 = 1, re = 5, r = 10. 80 Chapter 5 Piecewise Linear Continuous Approximation In this chapter, we w i l l look at the continuum model w i t h a piecewise linear, continuous approximation of the cubic. T h i s function will be qualitatively, and even quantitatively, similar to the cubic. T h i s piecewise linear function has roots at v = 0 and v = 1 and w i l l attain the same local m a x i m u m and local m i n i m u m at the same points as the cubic. The primary motivation for exploring this approximation is that there may be some internal functional structure in these model equations which is important i n the behavior of the solution that is not captured in the piecewise linear discontinuous approximation. We w i l l show that for certain parameter ranges, there are three steady states, two of which are stable in the space-clamped case, while the t h i r d is a saddle point. Furthermore, we show that there is, i n fact, an inner region of the solution, which does not exist i n the piecewise linear discontinuous approximation. We now begin the study of the piecewise linear continuous ( P W L C ) model, which is given by dv dt dw f(v) + j(w dw W 2 ~di~ where 7 > 0, K > 0, r > 0, and 0 < a < 1. 81 - v), K , (5.1) (5.2) Chapter 5. Piecewise Linear Continuous Approximation In order to write a formula for / explicitly, we first need to determine the local m a x i m u m / m i n i m u m of / and the points where they occur. Differentiating / w i t h respect to v, we find the roots of f'(v ) = -3'u , + 2(a + 1 > - a = 0 2 0 0 (5.3) to be 1 â€” \J a â€” a + 1 2 v - max u a + l W Â° - 2 â€” a + (5.4) (5 5) 1 g â€¢ G i v e n the restrictions on a, it is obvious that 0 < v min \Â°-Â°) < v max < 1. T h e local m i n i m u m and m a x i m u m of / are given by: fmin â€” /\'Umiri) i fmax â€” f{^max)i (5-6) (p-^) respectively. If we choose / ( 0 ) = 0 = / ( l ) , then we may write rriiv, { /(Â«) = -oo < v< mv + b, v m v-\-b , v 2 2 3 min < v< v min 3 v , , (5.8) max m a x <v, where fmin m = 1 _ x (5.9) â€”-, fmax m = m = 2 3 j 0 = h = 2 fmin ^\ n Umax ^min fmax , (5.10) i_ , -*â€¢ "max fmin Umax ~~ fmax^min â€” max J â„¢ f 1 x Umax 82 â€” ; mm u . x (5.11) , l ~\ (5.12) r n (5.13) Chapter 5. Since f min Piecewise Linear Continuous Approximation < 0 and f max > 0, we conclude that rri\ < 0, m > 0, b < 0, 6 > 0. T h i s can 2 2 3 be determined from the x - and y-intercepts of the different linear pieces of / . Now that we have an explicit formula for / , we can examine the spatially homogeneous steady states. 5.1 Spatially homogeneous steady states and stability We look for spatially homogeneous steady states, so we are looking for solutions to 0 = f^+jfa-y), (5.14) 0 = KV-(1 + K)W. (5.15) One obvious solution is the origin, v = 0 = w. The equation of the v nullcline is f(v) w = v , 7 = (5.16) P H . while the equation for the w nullcline is w = h(v) = S i n c e / ' ( u ) < 0 for ve(-oo,v ) min \J{v ,oo), ^v. then p (v) > 1 for ve(-co, max (5-17) v ) \J(v min , oo). max O n the other hand, ti (v) = ^-j- < 1. Thus, there will be three steady states i f and only if h{v ) max > p{v ). max (5.18) T h i s condition is equivalent to 7 < K, + 1 frnax^ V ma/J 83 ^ ^ Chapter 5. Piecewise Linear Continuous Approximation T o determine the middle steady state, (v ,v: ), 2 0 = (m v 0 = KV we look for solutions of 2 + b ) + j(w 2 2 2 2 â€” (1 + K)W2, - 2 (5.20) v ), 2 (5.21) which yields v = - w = - 2 6 (Â« + l) (K + l ) m -7' 2 (5.22) 2 b K 2 2 Since m > 2 > Umax (5.23) when condition (5.19) holds, we know that v > 0. If we substitute 2 K-j-1 ' v m for m , and 63 for b i n the above equations and use the fact that m = â€”63, then the 3 2 2 3 t h i r d steady state, (^3,^3), is given by b (K + l) 3 v 3 6 (Â« + (5.24) l)+7' 3 bK 3 w (5.25) b {K + 1) + 7' 3 3 T h e stability of the steady states (let us denote the steady state at the origin as (vi,w\)) can be determined by looking at the stability m a t r i x Ai = Its determinant, det(Ai), Since m,i,m 3 and (v ,w ) 3 3 < mi - j 7 K -(l+.re) and trace, tr(Ai) are 1,2,3, det(Ai) = â€”mj(l + Â«;)+7, tr(Ai) = m j ( l + K) â€” 1 â€” 7 â€” K,, 0, we know that det(Ai), (5.26) % = 1,2,3. det(A ) 3 > (5.27) (5.28) * = 1,2,3. 0 and tr{Ai),tr{A ) represent stable steady states. Conversely, since m 3 2 > < 0, so (vi,Wi) > - ^ r , we know that rfet(A2) < 0, and this steady state is a saddle point. Since there are two stable steady states separated by a saddle point i n the space-clamped case when condition (5.19) holds, we now inquire into the existence of traveling front solutions between the two steady states. 84 Chapter 5. 5.2 Piecewise Linear Continuous Approximation Traveling front solutions We now seek out traveling fronts with constant speed c which traverse a path from the steady state at (vi,Wi) to the one at (v , w ) as solutions to our P W L C model. A s i n the 3 3 previous chapter, we look only for solutions w i t h constant shape and make the reduction to the characteristic z = x- ct. (5.29) Substituting into (5.1) and (5.2), we get -cv' = f(v)+j(w-v), (5.30) -cw' = w" - â€” + -{v - w) r r (5.31) where ' denotes differentiation with respect to z. T h e relevant boundary conditions are l i m w(z) = uÂ»3, (5.32) l i m v(z) = v, 3 (5.33) l i m w(z) = 0, (5.34) \imv(z) = 0. (5.35) 2 â€” > â€” OO 2â€”> â€” O O 2 â€” > O Q 2â€”>OQ A s i n the P W L D model, there is no spatial dependence i n the r i g h t - h a n d side of our nonlinear O D E , and the associated boundary conditions are at Â± o o . Thus, without loss of generality, we enforce the condition that v{0) = v . min (5.36) Furthermore, since we expect our solution to be monotonic, we shall require V(zi) = where z\ < 0. 85 Vmax (5-37) Chapter 5. Piecewise Linear Continuous Approximation If we solve (5.31) for v, we get T K w 1 + K + cw (5.38) Differentiating the above equation with respect to z, we get T 1 it III w + cw ~~r~ ^ / (5.39) w We see that the boundary conditions on v become redundant since i f w satisfies the boundary conditions (5.32) and (5.34), then the boundary conditions for v are automatically satisfied. E l i m i n a t i n g v and v' i n (5.30) using (5.38) and (5.39), respectively, we get cw'" + [c 2 (7 - mi)]w" where i â€” 1 for ve(oo,v ), 7 - i min K 77lj + = 2 for ve(v i ,v ), m n max ^ + 7 ~ ' = ^ (5.40) and i = 3 for ue(w , 00). We ( 1 + r " ) m r maa; denote the characteristic polynomial of the above O D E by cpi where cpi(y) = v c 3 + [c 2 (7 m) 7 - m,i + y l + K y + 7 - (1 + K)m,i (5.41) Notice that cpi(0), cp (0) > 0 and cp'^O), cp' (0) < 0 (when c > 0), so b o t h of these 3 3 characteristic polynomials have one negative real root and two w i t h positive real part which may be complex. However, cp2{0) < 0, so it has one positive real root, but may have either two roots w i t h negative real part or two roots w i t h positive real part. The next step is to search for stationary solutions, that is, solutions for c = 0. T h e reasons for doing this is that it reduces the order of the O D E , so it is easier to solve. A l s o , it allows us to determine the boundary i n parameter space that corresponds to positive speed traveling front solutions. 5.2.1 Stationary front solutions Here we set c = 0 i n (5.40) to obtain [m-i â€” j)w -\ r 86 w = â€”. r (5.42) Chapter 5. Piecewise Linear Continuous Approximation Thus, we define A,; = UJ = Ij - + m,j(l mÂ») - T(J K) i = 1,3, (5.43) / m 2 ( l + K,) â€” 7 (5.44) T(7 - m, ) 2 and write down a solution for w d w(z) = I l t l e X l Z + Xlz i2 d ,i cos(coz) + d 2 s'm(ujz) + w , 2>2 d ^{z-zi) 3 e 0 < z < 00, di e- , + d -*3(*-*i) + w, 32e (5.45) z\ < z < 0, 2 3 - 0 0< ^ < In order to match the boundary condition (5.32), we require d 3>2 Z l . = 0, and i n order to match the boundary condition (5.34), we require d ^ = 0. We now apply C continuity 1 conditions on w, and the matching conditions (5.36) and (5.37). If we apply continuity and differentiability across z = 0, we get d\,2 ~ d ,i = - iod = 2 -Aid 1 ) 2 2>2 (5.46) w, 2 (5.47) 0, and if we do the same at z â€” z-i, we obtain d x ^3,1 - A d 3 3 ; 1 + 2t COS(UJZI) d ,iUJ [sm(cozi) 2 - d - d sin^^x) = sin^wzi)] = 0. 2 ] 2 2>2 w 2 w, 3 (5.48) (5.49) Now let us write down the solution for v using (5.38): v{z) - [UJ + i Â± ^ ] [d ,i cos(a;z) + d T 2 2 2 ; 2 87 sin(w2)] + v , z < z < 0, 2 x (5.50) Chapter 5. Piecewise Linear Continuous Approximation If we apply our matching conditions (5.36) and (5.37), we get 1+ K â€¢A? T K d-iAT + (5.51) (5.52) V 3 If we substitute i n for A] and A i n the above equations using (5.43), we get, 3 Vminil di: - mi) (5.53) 7 dA 3 Jmax = ^3 (5.54) "7- 7 - m, 3 We may substitute i n for rf i n (5.46) using (5.53) and solve for d i 1]2 2 Vmin{l d ,i -mi) 2 - yw 2 Similarly, using (5.53) and (5.47), we may solve for d, 2 2 ~ We only have to solve for d (5.55) 7 d 2>2 [7 - m )[7 - mi][7 - m i ( l + 2 7 3 ) 1 V m ( l 2 + K)\ re)-7 (5.56) and z\ using (5.48) and (5.49). If we let cos(a>Zi), (5.57) 2/2 sin(u;zi), (5.58) Â«1 ^3,1 + (5.59) w -w , 3 2 A c? ,i 3 a 2 3 (5.60) UJ then we may rewrite the problem as -d 2 t 2 d-2,2 Q.I d.2,1 Ci V2 (5.61) 2 w i t h solution d ,i 2 V2 d-2,1 + ^2,2 (d ,i 3 d,2,2 ( d 88 + w 3 3 , 1 + w w) 2 3 - w + 2 (5.62) ) - ^ ^ - Chapter 5. Piecewise Linear Continuous Approximation Enforcing the condition y\ + y\ â€” 1 gives us a solvability condition on the parameters. However, it turns out that there is an easier way of determining this restriction i n a certain case, and we derive the restriction this way. Once this condition is known, we can use the above derivation to obtain the full solution to the problem. T h e method we use is to integrate (5.30) and (5.31). F i r s t , m u l t i p l y both sides of (5.31) by w' and integrate from z â€” â€”oo to z = oo to obtain II 1 + re re ! ww Hâ€”vw i ; w w T dz = Q T (5.63) Using integration by parts and the fundamental theorem of calculus, the above expression can be simplified to: r {w'{z)f - (1 + re) (w(z)f + 2KV(Z)W(Z) rer / 2r v w dz = 0, (5.64) or if we use the boundary conditions u / ( Â± o o ) = 0, w(â€”oo) = w , w(oo) = 0, v(â€”oo) 3 v3, v(oo) = 0, and u>3 = we obtain V W dZ = ; (5.65) . 2(1 + re) Now m u l t i p l y (5.30) by v' and integrate both sides from z = â€” oo to Â£ = 00 to obtain 0 = f(v(z)) - yv(z) v (z) + jw(z)v (z)j dz. (5.66) Using (5.65) and the fundamental theorem of calculus, we get "3 0 7Â«3 f i V ) d V = 89 2(1 + re)' (5.67) Chapter 5. Piecewise Linear Continuous Approximation We can do the integral on the left i n three parts as follows: V3 ^ rv n mi f{v) dv = / f(v) d,v + f(v)dv+ f(v)dv rmn V3 V ma x m\v dv + / + b ) dv + (m v 2 63(1 â€” v) dv 2 0 m,i vl + + 63 [v 3 max 'Vmin] [ 2(Vmax v m 2 'W.3 + Vmax V ] 2 [/moi("raoi ~ Vmin) + 26 rnax ^mi?x) + frnin'Umax] + 63 [^3 1 - V Tl '"3 + V , r (5.68) Thus, going back to (5.67), we require 2 [/raai("mai Vmin) + fmin'Umax] + ^3 [^3 1 - V n ^3 + V 7Â«3 ri 2(1 +re) = 0. (5.69) If we let u â€” â€”fr and substitute i n v = r-^f-, we get 3 [fmaxi'Umax V i) m n + fmin'U'max. + 63 2 - vmax 63 b\u 63 63 + ti 63 + U 2(63 + u) 2 (5.70) If we m u l t i p l y the above equation by (6 + u) and collect powers of u, we get 2 3 au 2 + a\u (5.71) + 0,0 = 0, where 3 Ff â€” â€” &2 â€” CVmax ^3 ( [ / m a i ( " m o i 2 \_fmax(Vrnax V min ) + fmin'Umax + (1 ^ m m ) + fminVmax] ^ m i n ) + frninVmax ^moi) + 63 bV 3 (2, max 2 (5.72) > 2v max Vmax)] ^+ " m a x , (5.73) (5.74) Now ao, Qi, and 02 depend only upon a, and it can be shown that for 0 < a < | , an > 0, a i < 0, and a < 0, so there is only one positive root to this equation. Also, for a = |, 2 90 Chapter 5. Piecewise Linear Continuous Approximation a = 0 and thus there is one negative root and one zero root. W h a t this shows, i n essence, 0 is that there can be no positive speed traveling wave solutions for a > |. It can be shown that one part of the boundary corresponding to traveling wave solutions is determined by 7 + 1 -a-i ~ y/aj2a 4aoa 2 , 2 ' 1 ' , ' If we recall, the result for the P W L D model is that there were usually two values of 7 which corresponded to stationary front solutions. It turns out that there is another boundary for zero speed waves i n the P W L C model, but it has to be attained i n a different manner than the above result. The reason why it does not appear i n the integral method we just used is that it occurs as z\ â€”> 0, and thus v' attains a j u m p discontinuity which it did not previously have and the above analysis is incorrect. A l s o , when we tried to solve the problem exactly, we were operating under the assumption that there was an internal matching layer. A g a i n , as before, this second boundary emerges as a consequence of the discontinuity. T h e same methods and techniques that were used to solve the problem for the stationary fronts i n the P W L D model could be used here, i n principle. In order to find the other boundary, we first have to recognize that it occurs for small values of 7. In fact, we assume 7 < m unlike what we had tacitly assumed previously. 2 In this case we have lost our internal matching layer, so our solution for w becomes { d 1 2 e~ dz,ie X3Z X i Z , z>0, + w, (5.76) z<0. 3 We can apply our matching conditions, (5.46) and (5.47) to obtain di,2 -Aidi, 2 = d i+w , (5.77) = A d ,i- (5-78) 91 3i 3 3 3 Chapter 5. Piecewise Linear Continuous Approximation We solve the above system of equations to obtain = Xrip (5 79) Now, we use equation (5.38) to write down a solution for v A ] e~ 2 v(z) = { , K L T , z>0, XlZ L J (5.81) Here we cannot enforce matching condition (5.36), but we t r y to enforce the condition we d i d before i n the P W L D model l i m v(z) = v , (5.82) mm Z-S-0+ which is equivalent to , If we substitute for d l i 2 r i + Â« (5.83) and A] i n the above equation using (5.79) and (5.43), we get A37 min u (Ai + A 3 )(7 - m i ) (5.84) w 3 Thus we see that there is a second boundary condition for zero speed waves. T h e above equation could be solved explicitly to yield a lower bound for 7 i n terms of a and K. O u r next goal w i l l be to tackle the problem of positive speed traveling wave fronts since we have some idea of the parameter set we require. 5.2.2 Positive speed traveling fronts We are looking for solutions to equation (5.40) subject to the boundary conditions w(â€”oo) = u>3 and w(oo) = 0 w i t h v(0) = v min 92 and ' 0 ( ^ 1 ) = v max for some z\ < 0. Chapter 5. Piecewise Linear Continuous Approximation Let us denote by X j one of the roots of the characteristic polynomial cpi. W e know it that cpi and cp each have one negative real root and two roots w i t h positive real part. 3 We initially assume that a l l three roots are real and distinct. We also note that i n the l i m i t i n g case of zero speed solutions, cp had two imaginary roots and one positive real 2 root. Here we initially assume that cp has two complex conjugate roots w i t h negative 2 real part. Thus the solution for w is given by di,ie ' , 0 < z < oo, x, LZ w(z) = { e" [d ,i cos(cuz) + d , sin(u;z)] + d , e ' z 2 d ,2e 3 where A 1 : 1 2 A 3 ' - 2 ( z Z l ) + d ,3e 2 A 3 3 2 - 3 ( z - 3 A2 + w, 3Z z < z < 0, 2 + w, Z l ) -oo 3 < 0, p Â± LO are roots of cp with p < 0, A 2 2 ) 3 (5.85) x < z < > 0, and 0 < A 3 ] 2 z u < A 3 ) 3 . Now we apply our boundary conditions and matching conditions to p i n down some of these constants. A p p l y i n g C Â° , C , and C conditions at z â€” 0, we get 1 2 di,i - d , i - ^2,3 2 A i , i d i - pd ,i - cod 1: 2 2j2 A^di.i - pd 2 - A - 2pud 2}1 2>2 2 ] 3 d 2 ) 3 + io d 2 2y3 = w, (5.86) = 0, (5.87) = 0. (5.88) 2 A t z = zi, we obtain d , + d , + w 3 2 A3,2^3,2 3 3 3 + A , d 3 3 3 > 3 = e' = pe' [d ,i cos(o;zi) + d 1Z{ 2 2) IJ Zl + A 3 ) 3 d 3 i 3 = 2)3 2i d , e ^ 2 2>3 e ' A2 3Zl + w, 2 (5.89) llZx 3 A 3Zl 2 +Al d , e ' 2 3 A 2 2 COS(LOZI)] , [p sin(LOZi) tlZl 3 2 (5.90) 2 +d e 2>2 2 [p cos(w,zi) â€” 2pw d ^e 2 2 [â€”d i sin(wzi) + d , -\-Loe ' ^3,2^2,3 sin(a;zi)] + d [d i cos(u;zi) + d , sin(u;zi)] JZi +A 2>2 3 2 1 . 93 + S\TL(LOZX) 2PLOCOS(LOZI) â€” LO COS{LOZI)\ 2 â€” co sin(uz\)\ 2 (5.91) Chapter 5. Piecewise Linear Continuous Approximation Unfortunately, no easy formula exists for finding the solution of the d j i n terms of the i} Xij. In principle, i t is a linear system, but one that does not appear to simplify neatly as i n the P W L D model. Thus, we do not attempt to write down a solution for the d . itj We can write down a solution for v easily using (5.38) ' - ^ ( ^ i ) ^ ' -^e / i Z 1 . 1 0<z<oo, d , i [(M - to + cp - ^) 2 2 v(z) = <j -^'zd ,2 [{u - U J 2 2 k -I x + cp, - 2 -^fg(X2, )e- ^ 3 cos(uz) - uj(2p + c) s i n ^ ) ] 2 sin(u;z) + u (2p + c) cos(wz)] + v, z <z<0, 2 [d M>^)e 5 X 3 M z ~ z l ) Zl + d jg(\ , )e ^ -^] 3 3 3 x +v , z 3 -oo < z < z u (5.92) where g is as given previously i n (4.63). We can explicitly write down equations for our matching conditions (5.36) and (5.37) diMKi) = ~ V - ^ i (5-93) T d 3)2 0 ( A 2 ) + d3 3<-(A3,3) 3l 1 = K ( V 3 ~ V m Â° * \ (5.94) r Thus we are left w i t h the problem of solving 14 transcendental equations (the 8 listed above a n d the six for the eigenvalues). In general, it is difficult to give a sufficiently accurate i n i t i a l guess for the Newton method nonlinear equation solver that we are using to determine a solution to such a system. However, we showed previously how one can get two different zero speed solutions, one corresponding to a small value of 7, and one corresponding to a larger value of 7. T h e small 7 problem is a very difficult one to perturb off of, so we do not attempt to do that here. One can perturb off of the large 7 solution, but it requires a bit more work than i n the P W L D model as we don't have explicit expressions for the d^ i n terms of the eigenvalues. Here we need to use a bit of asymptotics to guess the appropriate scalings for the non-order 1 coefficients. 94 Chapter 5. Piecewise Linear Continuous Approximation In order to begin the process, we proceed as i n the P W L D model and let c = e <C 1 and assume that re, a, and r are fixed. Let 7 be expanded i n a power series i n e as =7 7 0 + e 7 l + O(e ). (5.95) 2 We also need asymptotic expansions of our steady states u>2 â€” W 2 , i + ui3 w ,i = (5.96) + 0(e ), (5.97) 2 + CW3.2 3 + 0(e ), 2 6 (re + l ) 2 m (re + 1) - 70 2 â„¢2,2 = + 1) jib (K 2 } 72> 7 â€” - ; ( 5 [m (re + 1) - 7 ]" 2 99 ) 0 b-iU + l) ^3,1 = , , ^ _/ , 6 (re + 1) + 7 0 7i6 (re + 1) 3 â„¢3,2 = 5.100 3 -77-7â€”r7- 72- ( 5 [& (re + l ) + n f 1 0 1 ) 7 3 Expansions of v and 113 can be attained by multiplying the equation for u> and 103, 2 2 respectively, by ^Â±1. In the next step, we seek to find expansions for the eigenvalues. T h e ones which were 0(1) i n the l i m i t e â€”>â€¢ 0 are easy to handle, and we deal w i t h them first. Thus, we expand Ai_i, co, and A 2 as follows 3; 1>1 - LO = 3,2 - 0(e ), (5.102) w 0(e ), (5.103) A 3,2 0(e ). (5.104) A l , l 2 ( 1 ) A w 2 2 ( L ) A Following the same techniques as before, we try to find expansions for the "missing eigenvalues" as in the P W L D model. Here, we can expand A , and A 2 95 3 3 ) 3 as we d i d in the Chapter 5. Piecewise Linear Continuous Approximation P W L D model for A 3 . Thus we expect A A ,s = 2 -f + >tf} + 0(e), A A, = 3 3 (1) (5.105) (i) -^ + \ l + 0(e). (5.106) { 2 2 We also notice that u. = 0 i n the l i m i t i n g zero speed wave case, so we expect to be able to expand \x as: p = eu {1) 5.3 +0(e ). (5.107) 2 Numerical Results 5.3.1 Varying a over its range In Figure 5-1, we see the fairly classical result that the speed is a monotonic decreasing function of a w i t h the zero-speed wave occurring before a = 0.5. In Figure 5.2, we see that the absolute value of z\ is a monotonic increasing function of a. T h i s is somewhat counterintuitive as one would expect that when a is decreased, the dendrite should be becoming more excitable, and this should translate into faster moving, steeper solutions. One should note that the absolute value of z\ corresponds to distance traveled w i t h i n the wave for the value of v to increase from v i m n However, the difference between v to v . max and v i , max m n 2y o -a + l / max J mm = Vmin 2 = u , (5.108) is a decreasing function of a. Thus, although z\ is increasing as a decreases, a greater portion of the total height achieved by the traveling front is being traversed, and the solution is not necessarily any less steep. 96 Chapter 5. Piecewise Linear Continuous Approximation 0.35 0 0.05 0.1 a 0.2 0.15 0.25 0.3 Figure 5.1: Speed of the wave as a function of a for 7 = 1.5, K = 5, r = 10. 5.3.2 Varying K over its range of values In Figure 5.3, we see roughly the same relationship between c and K that we saw i n the P W L D model, that is, there is something of a skewed hump which intuitively should asymptote at some positive value of c. In Figure 5.4, we have plotted z\ against K over the range corresponding to traveling front solutions. Here we see the relationship between z\ and K is directly opposite to that between c and K. One can immediately reason out that this is the case because when the dendrite is highly excitable, this should correspond to high speed, steep solutions. T h a t is, solutions in which c is large and the absolute value of z\ is small. Conversely, when the dendrite is not highly excitable, this should correspond to low speed, gently sloped solutions. T h a t 97 Chapter 5. Piecewise Linear Continuous Approximation Figure 5.2: z\ as a function of 7 = 1.5, K = 5, r = 10. is, solutions i n which c is small and the absolute value of z is large. x 5.3.3 Varying 7 over its range of values In Figure 5.5, we see the same relationship that we noted i n the P W L D model, that is, there appears to be a hump shaped relationship between 7 and c. T h i s is due to the balancing factors of the strength of local depolarization, voltage saturation, and current attenuation which are intricately linked to 7. T h e next two graphs, Figures 5.6 and 5.7, show where the the assumed functional form of the wave breaks down. W h e n 7 becomes small, the roots are no longer complex conjugates i n the inner matching region. Instead, they become negative real roots. In Figure 5.6, we see the complex portion of the root going to zero. In Figure 5.7, we see 98 Chapter 5. Piecewise Linear Continuous Approximation 0.25 C 0.2 \- 0.15 h o.i H 0.05 0 50 100 150 200 250 300 350 400 450 500 K F i g u r e 5.3: S p e e d of the wave as a f u n c t i o n for K for a = 0.1, 7 = 1.5, r = t h e e m e r g e n c e a n d s p l i t t i n g of the t w o r e a l r o o t s . 10. T h i s b i f u r c a t i o n was d i f f i c u l t t o get n u m e r i c a l l y . T h e p r o b l e m was t h a t n o n - s i m p l e r o o t s are t y p i c a l l y v e r y i l l - c o n d i t i o n e d . T h u s , w h i l e o n l y a n O(e) c h a n g e is m a d e i n the coefficients of the p o l y n o m i a l , there is a n 0(1) c h a n g e i n the r o o t s . T h e easiest w a y a r o u n d the p r o b l e m t h a t I f o u n d was t o j u m p c o m f o r t a b l y across the b i f u r c a t i o n , a n d p r o j e c t a l l o t h e r v a r i a b l e s across t h e j u m p a c c o r d i n g t o a s i m p l e l i n e a r a p p r o x i m a t i o n . T h e n go b a c k , a n d a c c u r a t e l y c a l c u l a t e the t w o n e g a t i v e r e a l r o o t s u s i n g M a p l e . O n e c a n h e a d b a c k t o w a r d s the b i f u r c a t i o n site, or m o v e o n i n the o t h e r d i r e c t i o n . I n F i g u r e 5.8, we have p l o t t e d a g r a p h of z\ a g a i n s t 7. H e r e we see t h a t as 7 goes t o its l o w e r c r i t i c a l v a l u e c o r r e s p o n d i n g t o z e r o - s p e e d waves, t h e i n n e r m a t c h i n g r e g i o n d i s a p p e a r s (i.e., z\ â€”> 0). T h i s c a n be u n d e r s t o o d q u i t e w e l l m e c h a n i s t i c a l l y . If 7 is s m a l l , 99 Chapter 5. Piecewise Linear' Continuous Approximation 45 - 40 - 35 - 50 100 150 K, 200 250 300 350 400 Figure 5.4: z as a function of K of a = 0.1, 7 = 1.5, r = 10. x there is very strong local depolarization so the spine head voltage should quickly j u m p to its equilibrium value. This corresponds to a slow-moving wave because it effectively chokes off the current source for the dendrite due to voltage saturation. Conversely, as 7 approaches its higher critical value corresponding to zero-speed waves, we see that the absolute value of z is increasing. This is due to the low excitability of the spines x which occurs because too much current is being lost to the dendrite. T h i s results i n a requirement for a long period of time for the spines to depolarize and thus a large absolute value of z . x 5.3.4 V a r y i n g r over its range of values In Figure 5.9, we have graphed A / T C against r to compensate for the factor involved i n the nondimensionalization and to make it easier to interpret graphically. We see that the 100 Chapter 5. Piecewise Linear Continuous Approximation c 1.2 7 Figure 5.5: Speed of the wave as a function of 7 a â€” 0.1, K = 5, r = 10. speed of the wave, c, is a monotonic increasing function of r , the nonlinear time scale, which appears to asymptote to some finite positive value just as i n the P W L D model. In Figure 5.10, we have plotted the relationship of z\ with r . T h e graph shows that the magnitude of Z\ is a monotonic decreasing function of r . T h i s result is intuitively obvious, as the wave should become steeper as the nonlinear dynamics are sped u p . 5.3.5 N u m e r i c a l simulation of the P D E model The analytical method we developed for numerically finding solutions to the system of O D E s that we derived was very valuable i n mapping out some of the relevant parameter ranges, and allowed us quickly to verify numerically some of our intuitive expectations from the P W L D and full H o d g k i n - H u x l e y models.' However, these traveling fronts are 101 Chapter 5. Piecewise Linear Continuous Approximation 0.5 r 0.2 h 0.1 h 0 I 0 1 1 0.1 1 0.2 1 0.3 1 0.4 7 I 0.5 1 0.6 l 0.7 l 0.8 I 0.9 I 1 Figure 5.6: LO over the range of 7 values for which there are complex conjugate roots for a = 0.1, K = 5, r = 10. not of much experimental importance unless they can be shown to correspond to stable solutions. L a c k i n g a proof of the stability of the system of P D E s , we instead decided to examine the solution numerically using P D E c o l . T h e program was r u n w i t h Neumann boundary conditions, and the initial guess was the computed solution from our analysis (slight perturbations to the initial guess were made and still gave the same result). T h e result is shown i n Figure 5.11. T h e traveling front appears to propagate to the right w i t h the theoretically computed speed without changing shape and thus acts as an important check of our analysis. 102 Chapter 5. Piecewise Linear Continuous Approximation -1.3 0.053 8 0.053 9 0.054 0.0541 0.0542 0.0543 0.0544 0.0545 0.054 6 0.0547 0.0548 0.0549 7 Figure 5.7: T h e real eigenvalues which exist on the other side of the bifurcation for a = 0.1, K â€” 5, T = 10. 103 Chapter 5. Piecewise Linear Continuous Approximation Chapter 5. Piecewise Linear Continuous Approximation 105 Chapter 5. Piecewise Linear Continuous Approximation Figure 5.10: Z\ as a function of r for 7 = 1, K = 5, r = 106 Chapter 5. Piecewise Linear Continuous Approximation x v Figure 5.11: T h i s is a simulation of the traveling front for a = 0.1, 7 = 1, K = 5, r = 10. 107 Chapter 6 The Full Cubic Model In this chapter I w i l l attempt to solve some of the dynamics for the full cubic model. T h i s system w i l l be seen to be similar to an F H N system w i t h reversed kinetics. T h i s reduced system parallels some of the ideas developed for the F H N equations by R i n z e l and Terman for bistable steady states. The m a i n thing that w i l l be seen here is that some of the steps from the analysis of the piecewise linear approximations have natural correlates i n the fully nonlinear model. However, there are a couple of important differences which w i l l be noted and w i l l be shown to be artifacts of the discontinuity of the linear interpolations. The form of the equations that we are studying is dv (6.1) di dw (6.2) ~dt where 7 > 0 , K > 0, r > 0, 0 < a < 1, and / is given by /(Â«) v{v â€” a)(l 108 â€” v). (6.3) Chapter 6. 6.1 The Full Cubic Model Spatially homogeneous steady states For spatially homogeneous steady states, we are looking for solutions to 0 = f( ) 0 = KV-(1 v 1 (6.4) -y), ( + w + (6.5) K)W. One obvious solution is the origin, v = 0 = w. To determine the existence of other steady states, we have to be a bit more thorough. The equation of the v nullcline is w =v â€” , f (6-6) 7 3 7 while the equation for the w nullcline is K, w = v, 1 = + (6.7) K h(v). Since / ' ( 0 ) < 0, we know that the slope of the v nullcline i n the (v, w)-plane is greater than 1 at the origin. However, ti (0) < 1, so the v nullcline always passes over the top of the w nullcline at the origin which is exactly the opposite of the case i n the F H N equations. In order to find the other steady states we see that the equilibria must satisfy P(v) ~ 1+K = 0, (6.8) or alternatively, v (v 2 V (a + l)v + a + -lâ€” 1+ 109 J KJ =0. (6.9) Chapter 6. The Full Cubic Model The non-zero roots of the above cubic are given by = I a+l + ^ a - l ) ' - ^ vs = MÂ« + l - A / ( a - l ) 2 1 " ' " V ~ ~' v 2 - â€” 1+ K (6.10) (6.11) " which represent the v components of the steady state. T h e corresponding w components are obtained by noting that = Wi 1 + -u,-, (6.12) i = 2, 3. K A n obvious condition on the existence of these roots is 47 1 + K, <(l-a)' (6.13) If we let v-i â€” 0 â€” wi, then we can determine the stability of the steady states from the eigenvalues of the m a t r i x f'{vi)-J A, 7 re Since trace(Ai) -(1 + i = 1,2,3. (6.14) K) < 0 and det(Ax) > 0 (since /'(0) = â€” a < 0), we know that the origin is always stable and can be shown to correspond to a stable node. If the inequality (6.13) holds, then we know that v is the second root of (6.8). A s this equation is a positive 2 cubic, we see that this cubic must have a negative slope at v , and hence 2 V {v ) < 2 KJ 1 (6.15) + K,' Thus, substituting for p (v ) using inequality (6.15) i n the equation for det(A ), 2 2 we see that det(A ) 2 = p (v )(l + K) - JK 2 < ^â€”â€” (l + Â«) < 0. 110 -JK (6.16) Chapter 6. The Full Cubic Model Thus, this equilibrium point must be a saddle point. Similarly, we can show that p'(v ) > (6.17) 3 X ~T and thus determine that tr(A ) 3 K < 0 and det(A ) > 0. Hence, this equilibrium point must 3 be stable. Here we have the classical case of a bistable steady state i n the phase plane. If, however, = (1 â€” a ) , the two roots v and v coalesce into one. In this case, one 2 2 3 can easily see that the determinant of the stability matrix should be zero, and we have a nonhyperbolic equilibrium. A center manifold reduction could be attempted here to determine some of the features of this equilibrium, but it is not of much concern here and won't be pursued any further. Since we now know that there are two stable steady states separated by a saddle point i n the space clamped case if inequality (6.13) holds, we now can inquire into the existence of traveling front solutions between the two stable steady states. 6.2 Traveling front solutions We seek traveling front solutions of the fully nonlinear equations w i t h constant speed c which traverse a path from the steady state at (vi, wi) to the one at (v , w ). 3 3 We proceed as i n the previous two chapters and look for solutions with constant shape in the traveling wave coordinate z = x- ct. (6.18) Substituting into (6.1) and (6.2) we get -cv = f(v)+j(w-v), â€”cw' = w â€” â€” + â€” (v â€” w), r 111 r (6.19) (6.20) Chapter 6. The Full Cubic Model where ' denotes differentiation with respect to z. T h e relevant boundary conditions are: lim w(z) (6.21) lim v(z) (6.22) 2â€” > â€” OO lim w(z) 0, (6.23) lim v(z) 0. (6.24) 2â€”â€¢OO There are four boundary conditions that we shall try to satisfy for this third-order problem, so the problem is overdetermined. A s i n the previous models, our nonlinear O D E is autonomous, and the associated boundary conditions are at oo. Thus, the problem is translation invariant. Since we know that for some value of z, say z , v(z ) = v , we may without loss of generality enforce the 0 0 2 condition v{0) = (6.25) v. 2 The next critical issue is determining the stability of the various equilibria in the traveling wave frame. Let u = iu', then we can use the fact that u = w" to write the above system as the system of three first-order nonlinear O D E s 1 1 V 1 w â€” â€” = u c v) - jtv], (6.27) 1 / ') 1 Ui (6.26) w - - v \ . c T T (6.28) j T h e stability matrix of each of the three roots is given by Ai 0 0 K, 1+ CT CT 1 K. 112 , Â« = 1,2,3. (6.29) Chapter 6. The Full Cubic Model T h e characteristic polynomials for the equilibria are given by cpi(X) = A 3 p'{vi) - A 2 p'(Vi)T + l + K CT 1 p'(^)(l + K ) - Â« 7 , CT . z = l,2,d. (6.30) Since p (0) = a + 7, we know that cpi(O) > 0 and cp'j < 0, so cpi has one negative real root and two roots w i t h positive real part. Thus, as z â€”> 00, we have a two-dimensional unstable manifold. G i v e n inequality (6.15), we assert that cp2(0) < 0 and cp' (0) < 0, so cp has one positive 2 real root and two roots w i t h negative real part. 2 Thus, as 2 -> â€”00, this equilibrium point has a two-dimensional unstable manifold, and it is unlikely that a trajectory from the equilibrium at (0, 0, 0) could enter this equilibrium. Similarly, given inequality (6.17), we assert that cp (0) > 0 and cp' (0) < 0, so c p 3 3 3 has one negative real root and two roots w i t h positive real part. Thus, as z â€”>â€¢ â€”00, this equilibrium has a two-dimensional stable manifold, so it seems plausible that there may be a path connecting it to the equilibrium at (0, 0, 0) for a certain speed, c. We notice a problem that we did not have previously. Before, we could explicitly state a general form of the solution by using the fact that the equations were linear w i t h constant coefficients i n the various regions. We then could use boundary conditions to try and determine certain coefficients, and the values of z which split up the axis to make everything consistent. Here we first develop a technique to show how one may determine the parameter range which corresponds to zero-speed waves. T h e n , we show how to obtain an implicit formula for the solution trajectory to the standing wave problem and show that the results are consistent with the analysis done on the previous problem. We verify the results numerically and show the similarity between it and the solution to the P W L C model. Lastly, we suggest a means to perturb off this solution i n order to determine the parametric dependence of speed. 113 Chapter 6. 6.2.1 The Full Cubic Model Using integration to determine zero speed wave solutions If we m u l t i p l y (6.20) by w and integrate both sides of the equation over (â€”00,00), we obtain (using integration by parts, the fundamental theorem of calculus, and the boundary conditions) j'C OO (w') dz 2 / = WW / J â€¢00 â€”c 1 '\2 K [W ) 1+ â€” Â« zr 2 WW -\â€”vw dz T K 1+ 2 W â€”VW T T W - KTV W 3 3 3 wv T If we use the above derivation to solve for wv (6.31) dz, dz. we can use it to derive an implicit wv'dz, equation for c. T h e first step is to multiply both sides of (6.19) by v and integrate over (â€”00, 00) to obtain â€¢OO fOO (v ) dz 2 / = I [(f(v) Jâ€” 00 -00 Â° r f(v) â€” jv) + 7u>] v dz (6.32) 7 '1 fÂ°Â° â€” â€”(v ) dv + 7 / wv dz. 2 2 J J_ 00 Now we make the substitution described above and isolate the terms involving c on the right to obtain -c / 00 [v') -00 2 + ^V) 2 dz = 7(1 + K)W2 f(v) dv + j2vj + 2K ^3 f(v) M ; dv + , 7 " 3 jv w 3 3 (6.33) .. 2(1 + K) Thus, we can obtain the following implicit expression for c (6.34) c = Clearly the denominator i n the above expression must always be positive a n d non-zero for any solution to the system of O D E s we are examining. Thus the speed of the wave equals zero when f{v)dv 7 2(1 114 + K) vi = 0. (6.35) Chapter 6. The Full Cubic Model If we solve the integral above and do a little bit of algebra we find that v l _ Â« a + T> V3+ 2 LX) 3 \ = 0 1+ K, J , (6 .36) Recalling the equation for v , (6.8), we can derive another constraint 3 vi - (a + l)v + 3 1 a + â€”!â€” 1 + K, = 0. (6.37) ' We subtract these two equations, yielding h +l)v -a. 7 1+ 3 a 3 (6.38) Substituting the above expression into (6.37), we obtain v = \{a + \). 3 3 (6.39) T h e n eliminating v i n (6.37), we get the tidy result: 3 7 1 + 2 9 K, f a - ] - ] (a-2). V 2^ (6.40) B y analogy, it is easy to see that the positive speed traveling wave fronts should exist i f the following inequality is satisfied 7 1 + < K, 2 9 (a - 0 (a - 2 ) . (6.41) Now we show a method to determine the profile of a zero-speed solution. 6.2.2 Calculating the profile of a zero-speed front In this section, we are searching for solutions to the nonlinear O D E s corresponding to c = 0. In this case, the O D E s reduce to 0 = - P ( T J ) + 7JJ, (6.42) 0 = u; (6.43) 7 T 115 w + -v. r Chapter 6. The Full Cubic Model One should notice that the simple change of variable z = -jU eliminates r from the problem. Thus, we shall just adopt the convention that r = 1 for the analysis that we shall do. Note that / dp a dv d p . i, i 2 (6.44) dp 9 a (6.45) Substituting these results into (6.43), we obtain the single nonlinear O D E ~v" + ^-f(v) (6.46) - (1 + K)P(V) + K^V = 0, 2 w i t h the associated boundary conditions given by (6.22) and (6.24), and the enforced condition (6.25). If we use the assumption that condition (6.40) holds, then we can show that KJV â€” (1 + K,)p(v) = KJV â€” (1 + K)V(V 2 = -(1 + =- ( 1+ â€” (a + l)v + a + l^v - (a + l)v + a + K)V 2 Â« ) Â» ( â€ž - i Â± i ) (v â€” 7), K7 i+V' (6.47) 2(a + l ) ' Now let us write d,p dv = 3 u - 2 ( a + l)w + a + 7, 2 = 3 a+1 o 1 (6.48) where Pl a + 7 116 (a + 1) 5 (6.49) Chapter 6. The Full Cubic Model Thus, . / dp 2 d^ = 6 [ a+ 1 V (6.50) - ~ 3 - There is a natural change of variables to try i n order to minimize the algebra (6.51) i ( a + l) ' z = (a + 1 (6.52) + Ky, A/1 (6.53) iy M a k i n g these substitutions into (6.46), we get 3(u + p 2 2 du ) ^ + 6u [dy dy + u(l - u ) = 0 2 2 (6.54) subject to the boundary conditions = 1, (6.55) u(0) = 0, (6.56) â€” 1. (6.57) lim u lim u = 2/->oo We can integrate (6.54) i n phase space. Define g = ^ . Rewrite the O D E as dg _ â€”6ug + u(u 2 dy â€” 1) 2 3(u + p ) 2 2 However, we know that dg/du â€” (dg/dy)/(du/dy), so we may conclude dg â€”6ug + u(u â€” 1) d,u 3g(u + p ) 2 (6.58) 2 2 2 (6.59) If we now rewrite the equation as 3(u + p )gdg + (6g u + u(l - u ))du = 0, 2 2 2 2 (6.60) then the O D E given above has the integrating factor p = 3(u + p ) 2 117 2 (6.61) Chapter 6. The Full Cubic Model If we now endeavor to integrate partially (the equation is now exact) we can show that the solution is given by 9 u + p ) g = + (f)(u) constant, d(f> (6.62) 3u(l-u )(u 2 du + 2 (6.63) p ), 2 or u 2 (v 2 + p)g 2 2 + ^ 2 [6p + 3(1 - pV 2 - 2u ] (6.64) = k 4 where k is some constant. T h e above equation is even i n u, so if (u, g) = (1, 0) is a point on the trajectory, then so is (u,g) = (â€”1, 0). T h e trajectory goes through these points if 1 ;i + 3 ) (6.65) = k. 2 P Now we may rewrite the equation as -(u 9 If we recall that g = 2 + p )g 2 = \(u -l) (2u 2 2 2 + l + 2 (6.66) 3p ). 2 we obtain the result du _ {u - l)x/2u 2 + l + 3p 2 dy 2 (6.67) 18(ti + p ) 2 2 A p p l y i n g the condition u(0) = 0 yields s 2 y = V18 + p )ds 2 (s -l)(2s 2 (6.68) + l + 3p )' 2 2 and integration gives y = 31og e y/i2)u + (u) gi + <5(l + p) 2 ,5g (-u) 2 atanh{ ) â€” 9i(u) 6g (u). 2 atanh(â€”7-r) (6.69) where 0i W = 92(u) = 5 ^2U + l + 2 3p + l + 2'u, 2 1 V 3 + 3p / 118 3p , 2 (6.70) (6.71) (6.72) 2 Chapter 6. The Full Cubic Model Now that we have shown how to generate implicit solutions to the problem for zerospeed waves, we can see that if we perturb one of the parameters, we should be able to predict the speed of the resulting wave analytically. T h i s is, however, beyond the scope of the present investigation and will be left as an open problem for someone else to pursue. T h e basic techniques are standard regular perturbation techniques, and the leading order solutions is stated above. The problem should simplify to finding the solution to some linear second-order differential equation which can be solved numerically. One thing that is somewhat striking about the above result is that it should perhaps generalize to other sets of equations. We now show that this is, i n fact, the case. 6.2.3 General method for getting solution trajectories to degenerate reactionâ€”diffusion traveling fronts The problem can be stated as follows 0 = -f) + 0 = w" - w + ipv, q v (6.73) W) (6.74) subject to the boundary conditions lim v = 'u , 3 (6.75) lim w = w, (6.76) lim v = 0, (6.77) lim w = 0, (6.78) zâ€”> â€”oo 3 zâ€”>â€”oo 2 - 4 OO zâ€”>oo where (v ,w ), 3 3 (0,0) represent spatially homogeneous stable steady states and ( t ^ u ^ ) represents a spatially homogeneous unstable steady state, tp > 0, g(0) = 0 and is generally a cubic i n profile. Here we also assume the auxiliary condition v(0) = v 2 119 Chapter 6. The Full Cubic Model We can proceed as before by noting that w â€” q(v), and thus i, dq 'N2^ Z 2 ( â€” v â€” 4- (v ) dv d,v w (6.79) 7 Substituting these results into (6.74), we get dq Â» dq dv dv 2 ' + I 7 ( ' ) -q\v) + il)v. ,; u (6.80) If we let Q{v) = 'ipv â€” q(v) and g = v , we again can simplify the problem by recalling that ^ = If we substitute these relationships into (6.80) , we get dv (6.81) 9 dv which may be rewritten as dq ' do dv dv 2 g~i dg+\-Â±g L (6.82) 2 + Q{v))d,v = 0. Now this equation is not exact, but we may determine an integrating factor. If we let M(g,v) = N(g,v) dq g-f, d,v d?q g + Q(y), dv 2 2 (6.83) (6.84) then we notice that M(g,v) dN d,M dg dv J (6.85) is independent of g. Thus, we know that there is an integrating factor of the form pi [M(g,v) \ dcj dv )\ (lV (6.86) T h e equation is now exact, and the general solution is given by (6.87) (6.88) 120 Chapter 6. The Full Cubic Model where k is some constant. Since 0(0) = 0 and l i r n ^ o o v â€” 0, we immediately see that k â€” 0. Now i n order for this to be possible, we also see that we require 9{v) > 0 for v i n the open interval (0, v%) and 0(1*3) = 0 to match the boundary condition at z = â€” 00. T h a t is, I"" ~Q(v)dv Jo dv = 0. - (6.89) We can apply integration by parts here and noting that Q(v ) = 0 = Q{0), we see that 3 we require f Jo VS q(v)^r-dv = 0. (6.90) dv If we follow this derivation for our problem, we w i l l end up w i t h the result: JK ~ = 0. 1.2(1 + *.) (6.91) j Q The fact that the speed of the wave goes to zero as r â€”> 00 is something of an anomaly. The actual physical speed of the wave goes to a constant, but i n the nondimensionalization of the problem, the distance was scaled by r so the unit distance goes to 00 as r goes to 00. Here we also see the classical results that the speed of the wave should go to zero as 7 a n d K become s m a l l . Physically, this makes sense, since as 7 â€”*â€¢ 0, we have a situation in which current through the dendritic shaft is unable to penetrate into the spine head, Thus spines ahead of the traveling wave cannot reach threshold and create current to continue feeding the wave. Similarly, the speed of the wave should go to zero as K â€”> 0 since there is no source to feed the dendritic shaft voltage. It should be noted here that these results do not agree w i t h the results from the P W L D model, the P W L C model, or the full H H model of Baer and Rinzel. T h e lack of agreement w i t h the former two can be shown to be due to a lack of sufficient continuity for the source term (i.e., f ) . The lack of agreement with the full H H equations is almost certainly due to the lack of a recovery term. Intuitively, one would expect the region i n parameter space corresponding 121 Chapter 6. The Full Cubic Model to traveling waves to contract i n each direction as a recovery term is added, that is, the parameter space for positive speed pulses for the F H N equations w i t h recovery should contain the parameter space for positive speed fronts for .the F H N equations without recovery. 6.3 Numerical simulation of the P D E We decided to attempt a numerical simulation of the full P D E model on P D E c o l to ensure that there were actual traveling front solutions i n our presumed parameter space. We ran the simulation using Neumann boundary conditions, a n d used a scaled version of a solution to the P W L C model for the same parameter set as an initial guess. T h e solution to the P W L C model was scaled so that it had the same height as the solution to the full cubic model, but no alterations i n the shape were made. T h e resulting wavefront solution is plotted i n Figure 6.1. One can see that the P W L C model gave a solution w i t h a fairly similar shape to the cubic model as the shape did not evolve very much with time. Furthermore, the speeds of the fronts were actually quite comparable. However, i t should be noted that the slope of the wave front i n the cubic case is steeper than the the slope of the P W L C wavefront as would be expected. 122 Chapter 6. The Full Cubic Model x Figure 6.1: T h i s is a simulation of the traveling front for a = 0.1, 7 = 1, K = 5, r = 10. 123 Chapter 7 Discussion 7.1 Theories of spine function The function of dendritic spines is a mystery that has long plagued neuroscientists. M a n y different theories have been put forth on their functions, but until recently, they have largely been untestable due to lack of appropriate experimental techniques. W i t h the advent of devices such as fluorescence microscopy and other advances i n laboratory equipment, some direct observation of spine function has been possible i n recent years. T h i s has intensified the interest i n the field amongst experimentalists and theoreticians alike. The m a i n goal of this thesis was to review some of the major theories and study mathematical models put forth on spine function. In order to understand spine function, one has to understand that spines are the m a i n site of reception of excitatory synaptic transmission. T h e exact reason for why this is the case has been debated for many years. There are two predominant theories w i t h i n the field of research. One theory holds that the primary reason is to allow for spatially localized regions i n which biochemical reactions mediating changes i n synaptic efficacy can be sequestered. T h i s was the primary focus of Chapter 2. T h e other major theory suggests that spines may have special electrical properties that allow them to modify the size of the E P S P at the axon hillock compared to a neuron without spines. T h i s is the focus of much of Chapter 3. We subsequently explored i n greater detail one particular 124 Chapter 7. Discussion continuum model of excitable spine heads and proceeded to derive an analogue model based on the F i t z h u g h - N a g u m o equations. T h e mathematical analysis of three different models was the focus of Chapters 4, 5, and 6. T h e main purpose behind these analogue models was that the forms of their equations allow for analytical results which cannot be attained by numerical simulation and may give a deeper understanding of the underlying phenomena. 7.2 7.2.1 Discussion of spines and L T P Spines and LTP In Chapter 2, we focused primarily on evidence suggesting a role for spines i n mediating a localized memory phenomena known as long-term potentiation ( L T P ) . W e introduced some of the important features of L T P such as cooperativity, associativity and i n p u t specificity. We explored three initial models suggesting how spines can play an important role i n L T P by allowing high localized concentrations of [ C a ] to develop w i t h i n the 2+ i spine head i n response to high frequency stimulation. This high level of intracellular calcium then could activate a number of calcium-dependent enzymes and unleash a biochemical cascade which eventually culminates i n an increased postsynaptic response when that synapse is subsequently activated. These theories focused primarily on the contributory roles of spine morphology, the presence of a specialized postsynaptic receptor known as a N M D A R which has a high calcium conductance and is largely inactive during low frequency activation but highly active during high frequency stimulation, and the presence of calcium buffers and pumps. These models went a long way i n demonstrating that the special features of a spine, its shape and size, may be critical i n creating a biochemical compartment specific to a particular synapse. We then proceeded to explore some ideas concerning calcium dynamics w i t h i n spines. The work of Woolf and Greer [118] showed that spines could develop localized calcium 125 Chapter 7. Discussion transients if they had long spine necks, high amounts of buffers, and the absence of calcium-induced calcium release mechanisms i n the spine neck. T h e unfortunate aspect of the above numerical result was that it failed to give one a true qualitative feeling about how a l l the different factors interplay to produce localized calcium transients i n the spine heads. Zador and K o c h managed to shed a great deal of light on the subject through a linearization of calcium dynamics through the use of asymptotics [121]. They showed that under certain conditions, the nonlinear calcium dynamics reduce to the cable equation and were able to identify analogues to the concepts of input resistance, time constant and space constant. They showed that the chemical input resistance of the spine head was much larger than the parent dendrite, which allows for much higher increases in [ C a ] , i n the spine head than the same input would cause on the parent dendrite. 2+ t Furthermore, the chemical space constant of the spine was also much smaller than the parent dendrite. T h i s means that much higher levels of [ C a ] j w i l l be seen i n the spine 2+ head than at the base of the spine head, which allows for spatial localization of calcium transients. Finally, they showed that the chemical time constant of the spine head was much smaller than the parent dendrite. T h i s only further amplifies the difference i n [ C a ] j between the spine head and parent dendrite during calcium transients. 2+ We proceeded to explore the temporal nature of calcium transients required i n L T P . It was shown that there may be an important role for calcium-induced calcium release processes to keep the level of [ C a ] i n the spine head elevated for long enough to allow 2+ i for the biochemical cascade controlling L T P induction to become sufficiently activated. Finally, we looked at the presumed final common pathway involved i n L T P induction, the activation of protein kinases. We focused on how C a M K I I ' s peculiar autophosphorylation capacity may be critical i n the induction of L T P , and briefly reviewed the results from a finite Markov chain model of C a M K I I function. We felt it was important to give the reader an understanding of L T P a n d its relationship 126 Chapter 7. Discussion to spine function as this is one of the major fields of interest i n neuroscience today. 7.2.2 T h e electrical properties of spines In Chapter 3, we explored a number of theories proposed for the unique electrical properties that the presence of spines may confer to neurons. Some of the more prominent ideas put forth i n the past include: â€¢ Spines mediate synaptic potential attenuation such that there is a large voltage gradient between the spine head and parent dendrite upon synaptic excitation. â€¢ Spines may allow for linear summation of E P S P s by decreasing nonlinear interactions w i t h neighboring spines. â€¢ T h e spine neck resistance may be a controller of synaptic weight. â€¢ Spines increase the low-pass filtering capacity of neurons simply by increasing the surface area of dendrites. â€¢ Spines effectively decrease the space constant and input resistance of dendrites while having little effect on the time constant. â€¢ Spines w i t h active channels are computationally richer than passive spines and may allow for a form of pseudosaltatory conduction i n dendrites. We finished the chapter by exploring a continuum model of active dendritic spines. T h e goal of this model was to simplify the problem of modeling spines by averaging them over the dendrite as a continuous density and thus significantly simplify the problem. Active continuous channels based on the H o d g k i n - H u x l e y equations were placed i n the spine head, and a thorough numerical investigation of changing various parameters such as spine neck resistance was made. This model went a long way i n allowing more detailed analysis of the possible role of active spine dynamics on dendritic action potentials. 127 Chapter 7. Discussion However, the model is still far too complicated to allow for the derivation of any significant numerical results. Instead, it must rely upon brute force numerics to simulate a solution for each individual parameter set. In order to obtain a better qualitative feeling for the nature of the various parameters i n dendritic action potentials, we decided to employ the simpler F i t z h u g h - N a g u m o d y n a m ics for the H o d g k i n - H u x l e y dynamics. This approach has proven to be very successful in allowing insights to be made into action potential generation in axons by maintaining the basic flavor of the equations, while drastically simplifying the functional form of the P D E s involved. Thus, we derived a continuum model based on active spines w i t h F i t z h u g h - N a g u m o - t y p e dynamics. We decided to simplify the model i n the manner of B e l l and Cosner [8] by removing the recovery variable as justified by asymptotics. T h i s reduced the problem to a third-order problem which is analogous to a F i t z h u g h - N a g u m o system w i t h bistable steady states, which was studied previously by R i n z e l and Keller [92]. We now compare and contrast the results from our analogue models. 7.3 7.3.1 Results for the 3 analogue models T h e piecewise linear discontinuous model In Chapter 4, we used the piecewise linear term â€”v + H(v â€” a) where H is a Heaviside function instead of the cubic which classically is used in the F i t z h u g h - N a g u m o equations. We used this equation as it is the simplest one that retains the salient features of a cubic and allows for an initial exploration of the behaviour of the system of P D E s . We started by deriving the conditions under which a bistable steady state exists, 7 1+ K < â€” â€¢ a We further demonstrated that both steady states are stable. 128 (7-1) Chapter 7. Discussion We then began to search for constant speed traveling front solutions that travel w i t h constant speed which go from one steady state to the other. We then proceeded to change the variables into the moving wave frame which allowed us to reduce the problem to a third-order system of O D E s with boundary conditions at z = Â±00. Since the problem was translation invariant, and we knew that there was some value of z, say z , 0 for which V(ZQ) = a, we arbitrarily imposed the condition that ZQ = 0. Since we were p r i m a r i l y interested i n determining the parameter range which corresponded to positive speed traveling front solutions, we can simplify the problem by mapping out the boundary of this region. T h a t is, we can attempt to study the region i n parameter space corresponding to zero-speed solutions. This has the effect of reducing the problem to a simple second-order O D E which can be solved explicitly. Moreover, we find that zero-speed waves correspond to T, "TTT = 2a ( - ) > 7 2 (1 + 7)(1 + 7 + K and positive speed solutions occur for the inequality > 2a. r-^ (1 + 7)(1 + 7 + K, (7.3) V ; Interestingly, it can be shown that there are no positive speed solutions for 0, > \. We then proceeded to examine positive speed traveling front solutions i n the appropriate parameter range. O u r technique was to try and solve the set of nonlinear equations by a Newton's method, but we needed a reasonable approximate solution initially for a given parameter set. A n initial guess for a solution corresponding to a slow moving wave was made by perturbation techniques. Having accomplished this, we then could explore the dependence of the shape and speed of the wave i n terms of the parameter set by holding three of the parameters fixed and altering the fourth using the method of continuation. Since we were mainly interested i n the speed of the wave, we w i l l not comment any further on the shape of the wave. 129 Chapter 7. Discussion We were most interested i n the effects of the parameters on the speed of the wave. We found that the speed is a monotonic decreasing function of a. T h i s is intuitively obvious since low values of a correspond to more highly excitable systems which should have faster propagating solutions. T h e speed of the wave has a skewed hump solution for K w i t h a lower value which corresponds to a zero-speed solution. This is due to the fact that K is a measurement of the extent to which spines can depolarize the dendritic shaft. However, there is a value of n corresponding to a m a x i m u m speed because increasing spine density beyond a certain limit creates a very large conductance load on the dendritic shaft which must be overcome to bring spines that are on the leading edge of the wave up to threshold. The graph of the speed of the wave against 7 also had a skewed hump appearance, but there were two values of 7 which corresponded to zero-speed solutions. If 7 was too small, the spine would achieve voltage saturation too quickly and not enough current would be delivered to the parent dendrite to bring the set of spines ahead of the front to threshold. Conversely, if 7 was too large, too much current would be lost to the parent dendrite, and the spine would be unable to reach voltage threshold. T h e graph of the speed of the wave against r was monotonic increasing. T h i s is not surprising since it is intuitively obvious that speeding up the dynamics of the nonlinearities w i t h i n the spine head should speed up the wave as well. This is seen, and there appears to be a definite m a x i m u m speed of the wave. Our technique of simplifying the original set of equations and parameters to more manageable sets allowed us to do some analytical work suggesting the appropriate balances i n parameters to achieve propagating solutions. T h i s gives one a better qualitative feel for the features of dendrites and spines required to allow for dendritic action potentials. T h i s is much more intuitive than studying each of the numerous parameters of the individual model w i t h time-consuming and computationally expensive nonlinear P D E s . 130 Chapter 7. 7.3.2 Discussion The piecewise linear continuous model In Chapter 5, we used a piecewise linear continuous ( P W L C ) approximation of the F i t z H u g h - N a g u m o cubic. This approximation, / , satisfied /(0) = 0 = / ( l ) , and / achieved the same m a x i m u m and m i n i m u m as the F i t z h u g h - N a g u m o cubic at the same points. The general aim in using this approximation was to attempt to reflect more accurately the true shape of the F H N cubic while still retaining the feature that the problem could be reduced down to solving a set of nonlinear algebraic equations. Most of the salient features of piecewise linear discontinuous ( P W L D ) model were retained i n the piecewise linear continuous model. There were again two stable steady states i n the traveling front frame so long as the inequality 7 1+ < frnox ^ ^ V K n is satisfied. T h i s is clearly analogous to the result in the piecewise linear discontinuous model. However, i n the P W L C model, there was a third steady state which is a saddle point. Thus, as in the P W L D model, we looked for traveling front solutions going from one stable steady state to the other. A s i n the P W L D model, we showed there were no traveling wave solutions for a > |, and that for a parameter set with a < ^, there are two values of 7 which correspond to zero-speed traveling fronts for fixed 7 and K, but only one value for a and AC when the other two parameters are set. Finally, the dependence of the speed of the wave on the different parameters was generally the same in the two models. One of the major differences between the two models is that the piecewise linear continuous model has an "inner matching region" that the P W L D model lacks, and this generates a new set of eigenvalues. T h i s creates a difference i n how the smaller value of 7 which corresponds to zero-speed waves, can be calculated. In the P W L C model, this cannot be accomplished directly as in the P W L D model, because the solution is 131 Chapter 7. Discussion degenerate. T h a t is, as 7 decreases towards its critical value, c â€”>â€¢ 0 and z\ â€”> 0. In this case, we can circumvent this problem and discover the lower value of 7 by recognizing the disappearance of the "internal matching layer" and the problem becomes similar to the P W L D model i n that there is a j u m p discontinuity i n v. Another dimension that has to be accounted for i n the P W L C model is the parametric dependence of z\ on a, 7, K and r . z\ is a monotonic decreasing function of a, meaning that as a decreases, the distance required for v to reach v increases. T h i s is to a certain max extent counterintuitive. However, it should be noted that z\ is the distance required for the value of v to go from v to v min max and _ 27a - a + 1 2 Vmax so the difference between v x V in m â€” ^ > V / and v in is a decreasing function of a for a < |. In terms ma rn of the relationship between z and K, we note that z becomes large as K goes its critical x x value which corresponds to zero-speed waves, develops a local m i n i m u m at roughly the same value of K for which the m a x i m u m speed is obtained, and then begins to increase thereafter as the speed begins to fall. It can easily be reasoned that the same line of reasoning which explained the relationship of speed to K i n terms of the excitability of the system applies to zi i n terms of the steepness of the solution. T h e relationship between z\ and 7 is a bit more interesting. For large values of 7, we see that the value of Zi decreases, which corresponds to the decreased excitability of the system due to excess current loss to the dendrite. However, we also noticed that the value of zi went to 0 as 7 approached its smaller critical value which corresponds to zero-speed waves. T h i s can be seen to be due to the overexcitability of the spines due to the fact that little current is lost to the dendrite. This results i n the rapid firing of spines which is reflected i n the small absolute value of z , and the small value of c due to the voltage saturation of the x spine choking off current transfer to the dendrite. In terms of r , we get the expected result that z\ is an increasing function of r which plateaus at some level. T h e reasoning 132 Chapter 7. Discussion here is related to an increased level of excitability as r increases which steepens the wave profile. 7.3.3 T h e cubic model The cubic model presents unique challenges that are lacking i n the P W L C and P W L D models. The foremost problem is the presence of a cubic nonlinearity. In the other models, we knew the general form of the solution would be a linear combination of exponentials i n different regions of space. This allowed us to reduce the problem from solving a series of O D E s to solving a system of nonlinear algebraic equations, for which a certain amount of analysis was possible. Our first step was to analyze the problem in phase space. A s i n the P W L C model, there are three steady states when <(!-)â€¢ (7-6) is satisfied. Furthermore, the first and t h i r d roots again can be shown to be stable, while the middle root corresponds to a saddle point. Thus, we again looked for solutions going from one stable steady state to another. We proceeded to look for zero-speed traveling wave solutions. B y using integration techniques, we demonstrated that zero-speed solutions exist when is satisfied. T h i s result is analogous to the upper limit value of 7 corresponding to zero-speed solutions i n the P W L C and P W L D models as positive speed solutions exist when 7 1+ K < -(a-)(a-2). 9V 2' 2 l 133 (7.8) Chapter 7. Discussion It also shows the same relationship i n K and a i n that there is some lower value of K and upper value of a which correspond to zero-speed waves. However, an important difference was that it was not immediately obvious whether there was a lower value of 7 which corresponded to zero-speed solutions. We were able to show i n a more general manner later that there were zero-speed solutions, but they were degenerate and occurred i n the l i m i t as 7 â€”Â» 0. T h e fact that the P W L C and P W L D models had zero-speed solutions for positive values of 7 is a consequence of the lack of sufficient continuity i n the equations. Finally, we calculated the profile of the zero-speed solutions by integrating i n phase space and demonstrated that this technique was applicable to a l l similar degenerate reactiondiffusion systems. T h e ability to calculate this profile gives us a technique to perturb off this solution using asymptotics to determine an approximate profile of a slow moving wave. 7.4 Future areas of investigation We reviewed the field of research of mathematical models of L T P induction, and clearly much remains to be done. T h e biggest challenge that remains to be worked out here are the exact mechanisms of regulation of a huge biochemical cascade of calcium-dependent protein kinases and phosphatases. A n interesting project here would be to look into viable forms of regulation that could explain the spatiotemporal requirements of calcium transients required to induce L T P . There is also much work that can be done i n following up theories of the electrical functions of dendritic spines. T h e two problems that could immediately be taken up are the utilization of asymptotic techniques to perturb off of the zero-speed wave profile that we have calculated to generate approximations of the profile of slow moving waves. Addi- tionally, reaction-diffusion systems w i t h one small diffusion coefficient a n d a functional form similar to our set of equations also could be handled using asymptotics to perturb 134 Chapter 7. Discussion off the degenerate solution profile that we have calculated. Lastly, an interesting problem to pursue would be to try to determine traveling pulse solutions by not excluding the recovery variable that we have dropped. T h i s problem is much more difficult as it greatly increases the dimension of the parameter space and increases the order of the system of O D E s involved from three to four. 7.5 Conclusion The field of mathematical neurobiology is developing rapidly and many new interesting problems surface every day. 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Long-term potentiation and mathematical analysis of electrical models of dendritic spines Thiara, Rajpal Singh 1998
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Title | Long-term potentiation and mathematical analysis of electrical models of dendritic spines |
Creator |
Thiara, Rajpal Singh |
Date Issued | 1998 |
Description | Dendritic spines are small evaginations of the dendrites of neurons first discovered late in the 19th century. Since their discovery, many theories have been put forth to explain the physiological role of the spine. However, only recently with the advent of new laboratory technology has data been available to test the various theories put forward. The two most compelling theories today are that spines are important mediators of a form of cellular memory known as long-term potentiation (LTP), and that spines may be involved in the conduction of regenerative electrical impulses within dendrites similar to action potentials in axons. We will review some of the major mathematical models put forth which attempt to explain the role spines may play in the induction of LTP. We will address the importance of calcium signals in LTP induction and suggest how the unique morphology of the spine may allow for transient, spatially localized increases in calcium within the spine head, but not elsewhere in the dendrite. This could help account for the associativity, cooperativity, and input specificity requirements of LTP. We will also review some of the major mathematical models on dendritic action potentials. These models generally assume the existence of voltage-gated ion channels with Hodgkin-Huxley (HH) type dynamics exist in the spine head. We will employ a continuum approach in which spines are modeled as having a certain uniform density. We will further make use of the FitzHugh-Nagumo (FHN) equations without recovery to approximate the HH equations. We will examine the new set of equations in the traveling wave frame and seek to determine how the various parameters influence the speed, and the shape of the traveling wave front solutions. We show that there is a certain balance between local excitation of the spine heads, and freedom for the electrical current to pass from spine head to dendrite required for traveling front solutions to exist. Furthermore, strict parameter spaces in which traveling front solutions exist are determined, as is the profile of the wave in the special case of a standing wave. |
Extent | 6338108 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-05-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080026 |
URI | http://hdl.handle.net/2429/8280 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1998-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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