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A study of synchrony and phaselocking through excitatory/inhibitory coupling Dawes, Adriana Tiamae 1999

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A S T U D Y OF S Y N C H R O N Y A N D P H A S E L O C K I N G T H R O U G H E X C I T A T O R Y / INHIBITORY C O U P L I N G by A D R I A N A T I A M A E DAWES B.Sc.H. (Mathematics and Physics) University of Toronto, 1995 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 OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF S C I E N C E 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 Applied Mathematics We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A November 1999 © Adriana Tiamae Dawes, 1999 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 refer-ence 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 Date Abstract This paper is an investigation of the possible behaviours that can arise when oscillators are coupled by excitatory / inhibitory coupling. In the case of two oscillators, this means that the action of the first oscillator inhibits the second oscillator, while the action of the second oscillator activates the first oscillator. Here the first oscillator would be called Vi while the second oscillator would be denoted Ve. This investigation is to clarify the interaction of two oscillators found to operate in a pituitary cell. Within the pituitary cell there is a voltage oscillator that is determined by the flow of currents across the plasma membrane, and a calcium oscillator which measures the change in calcium concentration due to the release of calcium from internal stores. Experimental work has demonstrated that these oscillators are coupled and the voltage oscillator activates the calcium oscillator while the calcium oscillator inhibits the action of the voltage oscillator. Realistic models of pituitary cells are quite complicated and difficult to understand. In order to study this coupling in more detail we can make use of simplified models coupled with excitatory / inhibitory synaptic currents. For this investigation, we will be using the simple models given by the Connor model for type , I oscillators and the Morris-Lecar model for type II oscillators. The oscillators are coupled by synaptic currents which can be either excitatory or inhibitory. The synaptic currents have many parameters that influence how the oscillators interact with each other. By varying these parameters, we will gain insight into how oscillators coupled in this manner behave. This will be accomplished by using computer simulations and data analysis. In general we find that oscillators that interact through excitatory / inhibitory coupling can exhibit many different and interesting behaviours including harmonic and asynchronous phase-locking, suppression of one or both oscillators and drifting. Of particular interest is the case of drifting, where the oscillators fire at the same time only after a long period of time and the time between action potentials can vary in both oscillators. This drifting behaviour could help to explain why voltage and calcium oscillators operate on a time scale of milliseconds and seconds respectively, while hormone release occurs on a time scale of minutes to hours. ii Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures v i Acknowledgements ix Chapter 1. Introduction and Statement of the Prob lem 1 1.1 Type I and Type II Oscillators 10 1.2 Dynamics of Type I and Type II Oscillators 12 Chapter 2. Effect of Coupl ing Strength 20 2.1 Type I Oscillators , 21 2.2 Type II Oscillators 26 Chapter 3. Effect of the T i m e Scale of Synaptic Coupl ing 34 3.1 Type I Oscillators 36 3.2 Type II Oscillators .38 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupl ing 41 4.1 Type I Oscillators 44 4.2 Type II Oscillators 51 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents 57 5.1 Type I Oscillators 60 5.2 Type II Oscillators 62 Chapter 6. L i -R inze l M o d e l of a Pi tui tary Ce l l 66 6.1 Dynamics of the model 71 6.2 Interaction of Voltage and Calcium Oscillators 73 6.3 Conclusion and Discussion 74 Bibl iography 80 A p p e n d i x A . M o d e l Equations and Parameter Values 83 A . l Type I Connor Model 83 A.2 Type II Morris-Lecar Model 84 A.3 Li-Rinzel Model 85 iii Table of Contents Appendix B . Reduction of the Connor Model by Equivalent Potentials 88 B . l Step 1: Conversion of gating variables into equivalent potentials 89 B.2 Step 2: Determining single potential ip to replace membrane potential 90 B.3 Step 3: Combining remaining equivalent potentials '. 92 B.4 Step 4: Determining the weighting coefficients 93 B.5 Step 5: Consistency and the reduced model 94 iv L i s t o f T a b l e s 2.1 Behaviour of type I and type II oscillators for different coupling strengths 21 4.1 Behaviour of type I oscillator with slow synaptic coupling for different threshold voltages. 43 4.2 Behaviour of type I oscillator with fast synaptic coupling for different threshold voltages. 43 4.3 Behaviour of type II oscillator with slow synaptic coupling for different threshold voltages. 43 4.4 Behaviour of type II oscillator with fast synaptic coupling for different threshold voltages. 44 5.1 Behaviour of type I oscillator with slow and fast synaptic coupling for different voltage sensitivities 58 5.2 Behaviour of type II oscillator with slow and fast synaptic coupling for different voltage sensitivities 59 6.1 Parameter conditions for type I model that give the corresponding behaviour. Unless indicated, parameter values are as given in table A . l 76 6.2 Parameter conditions for type II model that give the corresponding behaviour. Unless indicated, parameter values are as given in table A.2 77 A . l Type I Connor model parameter values 84 A.2 Type II Morris-Lecar model parameter values 85 A.3 Li-Rinzel model parameter values 87 v List of Figures 1.1 Location of pituitary gland and hypothalamus in human brain ([13]) 2 1.2 Representative diagram of currents across plasma membrane and endoplasmic reticulum. 3 1.3 Uncoupled type II oscillator, Morris-Lecar model. Parameter values as given in appendix A with applied current 7 = 80 pA. There is a regular oscillation with period T = 46.9 msec 7 1.4 Uncoupled type I oscillator, Connor model. Parameter values as given in appendix A with applied current I — 10 pA. There is a regular oscillation with period T = 40.9 msec. 8 1.5 An example of how to calculate the adjoint function, Z(t) 10 1.6 Single Spike and Phase Response Curve of type I oscillator, Connor model 11 1.7 Single Spike and Phase Response Curve of type II oscillator, Morris-Lecar model. . 11 1.8 Stable periodic orbit of reduced type I Connor model 14 1.9 Bifurcation diagram of reduced type I Connor model 15 1.10 Frequency diagram of reduced type I Connor model 16 1.11 Nullclines and stable periodic orbit for type II oscillator 17 1.12 Bifurcation diagram of type II Morris-Lecar model 18 1.13 Frequency diagram of type II Morris-Lecar model 19 2.1 Phase locking for type II oscillator 22 2.2 Phase locking for type I oscillator with instantaneous coupling 22 2.3 Drift of type I oscillator with slow synaptic current and weak coupling strength (gsyn=0.1). 23 2.4 Harmonic locking for type I oscillator with slow coupling and stronger (gsyn > 0.5) coupling strength 24 2.5 Excitatory and Inhibitory currents for type I oscillator with instantaneous coupling and weak (gsyn = 0.1) coupling strength 25 2.6 Excitatory and Inhibitory currents for type I oscillator with slow synaptic coupling and weak (gsyn = 0.1) coupling strength 26 2.7 Excitatory and Inhibitory currents for type I oscillator with slow coupling and strong (dsyn — 1) coupling strength 27 2.8 Cycling of the period of the oscillators in the Connor model for gsyn = 0.1 28 2.9 Cycling of the period of the oscillators in the Connor model for gsyn = 0.5 29 2.10 Cycling of the period of the oscillators in the Connor model for gsyn = 1 30 2.11 Phase space diagram of reduced model for gsyn = 0.05 (top figures) and gsyn = 0.7 (bottom figures) 31 2.12 Relative voltage difference between peaks in coupled type II oscillator 32 2.13 Relative time difference between peaks in coupled type II oscillator 32 2.14 A representative plot of coupled type II oscillators 33 3.1 Synaptic current of type II oscillator with a=l 35 3.2 Synaptic current of type II oscillator with a=100 36 3.3 Synaptic current of type II oscillator with instantaneous coupling 37 vi List of Figures 3.4 Comparison of phaselocked voltage oscillations of type I model for a = 1 and a = 10. 37 3.5 Coupling currents of type I oscillator, gsyn = 0.1 and a = 10 38 3.6 Expanded view of synaptic coupling currents of type I oscillator, gsyn — 0.1 and a = 10. 39 3.7 Relative voltage difference between peaks in coupled type II oscillator 40 3.8 Relative time difference between peaks in coupled type II oscillator 40 4.1 Effect of different values of Vt on S QO 42 4.2 Suppression of oscillator Ve in type I Connor model with strong coupling (gsyn=l), slow synaptic current and low threshold (Vt=-70 mV) 45 4.3 Harmonic phaselocking in type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and low threshold Vt=-40 mV 46 4.4 Periods of type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and high threshold 14=50 mV 47 4.5 Type I Connor model with strong coupling (gsyn >0.8), fast synaptic current and low threshold Vt=-30 mV 48 4.6 Type I Connor model with strong coupling (gsyn >0.8), fast synaptic current and low threshold Vt=-50 mV 49 4.7 Period of Type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and low threshold 14=-30 mV 50 4.8 Cycling of the period of type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and low threshold Vj=-70 mV 53 4.9 Drifting of type I Connor model with strong coupling {gsyn >0.8), slow synaptic current and high threshold 14=40 mV 54 4.10 Steady state of type II Morris-Lecar model with strong coupling (gsyn=l), slow synaptic current and low threshold Vj=-40 54 4.11 Subthreshold oscillations of Vi synchronized with the firing of oscillator Ve of type II Morris-Lecar model with strong coupling (gsyn=l), slow synaptic current and low thresh-old V(=-40 mV 55 4.12 Harmonic phaselocking of type II Morris-Lecar model with medium coupling (gSyn=0-5), fast synaptic current and low threshold Vj=-20 mV 55 4.13 Drifting of type II Morris-Lecar model with weak coupling (gsyn=0.1), fast synaptic cur-rent and low threshold Vj=-30 mV 56 5.1 Effect of different values of Vs on S QQ 57 5.2 Type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and voltage sensitivity V s=0.1 mV 62 5.3 Periods of drifting type I Connor model with medium coupling (gsyn=QA to 0.8), slow synaptic current and steepness 1^=0.1, 1, 2 and 4 mV 63 5.4 Periods of drifting type I Connor model with medium (gS2yn=0.5) and weak {gsyn—QA) coupling, fast synaptic current and steepness V s=10 mV 64 5.5 Type II Morris-Lecar model with medium coupling {gsyn=0-5), slow synaptic current and steepness V s =4 mV 65 vii List of Figures 6.1 Representative diagram of currents across plasma membrane (PM) and endoplasmic retic-ulum (ER) which affect calcium oscillation 69 6.2 Voltage oscillations of pituitary model when not coupled to the calcium oscillator. . 70 6.3 Calcium oscillations of pituitary model when not coupled to the voltage oscillator. . 70 6.4 Bifurcation diagram of pituitary model using calcium concentration as bifurcation pa-rameter 72 6.5 Coupled type I oscillators with parameter values corresponding to Li-Rinzel model. . 75 6.6 Coupled type II oscillators with parameter values corresponding to Li-Rinzel model. 78 6.7 Coupled voltage and calcium oscillators of the Li-Rinzel model 79 B . l Full Connor model of V and gating variables m, h, n, A, and B 96 B.2 Full Connor model in terms of equivalent potentials of V and gating variables vm, Vh, vn, VA, and VB 97 B.3 Comparison of the full and reduced Connor models 98 B.4 Comparison of the single spike and phase response curves for full and reduced Connor models 98 B.5 Comparison of the bifurcation diagrams for full and reduced Connor models 99 B.6 Magnification of the bifurcation diagram of full Connor model 100 B.7 Magnification of the bifurcation diagram of reduced Connor model 100 viii A c k n o w l e d g e m e n t s There are many people that made this work possible. Thanks to my supervisor Professor Yue-Xian L i for his encouragement and support. He has been an invaluable source of information and I appreciate the many years of experience he brings to my work. It has been a pleasure to work with him and I look forward to future collaborations. I am also indebted to Professor Leah Keshet and Professor Robert Miura for their advice and giving so generously of their time. Thanks to the University of British Columbia department of mathematics and the Institute of Applied Mathematics for giving me the opportunity to pursue my graduate studies. I owe a debt of gratitude to my parents Karen and Andrew Dawes for their inspiration and continued patience. Leo Comitale has been essential to this project and I thank him for his incomparable technical support. My thanks to everyone who has helped me in the course of my studies but who are too numerous to mention here. ix Chapter 1 Introduction and Statement of the Problem The pituitary gland is located at the base of the skull, just behind the nose in an area called the sella turcica, or the Turkish saddle, as shown in figure 1.1. The pituitary is considered the master gland of the endocrine system, and a variety of hormones are secreted by cells in the pituitary. There are many chemicals, called hormones, produced in the body. In general, these are defined as biologically active products which travel some distance from the site of secretion to the target tissue. Here we are interested in a particular cell called the gonadotroph, which is located in the anterior pituitary. Gonadotrophs are identified by their secretion of luteinizing hormone (LH) and follicular stimulating hormone (FSH). Once these hormones are released into the blood stream, they travel to the gonads (testes in men, ovaries in women) where they produce a specific response, which can also depend on the concentration of other hormones present in the blood stream. One important feature of the endocrine system is that a pulsatile hormone signal from the hypothalamus (figure 1.1) is required in order for the reproductive sys-tem to function. Exactly how this pulsatile signal is generated has not been determined. The role of the pituitary gland in regulating hormone release is very important and well understood. What is not clear is how individual gonadotrophs communicate in the pituitary to synchronize hormone release. This thesis is an investigation of synchrony among gonadotroph cells in the pituitary gland. In gonadotrophs, there are two oscillators, the cell (or plasma) membrane oscillator which is determined by its voltage, and the endoplasmic reticulum oscillator, determined by the flow 1 Chapter 1. Introduction and Statement of the Problem m\.-~ Hypo tha lamus — Pitui tary g l and Figure 1.1: Location of pituitary gland and hypothalamus in human brain ([13]). of calcium into the cell's interior, or cytosol. The endoplasmic reticulum is an intracellular compartment that stores calcium. An oscillator is something that behaves in a periodic man-ner. That is, some aspect of the oscillator changes with respect to time but after a finite time, repeats its previous behaviour. My particular interest is in studying, first, how the plasma membrane voltage oscillator and the endoplasmic reticulum oscillator in gonadotrophs interact with each other, and second, possible mechanisms of hormone release. It has been known for over a decade that pituitary cells are capable of producing action potentials. However, until a French research group was able to produce a slice preparation of the pituitary in 1997, it was believed that individual cells could not communicate with each other ([15]). The manner in which the cells are coupled has yet to be determined. Many excitable cells, for instance neurons, are known to exhibit oscillating voltage across their plasma membrane. Such signals are created by the activation and inactivation of ionic currents and lead to important neural signals called action potentials. Pituitary gonadotrophs have a 2 Chapter 1. Introduction and Statement of the Problem Figure 1.2: Representative diagram of currents across plasma membrane and endoplasmic retic-ulum. voltage oscillator, as well as a calcium oscillator, where calcium is stored in the endoplasmic reticulum and released into the cytosol in an oscillatory manner (figure 1.2). By interacting through the calcium concentration in the cytosol, the plasma membrane oscillator excites the intracellular calcium oscillator while the intracellular calcium oscillator inhibits the plasma membrane oscillator. In numerical simulations ([4]), the two nonidentical oscillators synchro-nize but the conditions under which they do so is not fully understood. The standard integrate and fire model ([4]) does not allow for synchrony under certain conditions so there must be a different type of interaction taking place. Coupled oscillators have been studied in great detail usually using strictly excitatory types of coupling. Biological oscillators are ubiquitous, for example in circadian rhythms, heart beats, ovulation cycles, action potentials, and so forth. Understanding how oscillators interact with each other is crucial for understanding basic physiological processes such as these. This type of coupling, excitatory / inhibitory, has not been considered before. Networks of excitatory and 3 Chapter 1. Introduction and Statement of the Problem inhibitory oscillators have been studied, but it was always assumed that each oscillator receives the same excitatory input as in [19] and [5] or, as in the case of [11] and [2], the coupling is symmetric, meaning the oscillators influence each other in the same way and the coupling is either strictly excitatory or strictly inhibitory. Often integrate-and-fire models are considered, where oscillations occur only when there is input from an outside source ([12]). This type of coupling is used in delay differential equations as in [1], [26] and [27], but the time scale is much larger than the one considered in this investigation. There are many models that describe the action of biological oscillators. Here, we are interested in voltage oscillators which describe excitable cells such as neurons. The classic Hodgkin-Huxley model for voltage-dependent membrane currents, [18], took data from experiments performed on giant squid axons and developed a mathematical model to accurately mimic the behaviour of the ionic currents under experimental conditions. This is a very powerful tool, as it is pos-sible to simulate the action of a neuron and to investigate properties that may not be directly observable. Both the Connor model and the Morris-Lecar model are based on the foundation built by Hodgkin and Huxley. In both models, the Vk term stands for membrane potential, where k = i denotes an inhibitory oscillator and k = e denotes an excitatory oscillator (when discussing these oscillators, Vjt is often used to denote the oscillator in question and not just the value of the membrane potential). The quantities Vion represent the reversal potentials of the given ions, and determine the direction of flow of the ionic current. We also have gi0n which represents the conductivity of the membrane to a particular ion. The functions m, n and h for example, are often called gating variables and represent the opening and closing states of membrane ion channels. The Connor model, [3], was developed to model the behaviour of the repetitive walking leg axons of crustaceans. The equations of the Connor model ([3] and [16]) are: 4 Chapter 1. Introduction and Statement of the Problem I - 9Nahi{Vi - VNa)mf - gK(Vi - VK)n\ - 9l(Vi - Vt) - gA(Vi - VAC)BiA\ -gSynS(Ve)(Vi -VNa) I - 9Nahe(Ve - VNa)m\ - gK(Ve - VK)n\ - gi{Ve - Vi) - gA(Ve - VAC)BeA\ -9synS{Vi)(Ve - VK). The only difference between the excitatory and inhibitory oscillator is in the synaptic coupling current term, given here as gSyns(Ve)(Vi — Vjva) for the inhibitory oscillator and gSyns(Vi)(Ve — VK) for the excitatory oscillator. The importance of the synaptic coupling current will be dis-cussed shortly. The equations for the model describe the change in voltage, V, of the membrane over time. Hodgkin and Huxley, and later, Connor et al, found several ionic currents that were predom-inantly responsible for the change in membrane voltage. The first term in the equations, 7, refers to the applied current which can be strictly controlled under experimental conditions. The second term, gNa,he{Ve — V/va)mg describes the flow of sodium currents across the plasma mem-brane. The permeability of the channels is given by gNa, while the activation and inactivation of the channels are determined by the functions m and h, respectively. The reversal potential VVa describes the direction of flow of the ionic current. The third term, gx(Vi — Vfc)nf, represents the action of the potassium current. As with the sodium current, the channel permeability is represented by gx- Here the channel has an activation function represented by n and the reversal potential is given as VK- The fourth term represents the leak current, gi(Vi — Vj). Here we only have permeability gi and a reversal potential, as the leak current is a passive process and so has no channels to open or close. The final regular term, gA(Vi — VAc)BiAf is called the A-current. It represents an inward current which is quickly activated, as represented by the activation function A, and slowly deactivated, as represented by the inactivation function B with permeability gA. dV ' dt ^dVe ' dt 5 Chapter 1. Introduction and Statement of the Problem The equations of the Morris-Lecar model ([21] and [9]) are: I + 9i(Vi - Vi) + gKw(VK - Vi) + gcam^Vca - Vi) - gsyns(Ve){Vi - VCa) I + 9i(Vt - Ve) + gKw(VK - Ve) + gcam-ooiVca - Ve) - gsynsiV)^ - VK) \{V){Woo{V)-w). As these equations are also based on the Hodgkin-Huxley model, they have many features which are identical to Connor model. These are applied current / , the leak current also given here a s 9i{Vi — Vi), and the potassium current gnw{VK — Vi). The only difference is that we have a calcium current instead of a sodium current. The calcium current given as gca^ooiVca — Vi) has permeability gca and a reversal potential Vca- However, it only has a channel activation term given by the function moo. This model was developed to conform to experimental observations of barnacle muscle fibers. dV 1 dt dVe 1 dt dw ~dJ Synaptic coupling is either governed by an ordinary differential equation that describes the gating of synaptic chemicals as follows: - = ak(V)(l - s) - Ps, (1.1) or, in the case of instantaneous synaptic transmission: The parameters a and B determine the time scaling of the ordinary differential equation. In other words, they control how quickly the value of s changes. The function k(V) represents the channel activation and is given by: 6 Chapter 1. Introduction and Statement of the Problem * { V ) i + e-(V-Vt)/vs-To analyze these models, I used the program XPPAUT written by Bard Ermentrout, [7]. The full models including all functions and associated parameter values can be found in appendix A. Whether a synapse is excitatory or inhibitory is determined by the reversal potential of the synaptic current. We call the oscillator with subscript i inhibitory since it exerts inhibitory action on the other oscillator. We call the oscillator with subscript e excitatory since it exerts an excitatory action on the other oscillator. In these models, oscillator Vi is inhibitory due to the inward sodium current in the type I model and the inward calcium current in the type II model. Oscillator Ve is excitatory due to the outward potassium current in the synaptic coupling term. Figure 1.3 shows the Morris-Lecar oscillator when there is no coupling present. The uncoupled Connor oscillator is shown in figure 1.4. 40 _40 I . 1 . 1 . i . 1 0 50 100 150 200 Time, msec Figure 1.3: Uncoupled type II oscillator, Morris-Lecar model. Parameter values as given in appendix A with applied current I = 80 pA. There is a regular oscillation with period T = 46.9 msec. The parameter values for both the Connor and Morris-Lecar models are determined experi-mentally. Since there is always some error involved with experimental measurements, these 7 Chapter 1. Introduction and Statement of the Problem > E of cn ro "o > " 0 50 100 150 200 Time, msec Figure 1.4: Uncoupled type I oscillator, Connor model. Parameter values as given in appendix A with applied current / = 10 pA. There is a regular oscillation with period T = 40.9 msec. parameter values are also subject to error. In addition, parameter values may be modified so that the model more accurately mimics experimental results. The synaptic current equations are also based on experimental work. However, to assess the effect of various parameter values, the parameters will be varied over a wide range. We are not so much interested in experimental accuracy, but rather in the general behaviour of the coupled oscillators, and how this behaviour changes as parameters are varied. To study this coupling in a simplified manner, I will be coupling oscillators given by the type I oscillator of the Connor model and the type II oscillator given by the Morris-Lecar equations. The oscillators of each model are coupled in such a manner that one is excitatory, denoted by oscillator Ve, the other inhibitory, denoted by Vj. I will be exploring two different types of synaptic couplings; slow and instantaneous. The synaptic current is the only place where the oscillators are coupled. When there is no synaptic current, the oscillators are uncoupled and no longer influencing each other. 8 Chapter 1. Introduction and Statement of the Problem With this investigation, we hope to gain a better understanding of how this type of coupling can affect the behaviour of oscillators. These oscillators are special, in that they oscillate without any input from other oscillators. In this way, we can see what effect this coupling has, not only on whether it can suppress the behaviour of an oscillator, but also more subtle effects such as changing the intrinsic period of the oscillator. In the following chapters we will investigate vari-ous properties of the synaptic current, and how that affects the action of the coupled oscillators. We begin by considering coupling strength which determines how large the synaptic current is for each oscillator. The smaller the coupling strength, the weaker the synaptic current. Thus it has less effect on the action of the oscillator. Next, we consider the effect of time scaling, which determines how quickly an oscillator can react to a changing synaptic current. Then we consider threshold voltage and voltage sensitivity, both of which help determine the voltage where the oscillator becomes sensitive to the action of the other oscillator. By varying these quantities, we can produce a range of behaviour in both the type I and type II oscillators. It is possible to produce six distinct behaviours under certain parameter conditions. Phase-locking is the most common behaviour when two oscillators fire at the same time relative to each other. They can fire at exactly the same time or there may be a time difference, called a phase lag, between the firing of the oscillators. The case that the time difference is the same at each cycle is called phaselocking. Harmonic phaselocking is the case in which one oscillator fires more than once before the other oscillator fires, but where the phase difference is the same at each cycle. Drifting is when there is no correlation between the firing of the oscillators. Interestingly, it is possible to have the interspike interval of the oscillators vary considerably during drifting. Suppression of oscillator Vi, Ve, or both occurs only for type II oscillators. 9 Chapter 1. Introduction and Statement of the Problem 1.1 Type I and Type II Oscillators In this paper I am looking at two different types of oscillators given by the Morris-Lecar and the Connor models. A phase response curve is a graphical representation of the response of an oscillator to a small perturbation in its phase at all points along the oscillator's cycle. A small 0.5 0.3 > 0.1 <B CO I -0.1 -0.3 0 10t 20 30 40, Time, msec Figure 1.5: An example of how to calculate the adjoint function, Z(t). perturbation corresponding to a small excitatory post-synaptic potential is applied at time t. This causes a shift in the phase of the oscillator. I show an example of this in figure 1.5, where it causes a small phase delay. Either the oscillator fires sooner, later, or at the same time as it would have without a perturbation. Whether the phase is advanced, delayed, or unchanged can depend on where the oscillator is in its cycle. If the oscillator's phase is advanced, delayed, or unchanged, the phase response curve will have a positive value, negative value, or zero value, respectively. The graph produced by this method is called a phase response curve, or adjoint function. A type I oscillator is defined as an oscillator that has a non negative phase response curve. This means that when the oscillator receives a weakly excitatory input, the phase of the oscillator is advanced, regardless of where it is in its cycle. An example of the phase response curve and its 10 Chapter 1. Introduction and Statement of the Problem corresponding single spike of a type I oscillator, the Connor model, is shown in figure 1.6. Time, msec o j 10 is 20 25 30 u Figure 1.6: Single Spike and Phase Response Curve of type I oscillator, Connor model. A type II oscillator is one which has positive, negative and zero components to its phase response curve. In this case, when the oscillator receives a slight phase advance, the reaction of the oscillator depends on where it is in its cycle. In particular, the oscillator may actually fire later than it would have without a perturbation, despite the fact that it received an excitatory input. This corresponds to the negative values of the phase response curve. A single spike and the phase response curve for the type II oscillator, the Morris-Lecar model, is shown in figure 1.7. Virtually all physiological current-based models are type II oscillators. Time, msec o j io 15 20 25 30 35 40 « Figure 1.7: Single Spike and Phase Response Curve of type II oscillator, Morris-Lecar model. 11 Chapter 1. Introduction and Statement of the Problem 1.2 Dynamics of Type I and Type II Oscillators Before we try to understand how the synaptic current influences the interaction between oscil-lators, we must first understand why these oscillators behave this way. This can be done by looking at various properties of the oscillators shown in bifurcation and phase plane diagrams. We already know why they are classified as type I and type II, but this does not explain why the oscillators oscillate or how these oscillations arise in the first place. We will investigate this using the Connor and Morris-Lecar models at the parameter values that will be used in the rest of this paper. The type I or Connor model is a complex model consisting of six differential equations. In order to investigate its dynamical properties, we must first reduce the model to a more manageable form. This is done using the method of reduction by equivalent potentials as proposed in [16]. The details of the calculations are given in appendix B. Using this method, we are able to restate the model in the following form: dt ~dt dvB (*2fh(<j>,1p) +Ol3fn{<l>,tp) +a^fA{<f>,1p), dt 12 Chapter 1. Introduction and Statement of the Problem Where: = 9NaH<t>?K^){<t>-VNa)+9K^)\(l>-VK) +gACA(i,)3B(vB)(cf>-VAc) + gi(<P-Vl), MM) = ^-(H^-hW), U ^ ) = ^ T T T T W ) - ^ ) ) , n'{ip) fA(M) = ^ ( i (^) - i (^) ) , fB{<t>A) = ^ - ( B ^ ) - B{vB)). B'{vB) Here the gating variables which were represented in this model by the functions m, h, n, A and B were restated in terms of a voltage or potential term. These equivalent potentials were then grouped according to whether they activated or inactivated channels and the time scale on which they acted. In this way, it is possible to restate the original model with fewer dimensions. However this does not simplify the model. The associated functions for the equivalent poten-tials given by F, fh, fn, fA and fB are more complicated than the original model equations. The weighting coefficients given by a, determine how much each equivalent potential influences the overall action of the oscillator. The motion of the oscillator can now be viewed in its phase space consisting of (f>, ip and VB as the three dimensions. The trajectory of the oscillator in this phase space is shown in figure 1.8. We can also determine numerically where the stable and unstable steady states are. The steady states, both stable and unstable, are determined by setting the model equations to zero and finding where they intersect. That is, we wish to know for which values of 4>, ip and vB do 13 Chapter 1. Introduction and Statement of the Problem we have: dxb ~dt <hp_ ~dt dvB dt 0 0 0. The surfaces created by these equations are called nullsurfaces. For this particular set of pa-rameters, there is only one steady state. It is unstable and occurs at the intersection of the nullsurfaces when <p — ip — VB = —36.3 mV. The implication of having only one stable state which is unstable means that for virtually all initial conditions, the oscillator will eventually fall into the stable periodic orbit (fig 1.8). Figure 1.8: Stable periodic orbit of reduced type I Connor model. Now that we have seen where the unstable steady state is and how the stable periodic orbit we are interested in travels with respect to cp, ip and VB, we can look at how oscillations arise in this model. This is done by looking at the bifurcation diagram showing how and when oscillations 14 Chapter 1. Introduction and Statement of the Problem •80 1-1 1 ' 1 — ' 1 1 0 50 100 150 200 250 300 Iapp Figure 1.9: Bifurcation diagram of reduced type I Connor model. arise according to a bifurcation parameter. Here, the bifurcation parameter for both the type I and type II model is the applied current, Iapp. In figure 1.9 we see the bifurcation diagram. The steady states are represented by lines, solid for stable and dashed for unstable. The peri-odic orbits are represented by circles, filled circles are stable periodic orbits while hollow circles are unstable periodic orbits. The upper branch of circles shows the upper voltage limit of the oscillations, and the lower branch shows the lower limit of the voltage oscillations. We can see the oscillations arise at the open end of the periodic branch at I = 12.9 pA. At the right most part of the branch, the oscillations terminate at a supercritical Hopf bifurcation. The bifurcation is called supercritical because the stable oscillations arise in the same direction as the steady state loses stability. The Hopf bifurcation occurs at I = 244.4 pA for the reduced model shown here. The oscillations that emerge at I = 12.9 pA and are unstable can not oscillate with an arbitrarily low frequency. The stable oscillations also emerge with a nonzero frequency. Figure 1.10 shows how the frequency varies as I is increased. It is clear from both figure 1.9 and 1.10 that as the applied current is increased, the frequency of oscillations increases 15 Chapter 1. Introduction and Statement of the Problem ( ) I 1 1 1 1 L_ 0 50 101) 150 200 250 Iapp Figure 1.10: Frequency diagram of reduced type I Connor model. while the amplitude of the oscillations decreases. The dynamics of the type II oscillator, or the Morris-Lecar model, are a little easier to analyze since the model is already in two dimensions. This analysis is based on the work of G. B. Ermentrout and J. Rinzel in [8]. Since we are viewing them on a plane, the nullclines and trajectories are much clearer. Using the parameter values as given in appendix A, the model exhibits repetitive firing. Nullclines are the first step in understanding the dynamics of this oscillator. These curves show where % = 0 and % = 0. These curves intersect at the steady states of the model. The steady states can be either stable or unstable. The nullclines and the stable periodic orbit are shown in figure 1.11. The limit cycle, or stable periodic orbit, is the trajectory of the oscillator when shown plotted in the phase plane of V vs. w. As seen on the figure, these nullclines intersect at only one point. The steady state represented there is unstable and its coordinates are given by V — 7.36 mV and w = 0.37. If we were to start the model exactly at that point, we would see a steady state, but if we were to start at any other point, no matter how close to the unsta-16 Chapter 1. Introduction and Statement of the Problem 0.5 V-nullcline w-nullcline 0.0 -50.0 -30.0 -10.0 Voltage, mV 10.0 30.0 Figure 1.11: Nullclines and stable periodic orbit for type II oscillator. ble steady state, the trajectory would spiral outwards until it settled on the stable periodic orbit. Next we consider the bifurcation diagram of this model. This shows the global dynamics of the system as they evolve according to the bifurcation parameter which is, in this case, the applied current Iapp. The bifurcation parameter determines whether the model oscillates or sits at a steady state. While other parameters can affect the stability of the model, in this case we are choosing the applied current as bifurcation parameter since this is strictly controlled under experimental conditions. The bifurcation diagram for the Morris-Lecar model is shown in figure 1.12 and the steady states are shown as a line. The stable steady states are the solid line, while the unstable steady states are the dotted line. Oscillations are shown as dots. Filled circles are stable oscillations, while the hollow circles are unstable oscillations. Again, these branches of circles show us the upper and lower voltages for the oscillatory state. Here the oscillations also begin at the open end of the branches of periodics. These oscillations arise at I = 39.96 pA and terminate at I = 97.79 pA. The branches terminate at a Hopf 17 Chapter 1. Introduction and Statement of the Problem Figure 1.12: Bifurcation diagram of type II Morris-Lecar model. bifurcation, although here we have a subcritical Hopf bifurcation unlike the type I oscillator, which terminated in a supercritical Hopf bifurcation. At this point the bifurcation is subcritical because a stable periodic orbit arises before the steady state loses stability. When we look at the frequency of the oscillations as / is increased (figure 1.13), we see another difference from the type I oscillator. In the type II model the oscillations of the steady state can emerge with zero frequency. This means that if we were to set I = 39.96 pA and run the simulation, we would have a limit cycle with an infinite period. This is known as a saddle node on an invariant circle. We can also see from figures 1.12 and 1.13 for the stable oscillations, the amplitude of the oscillations decreases while the frequency increases for increasing I. 18 Chapter 1. Introduction and Statement of the Problem Figure 1.13: Frequency diagram of type II Morris-Lecar model. 19 Chapter 2 E f f e c t o f C o u p l i n g S t r e n g t h I first wanted to see what effect the coupling strength would have on the dynamics of the cou-pled oscillators. By varying-the coupling strength gradually, it is possible to change the firing pattern of the oscillators. The different behaviours for type I and type II oscillators are summa-rized in table 2.1. For type II oscillators, where we always have phaselocking regardless of the coupling strength, the relationship between the height difference and time difference between spikes is summarized in figures 2.12 and 2.13. The coupling strength, denoted by gsyn in the voltage oscillator equations, determines to what extent one oscillator is affected by the other. In order to consider the synaptic coupling strength of Vi and Ve separately, we can write the voltage equations with additional subscripts as follows: where Ri = V^a for type I, Ri = VQQ, for type II and Re = VK for type I and type II oscillators. When the coupling strength is small, there is little interaction between the oscillators, but when gsyn is large, the firing of one oscillator has a significant impact on how and when the other oscillator fires. By doing numerous simulations for different parameter values, it can be seen that phaselocking occurs for the type II oscillator for all coupling strengths whether the coupling is fast or slow. 20 Chapter 2. Effect of Coupling Strength gsyn=0.1 - 0.5 9syn >0.5 Type I, fast current phaselocking phaselocking Type I, slow current drifting harmonic phaselocking Type II, fast current phaselocking phaselocking Type II, slow current phaselocking phaselocking Table 2.1: Behaviour of type I and type II oscillators for different coupling strengths. Phaselocking is when the two oscillators fire at the same time relative to each other. This means they can fire at exactly the same time, or there may be some time difference between the firings of each oscillator, but that time difference will be the same at each cycle. This is demonstrated in figure 2.1. The situation for the type I oscillator is somewhat different. For instantaneous coupling, phase locking occurs for all values of synaptic coupling, as shown in figure 2.2 for example. However, when the coupling is slow, we no longer have phase locking. Instead, we see two different types of behaviour depending on the coupling strength. For weak coupling, gsyn = 0.1 — 0.5, there is only drifting, and no correlation between the firings of the two oscillators. This behaviour is shown in figure 2.3. As we increase the coupling so that 9syn > 0.5 we begin to see harmonic phase locking as shown in figure 2.4. 2.1 Type I Oscillators As mentioned above, type I oscillators exhibit different behaviour depending on whether the coupling is fast or slow. Even within those coupling types, there is some difference according to whether the coupling is strong or weak, as we can see in figures 2.2, 2.3 and 2.4. In this section, except where indicated, the time scaling is kept constant at a = 1. The effect of changing the time scale will be discussed in a later section. 21 Chapter 2. Effect of Coupling Strength 5 0 . 0 30.0 10.0 -10 .0 -30.0 - 5 0 . 0 0.0 5 0 . 0 1 0 0 . 0 1 5 0 . 0 Figure 2.1: Phase locking for type II oscillator. 200.0 75.0 -25.0 -75.0 50.0 100.0 Time, msec 150.0 200.0 Figure 2.2: Phase locking for type I oscillator with instantaneous coupling. For the case of instantaneous coupling, the reason for phase locking is quite clear when we look at the synaptic currents. These are shown in figure 2.5. Clearly, the current corresponding to the excitatory coupling is much larger than the current corresponding to inhibitory coupling. This means that Vi is receiving a very large input from oscillator Ve while Ve is feeling very little influence from Vi. In other words, oscillator Ve is forcing Vi to spike when it fires. This can be seen in figure 2.2 where Ve entrains Vi- Regardless of the value of gsyn, oscillator Ve dominates oscillator Vi. This could explain why phaselocking occurs for these values of gsyn. 22 Chapter 2. Effect of Coupling Strength 7 5 . 0 2 5 . 0 - 2 5 . 0 ' " ' " O . O 1 0 0 . 0 2 0 0 . 0 3 0 0 . 0 4 0 0 . 0 T i m e , m s e c Figure 2.3: Drift of type I oscillator with slow synaptic current and weak coupling strength (gSyn=0-l)-When the oscillators are coupled using slow synaptic coupling, we can observe two different behaviours. For small values of gsyn, we find that the oscillators drift with respect to each other as in figure 2.3. As we increase the synaptic coupling strength, the drifting gives way to harmonic phase locking, as shown in figure 2.4. This can also be explained by looking at the excitatory and inhibitory currents. When the synaptic coupling strength gsyn is small, Iinh and Iexc can differ greatly, as in figure 2.6, and so the oscillators would influence each other in an inconsistent manner. This explains why we see drifting. As the coupling strength increases, Enh and Iexc gradually change until they exhibit regular periodic patterns. This can be seen in figure 2.7. Interestingly, when the type I oscillator is coupled with weak, slow synaptic coupling, the time between spikes in each oscillator varies. This variation in the timing between the action potentials of each oscillator becomes more pronounced, ranging over a much larger set of values. This can be seen by comparing figure 2.8 and figure 2.9 until phase-locking occurs, for the larger values of synaptic coupling. For weak coupling, shown in figure 2.8, the time between spikes follows a cycle that repeats roughly every 8-9 action potentials for oscillator Vi, and every 7-8 action potentials for oscillator Ve. When the simulations are run continuously for long periods 23 Chapter 2. Effect of Coupling Strength 7 5 . 0 2 5 . 0 > e CD cn m "o - 2 5 . 0 0 . 0 5 0 . 0 1 0 0 . 0 1 5 0 . 0 2 0 0 . 0 T i m e , m s e c Figure 2.4: Harmonic locking for type I oscillator with slow coupling and stronger (gsyn > 0.5) coupling strength. of time, the cycle returns to the point where the oscillators fire at roughly the same time. When this happens, the period starts cycling through various values until the oscillators fire together again and the cycle restarts. As the coupling strength increases, the oscillators fire simultaneously with more frequency, despite drifting. This is seen in figure 2.9 where the period of each oscillator is still cycling through various values, but the cycle resets more quickly. When the coupling strength is sufficiently high, harmonic phase locking occurs. This can also be seen in the plot of the periods shown in figure 2.10. Another way to explore this model is by reducing the equations to a lower dimensional system without destroying the fundamental dynamics. Following the method of Kepler, Abbott, and Marder as in [20], the uncoupled system can be reduced from a seven dimensional system to a three dimensional one. The detailed calculations can be found in appendix B. The new system 24 Chapter 2. Effect of Coupling Strength 5.0 o.o 100.0 ne, msec 150.0 200.0 Figure 2.5: Excitatory and Inhibitory currents for type I oscillator with instantaneous coupling and weak (gsyn = 0.1) coupling strength. in terms of equivalent potentials is represented by the following equations: dcf) dt ctolapp - ct0[gNarh{<f>) h(i>)((f) - VNa) + ^n(V') (</> - VK) +gAcAW3B(vB)(cf> - VAC) + 9I{4> ~ Vi)], ~dt = a 2 dip  dvB dt kh{4>) kB(</>) [h(4>)-hm + a3 K{4>) n'{tp) (h(<f>) -h(ip)) + CC4 kA(<f>) (Atf) - A(tl>)) B'(vB) (B(4>) - B(vB)) The difference in the periods is now more obvious when we view the phase space of the oscilla-tors, shown in figure 2.11. The transition between drifting and harmonic phase locking occurs for a lower value of gsyri in the reduced model than in the full model. In the full model, the transition occurs at roughly gsyn = 0.5, while for the reduced model, the transition occurs at roughly g s y n = 0.1. For weak coupling strength, the path of oscillator Vi assumes many values within a certain range. This corresponds to the period of the oscillator cycling through many values before the cycle resets. For greater coupling strengths, shown in the bottom plots, it can be seen that oscillator Vi follows two well defined paths which corresponds to the harmonic phase locking. The trajectory of oscillator Ve is only slightly affected by changes in the coupling 25 Chapter 2. Effect of Coupling Strength 2 -ex. "cr OJ Z3 o - 3 -- 8 I • 1 • 1 • ' • 1 0 50 100 150 200 Time, msec Figure 2.6: Excitatory and Inhibitory currents for type I oscillator with slow synaptic coupling and weak (gsyn = 0.1) coupling strength. strength. When the coupling strength is varied so that one is higher than the other, there is very little difference from when both synaptic coupling strengths are varied at the same time. When the synaptic current is slow, the coupling is always harmonic with Vi firing twice for every single spike of V e for almost all combinations of gsyni and gsyne- When the case of a fast synaptic current is considered, we see phaselocking for every combination of gsyni and gsyne-2.2 Type II Oscillators Type II oscillators have consistent behaviour for all values of gsyn- Even for very weak synaptic coupling, the oscillators phaselock. The difference in the height of the peaks and the relative time between the peaks differs according to both the synaptic coupling strength as well as the time scaling a. I have summarized the results of many simulations in figures 2.12 and 2.13. The manner in which the difference in voltage and time between the action potentials was obtained is represented graphically in figure 2.14. 26 Chapter 2. Effect of Coupling Strength 50.0 30.0 <c 10.0 O -10 .0 -30 .0 h 50.0 100.0 Time, msec Figure 2.7: Excitatory and Inhibitory currents for type I oscillator with slow coupling and strong (gsyn = 1) coupling strength. In the first plot, figure 2.12, the relative height between the peaks of Vi and Ve is shown for various coupling strengths and various time scalings. It can be seen that as the time scaling a is increased, the difference in voltage between the peaks increases. This can be explained by looking at the role of a in the equation for ^ given in equation (1.1). This can be rewritten in the form: ~ = a(k(V) + ^)(Soo-s). at a The time scaling a determines how quickly s varies in order to achieve the value of the function SOQ. Small values of a correspond to a slow relaxation, meaning that s takes longer to reach the value S Q O while for very large values of a, then s s^. Thus very large values of a in (1.1) gives the same results as using (1.2) as the coupling term in simulations. The effect of time scaling is discussed further in Chapter 3. Also, as the synaptic coupling strength gsyn increases, the difference in voltage between the peaks increases. This is entirely as expected, since both the synaptic coupling strength as well as the time scale, determine the degree of interaction between the oscillators. As gsyn is increased, the oscillators effect each other to a greater degree, and so the difference in peak heights becomes greater. In a similar way, increasing the time 27 Chapter 2. Effect of Coupling Strength O s c i l l a t o r i O s c i l l a t o r e 2 5 1 0 2 0 3 0 S p i k e N u m b e r 4 0 Figure 2.8: Cycling of the period of the oscillators in the Connor model for gsyn = 0.1 scale a allows the synaptic current to change more quickly, so that the oscillators interact more rapidly. This accounts for the greater difference in voltage between the peaks in this case. In fact, by using curve-fitting, an equation can be found that describes the relationship between the difference in the height of the peaks, dv, in terms of a and gsyn. The equation is as follows: ( "5+7 ) 9syn dv = \ 8 q + y . (2.1) \a+5A J + Ssyn In this way, given two of height difference, time scaling and coupling strength, it is now possible to calculate the third property without having to run simulations. I was able to numerically determine these numbers by first noticing that the curves in figure 2.12 closely resembled a Michaelis-Menten curve which has the form: ax x + 0 After determining the values of a and b that most closely matched these curves, I found that the graph of a and b also resembled Michaelis-Menten curves. After finding the numbers which most closely matched the curves for a and b, I was able to state an equation that closely matches the curves for dv and the only information required is gsyn and a. 28 Chapter 2. Effect of Coupling Strength eo so S 4 0 Q- 3 0 20 I O Oscillator i Oscillator e 10 20 Spike Number Figure 2.9: Cycling of the period of the oscillators in the Connor model for gsyn = 0.5. The second plot, figure 2.13, shows the relative time difference between the peaks, determined in the manner shown in figure 2.14. It comes as no surprise that regardless of time scaling, as the coupling strength gsyn increases, the relative time difference between the peaks becomes smaller. It is also interesting to note that as the time scaling a is increased, the time difference ,increases. In figure 2.13, it can be seen that the relative time difference between the peaks is longer for larger values of a for every value of gsyn. As the synaptic coupling strength increases, the relative time difference decreases, which means the oscillators are firing closer together. This relationship can also be expressed as an equation: dt = 5 - 8 5 a Q 51 I a+0.57 y , u ± / ysVn 13a a - 0 . 0 7 + 12.6 +g syn -2 .6a a + 2.3 + 6.8 (2.2) As before, I noticed the curves resembled Michaelis-Menten curves with a translation factor: y ax x + b + c. In exactly the same manner as for dv, I was able to determine values for a, b and c. These values also resembled Michaelis-Menten curves when plotted so by using the same method again I was able to determine an equation for dt that closely resembled the curves plotted directly 29 Chapter 2. Effect of Coupling Strength 6 0 50 30 20 Oscil lator i Oscil lator e IO Spike N u m b e r Figure 2.10: Cycling of the period of the oscillators in the Connor model for gsyn = 1 from simulation data. We can consider how much each synaptic coupling term contributes to the phaselocking of the oscillators by seeing how combinations of gsyni and gsyne affect the behaviour of the oscilla-tors. Interestingly, regardless of what combination was taken for gSyni and gsyne-, the oscillators always phaselocked. The only term that affected the behaviour was gsyni which caused the voltage difference between the peaks to be more pronounced regardless of the value of gsyne-30 Chapter 2. Effect of Coupling Strength 0.05 (top figures) and gsyn = 0.7 Figure 2.11: Phase space diagram of reduced model for g; (bottom figures). 31 Chapter 2. Effect of Coupling Strength Synaptic Coupling Strength, g. Figure 2.13: Relative time difference between peaks in coupled type II oscillator. 32 Chapter 2. Effect of Coupling Strength 33 Chapter 3 > Effect of the Time Scale of Synaptic Coupling In this chapter I consider changing the time scale a of the coupling, which is found in the equation for synaptic transmission. If we consider the synaptic gating variable in the form: <te _ sooiv) - s dt TS ' then by considering equation (1.1) we can see the time scaling TS is given as: _ 1 T s ~ a(k(V) + £y so by decreasing a, we increase the time scaling TS. This means that as a decreases, s takes longer to achieve the value of S Q O - By increasing a, s relaxes more quickly to S Q O - In this way, large values of a in (1.1) correspond to the instantaneous case (1.2). This can also be seen by considering the synaptic current for different values of a and comparing those to the instanta-neous case. In figure 3.1, with a=l in the slow case, we can see there is a difference between the slow and instantaneous current case shown in figure 3.3. However, in figure 3.2 where ct=100, there is no noticeable difference between the slow and instantaneous current shown in 3.3. For this chapter we will take very large values of a to be identical to the instantaneous current case. We can consider the synaptic transmission as (1.1), look at the effect of the time scaling a and how that affects the firing of Vi and Ve. The different behaviours observed for different values of time scaling can be found in table 2.1 for both type I and type II oscillators and for details on the height and timing difference of type II oscillators see figures 3.7 and 3.8. 34 Chapter 3. Effect of the Time Scale of Synaptic Coupling Later we will distinguish between the time scaling for oscillators Vi and Ve. We can write for oscillator Vf. _ 1 T s i ~ ai{k{Ve) + £) and for oscillator V e : _ 1 T s i ~ ^(k(Vi) + i e y l\ A -\ 1 { { \ 0 50 100 150 200 Time, msec Figure 3.1: Synaptic current of type II oscillator with a=l. For type I oscillators, there are three separate cases that must be considered: fast coupling where a takes on values greater than 100 so that the coupling appears to be instantaneous, slow coupling where a varies from 0.1 to 10 with weak coupling strength (gsyn=0-l to 0.8) and slow coupling with strong coupling strength (gsyn >0.8). In the first case, instantaneous coupling, varying the time scaling has no effect on the timing of the oscillators. They fire at the same rate and at the same height for all values of a. In the next case, slow coupling, with gsyn small, drifting still occurs but the pattern of firing is different for different coupling strengths. 35 Chapter 3. Effect of the Time Scale of Synaptic Coupling 0.8 0.6 £ 0.4 0.2 n n 50 n 100 Time, msec n 150 200 Figure 3.2: Synaptic current of type II oscillator with a=100. In the third case, slow coupling with strong coupling strength, we see harmonic phase locking. By varying the time scaling, we still see harmonic phase locking, but the spiking occurs with a slightly higher frequency as the time scale increases. For type II oscillators, the situation is much the same as it was for varying the coupling strength. That is, regardless of the time scaling, the oscillators will phaselock. However, unlike the type I oscillator, the time scaling effects the height, as well as the timing of the oscillators. 3.1 Type I Oscillators In the case of instantaneous coupling, varying the time scaling has no effect on the timing or height of the peaks as can be seen in figure 3.4. As with the synaptic coupling strength, this can be explained in terms of the coupling currents, ijn/j and Iexc. Recall figure 2.5 which showed the coupling currents for gsyn = 0.1 and a = 1. As a is increased, there is some change in the currents as can be seen in figure 3.5 where gsyn = 1 and a = 10. However, Iexc is still proportionally much larger than 1^. For instance, for gsyn = 0.1 and a = 1, the maximum absolute value of Iexc = 9.59 and the maximum absolute value of 1^ = 4.55. For gsyn = 0.1 and a = 10, the maximum absolute value of Iexc = 9.41 and the maximum absolute value of 36 Chapter 3. Effect of the Time Scale of Synaptic Coupling 0 -"1 --I. 1 1, \ I I L_, I i , I i_i • I V i 1 V I 0 50 100 150 200 Time, msec Figure 3.3: Synaptic current of type II oscillator with instantaneous coupling. link = 7.31. Also, as can be seen in figure 3.6, Iexc exerts an influence on oscillator Vi for a much longer period of time. Thus the phase locking is occurring because oscillator Ve is still forcing Vi. Figure 3.4: Comparison of phaselocked voltage oscillations of type I model for a = 1 and a = 10. In the case of slow coupling, we observe two very different behaviours depending on the coupling strength: drifting and phase locking. For weak coupling strength, we see drifting. However, the 37 Chapter 3. Effect of the Time Scale of Synaptic Coupling < CL ~ 3 g> o -2 h 50 100 Time, msec 150 200 Figure 3.5: Coupling currents of type I oscillator, gsyn = 0.1 and a = 10. transition from drifting to phase locking occurs at a lower value of gsyn as a is increased. For instance, when a = 1, the transition from drift to phase lock occurs between gsyn = 0.5 - 0.6, while for a = 10, the transition from drift to phase lock occurs between gsyn = 0.2 - 0.3. It is also interesting to consider what the effect might be of having one oscillator with a slow synaptic current and the other with a fast current. When the coupling strength was high, all combinations of oti and ae produced harmonic phaselocking. With medium coupling strength there was drifting when either CCJ or ae was low, meaning either oscillator Vi or Ve had a slow synaptic current, and when the coupling was weak the oscillators would only drift with respect to each other. 3.2 Type II Oscillators The situation for type II oscillators when changing the time scaling is very similar to the effect of changing the coupling strength. Regardless of the coupling strength and time scaling, the oscillators will always phase lock. The relationship between the height of the peaks and the 38 Chapter 3. Effect of the Time Scale of Synaptic Coupling < o 16 18 T i m e , m s e c 2 2 Figure 3.6: Expanded view of synaptic coupling currents of type I oscillator, gsyn =0.1 and a = 10. relative time difference between the spikes is shown in figures 3.7 and 3.8. The data used is the same as used for figures 2.12 and 2.13 and determined according to figure 2.14, but here the relationship is plotted with respect to time scaling a and not coupling strength gsyn- As was calculated in chapter 2, there is a definite relationship between the height and timing of the peaks and the time scaling and coupling strength as given by equations (2.1) and (2.2). When we change the time scaling so that one oscillator has a slow synaptic current while the other has a fast current, we still only see phaselocking. Although it may not be possible to come up with an equation to describe the difference when both ai and ae are varied in different directions, it is interesting that the phaselocking behaviour persists, regardless of the values of ai and ae. 39 Chapter 3. Effect of the Time Scale of Synaptic Coupling 30 0 q =4 1 -q =2.5 — ' £L*n.=2 -3 s y r , = 1 5 q =0.5 y s y n -q =0.1 I , I . I 1 I , I , I , I , I , I 0 2 4 6 8 10 Time scaling, a Figure 3.7: Relative voltage difference between peaks in coupled type II oscillator. 7 I , i , i , i , i , I 0 2 4 6 8 10 T ime scaling, oc Figure 3.8: Relative time difference between peaks in coupled type II oscillator. 40 C h a p t e r 4 Effects of the Threshold Voltage in Synaptic Coupling In this chapter we will be considering the effect of changing the threshold value in the synaptic current term. The range of behaviours that arise as a result of varying Vt are summarized in tables 4.1 and 4.2 for type I oscillators and in tables 4.3 and 4.4 for type II oscillators. The synaptic transmission equation (1.1) is given by: at a where Here the threshold is represented by Vj. Figure 4.1 shows how different values of Vt change the shape of SQO- By lowering the value of Vj, the synaptic channels open for a lower value of membrane voltage. If the threshold value is lowered below the range of values of the membrane potential of the oscillator, the oscillators are always coupled, the oscillators are always influenc-ing each other over their entire cycles. For the same reason, when the threshold value is much higher than possible values of the membrane potential, the oscillators are not interacting and are essentially uncoupled. When we wish to consider the individual effects of Vt, we can rewrite Soo(V) as follows: 41 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling o. S c o { V i ) = a + ^(l + e - ( V i - v « ) / ^ ) ' ^ S c o { Y e ) = a + 3(l + e-(ye-vte)/vsey (4-3) (4.4) Since s(Ve) appears in the equation for Vi, changing the value of Vj e will affect the sensitivity of oscillator V. Interesting and varied behaviour can be seen in both the type I and type II oscillators. The threshold values for the type I model vary between Vt = —70 and Vt = 60 while in the type II model they vary between Vt = —40 and Vt = 60. For the type I Connor model with slow coupling, at very low threshold values for the strong and medium coupling strengths (gsyn = 1, 0.5) we see very small oscillations in Ve. This then switches to a harmonic phaselock, where Vi spikes twice for each single spike of Ve and then drifting for higher values of Vj. For weak cou-pling, gsyn = 0.1, there is drifting for all values of Vt. For the Connor model with fast synaptic current, when the coupling strength is strong or medium, small oscillations are observed in Ve which are synchronized with Vi. For higher threshold values, Vt > —30, there is phaselocking. For weak coupling, the oscillators drift with their periods taking on a range of values until 42 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling V t=-70 mV to -40 mV V*=-40 mV to 20 mV V t=20 mV to 50 mV Vt >50 mV 9syn > 0.8 Ve suppressed harmonic phaselocking harmonic phaselocking drift gsyn=0A to 0.8 Ve suppressed harmonic phaselocking drift drift gsyn=0.1 to 0.4 drift drift drift drift Table 4.1: Behaviour of type I oscillator with slow synaptic coupling for different threshold voltages. V t=-70 mV to -40 mV Vt=-40 mV to -30 mV Vt >-30 mV 9syn >0.8 Ve suppressed phaselock phaselock 9syn=0A tO 0.8 Ve suppressed drift phaselock gsyn=0A to 0.4 drift drift phaselock Table 4.2: Behaviour of type I oscillator with fast synaptic coupling for different threshold voltages. V t=-50 mV to -30 mV V t=-30 mV to -20 mV Vi=-20 mV to -10 mV Vt >-10 mV gsyn > 0.8 Vi and Ve suppressed Vi and Ve suppressed phaselock phaselock gsyn=0A to 0.8 Vi suppressed phaselock phaselock phaselock gsyn=0A to 0.4 drift drift phaselock phaselock Table 4.3: Behaviour of type II oscillator with slow synaptic coupling for different threshold voltages. 43 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling V*=-50 mV to -30 mV Vi=-30 mV to -20 mV Vt=-20 mV to -10 mV Vt >-10 mV 9syn > 0.8 V{ and Ve suppressed Vi and Ve suppressed Ve suppressed c phaselock 9syn=0A to 0.8 Vi suppressed Vi suppressed harmonic phaselock phaselock gsyn=0.l to 0.4 drift drift drift phaselock Table 4.4: Behaviour of type II oscillator with fast synaptic coupling for different threshold voltages. Vt = 30 when phaselocking occurs. The behaviour for each coupling strength is similar for both the fast and slow coupling. For the type II Morris-Lecar model, the situation is similar across the slow and fast synaptic current cases. For strong coupling, gsyn = 1, both Ve and Vi are at a steady state and then phaselocking when Vt = —20. For medium coupling, gsyn = 0.5, there is initially harmonic phaselocking which leads to strict phaselocking. For the weak coupling, gsyn — 0.1, there is also initially harmonic phaselocking which leads to strict phaselocking. 4.1 Type I Oscillators In this section, I will look at the behaviour of the type I oscillator more closely while varying the threshold voltage. As both the fast and slow synaptic coupling display similar behaviour, I will discuss them at the same time. We can observe many different behaviours here, from drift to phaselock to harmonic phaselock. For strong coupling in the both the fast and slow synaptic current case when gsyn = 1, we begin with oscillators Vj, and Ve synchronized with Ve exhibiting very small subthreshold oscillations. In both cases, Ve takes values close to -65 mV, a very low voltage, while Vi ranges from -52 44 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling mV to 18 mV with fast oscillations having a period of 6.8 msec, as shown in figure 4.2. As Vt is increased, the range of Ve is extended a little further but the oscillations are still occurring over a very small voltage range and the period has increased to 10.6 msec in the slow synaptic current case and 10.8 msec in the fast case. At Vt—-50 mV, the period increases, as does the range of Ve, to 26.4 msec in the slow case and 25.1 msec in the fast case. These oscillations are still much faster than in the uncoupled case where the period T=40.9 msec. 30 10 > E -10 <D £ "o > -30 -50 -70 0 10 20 30 Time, msec Figure 4.2: Suppression of oscillator Ve in type I Connor model with strong coupling (gsyn=l), slow synaptic current and low threshold (Vt=-70 mV). When Vt=-40 mV, we see a switch in the behaviour to harmonic phaselocking where Vi is spik-ing twice to every one spike of Ve, as shown in figure 4.3. In the slow case, the period of Ve=56.8 msec the same as the combined period of the two spikes of oscillator Vj. In the fast case, the period of oscillator Ve is 60 msec, also the total period of oscillator Vi. Now as Vt increases, we begin to see some variation in the fast and slow coupling. In the slow case, we continue to see this harmonic phaselocking with Vi spiking twice to every one spike of 45 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling Figure 4.3: Harmonic phaselocking in type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and low threshold Vt=-40 mV. Ve. The period of Ve and Vi gradually increases as Vt increases until Vt—10 mV at which time the period decreases slightly for increasing values of Vt. For 14=50 mV when the oscillators are interacting only slightly, we see drifting with the oscillator's periods taking on a range of values in a cyclic manner. A plot of the period in terms of spike number showing the range of values they take on is shown in figure 4.4. When 14=60 mV we no longer have any coupling. In the fast coupling case, the harmonic phaselocking disappears quite quickly and we see strict one-to-one phaselocking for Vt=-30 mV (figure 4.5). At this point, the difference in time be-tween the peaks is very small and the voltage difference is dv=2 mV. This is determined as in figure 2.14. We continue to see this phaselocking for all values of Vt until there is no longer any coupling when Vj=60 mV. As Vt increases, dt increases and then stabilizes at dt=0.27 msec when Vf=10 mV, where it remains for higher values of Vt. However dv continues to decrease as Vt increases until it eventually becomes dv=0 when the oscillators are no longer coupled. 46 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling Figure 4.4: Periods of type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and high threshold Vt=50 mV. When we vary Vt in different directions, for instance Vu = —40 mV and Vte = 30 mV, we again find the same behaviour for both the fast and slow synaptic current cases. We usually see harmonic phaselocking unless Vu or Vte is small, in which case oscillator Ve is suppressed. When the coupling strength is decreased so that gSyn=0-5, we can see different behaviour be-tween the fast and slow synaptic current cases depending on the threshold voltage. In the slow current case, they do behave in similar manners and for Vt=-70 to -40 mV we see very small oscillations in Ve that are synchronized with Vi having a fast period of 8.8 msec (figure 4.6). the range of Ve increases slightly as Vt increases and the period increases to 23.3 msec in the slow case and 21.3 msec in the fast case. When Vt=-40 mV we see the first difference in behaviour with harmonic phaselocking for slow and drifting for fast current coupling. In the slow current case, when Vt=-40 mV we see harmonic phaselocking with Vi spiking twice to every spike of Ve. This occurs with a long period of 59 msec. As Vj increases we continue 47 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling Figure 4.5: Type I Connor model with strong coupling (gsyn >0.8), fast synaptic current and low threshold Vt=-30 mV. to see this harmonic phaselocking with the period gradually decreasing. This continues until Vt=20 mV where we begin to see drifting. The periods of both Ve and Vi cycle through a range of values which corresponds to the drift of the oscillators as shown in figure 4.7. We continue to see drifting until Vt=60 where the oscillators are no longer coupled. In the fast current case, we see drifting for Vi=-40 mV and then one-to-one phaselocking for all remaining values of Vt. As expected, Ve fires before Vi with Vi peaking at higher voltages than Ve. When phaselocking is first observed at Vj=-30, the time and voltage differences between the spikes are small. The time difference between the two oscillators reaches its maximum of 0.33 msec when Vt=-10 and decreases slowly for all further values of Vt until the oscillators are no longer coupled. The voltage difference continues to decrease steadily until it reaches dv—0. when the oscillators are not interacting. When we vary Vt in opposite directions we also find differences between the fast and slow 48 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling Figure 4.6: Type I Connor model with strong coupling (gsyn >0.8), fast synaptic current and low threshold Vt=-50 mV. synaptic current cases with medium coupling strength. In the slow current case, we have the same behaviour as we had with strong coupling. When either Vu or Vte is low, oscillator Ve is suppressed otherwise we see harmonic phaselocking. In the fast current case Ve is suppressed only when both Vu and Vj e are low and is phaselocked when either Vu or Vte is greater than -20 mV. When we consider weak coupling where gsyn = 0.1, we see differences between the fast and slow synaptic coupling. When the current is slow, we only see drifting for every value of Vt. For very low values of Vt, there is a large difference between the periods of the two oscillators. The period of Ve takes on values around 70 msec while the period of Vi takes on values around 16 msec as shown in figure 4.8. The period of Ve gradually decreases and the period of Vi increases until they take on a range of values around 42 msec for Ve and 32 msec for Vi which first occurs when Vt=-40 mV. Also as Vt increases, the number of spikes that occur between the simultaneous firing of both oscillators increases as shown in figure 4.9 where oscillator Vi 49 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling 57 52 47 TD O Q_ 27 h\ 22 h 17 \-12 0 5 10 S p i k e N u m b e r 15 20 Figure 4.7: Period of Type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and low threshold Vt=-30 mV. spikes 14 times for every 13 spikes of oscillator Ve. When Ssyn — 0.1 for the fast current case we initially see drifting as in the slow current case. When Vt=-30, we have one-to-one phaselocking with Ve firing before Vi with Vi having a higher voltage spike. As Vt increases, dt stays constant at 0.3 msec while dv decreases. It is interesting to note that regardless of the value of Vt, the period of the oscillators is constant at 40.9 msec, which is the period of the uncoupled oscillator. When we vary Vt in opposite directions we see different behaviour from the fast and slow synaptic current cases when they are coupled weakly. In the slow current case we see harmonic phaselocking only when Vu is low and otherwise we see harmonic phaselocking. In the fast cur-rent case, we see similar behaviour to the fast current case with medium coupling. Here for low values of either Vu or Vte we see drifting instead of suppression and for all other combinations of Vti and Vte we see harmonic phaselocking. 50 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling 4.2 Type II Oscillators Here we consider the behaviour of the type II oscillator when the threshold value Vt is varied from -40 mV to 60 mV. There are many similarities between both the fast and slow synaptic current cases. When the coupling is strong, gsyn = 1, both Vi and Ve go to steady state values as shown in figure 4.10. When Vt increases to -20 mV, both cases show phaselocking with Ve spiking before Vi. The time difference between the spikes decreases at first as Vj increases, but then increases for Vt > 0 mV while the height difference between the spikes continually decreases. In both cases, the period of the oscillators decreases until Vj=30 mV and then in-creases slightly until Vj=60 mV, when the oscillators are no longer coupled. When we vary the threshold value in different directions, we see the same behaviour for both the fast and slow current cases. When the coupling strength is high and either Vu or Vte is low, both oscillator Vi and Ve are suppressed. For all other combinations we see phaselocking. In the case of medium coupling, gsyn = 0.5, we see different behaviours until Vt=-10. For the case of the slow current, the oscillations in Vi are very small and exhibit three subthreshold oscillations for each spike of Ve (figure 4.11). This continues until Vt=-30 when Vi starts spiking twice for each spike of Ve. Then when Vj=-10 we see strict phaselocking that continues for all further values of Vt. For the fast current case, Vi is at a steady state value while Ve is oscillating. As Vt is increased to -50, Vi begins showing very small oscillations that spike three times to each spike of Ve. When Vj=-20, Vi is now fully spiking twice for each action potential of Ve, as seen in figure 4.12. When Vt=-10 we again see strict phaselocking that continues for all further values of Vt. In both the fast and slow currents cases, once the oscillators have phaselocked, the time difference between 51 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling the spikes increases as Vt increases, while the peak voltage difference steadily decreases. The period of the oscillators does not vary much as Vt is changed. When we vary Vt in different directions, we see the same behaviour for both the fast and slow current cases. Only when Vte is so low that it is influencing the synaptic current for oscillator Vi is the oscillator V. suppressed. For all other combinations of Vu and Vte, oscillators Vi and V e are phaselocked. When 9syn—0-1, we again see similarities in the behaviour of the fast and slow current cases. When Vt is very low, we see drifting with Vi spiking roughly 10 times for every 9 spikes of Ve in the slow current case and Vi spiking approximately 8 times for every 7 spikes of Ve in the fast current case as shown in figure 4.13. What is interesting about this case is that both oscillators have a fixed period that is different from each other. Oscillator Ve has a fixed period of 48.5 msec in the slow case, 48.86 msec in the fast case, while oscillator Vi has a fixed period of 43.5 msec in the slow case, 42.91 msec in the fast case. In both cases as Vt is increased we see a drifting pattern where the period of both oscillators is variable. When Vt=-10 we have phaselocking that persists for all higher values of Vt. The period of the oscillators in both cases remains at the same value regardless of the value of Vt while the time difference in the peaks increases and the peak voltage difference decreases as Vt increases. As with all other coupling strengths, we see the same behaviour for the fast and slow current cases when we vary Vt in opposing directions. When either Vu or Vj e is low we have drifting. Otherwise we have phaselocking. 52 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling Spike Number Figure 4.8: Cycling of the period of type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and low threshold Vt=-70 mV. 53 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling > E <g o > 100 200 300 400 Time, msec 500 600 Figure 4.9: Drifting of type I Connor model with strong coupling (gsyn >0.8), slow synaptic current and high threshold Vt=40 mV. 100 Time, msec 200 Figure 4.10: Steady state of type II Morris-Lecar model with strong coupling (gsyn—l), slow synaptic current and low threshold Vj=-40. 54 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling Figure 4.11: Subthreshold oscillations of Vi synchronized with the firing of oscillator Ve of type II Morris-Lecar model with strong coupling (gsyn=l), slow synaptic current and low threshold Vt=-40 mV. Figure 4.12: Harmonic phaselocking of type II Morris-Lecar model with medium coupling (dsyn=0-5), fast synaptic current and low threshold Vt=-20 mV. 55 Chapter 4. Effects of the Threshold Voltage in Synaptic Coupling -40 1 ^ ^ '-^  —1 '-^ ;—' —' —1—-—•—-—1 0 100 200 300 400 500 Time, msec Figure 4.13: Drifting of type II Morris-Lecar model with weak coupling (gsyn=0.1), fast synaptic current and low threshold Vj=-30 mV. 56 C h a p t e r 5 Effects of Voltage Sensitivity of Synaptic Currents In synaptic transmissions, the voltage sensitivity Vs, determines the sharpness of the oscillator interaction. The range of behaviours found by varying Vs are shown in table 5.1 for type I oscil-lators and in table 5.2 for type II oscillators. When Vs is small, the transition from decoupled to fully coupled occurs steeply at the threshold voltage Vt. For larger Vs, the transition is more gradual. Note how in figure 5.1, the curves intersect at Vt, 20 mV. T 0.8 0.6 0.4 0.2 -40 / / Vs=0.1 mV Vs=2 mV Vs=4 mV V=10mV -20 0 20 Voltage, mV 40 60 Figure 5.1: Effect of different values of Vs on sc The steepness Vs, whose units are voltage, appears in the exponential term e (v Vt)lVs for 57 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents V s=0.1 mV to 10 mV Vs >10 mV slow current, harmonic harmonic 9syn > 0.8 phaselock phaselock slow current, drift harmonic gsyn=0A to 0.8 phaselock slow current, drift drift gsyn=0.1 to 0.4 fast current, phaselock harmonic 9syn > 0.8 phaselock fast current, phaselock drift gsyn=0A to 0.8 fast current, phaselock drift Psyn=0.1 tO 0.4 Table 5.1: Behaviour of type I oscillator with slow and fast synaptic coupling for different voltage sensitivities. 58 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents Vs=0A mV to 10 mV Vs >10 mV slow current, 9syn > 0.8 phaselock phaselock slow current, gsyn=0A to 0.8 phaselock phaselock slow current, gsyn=0A to 0.4 phaselock phaselock fast current, 9syn > 0.8 phaselock V. and Ve suppressed fast current, gsyn=0A to 0.8 phaselock Vi suppressed fast current, gsyn=0A to 0.4 phaselock drift Table 5.2: Behaviour of type II oscillator with slow and fast synaptic coupling for different voltage sensitivities. 59 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents Soo{V) in the synaptic term of equation (4.1). In effect, the voltage controls the slope of s^. Voltage values of Vs are always positive so that SQO is monotonic increasing. If Vs was given a negative values, SQQ would be monotonic decreasing so that the oscillators would be interacting until they reached the threshold value at which point they would cease to interact. When we wish to consider separately the effect of the steepness on oscillator Vi and V e , we can rewrite the equations as in equation (4.2). By changing Vse we are affecting oscillator Vi while changing VSi affects oscillator Ve. Varying the threshold value has seemingly less effect than other factors such as coupling strength and threshold voltage. For the type I oscillator, the behaviour of the coupled oscillators, re-gardless of the coupling strength and whether the synaptic current in fast or slow, is almost identical for values of Vs in the range of 0.1 mV to 4 mV. In the slow current case, this means that we see harmonic phaselocking for strong coupling, and drift for medium and weak coupling strength. In the fast current case, we see phaselocking for all coupling strengths. When Vs is large, we can see different behaviour in some cases. For the type II oscillator we see phaselocking in both the fast and slow current case for Vs in the range of 0.1 mV to 4 mV. The only case that is effected by increasing the steepness is the fast current, where we see a steady state solution for strong coupling, harmonic phaselocking for medium coupling and drifting for the weak coupling case. 5.1 Type I Oscillators First, we consider the slow current case. When the coupling is strong, gsyn = 1, we see harmonic phaselocking with oscillator Vi spiking twice for every single spike of Ve as shown in figure 5.2. This behaviour holds for all values of Vs from 0.1 mV to 10 mV. The timing of the spikes and the overall period of the oscillators does not change for any value of Vs. When the coupling 60 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents is medium or weak, g3yn = 0.5 and 0.1, we see drifting for almost all values of Vs. The rate of drift and the manner in which the periods oscillate is similar for all these values of Vs. The periods for Vs=0.1, 1, 2 and 4 mV is shown in figure 5.3. The only difference is in the medium coupling case, gsyn = 0.5 when Vs = 10 mV. Here we see harmonic phaselocking with Vi spiking twice for every single spike of Ve. This is almost identical to the case where we lowered the threshold voltage Vt to -40 mV. This is not surprising when we consider the plot of SQO in figure 5.1. For very large values of V3, we can see how the oscillators can begin interacting at very low voltages. This would be the same as lowering the threshold voltage Vt. When we vary the steepness, Va, in different directions for Vi and Ve we see similar behaviour in the strong coupling case and the weak coupling case. In the strong coupling case, we have harmonic phaselocking with V. spiking twice for every spike of Ve, regardless of whether Vsi is low and Vse is high, or Vsi is high and Vse is low. Similarly, in the case of weak coupling we see drifting regardless of which Vs is low or high. In the case of medium coupling, we see harmonic phaselocking when Vsi is low and Vse is high. This is the same as the strong coupling case, but when Vsi is high and Vse is low we see drifting just as in the weak coupling case. In the fast current case, we see similar behaviour for all values of coupling strength when Vs varies from 0.1 mV to 4 mV. For gsyn=0.l, 0.5 and 1 we see phaselocking with oscillator Ve spiking slightly ahead of Vi. The time difference between the spikes is the same for all cou-pling strengths while the voltage difference increases as gsyn increases. When the steepness is increased, we see harmonic phaselocking for strong coupling and drifting for medium and weak coupling. The variation of the periods of the oscillators for Vs=10 and gSyn= 0.1 and 0.5 is shown in figure 5.4. These cases also correspond to the behaviour we observed when Vt is lowered to -40 mV. When Vs is varied so that one oscillator has a steep slope while the other has a shallow slope 61 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents 25 > E D) ro o > -25 1 • 1 1 1 — v, .... ve - 1 L L 0 50 100 150 200 Time, msec Figure 5.2: Type I Connor model with strong coupling ( g s y n >0.8), slow synaptic current and voltage sensitivity Vs=0.1 mV. we see the same behaviour, regardless of what the coupling strength is. In all these cases the oscillators phaselock. 5.2 Type II Oscillators The behaviour we see for type II oscillators is consistent for all values of Vs. In the slow synap-tic current case, we see phaselocking for all values of Vs as shown in figure 5.5. The voltage difference between the peaks changes where dv increases as Vs increases. The period of the oscillators also decreases with increasing Vs with T=46.2 msec when gsyn— 0.1, and T=42.2 msec when gsyn=l. When we vary the steepness in different directions for each oscillator, this has little effect on the behaviour of the oscillators. Regardless of the coupling strength and whether VSi is high and Vse is low or Vs{ is low and Vse is high, the oscillators always phaselock. 62 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents Vs=4 60 20 S p i k e N u m b e r 20 S p i k e N u m b e r 20 S p i k e N u m b e r Figure 5.3: Periods of drifting type I Connor model with medium coupling (gSyn=0A to 0.8), slow synaptic current and steepness Fs=0.1, 1, 2 and 4 mV. In the fast current case we also see phaselocking but only for values of Vs from 0.1 mV to 4 mV. As in the slow current case, the time difference between the peaks decreases as Vs in-creases while the voltage difference between the peaks increases and the period of the oscillator decreases. However, when we increase Vs to 10 mV we see a steady-state solution for strong coupling, harmonic phaselocking of three very small spikes of Vi for each spike of Ve in the case of medium coupling and drifting for weak coupling. If we look back to the observed behaviour when we varied Vt, we again see that this corresponds to the case when Vt=-40 mV. When we vary the steepness so that either Vsi is high and Vse is low or Vsi is low and Vse is high we see differences not only for each of those two cases but also for different coupling 63 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents S p i k e N u m b e r S p i k e N u m b e r Figure 5.4: Periods of drifting type I Connor model with medium {gsyn=0-o) and weak (<?s3/n=0.1) coupling, fast synaptic current and steepness Vs=10 mV. strengths. When gsyn is large, Vsi is low and Vse is high we see phaselocking while when Vsi is high and Vse is low we see harmonic phaselocking with Vi spiking twice for every single spike of Ve. This is due to the low value of Vse which means the synaptic current of oscillator Vi is activated at lower voltages of oscillator Ve allowing for a significant contribution to oscillator Vi. When the coupling strength is medium, we find for Vsi low and Vse high that we have harmonic phaselocking but now Ve is spiking three times to every action potential of Vi. This is again due to the low value of Vsi which contributes to the value of oscillator Ve. However, when VSi is-high and Vse is low and for both the fast and slow current cases with weak synaptic coupling, we see drifting. 64 Chapter 5. Effects of Voltage Sensitivity of Synaptic Currents Figure 5.5: Type II Morris-Lecar model with medium coupling (gSyn—0.5), slow synaptic current and steepness V s=4 mV. 65 C h a p t e r 6 L i -Rinze l Mode l of a Pi tui tary Cel l This chapter will consider a realistic model of a pituitary cell which consists of two oscillators, a plasma membrane voltage oscillator and an internal endoplasmic reticulum calcium oscillator. The model is discussed in more detail in references, [4], [22], [24] and [28]. This model is based on experimental results obtained from pituitary cell. Spontaneous action potentials had been observed in gonadotrophs and further investigation using calcium imaging and other techniques demonstrated a relationship between membrane voltage oscillations and intracellular calcium concentration. These and other observations led to the development of the current model. By considering how the two oscillators interact, we can relate the behaviour of the more compli-cated model to the simplified case we have been considering in previous chapters. In this way we wil l use the results from the simple models to better understand the action of the model of the pituitary cell. 66 Chapter 6. Li-Rinzel Model of a Pituitary Cell The equations for the Li-Rinzel pituitary cell model are as follows: dV 1 —rr = [lapp ~ leal — Icat - iKdr — IK{CO) — h) ox cm dq. < 7 o o ( V ) - q dt T g (V) , q = mL,mT,hT,n t t = {l+P{c-Td-) ^ j ^ - ^ - ^ r - ^ ^ + fPjpm -77 = a2{din - (c + din)h) dt dt = f P J p m with associated functions given in appendix A . The parameter values are shown in table A.3. The voltage oscillator across the plasma membrane represented by the equations for V, rriL, TTIT, iiT and n, is based on the classic Hodgkin-Huxley model of the giant squid axon. The various currents are as follows. There are two voltage-gated calcium channels, IcaL, a high-threshold voltage activated current and IcaT, a transient current. There is a delayed-rectifier potassium channel, IRCLT-, a calcium activated potassium current, IK(CO,)I a n d a leak current II. In order to better conform to experimental evidence, the reversal potential of these channels, except for the leak channel, is given by the Goldman-Hodgkin-Katz driving force, cf)Q(V). The calcium oscillator is also based on the Hodgkin-Huxley model with calcium taking the place of the voltage term. The equation for c represents the calcium concentration in the cytosol with contributions from both the endoplasmic reticulum as well as the flow of calcium across the plasma membrane. The gating equation h represents the activation of the calcium channel in the endoplasmic reticulum which allows calcium to be released from the cell's internal stores. The equation for a represents the change in calcium concentration inside and outside the cell. This is an important term that helps ensure the cell doesn't receive too much calcium during an action potential. Calcium is toxic to cells when too much is present in the cytosol and can't be taken up by the endoplasmic reticulum or pumped out of the cell. The calcium equations, c, 67 Chapter 6. Li-Rinzel Model of a Pituitary Cell h and a, are constructed to balance the flux of calcium across the plasma membrane with the calcium flux across the endoplasmic reticulum. The equation for calcium concentration can be rewritten as follows: dc — J r e l J f i l J i n J o u t j where J r e l = l + P , C + dac J f i l = V, ^ / i 3 ^ (a - Cpc), J i n = -Oi{IcaL + IcaT), c n p J o u t — V p ~ , , 7 l p • CnP + Kvv The flux of calcium across the endoplasmic reticulum into the cytosol is represented by the term Jrel. This release is through an IP3 receptor calcium release channel with a small contribution coming from a leak current. Jfu stands for the refilling of the endoplasmic reticulum calcium stores by sarcoplasmic and endoplasmic reticulum calcium-ATPase (SERCA) pumps. Both Jrei and Jfu represent the flux of calcium across the endoplasmic reticulum. Calcium influx across the plasma membrane into the cytosol is represented by Jin which is the action of voltage gated calcium channels while Jout represents transport of calcium out of the cytosol against its concentration gradient by plasma membrane calcium-ATPase pumps (see figure 6.1). There are some important assumptions that have been made in constructing this model. The most obvious is the lack of a spatial dependence. This means we are assuming that once calcium enters the cell, whatever is not buffered diffuses evenly and instantaneously through the entire cell. There are also the assumptions that ion channels are evenly distributed on the plasma membrane and that the endoplasmic reticulum is continuously distributed throughout the cell. The contribution of organelles such as mitochodria, also releasing calcium into the cytosol, has been disregarded. The extracellular medium is assumed to have a limitless supply of calcium. 68 Chapter 6. Li-Rinzel Model of a Pituitary Cell Figure 6.1: Representative diagram of currents across plasma membrane (PM) and endoplasmic reticulum (ER) which affect calcium oscillation. By introducing these assumptions we may no longer have exact agreement with experimental results but the dynamics of the system are preserved and are sufficiently close to observed be-haviour to provide us with insights into the dynamics of the system. Both the calcium and the voltage oscillators are capable of independent oscillations as shown in figure 6.2 and figure 6.3. It is clear in these plots that the calcium oscillations occur on a much slower time scale than the voltage oscillations. It is possible to change the rate of oscillation of both the calcium and voltage oscillators while still staying within reasonable physiological parameter ranges. The plasma membrane voltage oscillator is connected to the endoplasmic reticulum calcium oscillator though the calcium activated potassium current, IK(CO,) while the endoplasmic reticulum is connected to the plasma membrane oscillator through the J , n term which represents the calcium influx across the plasma membrane through voltage-gated calcium channels. In this way the plasma membrane excites the endoplasmic reticulum oscillator and the endoplasmic reticulum inhibits the plasma membrane oscillator. We can describe what happens during an action potential according to this model. First, the 69 Chapter 6. Li-Rinzel Model of a Pituitary Cell Figure 6.2: Voltage oscillations of pituitary model when not coupled to the calcium oscillator. 0.8 10 Time, sec Figure 6.3: Calcium oscillations of pituitary model when not coupled to the voltage oscillator. 70 Chapter 6. Li-Rinzel Model of a Pituitary Cell membrane voltage experiences a rapid depolarization. This causes the ion channels on the plasma membrane to open and current to flow across the membrane. This action excites the endoplasmic reticulum to release calcium from its stores into the cytosol. The elevated level of cytosolic calcium activates the calcium activated potassium channel, and causes a hyperpolar-ization of the plasma membrane. The calcium activated potassium channel slows the upstroke of the plasma membrane oscillator, while the calcium in the cytosol is slowly cleared out by endoplasmic reticulum uptake and plasma membrane pumps. 6.1 Dynamics of the model Although the model is a very complicated one, it is still possible to determine certain properties relating to its dynamics. For the type I and type II models, we used the applied current as the bifurcation parameter. Here it is the calcium concentration that determines much of the behaviour of the system and we will use that as our bifurcation parameter. The bifurcation diagram as produced by XPPAUT is shown in figure 6.4. There are several interesting features of this bifurcation diagram. For low calcium concentration we have two stable states, one that is oscillating between V = —52 mV and V = 5 represented by the filled circles in the diagram, and another that is a steady state at V = —37.2 mV repre-sented by the solid line. These two stable states are separated by an unstable oscillating state shown with hollow circles. If we increase the calcium concentration to approximately c = ±0.22 /iM, we see many stable and unstable states coexisting. We have the large stable oscillation, the smaller unstable oscillation as well as the stable steady state that were present for lower values of c, but now we also have a low unstable steady state, as well as a low stable steady state. For this narrow range of c values, we can see a number of different behaviours depending on our initial conditions. As the calcium concentration is increased even more, we lose the branch of periodics and simply have the low stable steady state. By looking at figure 6.3 we can see that the calcium concentration is on average less than 0.2 \xM. 71 Chapter 6. Li-Rinzel Model of a Pituitary Cell 0.0 0.2 0.4 0.6 0.8 Calcium Concentration Figure 6.4: Bifurcation diagram of pituitary model using calcium concentration as bifurcation parameter. 72 Chapter 6. Li-Rinzel Model of a Pituitary Cell 6.2 Interaction of Voltage and Calcium Oscillators As we've already seen, the voltage and calcium oscillators are coupled. The plasma membrane is coupled to the calcium oscillations through the calcium activated potassium current and the endoplasmic reticulum calcium oscillator is affected by the plasma membrane oscillator through the influx of calcium through voltage gated calcium channels. The voltage oscillator excites the calcium oscillator and the calcium oscillator inhibits the voltage oscillator. By taking a closer look at these interaction terms, we can understand how this model can relate to our previous investigation. The plasma membrane voltage oscillator is connected to the calcium oscillator through the term: cnc lK(Ca) = 9K(Ca) c n c + knc 4>K{V). The coupling strength would be represented by gK(Ca) here, and the time scaling would be very large as the current is instantaneous. The threshold value is given as kc while the steepness is nc. In our earlier investigation, this would correspond to the case for Ve where gsyne is small, the time scale ae is large, threshold Vu is very high and the steepness Vsi is moderate. In other words, it takes a high level of calcium to activate a small current, but it activates quickly. When we consider the endoplasmic reticulum calcium oscillator, we must consider the term J j n : Jin = -Oi{IcaL + lea?), where: ICaL = 9CaL m\ <f>c{V), ICaT = 9CaT m T hT 4>c{V). 73 Chapter 6. Li-Rinzel Model of a Pituitary Cell Here the synaptic coupling is represented by gcah and gcaT, the time scaling by TmT,(V) and TmT(V), the threshold voltage by VmL and Vmr and the steepness by kmL and kmT- This cor-responds to the case where Vi has gsyni large, time scaling ajj moderate, threshold Vte low and steepness Vse shallow. In this case, the oscillator activates at very low voltages, has a strong effect on the other oscillator and relaxes slowly to its final value. We can demonstrate that when we have the parameters according to those specifications, in the case of type I oscillators we have harmonic phaselocking with Vi spiking twice for every action potential of Ve as seen in figure 6.5. In the case of type II oscillators we have one-to-one phaselocking with oscillator Ve firing slightly before and higher than oscillator Vi as seen in figure 6.6. The interaction of the voltage and calcium oscillators is shown in figure 6.7. In this model we can see the voltage oscillator fires slightly before the calcium oscillator, precisely the behaviour we saw with the type II oscillators. It is also interesting to note that the oscillators are able to phaselock even though the uncoupled voltage oscillator has a period of 0.18 sec and the uncoupled calcium oscillator has a period of 5.64 sec. When they are coupled, both the voltage and the calcium oscillator have a period of 2.28 sec. 6.3 Conclusion and Discussion The purpose of this paper is to take a close look at how excitatory / inhibitory coupling can affect the behaviour of coupled oscillators. The intended goal in doing this is to obtain insights into more complicated models that use this particular type of coupling. This was accomplished by using two simple models, the Connor model for type I oscillators and the Morris-Lecar model for type II oscillators, and coupling them with synaptic currents. In the course of this investigation we learned some interesting facts. The simple models can produce six very different types of behaviour depending on certain parameter values. In table 74 Chapter 6. Li-Rinzel Model of a Pituitary Cell 80 40 > E cn 0 re "o > -40 -80 0 50 100 150 200 Time, msec Figure 6.5: Coupled type I oscillators with parameter values corresponding to Li-Rinzel model. 6.1 we see which parameter values cause the type I oscillators to behave in a particular way while table 6.2 shows the corresponding parameter values for all the different behaviours for the type II oscillator. In particular we observed suppression of one or both oscillators, phase-locking, both synchronous and asynchronous, harmonic phaselocking and drifting. Drifting was observed in both the type I and type II oscillators. The period of the oscillator during drifting was not stable but varied, often by a substantial amount. This variation in the period of the oscillator was itself periodic. We also saw how the simple model could reproduce the behaviour of the more complicated Li-Rinzel model under similar conditions. This investigation is important for several reasons. This type of coupling is not well understood and is not often used in models. If there is excitatory / inhibitory coupling it is generally through all-to-all coupling where every oscillator receives the same integrated excitatory input as all the other oscillators as in [19] and [5]. This is useful as a simplification when the number of oscillators is large but does not fully explore this coupling. This coupling is also used in delay-differential equations. That type of modeling is used for feedback systems and can pro-75 Chapter 6. Li-Rinzel Model of a Pituitary Cell Phaselock Harmonic Drift v; ve Vi and V e Phaselock Suppressed Suppressed Suppressed fast current, slow current, slow current, not not all (Jsyn gsyn >0.8 g s y n=0.1->0.4, observed observed fast current, slow current, slow current, for slow current, for 9syn >0.8, Vt >-40 mV gsyn >0.8 V* t=-40-»50 mV >0.8, Vt >50 mV any parameter gsyn >0.8, Vt=-70->-40 mV any parameter fast current, slow current, slow current, values slow current, values g s y n=0.1->0.8, Vt >-30 mV 0 a y n=O.l->O.4, V t =-40->20 mV 5 S 2 /n=0.4->0.8, V t >20 mV gayn=0A-*0.8, V t = - 7 0 ^ - 4 0 mV fast current, slow current, fast current, fast current, gsyn >0.8, V s=0.1->10 mV >0.8 all V , gsyn=0A->0.8, V t =-40 mV gsyn >0.4 y t =-70->-40 mV fast current, slow current, fast current, 0 s y n =O. l -»O.8 , V*s=0.1-»10 mV ffStfn=0.4-»0.8, F s >10 mV 0 s „ n =O.l->O.4, V t=-70-»-30 mV fast current, slow current, • ffsj/n >0.8, Vs >10 mV 0ayn=O.4-+O.8, Vs=0.1->10 mV slow current, 0 a y n=O.l->O.4, all V s fast current, gsyn=0A^0.8, Vs >10 mV Table 6.1: Parameter conditions for type I model that give the corresponding behaviour. Unless indicated, parameter values are as given in table A . l . 76 Chapter 6. Li-Rinzel Model of a Pituitary Cell Phaselock Harmonic Drift Vi Ve 14 and 14 Phaselock Suppressed Suppressed Suppressed fast current, fast current, slow current, slow current, fast current, slow current, all gsyn Vt >-10 mV gsyn=0A^0.8, Vt=-20 mV gsyn=OA-rOA, V t=-50-»-20mV gsyn=0A->0.8, Vt=-50->-30mV 9syn >0.8, Vt=-20 mV 9syn >0.8, Vi=-50->-20mV slow current, fast current, fast current, fast current, ^h gsyn all Vs g a y n=0.1-»0.4, I/t=-50->-10mV g3yn=0A-r0.8, Vt=-50-^-20mV ffsj/n >0.8, Vi=-50->-20mV slow current, fast current, fast current, fast current, 9syn >0-8, Vt >-20 mV gsyn=QA->QA Vs >10 mV 0S„„=O.4-»O.8, Vs >10 mV 9syn >0.8, Vs >-10 mV slow current, gsyn=0A-^0.8, Vt >-30 mV slow current, pS2/n=0.1-K).4, Vt >-20 mV fast current, all gsyn V;=0.1->10mV Table 6.2: Parameter conditions for type II model that give the corresponding behaviour. Unless indicated, parameter values are as given in table A.2. 77 Chapter 6. Li-Rinzel Model of a Pituitary Cell 80 60 40 > E 20 OJ D) CO O 0 > -20 -40 -60 0 50 100 150 200 Time, msec Figure 6.6: Coupled type II oscillators with parameter values corresponding to Li-Rinzel model. duce some interesting and meaningful results as seen in [27], [26] and [1]. However, the delay used is quite long, longer than the action of the oscillator, so any interaction that occurs takes place on a much longer time scale than the action of the oscillators themselves. Here we have investigated a little used and not well understood type of coupling where the interaction is very fast and direct. The observed behaviours may lead to more profound insight into the problem of how pituitary gonadotrophs synchronize hormone release. Experimental evidence shows that gonadotrophs exhibit a variation in their period between action potentials and that the range of periods between gonadotrophs can be quite large. It is also known that calcium oscillations are very important to hormone release. As we saw in this chapter with the realistic model of a go-nadotroph, voltage oscillations occur on a very fast time scale, on the order of milliseconds and calcium oscillations occur on a much slower time scale, on the order of seconds. Yet hormone release is known to occur on a time scale on the order of minutes to hours. It is unknown how gonadotrophs can release hormones on such a long time scale when the processes internal to it 78 Chapter 6. Li-Rinzel Model of a Pituitary Cell 40 20 0 -20 -40 -60 -80 -100 0 5 10 15 20 Time, sec Figure 6.7: Coupled voltage and calcium oscillators of the Li-Rinzel model, occur on much shorter time scales. A possible direction of investigation would be to look for drifting between the voltage and cal-cium oscillators in experimental situations, and to investigate existing models of gonadotrophs to see if they can produce drifting. 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W h e n e v e r figures are p r o d u c e d u s i n g different p a r a m e t e r va lues , t hose va lues t h a t have b e e n a l t e r e d are i n d i c a t e d i n the figure c a p t i o n . A . l Type I Connor Model I - 9Nah(V - VNa)m3 - gK(V - VK)n4 - gL(V - VL) - gA(V - VAC)BA3 aM(V)(l - m) - 6M{V)m aH(V)(l - h) - 6H(V)h aN(V)(l - n) - pN(V)n ( A o o ( y ) - A ) TA(V) TB(V) dV dt dh ~dl dn ~dl dA ~dt dB_ ~dt 83 Appendix A. Model Equations and Parameter Values Parameter Value Parameter Value c vNa VK VL 1 pF 55 mV -72 mV -17 mV I 9Na 9K 9L 10 pA ion n S on n S Z U mM n ^ n S mM Table A . l : Type I Connor model parameter values. with the associated functions: 1 - e-(V+29.7)/W <*h(V) = 0.07e-(v+4 8)/2 0 1 ^oo(V) 1 + e - ( V + 1 8 ) / 1 0 0.01(V + 45.7) an{V) - i _ e _ ( y + 4 5 . 7 ) / 1 0 pn(V) = 0 .125 e -^ + 5 5 - 7 ) / 8 0 "0.0761e(v+94-22)/31-84' 1 + e(V+1.17)/28.93 1.158 TA(V) = 0.3632 + Boo{V) = 1/3 1 _|_ e(y+55.96)/20.12 4 T B ( V ) - 1-24 + 1 4. e(V+53.3)/14.54 2.678 1 4. e(y+50)/16.027 Parameter values are shown in table A . l A.2 Type II Morris-Lecar Model C m l l = I + SL{VL-V)+gKw{VK-V) + gcam00(Vc ^ = ^(VXwcoW-w) 84 Appendix A. Model Equations and Parameter Values Parameter Value Parameter Value VK VCa VL VI Vz 20 pF -84 mV 120 mV -60 mV -1.2 mV 12 mV 0.0666667 I 9K 9Ca 9L v2 Vi 80 pA o nS ° mM A n S * mM 9 i~iS L mM 18 mV 17A mV Table A.2: Type II Morris-Lecar model parameter values. with the associated functions: moo(V) = 0.5 I 1 + tank [ ^ —— V2 V-V3 Woo(V) = 0.5 ^ 1 + tank XW(V) = cf) cosh Vi V-Vz 2V4 Parameter values are shown in table A.2 A.3 Li-Rinzel Model -TT — (Iapp — I'Cal — ICat ~ IK&T ~ ^fC(Ca) ~ ^LJ at cm dq Qoo{V) - q dt rq{V) ' dq qoo(V) - q q = mL,rnT dt , q = hT,n dc I ( c \ 3 \ c 2 l+p[ — — 7 - hs ){a-Cpc)- ver + f 3 c dt V '"\c + dacJ •" j ^ "v"> ""-klT+c* dh ~dl da ~dl a2{din - {c + din)h) m 85 Appendix A. Model Equations and Parameter Values with the associated functions: ICaL = 9CaLm2L(j)c{V) ICaT = gCaTniThT4>c{V) iKdr = 9Kdrn4>K{V) c n c lK{Ca) = 9K{Ca)cnc + k n c M V ) hToo(V) IL = 9L(V-VL) 3pm = -C({IcaL + ICaT) ~ jpump c n p Jpump — Cnv + kpp _ FV Mv) = V FV , q = C,K 1 — e Zq RT 9oo(V) = 1 y - y „ , q = mL,mT,n 1 + e k" V_KL_ 1+e khT Tq{y) = ~v=K 2v-vr , q = mL,mT e  kr + 2e  kr The parameter values are shown in table A.3. 86 Appendix A. Model Equations and Parameter Values Parameter Value Parameter Value Parameter Value Cm 0.0125 pF 30 u-s W 10 us VT -60 mV 22 mV ThT 15 us Tn 22.5 ps p 33.75 i I 0.02 I Cp 1.185 f 0.01 p 0.3 u.M « 2 2 - 5 - din 0.4 \iM dac 0.4 iiM Ver 1.2 k 0.18 a 4.144 nM^pA VP 12 np 2 kp 0A*(MM nc 4 7 pt '•° kc 0.6 jiM 9CaL 0 - 0 6 ^ 9CaT 9Kdr 0 1 — U ' A mM 9K(Ca) 9h 0.02 nS Co 2.5 mM Ci 0.0002 m M K0 5 m M Ki 140 m M ZCa 2 ZK 1 F RT 0-0275 Jfr VL -55 m F VmL -12 kmL 12 m F VmT -30 mV kmT 9 mV VhT -50 mV khT 4 m F Vn -5.1 kn 12.5 m F Table A.3: Li-Rinzel model parameter values. 87 A p p e n d i x B Reduction of the Connor Mode l by Equivalent Potentials This is a reduction of the Connor model according to the method proposed by Kepler, Abbott and Marder in [20]. This method of reduction preserves essential dynamics as well as the original biophysical parameters of the system in question, while significantly reducing the dimensional-ity. This is done using a combination of perturbation analysis and change of variables. This method is supposed to be valid for all conductance-based neuron models of the Hodgkin-Huxley type, although modifications are required if there is calcium dependence in the gating variables. In the case of this Connor model, we can reduce the six ordinary differential equation system to a three ordinary differential equation system. This allows us to examine the dynamics of the system in a three-dimensional phase space. 88 Appendix B. Reduction of the Connor Model by Equivalent Potentials B.l Step 1: Conversion of gating variables into equivalent po-tentials The original system of equations is as follows: ^ = ^ilapp- 9Nam3h{V - VNa) -gKn4(V-VK) - gACA3B{V-VAC) - 9l(V-VI)), dm — = ctm{l - m) - Umm, dh — = ah(l - h) - 3hh, dn — = a n ( l - n) - 0nn, dA — = aA(l-A)-6AA, ^ = aB{l-B)-BBB. First, the gating variables must be converted into the following form: ^- = kXi(V)(xi(V)-xl), where xt = m,h,n,A or B and kXi(V) = aXi(V) + BXi{V), kXi(V)£i(V) = aXi(V). Define the equivalent potentials as follows: Z~i {vXi) = Xi, where xf1 denotes the functional inverse. This is simply restating the gating variable as in the voltage clamp case so that when equilibrium is reached, vXi = Xi. By using the chain rule for differentiation we find: -it \ dvXi dxi X i [ V x i ) c i r ~ ~di' 89 Appendix B. Reduction of the Connor Model by Equivalent Potentials The gating variables in terms of equivalent potentials are: dvm km(V) dt m'(vm) dvh = kh{V) dt ~ h'(vh) dvn _ kn(V) dt n'(vn) dvA _ kA(V) dt ~ A'{vA) dvB _ kB(V) dt ~ B'(VB) (rh{V) - m{vm)), (h{V) - h(vh)), (n(V) - n(u„)) , (AiV) - A(vA)), (B(V)-B{VB))-Even at this point we can see the difference that restating the gating variables in terms of equivalent potentials makes. The first set of plots, figure B . l shows the full model and the gating variables while the second set of plots, figure B.2 shows how the membrane potential and the gating variables appear after they have been restated in terms of equivalent potentials. B.2 Step 2: Determining single potential tp to replace mem-brane potential In this step, we are combining the membrane potential with equivalent potential gating variables that satisfy certain conditions. This combination then replaces the membrane potential in the reduced model. First we determine which equivalent gating variables satisfy dvXi where F(V,{vXi}) = I(V,{vXt}) = 9Narh(vm)3h(vh)(V - VNa) + gKn{vn)\V - VK) + gACA(vA)3B(vB)(V - VAC) + gi(V - Vj) 90 Appendix B. Reduction of the Connor Model by Equivalent Potentials Thus, dvm dF dvh dF dvn dF dvA dvB = gNaZrh{vm) rh'(vm)h(vh)(V -VNa), = gNam{vmfh'{vh){V-VNa), = 9K^n(vnfn'{vn)(V -VK), = gACSA(vA)2A'(vA)B(vB)(V -VAC), dF = gACA(vA)3B'(vB)(V -VAC). Now, taking V = Vrest = — Q8mV and noticing that Ijy a is an inward current while IK and IAc are outward currents then rh'(vm) > 0, h'(vh) < 0, h'(vn) > 0, A'(VA) > 0 and B'(vB) < 0 and also V - VNA < 0, V - VK > 0 and V - VAC > 0. Now we know: <0, >0, dvm dF dvh dF n ^ — >0, dvn dF dvA dF dvB > 0 < 0. Next we must evaluate a condition on the rate constants kXi before we can begin grouping terms. The condition is: 1 dF ckXi dvXi < 1. (B.2) This is done by calculating kXi = aXi + BXi and to determine a;^(wXi), calculating the slope of Xioo when V = — 68mV. We find that, km = 4.91, m > m ) = 0.0045, 91 Appendix B. Reduction of the Connor Model by Equivalent Potentials kh = 0.117, h'(vh) = -0.031, kn = 0.178, n'(vn) =0.0145, kA = 0.073, A'(vA) = 0.005, kB = -0.129, B'{vB) = -0.022. Now checking condition (2) as above: 1 W l = 0.037, ckm dvm 1 dF ckh dvh 1 OF ckn dvn 1 dF ckA dvA 1 dF = 0.188, = 2.142, = 5.813, = 8.94. IckB dvB Clearly, only vm satisfies both (1) and (2). Thus V and vm will be combined to replace the membrane potential in the reduced system. B.3 Step 3: Combining remaining equivalent potentials Here we are going to group the remaining equivalent potentials to determine the number of equations in the final reduced system. These are grouped according to two criteria, the rate constants kXi should be roughly the same and the value of should have the same sign. We already know the values of kXi from the previous step so we need only calculate j^- for the remaining equivalent potentials. We find that: dF — = 0.022, ovh dF — = 0.382, dvn dF 7 T - = 0.422, dvA 92 Appendix B. Reduction of the Connor Model by Equivalent Potentials = -1.157. By looking at the values of kXi and the signs of we can see that Vh, vn and vA should be grouped together while vB is a grouping by itself. B.4 Step 4: Determining the weighting coefficients In this step, we determine the weighting coefficients in each group. Once these are known, we can combine the equivalent potentials to produce our reduced system. The weighting coefficients are always positive, and are constrained so that the sum of a from each grouping adds to one. To calculate the weighting coefficient cto of the original membrane potential, we use the following equation: 'dF dF dV dvm — - c f c — dV m dvm . dF dF ckm - a 0 [ — + dV dvr, n - i = 0 a0 = 0.934,2.104 To determine which root to choose, we recall the constraint that the weighted averages must add to one. Thus, ao = 0.934. Now we determine the weighting coefficients for Vh, vn and vA. It is not necessary to calculate a weighted average for the third group since it contains only vB. These are determined by the following equations: a.2 = a 3 = tt4 = dF dvh = 0.03, dF dvn = 0.46, dF dvA 0.51. dF dF dF dvh dvn dvA dF dF dF dvn dvn dvA dF | dF | dF dvh dvn dvA - l - l 93 Appendix B. Reduction of the Connor Model by Equivalent Potentials B.5 Step 5: Consistency and the reduced model This is the final step of the process. It is here that we check to make sure our reduced model is consistent with the original model. The consistency condition is given as: - l 5 > < i . We only need to carry out these calculations for Vh, vn and VA- We find that: \a2kh + a3kn + a4kA - kh\ [kh + kn + kA]~l = 0.015 < 1, \a2kh + a3kn + a4kA - kn\[kh + kn + kA]~l = 0.15 <C 1, \a2kh + a3kn + a4kA - kA\ [kh + kn + kA]~l = 0.13 <C 1. Therefore our reduced system is given as: dt ~dt d,VB dt ctvlapp - a0F((p,ip), a2fh{4>,ip) + otsfn(<P,^) +aAfA(<p,ip), fB(4>,rp). Written out fully: d<P dt C^ O Iapp - ao[gNam((p) h{^){4> - vNa) + gKn{ip) (<p - VK) +9ACA{ipfB{vB){cp - VAC) + gi(<t>- Vi)] dip kh(<P) dvB dt h'(iP) kB((P) (h(cP)-h(iP)) + « 3 kn(<P) + a4 kA{cp) A'(iP) (A(cP) - A(iP)) B'(vB) (B(CP) - B(VB)) We can also check other qualitative features to ensure that the reduced model is similar to the full model. In figure B.3, we can see that the spikes of the reduced model have the same shape 94 Appendix B. Reduction of the Connor Model by Equivalent Potentials and approximately the same peak to peak time as the full model. In figure B.4 we see that the type I nature of the oscillator has been preserved; the phase response curve is still largely nonnegative. The bifurcation diagram shows that the reduced model is very similar to the full model. However on closer examination (figure B .6 showing the full model, figure B .7 showing the reduced model), it can be seen that there is a small difference in the reduced model for a certain range of applied current. There is a small area of stability in the reduced model with a Hopf bifurcation that gives rise to an unstable oscillatory regime. This added bifurcation is unstable and has a small range, making it unlikely to affect the dynamics of the reduced system. Thus, the reduced model appears to faithfully reproduce the dynamics of the full system, while significantly reducing the number of dimensions required to analyse it. 95 Appendix B. Reduction of the Connor Model by Equivalent Potentials Appendix B. Reduction of the Connor Model by Equivalent Potentials 40 97 Appendix B. Reduction of the Connor Model by Equivalent Potentials Figure B.3: Comparison of the full and reduced Connor models. Time, msec n 5 10 is :o 25 30 35 40 43 Figure B.4: Comparison of the single spike and phase response curves for full and reduced Connor models. 98 Appendix B. Reduction of the Connor Model by Equivalent Potentials 100 , 150 Iapp l(K) , 150 Iapp Figure B.5: Comparison of the bifurcation diagrams for full and reduced Connor models. 99 Appendix B. Reduction of the Connor Model by Equivalent Potentials Figure B.6: Magnification of the bifurcation diagram of full Connor model. 7.2 7.4 7.6 7.8 8.2 8.4 8.fi 8.8 8 9 Figure B.7: Magnification of the bifurcation diagram of reduced Connor model. 100 

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