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Analysis of electrophysiological models of spontaneous secondary spiking and triggered activity Enns-Ruttan, Jennifer Sylvia 1998

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A N A L Y S I S O F E L E C T R O P H Y S I O L O G I C A L M O D E L S O F S P O N T A N E O U S S E C O N D A R Y S P I K I N G A N D T R I G G E R E D A C T I V I T Y By J E N N I F E R SYLVIA E N N S - R U T T A N B . A . S c , Simon Fraser University, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1998 © J E N N I F E R SYLVIA E N N S - R U T T A N , 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M<<\W^mccri CS The University of British Columbia Vancouver, Canada Date CjC+ober- 13 , ) f?o DE-6 (2/88) Abstract We have examined two mathematical models describing the electrophysiology of a neuron and a cardiac cell, respectively, which exhibit an unusual response to high frequency stimulation. For certain parameter sets, both models behave qualitatively like the classic Hodgkin-Huxley equations for a squid giant axon; several brief depolarizing current pulses give rise to a corresponding number of action potentials followed by a return to rest. However, if the parameters are adjusted to reflect certain experimental conditions, a few "spontaneous" action potentials sometimes follow the directly induced action potentials. The number of spontaneous action potentials depends on the number and frequency of action potentials in the original spike train. Our objective was to gain a qualitative understanding of the mechanisms involved in this phenomenon and the effects of certain experimental interventions in promoting or suppressing the occurrence of the spontaneous action potentials. We first studied the Kepler and Marder (KM) model of spontaneous secondary spiking in a crab neuron. Then we examined the DiFrancesco-Noble (DN) equations for a mam-malian cardiac Purkinje fiber which can exhibit a behaviour analogous to spontaneous secondary spiking, referred to in cardiac literature as triggered activity. Using a combi-nation of bifurcation analysis and numerical computation, we showed that spontaneous action potentials are likely to occur in'both models when a critical bifurcation parame-ter is just to the left of a saddle-node of periodics (SNP) bifurcation. In the K M model, the neurotransmitter, serotonin, promoted spontaneous secondary spiking by shifting the bifurcation parameter closer to the SNP. Similarly, in the D N equations, application of digitalis increased the bifurcation parameter (intracellular [Na+]) while high extracellular n [Ca 2 +] shifted the SNP to a lower value, both effects promoting triggered activity. Both models consist of an excitable subsystem and another subsystem that can build up slowly in response to action potentials. Spontaneous action potentials result from the bidirectional feedback between the two subsystems. By simplifying the K M model, we showed that a 3D model can exhibit spontaneous action potentials and that while the shape of the action potentials is unimportant, the relative time constants of the two subsystems are crucial. 111 Table of Contents Abs t rac t i i L is t of Tables v i i i L i s t of Figures i x Acknowledgments x v i 1 I N T R O D U C T I O N 1 1.1 Spontaneous Secondary Spiking 4 1.2 Cardiac Triggered Activity 6 2 S P O N T A N E O U S S E C O N D A R Y S P I K I N G 10 2.1 Experimental Evidence for Spontaneous Secondary Spiking 11 2.2 Kepler and Marder's Model 13 2.3 Spatial and Temporal Behaviour of Kepler and Marder's Model 16 2.4 Reduction of Kepler and Marder's Model 22 2.4.1 Reduction Using a FitzHugh-Nagumo Approximation 24 2.4.2 Reduction Using the Method of Equivalent Potentials 25 2.5 Analysis of the Fast Subsystem Using the Slow Inward Current as a Pa-rameter 27 2.5.1 Analysis of the zFN Model Assuming Constant Iz 29 2.5.2 Analysis of the M E P Model Assuming Constant Iz 37 2.6 Effect of the Slow Inward Current in the Full 3D and 5D Models 39 i v 2.6.1 The Role oil, 40 2.6.2 The Role of ks 52 2.7 Conclusions 75 3 C A R D I A C T R I G G E R E D A C T I V I T Y A N D T H E D I F R A N C E S C O -N O B L E E Q U A T I O N S 80 3.1 Introduction to Cardiac Electrophysiology 81 3.1.1 Introduction to the Biology of the Heart 81 3.1.2 Electrophysiology of a Cardiac Cell 84 3.1.3 Electrophysiology and Phases of the Cardiac Action Potential . . 85 3.1.4 Afterdepolarizations and Triggered Activity 87 3.2 Theoretical Effects of Overdrive and C a 2 + Loading 89 3.2.1 Normal Effects of Overdrive (Overdrive Suppression) 89 3.2.2 Calcium Overload and Triggered Activity 91 3.3 Experimental Interventions for Promoting and Suppressing Delayed Af-terdepolarizations and Triggered Activity 93 3.3.1 High Extracellular C a 2 + Concentrations 93 3.3.2 Application of Catecholamines 94 3.3.3 Inhibition of the Na-K Pump 95 3.3.4 Altering the Sarcoplasmic Reticulum Dynamics 95 3.3.5 Delayed Afterdepolarizations and Cardiac Arrhythmias 96 3.4 Modeling Studies of Cardiac Cells and Triggered Activity 97 3.4.1 The DiFrancesco-Noble Model and Its Successors 98 3.4.2 The Luo-Rudy Models 98 3.4.3 Analysis and Simulations of the Models 99 3.5 Introduction to the DiFrancesco-Noble Equations 100 v 3.5.1 What the DiFrancesco-Noble Equations Include 101 3.5.2 How the Resting Membrane Potential and Concentrations Are Set 102 3.5.3 Ionic Events During an Action Potential 105 3.6 Reducing the Dimension of the DN Equations 107 3.6.1 Treating Slow Variables as Parameters 108 3.6.2 Fixing Variables That Do Not Qualitatively Alter the Dynamics of Interest I l l 3.6.3 Viewing the DiFrancesco-Noble Equations as Two Coupled Oscil-lators 112 3.7 Varghese and Winslow's Results 114 3.7.1 Subcellular Oscillator 114 3.7.2 Coupled System 117 3.8 Normal and Calcium-Driven Firing: Which Oscillator Drives Which? . . 118 4 R E S U L T S F O R T H E D I F R A N C E S C O - N O B L E E Q U A T I O N S 125 4.1 Subcellular Oscillator 125 4.1.1 Simplifying the Equations 126 4.1.2 Adjusting the Parameters to Simulate Experimental Interventions 127 4.1.3 Subcellular Oscillations and Calcium-Induced Calcium Release . . 131 4.2 Simulating Experimental Interventions in the Pacemaking System . . . . 133 4.2.1 Normal Behaviour 134 4.2.2 The Effects of Digitalis Poisoning and High Ca 0 134 4.2.3 The Effects of Overdrive 143 4.3 Simulating Experimental Interventions in the Quiescent System 148 4.3.1 Normal Behaviour 148 4.3.2 The Effects of Digitalis Poisoning and High Ca 0 150 v i 4.3.3 The Effects of Prolonged Overdrive 160 4.4 Examination of the Effects of Changes in K t 162 4.5 Conclusions 166 5 C O N C L U S I O N S A N D D I S C U S S I O N 172 5.1 Comparison of the K M Model and the D N Equations 172 5.2 Biological Implications 176 5.3 Cardiac C a 2 + Dynamics and the DN Equations 178 Bib l iography 181 Appendices 188 A K E P L E R A N D M A R D E R ' S M O D E L 188 B G U I D E T O B I F U R C A T I O N D I A G R A M S 190 C T H E D I F R A N C E S C O - N O B L E E Q U A T I O N S 195 C l The Equations 195 C.2 Simulations 202 D B I F U R C A T I O N D I A G R A M S F O R T H E P A C E M A K I N G D I F R A N -C E S C O - N O B L E E Q U A T I O N S F O R V A R I O U S C a D A N D A N a K 207 E B I F U R C A T I O N D I A G R A M S F O R T H E Q U I E S C E N T D I F R A N C E S -C O - N O B L E E Q U A T I O N S F O R V A R I O U S C a 0 A N D A N a K 211 vii L i s t of T a b l e s 3.1 Magnitude of membrane currents at steady state in the D N equations . . 104 4.1 Behaviour of pacemaking D N equations depends on Ca 0 and A^aK • • • 143 4.2 Behaviour of quiescent D N equations depends on C a 0 and A^aK 151 C l Currents in the D N equations 197 C.2 Reversal potentials in the D N equations 198 C.3 Definitions of a and (3 for the gating variables in the D N equations . . . 198 C.4 Standard parameter set for the D N equations 199 C.5 Initial conditions for the quiescent and pacemaking versions of the D N equations 202 vm Lis t of Figures 1.1 Example of spontaneous secondary spiking 3 2.1 Membrane potential along axon at several time steps 17 2.2 Response of space-clamped axon to two current pulses 18 2.3 Peaks of action potentials as a function of distance and time for high and low i , 20 2.4 Peaks of action potentials as a function of distance and time when two action potentials are initiated at each end of neuron . 22 2.5 Reduction of Kepler and Marder's equations from 5D to 3D 23 2.6 Monotonicity of the gating function x^v) ensures invertibility 26 2.7 Comparison of action potentials produced by the K M and M E P models . 28 2.8 Nullclines for excitable and bistable FitzHugh-Nagumo equations . . . . 30 2.9 Stability of fixed points in the FitzHugh-Nagumo equations 33 2.10 Hopf bifurcation points in FitzHugh-Nagumo equations for different a . . 33 2.11 Response of F N equations to sudden steps in Iz 35 2.12 Bifurcation diagram of the F N equations using Is as a bifurcation parameter 36 2.13 Nullsurfaces and nullclines in the M E P model 38 2.14 Bifurcation diagrams using Iz as bifurcation parameter for M E P and K M models 39 2.15 The fixed points of the zFN equations depend on J s 41 2.16 Fixed points of the zFN equations as a function of Is 44 ix 2.17 Bifurcation diagram for the zFN equations using Is as the bifurcation parameter . . . 46 2.18 Bifurcation diagrams for 5D K M model and 3D M E P model using Is as the bifurcation parameter 48 2.19 Responses of M E P model for different values of Is and applied current frequency / 50 2.20 Responses of zFN model for different values of Is and applied current frequency / 51 2.21 Bifurcation diagrams for 3D zFN model using Is as the bifurcation param-eter for large ks 54 2.22 Nullclines for the 2D zFN equations with ks = er for various Is . . . . . . 56 2.23 Bifurcation diagram for the degenerate z F N equations where ks — er using Is as the bifurcation parameter 57 2.24 Bifurcation diagram for the zFN equations using Is as the bifurcation parameter for small ks 59 2.25 Duty cycle and period of tonic firing in the F N equations 61 2.26 Predicted value of z as a function of Is during continuous spontaneous secondary spiking in the zFN equations 62 2.27 Asymptotic and current values of z dictated by Iz 63 2.28 Simulation of zFN equations with small ks: continuous spontaneous sec-ondary spiking 64 2.29 Simulation of zFN equations with small ks: no spontaneous secondary spiking following low frequency stimulation 65 2.30 Simulation of zFN equations with small ks: no spontaneous secondary spiking after short drive 66 x 2.31 Simulation of zFN equations with small ks: a few spontaneous secondary spikes 67 2.32 Simulation of zFN equations with small ks: more spontaneous secondary spikes following long drive 68 2.33 Simulation of zFN equations with small ks: the most spontaneous sec-ondary spikes follow long, high frequency drives 69 2.34 Bifurcation diagrams for 5D K M model with large and small ks values using Is as the bifurcation parameter 72 2.35 Bifurcation diagrams for 3D M E P model with large and small ks values using Is as the bifurcation parameter 73 2.36 Duty cycle and period of tonic firing in the fast subsystem of the K M and M E P equations 74 2.37 Predicted value of z as a function of Is during continuous spontaneous secondary spiking in the K M and M E P equations 74 3.1 Electrical conduction system of the heart 82 3.2 Typical ionic concentrations and currents found in a cardiac cell 85 3.3 Phases of typical cardiac action potentials 86 3.4 Sketches of early and delayed afterdepolarizations 88 3.5 DADs and triggered activity in a canine cardiac Purkinje fiber exposed to digitalis 96 3.6 Schematic view of the DiFrancesco-Noble equations 101 3.7 Steady state values of C a u p and Ca r e / as functions of Ca8- 105 3.8 Illustration of the complexity of interactions in the full 16D D N equations 108 3.9 "Steady state" values of Na8- and K; in quiescent and pacing fibers for various A^aK and If 109 xi 3.10 Response of Na; and K; to overdrive '. 110 3.11 Illustration of the reduced complexity of the 12D approximation to the D N equations 113 3.12 Varghese and Winslow's subcellular oscillator 115 3.13 Bifurcation diagrams for Varghese and Winslow's subcellular oscillator using Na; as the bifurcation parameter 116 3.14 Two-parameter curve of Hopf bifurcations in the f-Na, parameter space . 116 3.15 Defining the regions in the bifurcation diagram for the D N equations as a function of the bifurcation parameter Na; 119 3.16 Simulations of the transition between normal and calcium-driven tonic firing in the D N equations 120 3.17 Simulations of the transition from calcium-driven tonic firing to subthresh-old oscillations in the D N equations 121 3.18 Comparison of subthreshold Ca; oscillations in the isolated subcellular oscillator and the coupled system 123 4.1 Subthreshold subcellular oscillator 127 4.2 Two-parameter curves of Hopf bifurcations for the voltage-clamped sub-cellular oscillator in the u-Na8- parameter space with and without p and IsiCa 128 4.3 The intersection of hca with 2Ipjaca determines the steady state value of Ca; 129 4.4 Bifurcation diagram for the voltage-clamped subcellular oscillator as a function of the bifurcation parameter Na; for Ca0=2.5 and 8.0 m M . . . 130 4.5 Two-parameter curve of Hopf bifurcations for the voltage-clamped subcel-lular oscillator in the Ca 0-Na; parameter space 131 xii 4.6 Simulation of stable oscillations in the subcellular oscillator 132 4.7 Simulation of the transient response of the subcellular oscillator to an initially high C a u p 133 4.8 Bifurcation diagram as a function of the bifurcation parameter Na,- for pacemaking D N equations under normal conditions 135 4.9 Simulation of pacemaking D N equations under normal conditions . . . . 136 4.10 Bifurcation diagrams as a function of the bifurcation parameter Naj for pacemaking version of D N equations at various A^aK and Ca G 137 4.11 Simulation of pacemaking D N equations with digitalis poisoning 139 4.12 Simulation of pacemaking DN equations with intermediate and high Ca 0 140 4.13 Comparison of subthreshold Ca t oscillations of isolated and coupled system for pacemaking D N equations with Cao=8.0 m M 142 4.14 Simulation of effect of prolonged overdrive on Na; in pacemaking D N equa-tions 144 4.15 Simulation of effect of prolonged overdrive on pacemaking D N equations with Ca 0=3.5 m M 145 4.16 Simulation of pacemaking D N equations with Ca 0=3.5 m M in response to short overdrive 147 4.17 Bifurcation diagram as a function of the bifurcation parameter Na; for quiescent version of D N equations under normal conditions 149 4.18 Simulation of quiescent D N equations under normal conditions in response to a train of stimuli 150 4.19 Bifurcation diagrams as a function of the bifurcation parameter Na; for quiescent version of D N equations at various Ca 0 152 4.20 Simulation of quiescent D N equations with CaD=4.5 m M in response to overdrive 154 xiii 4.21 Simulation of quiescent D N equations with Cao=6.0 m M in response to overdrive 155 4.22 Simulation of quiescent D N equations with Cao=7.0 m M in response to overdrive 157 4.23 Bifurcation diagrams as a function of the bifurcation parameter Na t for quiescent version of D N equations at various AjvaA' and C a 0 158 4.24 Simulation of quiescent D N equations with digitalis poisoning in response to a train of stimuli 159 4.25 Simulation of quiescent D N equations with Ca 0=4.5 m M and A/vaA"=0.8 in response to a train of stimuli 160 4.26 Simulation of effect of prolonged overdrive on quiescent D N equations with CaG=4.5 m M 161 4.27 Two-parameter continuation in (Na,-, K t ) of critical fixed points of quies-cent version of DN equations 163 4.28 Bifurcation diagrams as a function of the bifurcation parameter Na, for quiescent version of D N equations at normal and low K; 165 A. l Voltage dependence of the activation curves and time constants in Kepler and Marder's model 189 B. l Typical one-parameter bifurcation diagrams 192 B. 2 Typical two-parameter bifurcation diagram 194 C l Activation curve f2oo and time constant r/ 2 of f2 in the D N equations . . 200 C. 2 Activation curves of gating variables in the D N equations 200 C.3 Time constants of slow gating variables in the D N equations 201 C.4 Time constants of faster gating variables in the DN equations 201 xiv C.5 Simulation of the full D N equations 203 C.5 Simulation of the full D N equations (continued) 204 C.5 Simulation of the full D N equations (continued) 205 C. 5 Simulation of the full D N equations (continued) 206 D. l Bifurcation diagram as a function of the bifurcation parameter Na; for pacemaking D N equations with CaQ=2.5 m M and AwaA'=0.85 208 D.2 Bifurcation diagram as a function of the bifurcation parameter Na; for pacemaking D N equations with Ca 0=3.5 m M and A^ a^-=1.0 209 D. 3 Bifurcation diagram as a function of the bifurcation parameter Na; for pacemaking D N equations with Cao=8.0 m M and ANaK=l.O 210 E. l Bifurcation diagram as a function of the bifurcation parameter Na; for quiescent D N equations with CaG=4.5 m M and ^4^^=1.0 212 E.2 Bifurcation diagram as a function of the bifurcation parameter Na; for quiescent D N equations with Cao=6.0 m M and ANaK=1.0 213 E.3 Bifurcation diagram as a function of the bifurcation parameter Na; for quiescent D N equations with Cao=7.0 m M and A^aK=1.0 214 E.4 Bifurcation diagram as a function of the bifurcation parameter Na; for quiescent D N equations with Ca0=2.5 m M and A/vaA"=0.8 215 E.5 Bifurcation diagram as a function of the bifurcation parameter Na; for quiescent D N equations with Ca0=4.5 m M and AjvaK'=0.8 216 xv Acknowledgements I would like to thank my supervisor, Robert Miura, for his advice and support during my six years of graduate school. I would also like to acknowledge the contributions of my thesis committee (Leah Edelstein-Keshet, Wayne Nagata, Ernest Puil , Michael Walker, Michael Ward), as well as my university examiners (Yue-Xian L i , Peter Lawrence). In addition, I would like to thank my external examiner, Raimond Winslow, and his col-league, Tony Varghese, for their helpful suggestions and encouragement regarding my research on cardiac triggered activity. I also want to thank Tom Kepler for helping me to get started on my spontaneous spiking research. I would like to acknowledge NSERC and the Faculty of Graduate Studies at U B C for providing the financial support that made this work possible. I also would like to thank the director of the Institute of Applied Mathematics, Uri Ascher, and the computer systems manager, Julie Ranada, for making sure we had a good place to work. I want to thank Judy Maxwell, the administrative secretary, for her constant empathy and support. I also want to thank a long list of I A M students, who shared the lab with me, for their technical support and their friendship: Pete New-bury, Athan Spiros, David Iron, John (Marty) Anderies, Michele Titcombe, Greg Lewis, Lonxiang Dai, and especially Gerda de Vries, Lynn van Cofler, and Patrick Doran-Wu. I would also like to acknowledge some special friends, Karen and Steve. Lastly, and most importantly, I would like to thank my family for always being there for me. Thanks Mom, Dad, Heather, Russell, and Nicole! Thanks also to my husband's family, especially Floyd and Gail, for accepting me. And most of all, thanks to my husband, Rob, for his endless support and understanding. x v i Chapter 1 I N T R O D U C T I O N Living cells are enclosed by a cell membrane or sarcolemma, which separates the outside or extracellular space from the inside of the cell, known as the intracellular space. The cell membrane controls what enters or leaves the cell. Embedded in the cell membrane are proteins, known as ion channels, which allow certain types of ions, such as N a + , K + , C a 2 + , and C l - , to pass through the cell membrane. Because the phospholipid bilayer, of which the cell membrane is composed, separates charge, an electrical potential develops across the cell membrane as a result of the movement of the ions. This membrane potential can regulate the conductance of some ion channels, so that these ion channels open and close both with time and voltage. Ions are driven through these ion channels as a result of ion concentration gradients and electric potential differences across the cell membrane. Normally, the membrane potential is at a steady state called the resting potential, and we say the cell is quiescent. At this rest state, the membrane potential is negative inside the cell relative to the outside. In an excitable cell, the opening and closing of ion channels can lead to a sequence of changes in the membrane currents which can result in a fast spike in the membrane potential, known as an action potential. An action potential can occur when an excitatory stimulus (a brief inward current pulse) depolarizes (raises) the membrane potential past a certain threshold voltage. A sharp upstroke in the membrane potential occurs due to a net inward current. This is followed by a plateau at a high membrane potential where the inward and outward currents balance, and then 1 Chapter 1. INTRODUCTION 2 a sharp repolarization (reduction in membrane potential) back to resting potential due to the eventual dominance of the outward currents. Sometimes the membrane potential undershoots the resting potential and we say that the cell is hyperpolarized before it returns to resting potential. If the cell continuously generates action potentials, we say that the cell exhibits tonic firing. Act ion potentials can propagate as a wave of depolarization along the cell membrane. The Hodgkin-Huxley equations model one of the simplest neuronal structures possible: the squid giant axon. These classical equations often are thought of as the prototype for an excitable cell. Consisting of equations describing a fast inward N a + current, a delayed outward K + current, and a less significant leakage current, they define a typical action potential wi th its fast upstroke, plateau, and rapid recovery to resting potential. The simple geometry and distribution of ion channels in the squid giant axon allows this action potential to propagate from one end to the other without changing shape. However, the wide variety of ionic currents, geometries, inhomogeneities, and connectivities found in other types of excitable cells lead to far more complex electrophysiological behaviour, including bursting, bistable behaviour, synchronization, and a variety of signal processing functions such as integration of incoming signals. In the space-clamped squid giant axon, several brief depolarizing pulses wi l l result in a corresponding number of action potentials followed by a return to resting potential. In this thesis, we examine two biological models in which one or more action potentials appear to arise spontaneously following such a train of action potentials. The number of spontaneous action potentials is highly dependent on the number of action potentials in the in i t ia l spike train as well as the frequency of stimulation. Figure 1.1 is an example of the frequency dependent behaviour discussed here. In all three panels, two short depolarizing pulses lead to two directly induced action potentials. When the time lag between pulses is relatively long (panel (a)), a small depolarizing Chapter 1. INTRODUCTION 3 hump, known as a delayed afterdepolarization ( D A D ) , follows the two action potentials but then the system returns to rest. When the time lag is reduced (the frequency of cur-rent injection is increased) as shown in panel (b), three "spontaneous secondary spikes" arise as a result of a larger D A D , followed by a D A D and a return to rest. If certain parameters of the system are adjusted, the spontaneous secondary spiking can persist indefinitely, as shown in panel (c). Figure 1.1: Example of the frequency dependence of spontaneous secondary spiking. In each panel, two small depolarizing current pulses induce two action potentials, (a) Low frequency stimulation leads to a D A D followed by a return to rest, (b) Higher frequency stimulation leads to three spontaneous secondary spikes, followed by a D A D and a return to rest, (c) Changing the parameters can result in spontaneous secondary spiking that lasts indefinitely. We examine mathematical models of two biological systems that exhibit this frequency dependent type of behaviour. The first example comes from a model developed by Kepler and Marder [36] to describe some "extra" spikes arising in the stomatogastric ganglion of a crab. They coined this behaviour, "spontaneous secondary spiking". The second example comes from cardiac electrophysiology. The "extra" spikes are referred to as Chapter 1. INTRODUCTION 4 triggered activity. In the following two sections, we introduce each of these phenomena in turn and outline what is to come in the remaining chapters. 1.1 Spontaneous Secondary Spik ing The first example of this fascinating frequency dependent phenomenon was revealed by some experimental findings of Meyrand et al. [45] who studied the lateral gastric (LG) axon in the stomatogastric nervous system of the crab Cancer borealis. They discovered that under certain conditions, action potentials traveling along the axon of an L G neuron can spontaneously initiate additional action potentials at peripheral sites up to 2 cm away from the stomatogastric ganglion. These action potentials would travel antidromically towards the soma and orthodromically towards the gastric mill muscles. They were shown to modify the contraction of some of the muscles. Meyrand et al showed that the number of peripherally initiated action potentials was strongly influenced by the presence of the neurotransmitter, serotonin, as well as an imposed depolarization of the soma. These extra action potentials are very interesting from the point of view of feedback and control of the gastric mill muscles. The exact mechanism by which these peripherally initiated action potentials occur is not known. However, Kepler and Marder [36] set up a "minimal" model to elucidate one possible mechanism suggested by Meyrand et al. [45]. The key idea was that a slow inward current, possibly a slow C a 2 + current or a Ca2 +-dependent current, might be initiated by incoming action potentials in the peripheral spike initiation zone. Thus, their minimal model consisted of a slow inward current plus a traditional Hodgkin-Huxley type system of equations representing a fast inward N a + current, a delayed rectifier K + current, and a leakage current. They demonstrated through numerical simulations that this model does indeed give rise to what they coined "spontaneous secondary spikes". The model Chapter 1. INTRODUCTION 5 exhibits a very interesting frequency dependence on incoming signals and is responsive to a "serotonin concentration" through adjustment of a particular conductance parameter. While Kepler and Marder's model exhibits some very interesting spatial behaviour, in studying their model, we have mostly limited our study to its space-clamped behaviour. However, a good understanding of the space-clamped behaviour provides some insight into the spatial behaviour since the distance between action potentials in the spatial model is, in a certain sense, analogous to the time lag between action potentials in the space-clamped model. In Chapter 2, we examine the ionic mechanism of spontaneous secondary spiking. We begin by reviewing the experimental evidence for spontaneous secondary spiking. We then present Kepler and Marder's model and discuss its temporal and spatial behaviour [36]. We review the results of Kepler and Marder's work, which provide the basis for our own work. Kepler and Marder's five-dimensional model consists of four differential equations describing a Hodgkin-Huxley type excitable cell plus an additional equation describing the gating conductance for the slow inward current. The objective of this research is to understand how the slow inward current modulates the excitable system to yield spontaneous secondary spiking and how the model's parameters affect its behaviour. We begin our analysis by showing that the five-dimensional model can be reduced to three dimensions while still preserving the essential behaviour of spontaneous secondary spiking. The simplest three-dimensional approximation is achieved by replacing the four-dimensional Hodgkin-Huxley equations by the two-dimensional FitzHugh-Nagumo equa-tions. While the FitzHugh-Nagumo equations are little more than a caricature of the Hodgkin-Huxley equations, the new three-dimensional system can still exhibit sponta-neous secondary spiking given appropriate choices of the parameters. A quantitatively more accurate model is produced by reducing the dimension of the Hodgkin-Huxley equations more formally by Kepler et aVs method of equivalent potentials [35]. Chapter 1. INTRODUCTION 6 We take advantage of the fact that the slow inward current has a much slower time constant than the time constants of the excitable subsystem. By assuming for the moment that the slow current is so slow (i.e., quasi-steady) that it can be treated as a parameter, we can further reduce the three-dimensional systems to two dimensions. This enables us to perform a combination of phase-plane analysis, bifurcation analysis, and simulation to gain insight into the mechanism behind spontaneous secondary spiking. While the analysis above is somewhat illuminating, the time constant of the slow inward current is actually quite important. In particular, if it is too slow, spontaneous secondary spiking is not likely to occur after spike trains of moderate duration. Alter-natively, we show using the FitzHugh-Nagumo system that if the time constant is too fast, the system becomes degenerate and the resulting two-dimensional subsystem also cannot exhibit spontaneous secondary spiking. We also examine the effect of the maxi-mum amplitude of the slow inward current on the occurrence of spontaneous secondary spiking. Using the numerical bifurcation package AUT086, we predict when spontaneous secondary spiking is likely to occur. 1.2 C a r d i a c T r i g g e r e d A c t i v i t y Our motivation in studying a phenomenon in cardiac electrophysiology known as "trig-gered activity", is the striking similarity to Kepler and Marder's "spontaneous secondary spiking". A number of cardiac systems, including the simian mitral valve [78], the canine coronary sinus [79], and the sheep [2] and canine [19, 20, 21, 22, 23, 66] Purkinje fibers exhibit "extra" or "spontaneous" action potentials in response to high frequency or pro-longed stimulation. In the experiments of Wit et al. (see [80], Figure 11), for instance, a simian mitral valve was stimulated fifteen times at several different cycle lengths or frequencies. As in the case of spontaneous secondary spiking, low frequency stimulation Chapter 1. INTRODUCTION 7 fails to evoke secondary spikes. However, as the frequency of stimulation increases, DADs grow in amplitude and eventually trigger secondary spikes. An even higher frequency of stimulation results in sustained secondary spiking. According to Cranefield and Aronson [10], "a triggerable but quiescent fiber remains quiescent until it has been excited by a locally evoked action potential or by an action po-tential that propagates into the fiber from a different site". This is completely analogous to Kepler and Marder's model [36] of spontaneous secondary spiking. While spontaneous secondary spiking is thought to perform an important signal pro-cessing function by influencing how long certain muscles contract [45], cardiac triggered activity involving DADs is thought to be associated with cardiac arrhythmias rather than normal behaviour [1, 5, 10, 32]. The relatively simple model for spontaneous secondary spiking is not adequate for describing cardiac triggered activity. We will see that triggered activity results from the interaction of two subsystems: the membrane oscillator which accounts for the currents in the cell membrane that give rise to the action potential, and the subcellular oscillator which describes the dynamics of C a 2 + inside the cell. In Chapter 3, we provide the necessary biological background necessary for under-standing cardiac triggered activity. We explain how C a 2 + can build up in the cell, result-ing in a condition known as calcium overload which can destabilize the C a 2 + subsystem and ultimately lead to triggered activity. We discuss experimental interventions which are known to promote and suppress this unusual behaviour. Then we review prior modeling work relevant to triggered activity. In the latter part of Chapter 3, we discuss one particular model, known as the Di-Francesco-Noble equations, which describes a mammalian cardiac Purkinje fiber. While this model has a number of drawbacks (which we will discuss later), it is one of the simplest models which contains the necessary components for yielding triggered activity. We examine the dynamics of this model and reduce its dimension where possible to make Chapter 1. INTRODUCTION 8 it easier to analyze. Then we review some important and relevant work by Varghese and Winslow [68, 69] which provides the basis for the remainder of our analysis. Lastly, we carefully define the difference between normal tonic firing and another type of tonic firing, known as calcium-driven firing, since it is crucial to understanding the mechanism of triggered activity. Digitalis glycosides have been used for centuries in the treatment of congestive heart failure and other ailments [18, 34]. Yet, digitalis must be administered carefully since an incorrect dosage can lead to digitalis toxicity. Digitalis toxicity has been associated with certain cardiac arrhythmias in vivo. At the same time, digitalis has been shown to promote DADs and triggered activity in vitro. In Chapter 4, we examine the effects of various experimental interventions, including application of digitalis, in promoting triggered activity in the DiFrancesco-Noble equa-tions. The DiFrancesco-Noble equations can exhibit either a pacemaking mode (slow tonic firing) or a quiescent mode depending on its parameters. If the pacemaker is driven at a frequency higher than its own intrinsic frequency, we say that it is overdriven. Under the right circumstances, driving the DiFrancesco-Noble equations at high frequency or for a prolonged period of time may lead to calcium-driven firing, which is referred to as triggered activity or overdrive excitation. We look at the effects of applying a high extracellular C a 2 + concentration or digitalis (which inhibits the Na-K pump activity) in combination with high frequency or lengthy pacing. Using a combination of numerical bifurcation analysis and simulation, we demonstrate why triggered activity is more likely to occur under some conditions than others. Lastly, we discuss some of the deficiencies of the DiFrancesco-Noble equations. Finally, in Chapter 5, we discuss some of the similarities and differences between spontaneous secondary spiking and triggered activity. We discuss the biological impli-cations of these two phenomena and the importance of our results. Lastly, we discuss Chapter 1. INTRODUCTION 9 the state of current modeling efforts in the area of cardiac triggered activity and suggest some areas of future work. Chapter 2 S P O N T A N E O U S S E C O N D A R Y S P I K I N G In this chapter, we examine Kepler and Marder's model of spontaneous secondary spiking [36]. This "minimal" model was set up to explain one possible mechanism for the seem-ingly spontaneous initiation of action potentials at a peripheral site on the lateral gastric axon in the stomatogastric nervous system of a crab. It consists of a four-dimensional Hodgkin-Huxley type system of equations describing an action potential plus an addi-tional slow inward current. Our objective is to understand how the slow inward current and the faster subsys-tem describing the action potential dynamics interact to produce spontaneous secondary spiking. In particular, we determine how certain critical parameters describing the slow inward current determine whether or not spontaneous secondary spiking can occur. We also show that spontaneous secondary spiking does not require a five-dimensional model, and that three dimensions are sufficient. We begin in Section 2.1 by reviewing Meyrand et a/.'s experimental evidence for spontaneous secondary spiking [45], upon which Kepler and Marder's model is based. Then in Section 2.2, we describe Kepler and Marder's model. We discuss their results, including the spatial and temporal behaviour of the model, in Section 2.3. In order to better understand the crucial elements which give rise to spontaneous secondary spiking, we reduce Kepler and Marder's five-dimensional (5D) model to two three-dimensional (3D) models, which behave qualitatively the same as the original 5D model. In Section 2.4, we replace the 4D Hodgkin-Huxley type system by the much 10 Chapter 2. SPONTANEO US SECONDARY SPIKING 11 s i m p l e r 2 D F i t z H u g h - N a g u m o equat ions to get a 3 D m o d e l w h i c h we c a l l the z F N m o d e l . T o get a m o r e q u a n t i t a t i v e l y accurate a p p r o x i m a t i o n , we also use a r e d u c t i o n scheme deve loped by K e p l e r , A b b o t t , a n d M a r d e r [35] ca l l ed the m e t h o d of equiva len t po ten t ia l s ( M E P ) to reduce the 4 D H o d g k i n - H u x l e y sys t em to 2 D , r e su l t ing i n a 3 D m o d e l (the M E P m o d e l ) . T h e r e su l t i ng 3 D models of spontaneous secondary s p i k i n g are t h e n a n a l y z e d by e x p l o i t i n g the difference i n the t i m e scales of the fast ac t ion p o t e n t i a l d y n a m i c s of the 2 D subsys tems and the slower d y n a m i c s of the slow i n w a r d current . In Sec t i on 2.5, we t reat the s low i n w a r d current as a pa ramete r i n the 3 D z F N and M E P mode l s . T h i s a l lows us to use a c o m b i n a t i o n of phase-plane analysis and b i fu rca t ion analys is on the r e su l t i ng 2 D systems to ga in an unde r s t and ing of w h y spontaneous secondary s p i k i n g occurs . T h e n i n Sec t ion 2.6, we r e t u r n to the fu l l 3 D and 5 D mode l s . W e e x a m i n e the b i f u r c a t i o n s t ruc tu re of these mode ls to see under wha t cond i t ions spontaneous secondary s p i k i n g is l i k e l y to occur . In pa r t i cu l a r , we e x a m i n e the role of two c r u c i a l parameters desc r ib ing the s low i n w a r d current , n a m e l y i ts m a x i m u m a m p l i t u d e and rate constant , to see how these inf luence the m o d e l s ' behaviours . F i n a l l y , i n Sec t ion 2.7, we s u m m a r i z e our resul ts for these mode l s . 2.1 Experimental Evidence for Spontaneous Secondary Spiking M e y r a n d et al. [45] s tud ied the l a te ra l gas t r ic ( L G ) axon w h i c h innervates the gas t r ic m i l l muscles (gm5b, gm6 and gm8a,b) tha t con t ro l the gast r ic m i l l , the par t of the crab i n v o l v e d i n m a c e r a t i o n (chewing) of food. T h e movement of these muscles is con t ro l l ed by the gas t r ic r h y t h m , w h i c h has an ac t ive phase where h i g h frequency a c t i o n po ten t ia l s r ide o n top of a slow depo l a r i z a t i on wave, fo l lowed by a h y p e r p o l a r i z e d si lent phase. M e y r a n d Chapter 2. SPONTANEO US SECONDARY SPIKING 12 et al noticed that when the muscles were attached to the L G axon, extra low frequency action potentials appeared during the hyperpolarized phase. Because the hyperpolarized phase is about -65 mV and the threshold for normal spike initiation in the soma is about -45 mV, they concluded that these action potentials must have been initiated at some peripheral site on the axon. By placing several electrodes along the length of the axon, they determined that the peripheral spike initiation site was in a portion of the lateral gastric neuron called the lateral ventricular nerve. The action potentials initiated here propagated both orthodromically towards the muscles and antidromically towards the soma. When the muscles were removed from the L G axon, the peripheral spike initiation zone was no longer active. However, application of serotonin to the peripheral spike initiation zone reactivated it. To understand the effects of serotonin and of the depolarization in the soma, Meyrand et al suppressed the gastric rhythm by blocking impulse conduction into the stomatogastric nerve using a sucrose well [45]. At a fixed serotonin concentration, they depolarized the soma several times by current injection to the potential normally seen during the gastric rhythm. After the first depolarization, no peripherally initiated spikes were seen. However, after several depolarizations, the secondary spikes began to appear again. They found that the longer the duration of the depolarization, the more peripheral spikes were initiated. Serotonin also profoundly affected the number of peripheral spikes initiated. For high concentrations of serotonin, relatively short depolarization pulses were required to elicit secondary spiking. In fact, for very high concentrations of serotonin, the peripheral spiking could last for minutes after the instigation of only a single action potential from the soma. For low serotonin concentrations, lengthy depolarizations in the soma were required to get peripheral spiking. Chapter 2. SPONTANEO US SECONDARY SPIKING 13 One can speculate on the functional significance of the peripheral spiking. The exper-imental evidence suggests the possibility of a mechanism involving serotonin for feedback between the contracting muscles and the peripheral spike init iation zone. Whi le the exact feedback mechanism is unknown, the effect of the peripherally initiated spikes on some of the muscles is known. Meyrand et al. demonstrated that the low frequency peripherally init iated spikes serve to lengthen the time of contraction of the gm5b muscles. They point out that such a lengthened contraction may allow the crab to clear food out of its gastric m i l l . 2.2 Kepler and Marder's Model To explore one possible mechanism for the init iation of peripheral spikes observed by Meyrand et al, Kepler and Marder [36] set up a minimal model. They chose to use Hodgkin-Huxley type dynamics to represent a fast inward N a + current, a slow rectifier K + current, and a leakage current in their model of the peripheral spike ini t iat ion zone. Since Meyrand et a/.'s experiments were performed on crabs, they chose the parameters suggested by Connor et al [9] for crustacean axons. Based on their experimental findings, Meyrand et al suggested that the peripheral spike init iat ion zone might exhibit an additional slow inward current, such as an inward C a 2 + current or a C a 2 +-dependent inward current. The current was assumed to be slow because several depolarizations were often required before its effect could be seen. Not knowing the exact dynamics of this hypothetical inward current, Kepler and Marder assumed that its gating conductance z would obey a relaxation equation of the form where v is the membrane potential and ks is a slow rate constant. Typica l ly 9(v — Vj) dt ks[9{v - VT) - z] Chapter 2. SPONTANEO US SECONDARY SPIKING 14 would be a sigmoidal curve. Kepler and Marder chose to approximate it by a step function 1 v - VT > 0, 9{v-VT) = (2.2) 0 v - VT < 0, for computational convenience^ The discontinuity at v = VT can cause difficulties with certain calculations that require smoother equations. Therefore, we have chosen to use the continuous function 9(v - VT) = 0.5(1 + tanh[c(u - VT)]), (2.3) which is a very good approximation of the step function for large values of c. This choice makes little difference to the results and is not important anyhow due to the qualitative nature of this model. Since spiking in the soma seemed to be required for peripheral spike initiation, Kepler and Marder assumed that the slow inward current would activate only above the spike threshold of -45 mV for the soma. Thus, they chose Vj > —45 mV. They found that the model was not particularly sensitive to the choice of VT-In order to account for the dependence of the peripheral spiking on serotonin, the parameter Is, describing the maximum possible amplitude of the slow inward current, is assumed to be a graded function of the serotonin concentration. Thus, high and low values of Is correspond to high and low values of serotonin concentration, respectively. However, the nature of the hypothetical slow inward current and how serotonin affects its amplitude or time course are unknown. The serotonin-sensitive parameters could be ks or VT instead of, or in addition to Is. Further experimental work is needed to verify or improve this model. The model proposed by Kepler and Marder, which we shall refer to as the K M model, is given by the system of equations: dv d2v C— + IHH(v,m,h,n) - zIs-j— = Iapp(t), (2.4a) Chapter 2. SPONTANEO US SECONDARY SPIKING 15 dm _ m c o ( f ) - m at Tm(v) § = ks[9(v-VT)-z}y (2-4e) where / / m = 9Nam3h(v - u;va) + gKn4(v - vK) + 9L{V - vL). (2.4f) Equation (2.4a) represents the effects of all the currents on the membrane potential v on the finite portion of the axon capable of exhibiting spontaneous secondary spiking. The first term is the capacitive effect of the membrane. The second term, IHH, includes Hodgkin-Huxley style expressions for the N a + , K + , and leakage currents, which are mod-ulated by the gating conductances m, h, and n, as indicated in equation (2.4f). The third term in equation (2.4a), consisting of an amplitude parameter IS and a gating conduc-tance £, represents the slow inward current. The next term in equation (2.4a) represents the diffusive effects that allow action potentials to propagate and the final term, Iapp(t), represents an applied (injected) current. When the diffusive term 7 § | | is removed, the model becomes a system of ordinary differential equations with no spatial variation and is said to be space-clamped. This is a mathematical convenience that allows us to study the interactions of the currents without the added complexity of the spatial variations. (Space-clamping is also a useful technique in some experiments since it makes it easier to measure the ionic currents.) The equations (2.4b) to (2.4e) are relaxation equations representing the voltage and time dependencies of the four gating variables. The full model, including the voltage dependencies of the activation curves ( m o o , ^ c o , ^ c o ) a n d the time constants ( r m , T ^ , r n ) , is given in Appendix A. Chapter 2. SPONTANEO US SECONDARY SPIKING 16 2.3 Spat ial and Tempora l Behaviour of Kep le r and Marder ' s M o d e l In this section, we discuss the spatial and temporal behaviour exhibited by the model proposed and studied by Kepler and Marder [36] and review their major findings, which serve as the basis for our work. Kepler and Marder demonstrated numerically that their model of the peripheral spike initiation zone initiated secondary spikes. As shown in the simulation1 of Figure 2.1, they applied two consecutive rectangular current pulses IaPp(t) to one end of the axon. Each current pulse initiates one action potential which propagates away from the stimulus. Using dispersion analysis, Kepler and Marder showed that, unlike the Hodgkin-Huxley case where the leading action potential propagates more quickly than subsequent action potentials, the slow inward current activated by the first action potential facilitates the speed of the action potentials following it. Thus the second action potential will slowly catch up to the first action potential until the refractoriness from the preceding action potential negates the velocity facilitation of the slow inward current and a stable spacing is reached. When the two action potentials are sufficiently close together, a third spike is initiated spontaneously behind the first two, and then splits into orthodromic and antidromic action potentials. The orthodromic spike starts to catch up to the orthodromic spike train, and the process repeats itself. Kepler and Marder also demonstrated some very interesting behaviour of the space-clamped model. Kepler and Marder demonstrated that if two current pulses are applied to the space-clamped model axon, then three qualitatively different outcomes can be observed. Simulations2 of these three possible outcomes are shown in Figure 2.2. For sufficiently low Is (low serotonin concentration) and applied current frequency, only the 1 Al l of the space-time simulations were implemented in the programming language C using a modified Crank-Nicolson method. The boundary condition used at each end was the outgoing wave condition, Vt + cvx = 0, where c is the approximate speed of propagation of the action potentials. 2 Al l of the simulations of systems of ordinary differential equations were done using the numerical package LSODE (Livermore Solver for Ordinary Differential Equations [29]). Chapter 2. SPONTANEOUS SECONDARY SPIKING 17 Figure 2.1: The membrane potential v as a function of axonal distance x along Kepler and Marder's model axon with Is =125 and 7=0.1 at time steps, 1 ms apart. Two rectangular current pulses of amplitude 100 uA and duration 1 ms are applied between x=0 and x—1 with a frequency of 65 Hz starting at time 2=2. (Based on Fig. 5, Kepler and Marder [36]). two directly initiated action potentials are observed. (This type of behaviour is typical of a squid axon type model lacking the slow inward current.) For a higher applied current frequency, several secondary spikes may occur. If Is is sufficiently high (Figure 2.2c), the two directly initiated action potentials may be followed by continuous secondary spiking Chapter 2. SPONTANEO US SECONDARY SPIKING 18 (a) v (mV) (b) v (mV) eo t^-o so o • so I^-O SO • so eo 40 so o -so —'1 c > -eo -so - | 7 S=125 M /=65 Hz *< i -3-0 eo eo -i oo -i so -i -*i < > i «:>c > £ (msec) t (msec) (c) u (mV) eo 40 I— SO | — o •so 4^-0 eo so J 5=125 M /=100 Hz *o SO "1 oo -i ^o -i -^ -O "I G O -fl / a =150 /zA /=100 Hz so -*o eo &o -i oo -iso -140 -1 eo t (msec) Figure 2.2: The response of Kepler and Marder's space-clamped model axon when two rectan-gular current pulses of amplitude 100 fiA and duration 1.25 msec are applied. The qualitative response depends on the amplitude Is of the z-current and the frequency / of the applied cur-rent pulses. When /=65 Hz, the lag between the beginning of the first and second current pulse is 15.4 msec and when /=100 Hz, the lag is 10.0 msec. (Based on Fig. 1, Kepler and Marder [36]) which wi l l go on indefinitely unless reset by a hyperpolarizing pulse. To examine how the Hodgkin-Huxley currents produce continuous secondary spiking, Kepler and Marder plotted the frequency of tonic firing for the Hodgkin-Huxley equations as a function of applied inward current. On this graph, they superimposed the slow inward current that would result from tonic firing at a given frequency (see [36] for Chapter 2. SPONTANEOUS SECONDARY SPIKING 19 details) for several different values of the amplitude parameter Is. They suggested that the intersections of the two curves for each given value of Is might represent possible steady states of the model. Based on this numerical evidence, they predicted that the only stable states for low values of Is would be the rest state. For higher values of Is, however, they found three intersections, one corresponding to rest, one corresponding to high frequency tonic firing, and the third corresponding to an unstable fixed point at an intermediate frequency. They stated that the intermediate frequency was the minimum frequency of stimulation required to initiate continuous secondary spiking. Because ks is small but finite, the slow inward current is not constant but fluctuates slightly. Therefore, the above analysis does not prove that the intersections of the curves are in fact fixed points of the system. In our work, we use bifurcation analysis to demonstrate that Kepler and Marder's predicted fixed points do exist in particular ranges of Is. Furthermore, we examine the effects on the model's behaviour of changing the time constant of the slow inward current. We examine the behaviour of the space-clamped model in detail in subsequent sec-tions. In the meantime, it is important to realize that the slow inward current builds up during an action potential and decays between action potentials. If the inward current gets large enough, it can initiate a spontaneous secondary spike. Keeping this in mind, we can recognize that the buildup of the slow inward current over the time course of several action potentials is analogous to its buildup in space during a train of propagating action potentials. Thus, when the model is not space-clamped, a train of action potentials must achieve a certain minimum spatial "frequency" before a spontaneous secondary spike can arise behind it [36]. Kepler and Marder noted that for low Is, the secondary spike does not arise close enough to the preceding spike train to immediately initiate another secondary spike. Since the secondary spike must propagate a small distance in order to catch up to the Chapter 2. SPONTANEO US SECONDARY SPIKING 20 X X Figure 2.3: Tracings of the peaks of the action potentials as a function of distance x along the axon and time t for (a) low Is (J s=110.7 uA) and (b) high Is ( J s = 135.3 fiA). At an Is large enough to allow regenerative firing in the space-clamped model, the spike generation zone ceases to move, as in case (b). (The applied current Iapp{f) is turned on at t—2 ms and 2=10 ms (freq=125 Hz) for 1 ms with an amplitude of 100 uA between x=10 and x = l l . preceding spike train before another secondary spike is initiated, the point at which the spontaneous secondary spike is initiated, called the spike generation zone, also propagates forward. The speed at which the spike generation zone moves depends on the difference between the spatial frequency between the last two action potentials in the spike train required to initiate a secondary spike and the spatial frequency (spacing) at which the secondary spike ini t ia l ly arises [36]. Since the spatial frequency at which the secondary spike arises increases with 7 5, the speed of propagation of the spike generation zone decreases unti l it stops moving altogether. This occurs when Is is sufficiently large to allow regenerative spontaneous secondary spiking in the space-clamped model [36]. Thus, an increase in Is increases the number of secondary spikes and hence the total number of spikes in the spike train impinging on the muscles. Figure 2.3 shows tracings of the peaks of the action potentials during numerical simulations for a high and a low value of Is. Serotonin may be able to control the duration of contraction of some of the muscles by influencing the number of secondary spikes that arise in the L G axon. Chapter 2. SPONTANEOUS SECONDARY SPIKING 21 The frequency of the initial spike train is also important on a finite axon, since it determines where the spike generation zone first arises, and hence how many secondary spikes are initiated before the end of the axon [36]. The secondary spiking would continue indefinitely on an infinite axon, but on a finite axon, it stops when the secondary spiking zone propagates off the end of the axon or out of the zone where the slow inward current is expressed. Kepler and Marder termed the process whereby higher frequency incoming signals result in more spontaneous secondary spikes "frequency to duration transduction" [36]. Because of the analogy between the spatial and temporal behaviour, we have chosen to spend most of our effort to understand the space-clamped version of Kepler and Marder's model. Before discussing this space-clamped model, we should mention that the phenomenology exhibited by the full model in one or more spatial dimensions is extremely rich. Some very complicated patterns can arise due to the interactions of colliding pulses or refractory regions. For instance, in Figure 2.4 we show what happens when two action potentials are initiated near each end of the axon. In this case, secondary spikes arise as before. In addition, colliding pulses annihilate each other. Because the axon is relatively short and pulses can propagate "off the end" of the axon, the action potentials eventually end up annihilating each other faster than new action potentials arise and activity dies out. However, for a longer axon, it would take much longer for activity to cease because there would be room for additional secondary spikes to arise. Adding another spatial dimension would increase the complexity further, especially when geometry or inhomogeneities are factored in. Chapter 2. SPONTANEOUS SECONDARY SPIKING 22 200 x Figure 2.4: Tracing of the peaks of the action potentials as a function of distance x along the axon and time t when J5=110.7 tiA. Two action potentials are initiated near each end of the axon. (Action potentials are initiated by current pulses of amplitude 100 /iA and duration 1 ms between x=10 and x = l l near the left end and £=89 and £=90 near the right end. The current pulses are applied at both positions at time t=2 ms and 2=8 ms.) 2.4 Reduc t ion of Kep le r and Marder 's M o d e l The Kepler and Marder (KM) model is a "minimal" model and is not meant to be quan-titatively or even biologically accurate. Rather, it illustrates one possible spontaneous spiking mechanism. It reveals the interplay between Hodgkin-Huxley-type dynamics for an action potential and a slow inward current. In the following work, we seek reduc-tions of the space-clamped K M model which illustrate the qualitative mechanisms of the system and are easy to analyze. The key assumption of our analysis is that the K M model can be split up into a fast subsystem representing the dynamics of the action potential and a slow subsystem consisting of the slow inward current. Thus the five-dimensional K M model can be split up into a fast four-dimensional and a slow one-dimensional subsystem as shown in Figure 2.5. Our objective is to reduce the four-dimensional fast subsystem to a two-dimensional subsystem. If we make the assumption that the slow subsystem has a much larger time Chapter 2. SPONTANEOUS SECONDARY SPIKING 23 constant than any of the time constants in the fast subsystem, then we can approximate the slow variable as a parameter, reducing the whole system to two dimensions. This is advantageous because it allows us to use phase-plane analysis to get a qualitative understanding of the interactions between the fast and slow subsystems. dm ~dt dh ~dt dn dt dz Full 5D System C— = -IHH(v,m,h,n) ~T~Z-£s ~~T~ -^ app(^ ) m T m [ V ) Th{v) : _ . J __Tn(V) ks[0(v-VT)-z} Reduced 3D System dV dt = F(V,U)\ +zls + lapp(t) Figure 2.5: Kepler and Marder's equations can be viewed as a 4D fast subsystem (dashed box) and a ID slow subsystem. Reduction of the 4D subsystem to a 2D subsystem (dashed box) allows the use of phase-plane analysis by viewing the slow z variable as a parameter. A number of methods have been used to reduce the Hodgkin-Huxley equations to a two-dimensional approximation. One of these approximations is the FitzHugh-Nagumo equations [24]. Other methods take advantage of empirical relationships between some of the variables. For instance, Krinskii and Kokoz [37] and Rinzel [55] assume that m is an instantaneous function of v and that some linear combination of h and n is approximately constant. Kepler et a/.'s method of equivalent potentials [35] is a general method for reducing the order of systems of conductance-based equations. In the following two sections we will discuss the two three-dimensional approximations of Kepler and Marder's Chapter 2. SPONTANEOUS SECONDARY SPIKING 24 model using the FitzHugh-Nagumo equations and the method of equivalent potentials. 2.4.1 Reduc t ion Us ing a F i t zHugh-Nagumo A p p r o x i m a t i o n FitzHugh [24] suggested a useful two-dimensional approximation of the four-dimensional Hodgkin-Huxley equations which exhibits many of the properties of the full Hodgkin-Huxley system and yet lends itself to analytical techniques and phase-plane analysis. These equations, also used by Nagumo [48], are called the FitzHugh-Nagumo equations. Replacing the Hodgkin-Huxley variables v, m, / i , and n by the FitzHugh-Nagumo vari-ables, a potential variable V and a recovery variable W, in the K M model gives the zFN model: ~ = -f(V;a)-W + zIs + Iapp(t), (2.5a) ^ = e(V-rW\ . (2.5b) d z ks[6(y-VT)-z], (2.5c) dt w here f(V;a) = V(V-l)(V-a) (2.5d) and 6{V - VT) = 0.5(1 + tanh(c(V - VT))). (2.5e) We refer to this set of equations as the zFN model because it contains the FitzHugh-Nagumo (FN) equations plus an extra variable z. The zFN model is a caricature of the original K M model. For more information on how the FitzHugh-Nagumo equations relate to the Hodgkin-Huxley equations, see Rinzel [54]. In order to reproduce the type of behaviour seen in the K M model, we have chosen the following parameters for the zFN model: a=0.1, r=2.5, e=0.01, ks — 0.005, Vx=0.3, and c=55. This parameter set will Chapter 2. SPONTANEOUS SECONDARY SPIKING 25 be referred to as the standard parameter set for the zFN model. The reasons for this choice of parameters will be discussed in detail in later sections. Choosing such a small value for e slows down the dynamics considerably. For plotting purposes, the dynamics of the z F N equations have been sped up by multiplying the right-hand sides of equations (2.5a-c) by a factor of r=10. 2.4.2 Reduc t ion Us ing the M e t h o d of Equivalent Potentials Kepler et al. [35] have created a "recipe" for systematically reducing conductance-based models like the Hodgkin-Huxley equations to lower-dimensional approximations whose parameters can be related directly to the original system of equations. These models retain a certain amount of realism which the FitzHugh-Nagumo equations lack. This method, called the method of equivalent potentials, transforms the gating variables into "equivalent potentials" so that all the variables in the equations are dimensionally equiv-alent. These transformed variables are then placed into groups exhibiting similar be-haviour and replaced by a weighted average of all the members of the group, found using a perturbation method. We have used a simplified version of this method to replace the four-dimensional Hodgkin-Huxley equations in the K M model by a two-dimensional approximation. The conductance equations for m, h, and n all have the form Since X c o ( ' y ) is a monotonic function of v, we can make the exact transformation x = xoo(vx), as illustrated in Figure 2.6. Using the chain rule, equation (2.6) can be rewritten in terms of the new equivalent potential v x : d x (2.6) Chapter 2. SPONTANEO US SECONDARY SPIKING 26 x=x00(v; Vx V Figure 2.6: The monotonicity of the gating function x^iy) ensures the invertibility of the transformation x = x^Vx). Therefore, the Hodgkin-Huxley equations can be rewritten in terms of the equivalent potentials v, vm, Vh, and vn: dv C— = -lHH{v,vm,vh,vn), (2.8a) dvm _ mcojv) - mco{vm) dt r m ( u ) m ^ ( u m ) dvn _ noo(u) - noo(u n ) dt Tn(v)n'^(vn) dvh _ h^v) - hoojvh) where (2.8b) (2.8c) (2.8d) IHH = gNam^Vmjh^Vh)^ - vNa) + QKn^v^v - vK) + gL(v - vL). (2.8e) The change of variables reveals that this fourth-order system does not fully utilize all four degrees of freedom. In fact, v and vm have a very similar time evolution, as Chapter 2. SPONTANEOUS SECONDARY SPIKING 27 do Vh, and vn. According to the method of equivalent potentials, we could calculate a representative equivalent potential for each pair (v,vm) and ( v h , v n ) using an optimized weighted average. However, we are not looking for accuracy and would prefer the simplest approximation possible. Therefore, we have simply chosen to replace vm by v and vn by Vh-We will name these new representative equivalent potentials V and U, respectively. This reduction produces action potentials whose time courses agree well with simulations of the full system, as shown in Figure 2.7. In this way we have obtained a two-dimensional reduction whose parameters are fully determined by the original system of equations. Reinstating the slow inward current, we get the following three-dimensional model, which we will refer to as the M E P model: °ld = - / M ^ y ' t / ) + z / * + /^(t)> (2-9a) d u _ M ^ O - M * 7 ) J = k.[0(V - VT) - z], _ (2.9c) where IMEP(V, U) = gNamKV^U^V - vNa) + gKnl(U)(V - vK) + gL(V - vL) (2.9d) and 0(V-VT) = 0.5(1 + t anh[c (V-V r ) ] ) . (2.9e) 2 .5 Ana lys i s of the Fast Subsystem Us ing the Slow Inward Current as a Parameter The goal of this section is to give a graphical demonstration of the mechanism of sponta-neous secondary spiking. We hope to provide an intuitive understanding of the qualitative Chapter 2. SPONTANEO US SECONDARY SPIKING 28 60 v,V (mV) -60 h -20 h -40 h 40 h 20 h 0 h -80 0 5 10 15 20 t (msec) Figure 2.7: Action potentials produced by the K M and M E P models. behaviour of the K M model's responses to various types of stimuli. To do this we will compare the behaviour of the full three-dimensional zFN and M E P models with the dynamics of their fast subsystems given various fixed values of the slow inward current Iz = zls. This, technique is particularly useful for demonstrating the behaviour of the zFN equations when ks is very small, or, equivalently, when the inward current is very slow. We will present the results of the zFN model first, since the dynamics of the FitzHugh-Nagumo equations are already well understood. The simplicity of the FitzHugh-Nagumo equations allows the use of analytical techniques from bifurcation theory 3 to gain insight into the parameter dependence of the solutions. The analysis of the M E P model to follow will rely more heavily on numerical techniques due to the increased complexity of the equations. 3Readers unfamiliar with basic terminology from dynamical systems theory and the concept of bifur-cation diagrams may wish to review Appendix B (Guide to Bifurcation Diagrams) at this point. Chapter 2. SPONTANEOUS SECONDARY SPIKING 29 2.5.1 Analys i s of the z F N M o d e l Assuming Constant Iz To begin the analysis of the zFN model, we assume that z is so slow, or equivalently, that ks is so small that we can approximate Iz = zls at any particular moment as a constant. This leaves the FitzHugh-Nagumo equations, where Iz is simply a parameter. The u-nullcline, the line where v = 0, is a cubic w = —f(v; a) + Iz which is raised or lowered by Iz. The tonullcline (w = 0) is a straight line w = v/r, independent of Iz. These nullclines separate regions of increasing and decreasing v and w, respectively, and define fixed points of the system where they intersect. The FitzHugh-Nagumo model is known to exhibit two types of behaviour [47], de-pending on the relative slopes of the nullclines where they intersect: bistability and excitability. The two possible configurations are shown in Figure 2.8. The configuration shown in Figure 2.8a has a single fixed point which may act as a sink or a source de-pending on the value of Iz. For low Iz, the fixed point tends to be a stable equilibrium. If the system is perturbed sufficiently far away from this equilibrium, for example by an injected current pulse, it will respond with a large excursion in membrane potential followed by a return to equilibrium. We say that the system has been "excited" by the current pulse and refer to this kind of system as an "excitable" system. The configuration in Figure 2.8b, on the other hand, has three fixed points, the middle one being a sad-dle. This system can have a low and a high resting potential for a given Iz, and, hence, is referred to as a bistable system. To mimic the behaviour of a Hodgkin-Huxley-type system which is known to be excitable, we want to find conditions which ensure that the FitzHugh-Nagumo equations will be excitable rather than bistable. In the next section, (2.10a) dw (2.10b) Chapter 2. SPONTANEO US SECONDARY SPIKING 30 we wi l l find these conditions analytically and calculate the stability of the single fixed point [3, 16, 47]. (a) 0.3 0.2 -0.1 1 1 1 1 A 1 - v = 0 / — r- w = 0 — • / -A / — _ \ / \ ' — \ ' / \ ' / N . / / — / — 1/ 1 1 1 1 1 -0.2 0 0.2 0.4 0.6 0.8 1 v (b) J I I I i i -0.2 0 0.2 0.4 0.6 0.8 1 v Figure 2.8: Two configurations of the nullclines of the FitzHugh-Nagumo equations correspond-ing to (a) excitable (single fixed point) and (b) bistable (3 fixed points) behaviour. (Parameters: (a) a=0.25, r=2.0, i z =0.0 and (b) a=0.25, r=10.0, Iz=0.0) Analytical Calculation of Stability The fixed points (vss,wss) of the FitzHugh-Nagumo equations occur when ^ = 0 and dw dt = 0. Thus vss is defined implic i t ly as a function of Iz by the equation —f(vss;a) vss/r + Iz = 0, and then wss is given by wss = vss/r [3]. Let's examine the stability of the fixed points (vss,wss) in terms of the slopes of the two nullclines. The slope of the lo-nullcline is K If we denote the u-nullcline by w = N(v; a, Iz) = —f(v; a) + Iz, then the slope is given by Nv(v; a) = —fv(v, a), which is now independent of Iz. Equations (2.10) can be rewritten as dv — = N(v;a,Iz)-w, (2.11a) dw — = ev — erw. (2.11b) Chapter 2. SPONTANEO US SECONDARY SPIKING 31 The Jacobian of (2.11) at the fixed point (vss,wss) corresponding to Iz is / Nv(vss;a) -1 \ ( -fv(v„;a) - 1 \ J{vss,ivss) - = . (2.12) \ e — erj y e — erj The eigenvalues are determined from ] J(vss, wss) - XI\ = (Nv - X)(-er - A) + e = 0, (2.13) or A 2 + ( — Nv + er)\ + e(l — rNv) = 0. (2.14) If we let f3 = Trace(J) = Nv — er and 7 = determinant(J) = e(l — rNv), then the eigenvalues are given by A = (l/2)[/? ± ^ 2 - 4 7 ] , (2.15) so that A = (l/2)[(Nv - er) ± ^{er + Nv)* - 4e] = (l/2)[-(er + /„) ± ^(er - fvf - 4e]. (2.16) Specific conditions on 7 and determine whether the fixed point (vss,wss) is a source, sink, or saddle [16]. If 7 < 0, then the fixed point is a saddle. Since e is positive, this is equivalent to the condition Nv > K If 7 > 0, then the fixed point is a sink if /? < 0 and a source if /3 > 0. This means that the fixed point is a sink if Nv < ^ and Nv < er. It is a source if er < Nv < -. As mentioned earlier, in order to get an excitable system, we require that the Fitz-Hugh-Nagumo equations have a single fixed point for any given value of Iz. As Iz increases, the fixed point changes from a sink to a source and back again via Hopf bifurcations. The Hopf bifurcations give rise to limit cycles, which we can view as tonic firing. In order to avoid having any fixed points which are saddles, we must ensure that Nv(vss;a) < ^ for any vss. The maximum slope NVmax of the u-nullcline Chapter 2. SPONTANEO US SECONDARY SPIKING 32 is NVmax — |(| - a)2 + | , occurring at v = Thus, we must restrict r such that | ( | - a) 2 + | < ^ or, equivalently, that r < 1 + a 3 2 _ a [3]. To ensure that this is true for all small a (0 < a < 0.5), we can make sure that r < 3. This statement says that the slope of the io-nullcline is greater than the slope of the u-nullcline at any point, regardless of our choice of a. A second restriction on the parameters is obtained by requiring that some of the fixed points be sources. Since the equation er < Nv < ± must be satisfied for some of the fixed points, we must ensure that er < i or e < At Nv = er, both eigenvalues A = ±i^e(l — er2) are imaginary and the fixed point vss is given by v%B = (l/3)[(a + 1) ± Va2 - a + 1 - 3er]. (2.17) As vss crosses v%B , the Hopf bifurcation points, the fixed point changes from a source to a sink or vice versa. Comparing v^B with the maximum and minimum u ± of the u-nullcline, = ( l / 3 ) [ ( a + l ) ± V a 2 - a + l ] , (2.18) we see that if er is small, then most of the positive slope of the u-nullcline is unstable, as illustrated in Figure 2.9. In the limit as e approaches 0, the Hopf bifurcation points occur at the maximum and minimum of the u-nullcline, as shown in Figure 2.10a. As er increases, the unstable region shrinks (Figure 2.10b). When er > | , the unstable region disappears at intermediate values of a, as shown in Figure 2.10d. Then as er exceeds | , the unstable region ceases to exist for any a. These analytical results indicate that we should restrict r < 3 and er < 1/r in order to get an excitable system. In addition, small er and small a tend to give a wider unstable region, which may make it easier to observe the secondary spiking. Chapter 2. SPONTANEOUS SECONDARY SPIKING \ stable unstable stable \ steady steady steady \ states states states Wss V VHB VHB V+ Figure 2.9: Stability of fixed points (vss,wss) in different regions of the f-nullcline. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 a a Figure 2.10: v^B (solid line) and (dotted line) as a function of a for a) er = 0, b) er = c) er = 0.25, d) er = 0.26. There are no Hopf bifurcation points for any a when er > 1/3. Chapter 2. SPONTANEOUS SECONDARY SPIKING 34 N u m e r i c a l Results Based on the previous analysis, we can expect three qualitatively different responses to suddenly switching Iz on, as shown in Figure 2.11. For low Iz, an action potential will be followed by a return to steady state at a low membrane potential. For medium Iz, tonic firing of action potentials will occur. For very high Iz, the system will approach an elevated steady state. Notice the damped oscillation following the action potential in Figure 2.11a. We could damp out this oscillation more rapidly by choosing smaller r or smaller e, so that the response would look more like that of the Hodgkin-Huxley equations. However, we will see later that such a choice is not appropriate in the context of the full z F N equations. While we have calculated the direction of the Hopf bifurcation (supercritical or sub-critical) analytically as Iz is increased, the resulting analytical expressions are too com-plicated to be illuminating. (Numerical evaluation of these expressions for specific pa-rameter sets, however, correctly predicted the directions of the Hopf bifurcations.) In addition, the calculation only gives local information near the Hopf bifurcation points. The numerical bifurcation analysis package AUT086, on the other hand, uses a contin-uation algorithm to obtain a more global picture, including the fixed points, and the maxima, minima, and periods of the limit cycles. Figure 2.12 shows the bifurcation diagram calculated by AUT086 for the FitzHugh-Nagumo equations using Iz as the bi-furcation parameter. A fairly small e has been chosen to mimic the strong all or nothing response of the Hodgkin-Huxley equations. A larger value of e would result in a softer onset of tonic firing in which the magnitude of the oscillations would slowly grow larger with increasing Iz. If we now recall that Iz is not actually fixed but slowly varying, continuous secondary spiking can be explained qualitatively in terms of the zFN model as follows. The first Chapter 2. SPONTANEOUS SECONDARY SPIKING 35 -0.5 0 0.5 1 1.5 v (b) 0.4 w 1.2 0 20 40 60 80 100 t - 1 1 1 1 -ft 0.8 0.4 h 0 -fl -0.5 0 0.5 1 1.5 v (c) 0.4 w 1.2 0.8 0.4 0 0 20 40 60 80 100 t ~l 1 tL_l I I l _ J -0.5 0 0.5 1 1.5 v 0 20 40 60 80 100 t Figure 2.11: Response of the F N equations (a = 0.10, r = 2.5, e = 0.01, r = 10) to sudden steps in Iz starting at Iz = 0: (a) low resting potential ( J z = 0.02), (b) tonic firing (72 = 0.08), and (c) high resting potential [Iz = 0.18). The first column shows (v(t),w(t)) (solid line) superimposed on the cubic u-nullcline and the straight w-nullcline (dashed lines). The second column shows v(t). applied current pulse causes an action potential which in turn causes z to slowly increase. When the voltage drops below the voltage threshold VT, Z wi l l slowly decrease back towards zero. If a second current pulse is applied, another action potential w i l l cause z to increase again. If the two action potentials are close enough together, z w i l l have a "headstart" when the second action potential occurs and wi l l rise even further the second Chapter 2. SPONTANEOUS SECONDARY SPIKING 36 0.16 0.18 /* Iz Figure 2.12: Bifurcation diagram of the F N equations (a = 0.10, r = 2.5, e = 0.01, r = 10) calculated by A U T 0 8 6 for equations (2.10) using Iz as the bifurcation parameter: (a) steady states and maxima and minima of v and (b) period of limit cycles. (Stable and unstable steady states are indicated by solid and dashed lines, respectively. Filled circles correspond to stable limit cycles, while open circles correspond to unstable limit cycles. For more information on bifurcation diagrams, see Appendix B.) time. If the z current rises high enough, the u-nullcline may be high enough to put the fixed point in the unstable region. As a result the trajectory may begin to oscillate about the unstable fixed point. If z is moving slowly enough, the fixed point wi l l stay in the unstable region and continuous spontaneous secondary spiking can occur. O n the other hand, if z is moving too quickly, z wi l l decay fast enough to bring the fixed point out of the unstable region, allowing only a few secondary spikes or none at al l , before the voltage decays to zero. A fine balance is required for continuous secondary spiking. The natural frequency of the fast subsystem must be sufficiently high that Iz does not decay enough in between action potentials to place the system in an inexcitable regime. On the other hand, if the frequency is too high, Iz may continue to increase, unti l it reaches its max imum Is. In this case, the system can reach a stable elevated resting potential, depending on the Chapter 2. SPONTANEO US SECONDARY SPIKING 37 magnitude of Is. The balancing act involving the time constants of the fast subsystem, the slow rate constant ks, and the magnitude Is will be explored for the full system in a later section. 2.5.2 Analys i s of the M E P M o d e l Assuming Constant Iz The M E P model is more complicated than the zFN model and, hence, requires more numerical rather than analytical techniques. As before, if we treat Iz as a parameter, then we can examine the U and V nullclines of the resulting two-dimensional model. The £/-nullcline is defined by the straight line V = U. The V-nullcline is defined implicitly by the equation, gvamKV^iUXV - vNa) + g-KnKUXV - vK) + gL(V - vL) - Iz = 0. (2.19) This equation can be plotted as a surface in terms of (V,U,IZ) as shown in Figure 2.13a. Taking "slices" of this surface for particular values of Iz gives the V-nullclines in Figure 2.13b. These nullclines share certain similarities with the nullclines of the FitzHugh-Nagumo equations in Figure 2.11. In particular, there is always a single fixed point which shifts as Iz changes. However, in the case of the M E P equations, an increase in Iz merely deforms the left-hand branch rather than raising the whole nullcline. In addition, the shapes of the nullclines at Iz = 0 differ substantially. A hyperpolarizing pulse would tend to invoke an action potential in the FitzHugh-Nagumo equations (postinhibitory rebound) while the M E P equations would merely hyperpolarize and then return to rest. The bifurcation diagram with Iz as the bifurcation parameter was calculated using AUT086 for the M E P model and the original K M model. The bifurcation diagrams of the two models are qualitatively similar, although they differ somewhat quantitatively due to the approximations made in the reduction process. Since spontaneous secondary spiking involves a transition from a low resting potential to tonic firing, the behaviour Chapter 2. SPONTANEO US SECONDARY SPIKING 38 Figure 2.13: (a) F-nullsurface and (b) U nullcline (straight line) and V nullclines for several values of Iz in the M E P model. Note that the F-nullcline is deformed upwards as Iz increases (Iz =0.0, 0.5, 0.137 (Hopf bifurcation point), 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 4.96 (Hopf bifurcation point), and 5.5), while the [/-nullcline is a straight line independent of Iz. of the models near the left Hopf bifurcation point is the most important. The M E P and K M models have a hard (abrupt) bifurcation at the left end and a softer bifurcation at the right end, as shown in Figure 2.14. The hard bifurcation at the left Hopf bifurcation point means that the spontaneous secondary spikes wi l l have an all or nothing response. The Fi tzHugh-Nagumo equations, on the other hand, have either two hard bifurcations or two soft bifurcations depending on the choice of e. The behaviour near the right Hopf bifurcation point is not particularly important in studies of spontaneous secondary spiking. Due to the similarities between the M E P and z F N models, we see that the qualitative explanation of spontaneous secondary spiking advanced for the z F N model also holds for Chapter 2. SPONTANEO US SECONDARY SPIKING 39 (JJ) V Period t ' ' Iz Figure 2.14: Bifurcation diagram calculated by A U T 0 8 6 using bifurcation parameter Iz: (a) M E P model and (b) K M model. Projection onto V and period are shown. the M E P model. The chief difference between the two models is that the M E P model does not exhibit a large undershoot in membrane potential after each action potential. 2.6 Effect of the Slow Inward Current in the F u l l 3D and 5D Mode l s In the previous section, we gained a qualitative understanding of spontaneous secondary spiking by assuming that Iz was sufficiently slow that we could treat it as a parameter. Intuitively, we see that continuous spontaneous secondary spiking occurs when the fre-quency of firing is high enough that z grows as much as it decays every cycle. However, it is not clear from the previous analysis how large Is must be for this to occur, nor is it clear how slow the rate constant ks must be. In the following two sections we discuss the Chapter 2. SPONTANEOUS SECONDARY SPIKING 40 role of these two very important parameters of the slow inward current in determining when spontaneous secondary spiking may occur. 2.6.1 The Role of Is As mentioned before, Is is the maximum amplitude of the current I 2 , corresponding to the case 2 = 1. However, z can take on intermediate values so it is difficult to predict whether spontaneous secondary spiking will occur based on the value of /<,. To understand the effect of the amplitude of Is, we examine its effect on the stability of the full 3D and 5D models. We first calculate the position and stability of the steady states analytically in the 3D z F N model. We then use AUT086 to obtain a more global picture of all of the models, including the steady states and limit cycles corresponding to different values of the bifurcation parameter Is. We find that the bifurcation diagrams obtained using Is as the bifurcation parameter in the zFN, M E P , and K M models are qualitatively similar. These bifurcation diagrams allow us to predict when spontaneous secondary spiking is likely to occur based on the value of Is. Calcu la t ing The Steady States and their Stabi l i ty in the z F N M o d e l The nullsurfaces of the zFN equations are defined by the three equations u = 0 : w =-f(v;a) + zls, (2.20a) w = 0 : w = -, (2.20b) r z = 0 : z = 9(v-VT). (2.20c) The fixed points of the zFN equations occur at the points where all three nullsurfaces intersect. These points are difficult to see in three dimensions. It is easier to see the effects of Is on the fixed points if we combine the two equations (2.20a) and (2.20c) Chapter 2. SPONTANEOUS SECONDARY SPIKING 41 w -0.4 0 0.4 0.8 1.2 v (d) IS(SN) < Is < L(HB) 0.4 w 0.3 0.2 0.1 -—1—i 1 1 1 1 1 i -1 1 / V 1 1 1 1 1 1 -0.4 0 0.4 0.8 1.2 v w w (b) 0 < Is < Is{SN) 0.4 -0.4 0 0.4 0.8 1.2 v (e) Is = IS(HB) n—i—i—i—i—r -0.4 0 0.4 0.8 1.2 v w w (c) Is = IS(SN) ~\ i i i i r~7 n -0.4 0 0.4 0.8 1.2 v (f) Is > IS(HB) 0.4 0.3 0.2 0.1 0 i 1 1 ~ I \ \ _ \ 1 1 I . J A 1 ( / ^ i / i v 1 / i -/ 1 1 1 1 / — T h •0.4 0 . 0.4 0.8 1.2 v Figure 2.15: The fixed points of the zFN equations occur at the intersections of the straight line describing w = 0 (solid line) and the piecewise cubic function describing v = z = 0 (dashed line). The right branch of the latter function rises as IS increases, as shown here for (a) IS = 0, (b) IS = 0.5 /^5^), (c) IS = Is(SN), ( d ) !s = 1-31,(5^), (e) L = IA(HB), a n d ( f) Is = l - 8 7 s ( f f B ) . (zFN equations using a step function for 9(v - VT): a = 0.1, r — 2.5, e = 0.01, VT = 0.3 .) describing the v and z nullsurfaces. Then we look for intersections in the v-w plane of the two lines described by v = z = 0 : w =-f(v;a) +IS9(v-VT), w = 0 v w — — r (2.21a) (2.21b) as illustrated in Figure 2.15. We have chosen to use the continuous function 9{v — VT) = 0.5(1 + tanh[c(u — VT)] Chapter 2. SPONTANEO US SECONDARY SPIKING 42 to describe the activation curve for z for the purpose of some numerical computations. However, it is easier to analyse the zFN equations in the limit of very large c when 9 approximates a step function. When this is the case, the curve described by equation (2.21a) consists of three distinct pieces. The left branch is described by w = —f(v; a) since 9{y — VT) = 0 for v < VT- The right branch is described by w — —f(v; a) + IS since 9{v — VT) = 1 for v > VT. In the limit of very large c, the near vertical middle branch connecting the left and right branches can be described by v = VT-As shown in Figure 2.15, the number of fixed points depends on the value of IS. We see from Figure 2.15b that for low IS, there is only a single fixed point corresponding to the rest state of the system. Since z = 9{y — VT) = 0 on the left branch, this fixed point occurs when v = w — z = 0. This fixed point persists for all IS and represents quiescence in the zFN equations. As IS increases past a critical value IS(SN)> a P a i r of fixed points arises via a saddle-node bifurcation, as shown in Figure 2.15d. IS(SN) I S the value of IS required so that the right branch of (2.21a) (io = —f(v; a) + IS) intersects with the line described by equation (2.21b) (w = v/r) at v = VT (Figure 2.15c). Thus, I S ( S N ) = VT/r + f(VT; a). For IS > 7 5 (5AT) , there are two fixed points. At the fixed point on the right branch of equation (2.21a), z = 1 and w = v/r where v is defined implicitly by v/r = —f(v\ a) + IS-On the middle branch, the fixed point occurs when zls = Is(SN)> s o that z = IS(SN)/IS, v = VT, and w = Vr/r. To find the stability of the fixed points, we calculate the Jacobian of the zFN equations (2.5a-c) at the fixed point (vss, wss, zss). J(vss) = ^ -fv{v„;a) - 1 IS ^ e — er 0 \ ks9v(vss - VT) 0 -ks j (2.22) Chapter 2. SPONTANEOUS SECONDARY SPIKING 43 When v 7^  Vr, 9v(v — VT) = 0 so the Jacobian reduces to / -fv(vss;a) - 1 IS \ J{vss) = 6 -er 0 (2.23) 0 K J The eigenvalues are determined by setting J(vss) - AI | = (-ks - X)((-fv(vss; a) - X)(-er - A) + e) = 0. (2.24) One eigenvalue is A = — ks, which is negative for all ks under consideration and, hence, does not destabilize the system. The remaining two eigenvalues satisfy the same equation (2.13) that we already saw in our analysis of the 2-D FitzHugh-Nagumo equations. Thus, from our previous analysis, we know the fixed point v = w = z = 0is stable. VT is between the two Hopf bifurcation points found earlier and v > Vr on the right branch where z = 1. Therefore, we expect to see a single Hopf bifurcation point at IS(HB) = VHBIT + f{vHB',a) where vHB = (l/3)[(a + 1) + \la? — a + 1 — 3er]. The fixed point on the right branch is, therefore, a source for IS < Is(HB) (Figure 2.15e) and a sink for h > IS(HB) (Figure 2.15f). Figure 2.16 illustrates the dependence of the steady states on Is, and indicates the positions of the Hopf bifurcation point and the saddle-node point corresponding to 7S(S7V)-The stability of the fixed points on the branch where vss — VT is more difficult to analyse than on the other branches because the term ks9v(vss — VT) in equation (2.22) does not disappear. Instead, it reduces to ks9v(0). To find the eigenvalues, we set the determinant fv{VT;a)-X - 1 \J(VT)-XI\ e er-X 0 (2.25) ks9v(0) 0 ks - A Chapter 2. SPONTANEO US SECONDARY SPIKING 44 (a) (b) w 0 0.05 0.1 0.15 0.2 I, 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 I, Figure 2.16: Fixed points of the z F N equations as a function of IS. The values of (a) v, (b) ru, and (c) z at the fixed points are shown. The positions of the Hopf bifurcation point (HB) and the saddle-node point (SN) corresponding to IS = Is(SN) a r e also indicated. The fixed points are stable on the branch where v = w = z = 0 and to the right of the HB point when z = 1. To the left of the HB point and on the branch where v = VT, the system is unstable. (zFN equations using a step function for Q(y - VT): a = 0.1, r = 2.5, e = 0.01, Vr = 0.3.) equal to zero. Using expansion by minors and rearranging, we see that the eigenvalues are determined by the equation A 3 + AX2 + BX + C = 0 (2.26) where A = er + fv(VT;a) + ks, B = [erfv(Vr,a) + e + kser + kJv(VT;a)}-ksIs0v(O), C = [kserfv(VT; a) + kse] - ksIser9v(0). (2.27a) (2.27b) (2.27c) Rather than solving for the eigenvalues A directly, we apply the Routh-Hurwitz crite-rion [61]. We must first establish the signs of A, B, and C in equation (2.26). Since VT is between the two Hopf bifurcation points found in our analysis of the FitzHugh-Nagumo equations, it lies on the part of the cubic with positive slope, so fv(Vr] a) > 0. Since all of the parameters (e, r, ks, Is) are also positive, we know that A > 0. B and C each Chapter 2. SPONTANEOUS SECONDARY SPIKING 45 consist of a set of finite positive terms and one negative term containing 6V(0). Since we have assumed that 6v(v — Vr) approximates a step function, we know that when v = Vr, 8v(v — Vr) has a positive, nearly vertical slope. Therefore, 0„(O) approaches +00, so B < 0 and C < 0. A similar analysis shows that AB — C < 0. Following the method used in [61] to determine the stability of our system, we con-struct the Routh table as follows: A 3 I B . A 2 A C (2.28) A 1 (AB - C)/A A 0 C The Routh-Hurwitz criterion states that the number of roots of the characteristic poly-nomial (equation (2.26)) with positive real parts is equal to the number of changes in sign of the first column of the Routh. table [61]. Since 1 > 0, A > 0, (AB - C)/A < 0, and C < 0, there is only one sign change in the first column of the table (2.28). Hence, there is one root (eigenvalue) with positive real part. This means that when v = Vr, the fixed point is unstable. Numerical Bifurcation Analysis Using A U T O 8 6 The analysis in the previous section only gave us information about the steady states. However, in order to understand spontaneous secondary spiking, we need to find out when large amplitude limit cycles (tonic firing) can occur. To get a more global overview of the behaviour of the equations, we calculate bifurcation diagrams using the numerical bifurcation analysis package A U T O 8 6 [13]. These diagrams concisely summarize all the possible limiting behaviours of the system, including stable and unstable limit cycles or steady states. Chapter 2. SPONTANEOUS SECONDARY SPIKING 46 We begin by calculating the bifurcation diagram for the z F N equations using ls as the bifurcation parameter. In order to avoid potential numerical difficulties caused by discontinuities, we choose to use the smooth function (equation (2.3)) with c = 55 to describe 9. The results of this computation are given in Figure 2.17. The position and stability of the steady states calculated by AUT086 look quite similar to those calculated analytically in the previous section (Figure 2.16). The values Figure 2.17: Bifurcation diagram for the 3D zFN equations using IS as the bifurcation param-eter. The projections of the steady states and limit cycles onto v, w, and z and the period T of the limit cycles are shown. A pair of fixed points arises at the saddle-node point (SN), while a pair of limit cycles arises at the saddle-node of periodics (SNP) when IS=IS(SNP)=®A158. (zFN equations using the smooth function (equation (2.3)) for 6(v - VT): a = 0.1, r = 2.5, e = 0.01, ks = 0.005, VT = 0.3, c = 55, r = 10.) Chapter 2. SPONTANEOUS SECONDARY SPIKING 47 of IS at the H o p f b i fu rca t ion poin ts found a n a l y t i c a l l y and n u m e r i c a l l y agree to four d e c i m a l places (IS(HB) — 0.1424). Because we have chosen to use a s m o o t h func t ion ra ther t h a n a step func t ion to descr ibe 9, the corner at the saddle-node po in t is r o u n d e d ins tead of sharp. A s a resul t , the value of IS at the saddle-node po in t c a l c u l a t e d by A U T 0 8 6 o n l y agrees w i t h our p r ed i c t ed value to one signif icant figure (IS(SN) = 0.08). A s one w o u l d expect , a l i m i t cyc le arises at the H o p f b i fu rca t ion p o i n t . W h a t is not obv ious f r o m our p rev ious analys is is tha t the l i m i t c y c l e grows i n a m p l i t u d e , even-t u a l l y g i v i n g rise to s table large a m p l i t u d e osc i l l a t ions . Before c o m m e n t i n g fur ther on the branches of l i m i t cycles i n F i g u r e 2.17, we c o m p u t e the co r respond ing b i f u r c a t i o n d iag rams for the K M and M E P equat ions . F i g u r e 2.18 compares the b i fu rca t ion d i a g r a m of the o r i g i n a l 5 D K M m o d e l w i t h tha t of the 3 D M E P m o d e l for the b i fu rca t ion paramete r IS. T h e branches of fixed po in t s are q u a l i t a t i v e l y s i m i l a r and resemble those i n the z F N m o d e l . C a r e f u l i n s p e c t i o n reveals tha t the s table branches of l i m i t cycles for the K M and M E P mode l s are also q u a l i t a t i v e l y s i m i l a r . T h e o n l y m a j o r difference between the two sets of b i fu r ca t i on d iag rams is i n the shape of the branches of uns table l i m i t cycles . N u m e r i c a l s i m u l a t i o n suggests tha t the differences i n the uns tab le branches of l i m i t cycles are r e l a t i ve ly u n i m p o r t a n t w h e n s t u d y i n g spontaneous secondary s p i k i n g . T h e b i fu rca t ion d i a g r a m for the z F N equat ions shares the same q u a l i t a t i v e features as for the K M and M E P mode l s . T h e shape of the branches of the l i m i t cycles differs s l igh t ly because of the different d y n a m i c s of the exc i t ab l e subsys t em discussed earl ier . F r o m these b i fu rca t ion d iagrams , we see tha t each sys t em has a s table res t ing p o t e n t i a l co r r e spond ing to z=0, regardless of the value of IS. T h i s is the quiescent m o d e . A n a d d i t i o n a l b r a n c h of s teady states is seen at an elevated m e m b r a n e p o t e n t i a l . A b r a n c h of l i m i t cycles arises f r o m this b r a n c h v i a a H o p f b i fu rca t ion . A f t e r a couple of t u r n i n g po in t s i n w h i c h the s t a b i l i t y of the l i m i t cycles changes, we see a b r a n c h of s table large Chapter 2. SPONTANEOUS SECONDARY SPIKING 48 (a) K M Model (b) M E P Model 1 2 3 4 5 6 7 ( / /A) 0 1 2 3 4 5 6 7 Is/55 (fiA) 1 2 3 4 5 6 7 Is/55 (fiA) 0 1 2 3 4 5 6 7 Is/55 ( M ) Figure 2.18: Bifurcation diagrams for (a) the original 5D K M model and (b) the 3D M E P model using ls as the bifurcation parameter. The projections of the steady states and limit cycles onto v and z and the period T of the limit cycles are shown. (Note that v and ls have been scaled by a factor of 55 to make them order one.) The saddle-nodes of periodics, marked SNP, occur at J,=2.383*55.0=131.065 ftA in the K M model and Is=3.859*55.0=212.245 uA in the M E P model. Chapter 2. SPONTANEO US SECONDARY SPIKING 49 amplitude oscillations. This branch corresponds to tonic firing wi th oscillations in z at intermediate levels. It is interesting to note that for high values of Is, the system can act as a bistable switch between a high and a low resting potential. A t intermediate values of Is, the system is, in fact, tristable, with two stable steady states and a stable l imi t cycle or two stable l imi t cycles and a stable steady state. This mathematically interesting behaviour is biologically implausible in this particular study since it is unlikely that the neuron would exhibit a stable elevated resting potential, so we should restrict ourselves to relatively low values of Is. The values of Is that we are interested in are those near or below the saddle-node of periodics, marked S N P in Figure 2.18, where the stable large amplitude l imi t cycles first arise. To the right of the saddle-node of periodics, the system has two stable modes: quies-cence or tonic firing. The normally quiescent system may be perturbed by appropriate stimulation so that it wi l l respond transiently with a few spontaneous secondary spikes followed by a return to quiescence or switch to tonic firing mode. A n example of the latter was seen for the K M model in Figure 2.2c in which Is was set at 150 pA, well above the saddle-node of periodics at J s=131 fj,A. For very low /<,, the slow inward current Iz is negligible so the system behaves like a normal Hodgkin-Huxley type system in which each induced action potential is followed by quiescence. However, for Is just to the left of the saddle-node of periodics, the system can exhibit transient behaviour consisting of a few spontaneous secondary spikes followed by quiescence. As the system is driven, z builds up, as discussed earlier. If Is is sufficiently large, then Iz becomes large enough to place the excitable subsystem in the tonic firing regime. However, the frequency of firing is insufficient to maintain z at its elevated level, so Iz decays unti l the system returns to rest. The number of spontaneous secondary spikes depends on how much Iz builds up. When 7S=125 uA in the K M model, which is Chapter 2. SPONTANEOUS SECONDARY SPIKING 50 below the saddle-node of periodics at 7S=131 fiA, we saw in Figure 2.2a that insufficient stimulation may not lead to any spontaneous secondary spikes. Increasing the number or frequency of st imuli can lead to enhanced buildup of Iz and, hence, more spontaneous secondary spikes, as shown in Figure 2.2b. In Figures 2.19 and 2.20, we show that the M E P and z F N models can exhibit the (a) v (mV)_ J 5=205 uA f=100 Hz 1 O O 1 S O 1 -1 ( > 1 fit ) t (msec) J s =205 uA f=150 Hz O S O - 4 - 0 J O -I O O 1 S O "I - * 0 -I G O t (msec) ) -1 o . s — l\ . . . \ i — L (mV)_ Q I - -i c 3 J !X s< y y o y e y S O • Q O y -i o o "y -1 3 O y -i -= y 4 - 0 >^  J,=215 p A f=150 Hz t (msec) Figure 2.19: The responses of V (solid line), U (lower dashed line), and z (upper dashed line) in the M E P model when two current pulses of amplitude 55 /LtA and duration 1 msec are applied for different values of IS and applied current frequency / : (a) IS = 205^tA < Is(sNP)i / = 100, (b) IS = 205fxA < Is(SNP), f = 15°, and (c) IS = 21b/j,A > Is(SNP), f = 150. The saddle-node of periodics occurs when IS = Is(sNP) = 212uA. (Note: V has been scaled by a factor of vpja = 55, so that it is order(l).) Chapter 2. SPONTANEO US SECONDARY SPIKING 51 same types of qualitative behaviour as the original K M model. If IS < IS(SNP), then the models can respond to a train of stimuli with either a direct return to quiescence (Figures 2.19a and 2.20a) or a few spontaneous secondary spikes followed by a return to quiescence (Figures 2.19b and 2.20b). For IS > Is(SNP), the models can respond in (a) O 7 7 7 J s=0.11 f=100 (b) o t >. ' 1 r-1/1/1/ 7S=0.11 f=130 (c) o o _ ; : ' - o . 4 I,=0.12 f=130 Figure 2.20: The responses of v (sohd line), w (lower dashed line) and z (upper dashed line) in the z F N model when four current pulses of amplitude 0.1 and duration 0.5 are applied for different values of IS and applied current frequency / : (a) ls = 0.11 < IS(SNP), / = 100, (b) IS = 0.11 < Is(SNP), f = 130, and (c) IS = 0.12 > IS(SNP), f = 130. (Note: if the unit of time in these dimensionless equations is considered to be msec, then the unit of frequency shown here would be Hz.) The saddle-node of periodics occurs when IS = IS<SNP) = 0.116. (zFN equations using the smooth function (equation (2.3)) for 6{y — VT): a = 0.1, r = 2.5, e = 0.01, ks = 0.005, VT = 0.3, c = 55, r = 10.) Chapter 2. SPONTANEO US SECONDARY SPIKING 52 the same way or, more likely, with tonic firing that lasts indefinitely (Figures 2.19c and 2.20c). Based on the results of the original model and its two different reductions, we can conclude that the phenomenon of spontaneous secondary spiking is generic. The key ingredients are a slow inward current and an excitable system that can have either a stable steady state or a stable oscillation. The exact shape of the action potential is unimportant, but the relative time constants of the system are crucial. In the next section, we will use the relatively simple zFN equations to explore the importance of the relative magnitude of the rate constant ks. 2.6.2 The Role of ks In our preliminary attempt to understand spontaneous secondary spiking, we assumed that the slow inward current was much slower than the dynamics of the excitable sub-system. While spontaneous secondary spiking does occur when this is true, a very slow rate constant ks is not a necessary condition. In the following sections we will study the effects of the magnitude of ks on the behaviour of the models. We will demonstrate that the bifurcation structure and hence the behaviour of the models changes when ks exceeds a particular value. We also will show that when ks is too small, spontaneous secondary spiking is unlikely to occur. The Effect of Large ks on the Bifurcat ion Structure of the z F N Equat ions In an attempt to answer the question of how slow the slow inward current must be, the bifurcation diagrams of the zFN equations are plotted using the bifurcation parameter Is for several values of ks. In order for spontaneous secondary spiking to occur, the bifurcation diagram should look qualitatively like the one shown in Figure 2.17. In this diagram, a Hopf bifurcation gives rise to a branch of limit cycles, which, after a few Chapter 2. SPONTANEO US SECONDARY SPIKING 53 turning points, leads to a branch of stable large amplitude oscillations. At the left end of this branch is the saddle-node of periodics, near which spontaneous secondary spiking tends to occur. Such behaviour occurs when ks << er. When ks increases towards er, the bifurcation diagram changes drastically, as shown in Figure 2.21. The branch of limit cycles emanating from the Hopf bifurcation eventu-ally appears to run into the unstable fixed point at v = VT- The fact that the period suddenly starts increasing rapidly suggests that this is a homoclinic connection. Thus, instead of giving rise to stable large amplitude limit cycles, the branch simply terminates. We will see in the next section that if ks = er, the three-dimensional zFN equations col-lapse to two dimensions. Thus, for ks near er, the zFN equations don't have enough degrees of freedom to exhibit spontaneous secondary spiking, which is an inherently three-dimensional phenomenon. It requires a two-dimensional excitable system at the minimum plus another dimension which contains its memory of the past (the variable z). Analys is of the z F N Equations when ks = er To understand the behaviour of the system when the slow inward current has approxi-mately the same time constant as the recovery variable of the action potential, we follow Hindmarsh and Rose's method of reducing the order of the system [30]. In particular, if ks = er, we make the transformation Iz = zls and rewrite the zFN equations as (2.29a) dW er(V/r - W), (2.29b) dt dh dt er[Is6(V-VT)-Iz}. Chapter 2. SPONTANEOUS SECONDARY SPIKING 54 Figure 2.21: Bifurcation diagrams for the 3D zFN model using Is as the bifurcation parameter for (a) ks = 0.0375 = l.ber, (b) ks = 0.025 = er, and (c) ks = 0.01 = er/2.5. The projections onto v, w, and z, and the period T are shown. (zFN equations using the smooth function (equation (2.3)) for 6(v - VT): a = 0.1, r = 2.5, e = 0.01, VT = 0.3, c = 55, r = 10.) Chapter 2. SPONTANEO US SECONDARY SPIKING 55 This allows us to define a new variable, Y = W — IZ, and combine the last two equations, reducing the system to two dimensions: DJL = _V(V-l)(V-a)-Y, (2.30a) dY — = E R ( V / r - I a 9 ( V - V T ) - Y ) . (2.30b) The nullclines of this system are defined by T> = 0 : Y = -f(V;a), (2.31a) Y = 0 : Y = j - I,9{V - VT). (2.31b) The V-nullcline is a cubic. The F-nullcline is a piecewise linear function, as shown in Figure 2.22, consisting of three branches described by Y = V/r for V < VT, Y = V/r — IS for v > VT, and a vertical branch connecting the two at V = VT-Let (VSS,YSS) denote the fixed points of the system (2.30), namely the simultaneous solution of equations (2.3.1a,b) such that V = Y = 0. As in the 3D system of Section 2.6.1, Vss = Yss = 0 is a fixed point regardless of the value of IS, since VT > 0 by definition and, hence, 9(v — Vr) = 0. As IS increases past a particular value IS(SN) = Vr/r + f(Vr', a), a pair of fixed points arises via a saddle-node bifurcation. The fixed point where the middle branch of the Y-nullcline intersects the y-nullcline is described by Vss = Vr and Yss = —f(Vr;a). For Vss > Vr, the fixed point is described implicitly by 0 = f(Vss; a) + Vss/r - IS and by Yss = -f{Vss; a). To find the stability of the fixed points, we calculate the Jacobian of the system (2.30) at the fixed point (VSS,YSS): J(VSS) ( -fv{Vss;a) - 1 ^ V e - erIs6v(Vss - VT) -er J The eigenvalues are determined by setting (2.32) \J{VSS) - \I\ = (-fv(Vss; a) - X)(-er - A) + e + erIs9v(Vss - VT) = 0, (2.33) Chapter 2. SPONTANEOUS SECONDARY SPIKING 56 (a) IA = 0 (b) 0 < IS < IS(SN) (c) Is — IS(SN) Figure 2.22: The fixed points of the 2D reduction of the z F N equations for ks = cr occur at the intersections of the cubic function describing V = 0 (dashed line) and the piecewise linear function describing Y = 0 (solid line). The right branch of the latter function drops as ls increases, as shown here for (a) ls = 0, (b) ls = 0.5IS(SN): ( c) IS = Is(SN)i (d) Is = 1.3Is(SN)i (e) IS — IS(HB)I a n d (f) Is = ^-8Is(HB)- (zFN equations using a step function for 9(v — VT)'-a = 0.1, r = 2.5, e = 0.01, VT = 0.3.) which reduces to \J(VSS) - A / | = (-fV{VSS; a) - \)(-er - X) + t = 0 (2.34) if VSS 7^  Vr, since then 6V(VSS — Vr) = 0. This equation is exactly the same as the stability equation (2.13) for the FitzHugh-Nagumo equations. Therefore, by the same reasoning as in Section 2.5.1, the fixed point VSS = Yss = 0 is a sink. Similarly, the fixed point for T^ s > Vr is a source if Vss < VHB and a sink if VSS > VJJB where VJJB — Chapter 2. SPONTANEOUS SECONDARY SPIKING 57 ( l / 3 ) [ ( a + l ) + V a 2 - a + l - 3 e r ] . When Vss = VT,-y = det J = er/„(V T ; a) + e - e r / A ( 0 ) . Since we are assuming that 9 is a step function, the term IS9V(0) is very large, so 7 approaches —erls9v(0), which is clearly less than zero. Thus, the middle fixed point where Vss = Vr must be a saddle. Figure 2.23 shows the bifurcation diagram calculated using A U T O 8 6 for the degener-ate z F N equations using Is as the bifurcation parameter. Figures 2.23a and 2.23d are the 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 1 0.8 V0, 0.4 0.2 0 H C 0.142 0.1425 0.143 0.1435 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Figure 2.23: Bifurcation diagram for the degenerate z F N equations where ks = er using Is as the bifurcation parameter. The projections of the steady states and hmit cycles onto V and Y and the period T of the limit cycles are shown. (Note that (c) is an expanded view of (a).) A branch of unstable Hmit cycles arises at the Hopf bifurcation point (HB) and terminates in a homoclinic connection (HC). (Degenerate zFN equations using the smooth function (equation (2.3)) for 6(v - VT): a = 0.1, r = 2.5, e = 0.01, ks = er = 0.025, VT = 0.3, c = 55, r = 10.) Chapter 2. SPONTANEOUS SECONDARY SPIKING 58 same as in Figure 2.21b since the 2D and 3D systems are equivalent when ks = er. The fixed points appear as expected from our previous analysis except for a rounding of the corners due to the smooth approximation for 6{y — VT). At the Hopf bifurcation, labelled HB in Figure 2.23a, a branch of unstable limit cycles arises. The Hopf bifurcation occurs at V s s=0.6713 and Js=0.1424, in agreement with our theoretical predictions. Zooming in on this branch in Figure 2.23c, we see that the limit cycle increases in amplitude with increasing Is as it circles the stable fixed point to the right of the Hopf bifurcation. At Is(HC) = 0.1433, the limit cycle collides with the saddle. The period (Figure 2.23d) tends towards infinity, suggesting a homoclinic connection. Instead of giving rise to a branch of large amplitude stable limit cycles, the branch of unstable limit cycles simply termi-nates. Clearly, this behaviour is very different from the behaviour exhibited by Kepler and Marder's original model. Thus we conclude that the rate constant ks of the slow inward current must necessarily be slower than er, which is related to the rate of recovery of the action potential. The Effect of Smal l ks on the z F N Equations The examples of spontaneous secondary spiking that we have seen so far all involved models with relatively large ks values. In these models, z oscillated substantially, as seen, for instance, in the bifurcation diagram of Figure 2.17. The result was that Iz built up relatively quickly so that spontaneous secondary spiking could occur after only a few action potentials. As ks is reduced, z changes more slowly. As we can see in Figure 2.24, the oscillations in z during tonic firing are much smaller when ks is small. Comparing this with Figure 2.17, we also see that the saddle-node of periodics occurs at a lower Is. Thus, continuous spontaneous secondary spiking is possible at lower Is values when ks is small. The positions of the steady states and the Hopf bifurcation point are unchanged by a change Chapter 2. SPONTANEO US SECONDARY SPIKING 59 in ks. Because z changes so slowly, we can view the system with small ks as a fast subsystem with a slow modulating inward current, as discussed in Section 2.5. Our previous analysis suggested that we could expect spontaneous secondary spiking if IZ reached the minimum value IZMIN for which the fast subsystem first exhibited tonic firing. Referring to Figure 2.12, we see that this is just below the Hopf bifurcation point since the Hopf bifurcation is subcritical. Thus we can approximate the minimum value IZMIN by Iz{vHBmtn) where Is Is Figure 2.24: Bifurcation diagram for the 3D zFN equations using IS as the bifurcation pa-rameter when ks is small. The projections of the steady states and limit cycles onto v, w, and z and the period T of the limit cycles are shown. The saddle-node of periodics (SNP) occurs when IS = IS(SNP) — 0.09712. (zFN equations using the smooth function (equation 2.3) for 0{v - VT): a = 0.1, r = 2.5, e = 0.01, ks = 0.001, VT = 0.3, c = 55, r = 10.) Chapter 2. SPONTANEO US SECONDARY SPIKING 60 Iz{v) = v(v - a)(v - 1) + 7 and vHBmin = (l/3)[(a + 1) - V'a2 - a + 1 - 3er]. For a given value of J s , we would expect spontaneous secondary spiking to occur only if z achieves a minimum value of z = Iz IIs. Kepler and Marder predicted that, in the limit of very small ks, the asymptotic value of z as the system is driven will be equal to the duty cycle (the spike width divided by the period of the oscillation) [36]. We first derive this prediction and then discuss how it can be used to determine when spontaneous secondary spiking can occur. Let us denote the time for which v > VT by ton and the time for which v < Vr by t0ff- Then the duty cycle is ton/(ton + t0ff). If ks is very small, then we can assume that z does not change appreciably over one cycle. If z is approximately equal to z0, then ks(l - z0) v > VT t (2.35) -ksz0 v <VT Thus the change in z over one cycle is A z PS ks(l — zo)ton — kszot0ff. Setting Az = 0 is equivalent to the condition z0 = ton/(ton + t0ff). If ZQ > ton/(ton + t0ff), then Az < 0, so z must decrease. Similarly, if ZQ < ton/(ton + t 0 / / ) , then Az > 0, so z must increase. Thus, z asymptotically approaches the duty cycle when ks is small. This means that we know how big z will be after driving the system for a long time with a particular duty cycle. If the duty cycle is less than IZrn,n/Is, we predict that z will never reach the minimum value necessary for spontaneous secondary spiking. Since neither z nor the duty cycle can exceed one, there is a lower bound on Is, namely Is = IZmin, where spontaneous secondary spiking will not occur, regardless of the rate of stimulation. As Is increases, the minimum z and hence duty cycle required to achieve spontaneous secondary spiking decreases, so spontaneous secondary spiking is more likely. Whether spontaneous secondary spiking occurs and how many spikes occur depends Chapter 2. SPONTANEOUS SECONDARY SPIKING 61 on how much IZ builds up. How much IZ builds up depends on two factors. The duty cycle of the drive determines the asymptotic value of z, so that the maximum value of IZ is just IZ(MAX) = 7S(duty cycle). How long the drive is determines whether i ~ achieves this maximum value. Thus, the longer the drive, the bigger IZ will get until it reaches its maximum. To understand why IZ decays when IS < IS(SNP), let's examine the duty cycle of the two-dimensional fast subsystem as a function of applied current IZ, as shown in Figure 2.25. Notice that the duration of the action potential in the FitzHugh-Nagumo equations increases substantially with IZ. The result is that the duty cycle increases monotonically with IZ. In the limit of low k s , the current value of IZ dictates the duty cycle. The duty cycle, in turn, dictates the asymptotic value of z. T, toni toff (ms) I - 4 -1 2 -1 O S ^ ^ ^ ^ ^ I • 1 T -ton y - ^ — ^ ^ D C -• • • I • -o o . o 2 o . o 4 o . o e DC O - O S L o . -i o . - i s o 1 o . - i Figure 2.25: The duty cycle (DC=ton/(ton + 20//)) of tonic firing increases monotonically as a function of applied current Iz in the FitzHugh-Nagumo equations. The period T and the times ton and t0ff for which the voltage is above and below Vr=0.3, respectively, also are shown (a = 0.1, r = 2.5, e = 0.01, r = 10). Only when the current value of IZ equals the product of IS and the duty cycle, will the system exhibit a stable oscillation. Thus, we can predict the values of IS for which stable spontaneous secondary spiking can occur using the duty cycle from Figure 2.25 and the relationship, i s = i z / (duty cycle). The resulting z versus IS bifurcation diagram Chapter 2. SPONTANEO US SECONDARY SPIKING 62 (Figure 2.26), which is valid in the limit as ks approaches zero, agrees reasonably well with Figure 2.24c, where ks is small but finite. Figure 2.26 tells us that for very small ks, continuous spontaneous secondary spiking only occurs for IS between / s(S;VPI)=0.093 and 7 s(SJVP2)=0-195. 1 0.8 0.6 Z 0.4 0.2 0 0 0.05 0.1 0.15 0.2 Is Figure 2.26: Values of z as a function of Is during continuous spontaneous secondary spiking in the zFN equations predicted from the duty cycle in Figure 2.25 in the limit of small ks (a = 0.1, r = 2.5, 6 = 0.01, T = 10, VT = 0.3). Figure 2.27 shows the duty cycle of tonic firing in the FitzHugh-Nagumo equations as a function of IZ. It also shows the actual value of z corresponding to a particular value of IZ given various values of I S , using the relationship z = I Z / I S . Based on the facts that z asymptotically approaches the duty cycle dictated by the current IZ value and that the duty cycle is monotonically increasing with I Z , we see that if IS < IS(SNPI), then z will eventually decay back to 0. On the other hand, if IS(SNPI) < IS < IS(SNP2), then the two curves intersect to give a stable steady state corresponding to continuous spontaneous secondary spiking. In Figures 2.28 to 2.33 we examine the effects of different pacing protocols on the z F N equations for a particular parameter set. Panel (a) of each of these figures shows simulations of v, z, and sometimes w as functions of time. Panel (b) shows the trajectory of the voltage v as a function of the slow inward current IZ. It is superimposed on the v Chapter 2. SPONTANEO US SECONDARY SPIKING 63 -1 o . s o . s 0 . 0 4 / 0.093.,- 0.14 . . - -" 0 . 1 9 5 ^ . - ' , . , - - - " 0 . 2 4 0 . 4 -o . s o • • i O 0 . 0 2 0 . 0 4 O . O S O . O S O . I 0 . 1 2 0 . 1 4 O . I G 0 . 1 S O S Figure 2.27: z asymptotically approaches the duty cycle dictated by its current IZ value (curved line). The actual value of z given its current IZ value depends on IS according to the equation z = IZ/IS (straight lines shown here for several values of IA). Continuous spontaneous secondary spiking is only possible when IS(SNPI) = 0.093 < IS < IS(SNP2) = 0.195. (a = 0.1, r — 2.5, c = 0.01, r = 10, VT = 0.3). versus IZ bifurcation diagram of the two-dimensional fast subsystem (FitzHugh-Nagumo equations) where IZ is assumed to be a constant applied current. Panel (c) shows the trajectory in v-w phase space, superimposed on the nullclines of the fast subsystem. The u-nullcline is shown for Iz=0 and IZ = Iz{vHBmin)-For the parameter set used in the bifurcation diagram in Figure 2.24, the saddle-node of periodics at which continuous spontaneous secondary spiking first arises in the zFN equations is at IS = IS(SNP) — 0.09712. This value is slightly larger than the value of 0.093 predicted in the limit of very small ks. The first Hopf bifurcation point where limit cycles arise in the two-dimensional fast subsystem occurs when VHBm,n = 0.06207 and Iz(vHBmin)=0.02704. Thus, we expect that lz must build up to a minimum value of approximately 0.027 before spontaneous secondary spiking will occur. Since the duty cycle and, hence, the asymptotic value of z are less than one during drive, spontaneous secondary spiking only can occur in the zFN equations if IS > Iz(min) = 0.027. For 0.027 < IS < 0.097, any spontaneous secondary spiking will be followed by a return to Chapter 2. SPONTANEO US SECONDARY SPIKING 64 -0_4 > 1 i  1 1 1 1 1 1 I I o so - i OO "I SO SOO SSO 300 3SO -*00 «*so soo t - C 4 - C 2 O C 2 0 .4 0 . 6 0 .8 1 1 .2 1 .4 Figure 2.28: Simulation of the zFN equations with small ks and J s > IS(SNP)- 20 current pulses of amplitude 0.1 and duration 0.5 are applied with a frequency of 100 Hz. When the drive ceases, the system exhibits continuous spontaneous secondary spiking, (a = 0.1, r = 2.5, e = 0.01, r = 10, IS = 0.100, ks = 0.001, VT = 0.3, c = 55) Figure 2.29: Simulation of the z F N equations with small ks and IS < IS(SNP)- Spontaneous secondary spiking fails to occur when the rate of stimulation is inadequate, even after very long drives. 50 current pulses of amplitude 0.1 and duration 0.5 are applied with a frequency of 70 Hz. When the drive ceases, the system returns to rest without exhibiting any spontaneous secondary spikes, (a = 0.1, r = 2.5, e = 0.01, r = 10, IS = 0.090, ks = 0.001, VT = 0.3, c = 55) Chapter 2. SPONTANEOUS SECONDARY SPIKING 66 V, w, z -I O - S o - o . s - < :> _ ^1 o - o . s - 0 . 4 steady states ill I lit cycles o . o s 0 . 0 s 0 . 2 0.1 5 0.1 0 . 0 5 H O - 0 . 4 - 0 . 2 O 0 . 2 0 .4 0 .6 0 .8 1 .2 1 .4 Figure 2.30: Simulation of the zFN equations with small ks and IS < IS(SNP)- Spontaneous secondary spiking fails to occur when the system is not driven for along enough time. 10 current pulses of amplitude 0.1 and duration 0.5 are applied with a frequency of 100 Hz. When the drive ceases, the system returns to rest without exhibiting any spontaneous secondary spikes, (a = 0.1, r = 2.5, e = 0.01, r = 10, IS = 0.090, ks = 0.001, VT = 0.3, c = 55) Chapter 2. SPONTANEO US SECONDARY SPIKING 67 - O * 1 ' • 1 1 1 1 1 i I O S O 1 O O -| £ 3 0 ^ O O S S O 3 0 0 : 3 S O - 4 - 0 0 - 4 - S O S O O t - 0 . 4 - 0 . 2 O 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 F i gure 2.31: Simulation of the z F N equations with small ks and IS < IS(SNP)- 25 current pulses of amplitude 0.1 and duration 0.5 are applied with a frequency of 100 Hz. When the drive ceases, the system exhibits 4 spontaneous secondary spikes followed by a return to rest, (a = 0.1, r = 2.5, c = 0.01, r = 10, I, = 0.090, ks = 0.001, VT - 0.3, c = 55) Chapter 2. SPONTANEOUS SECONDARY SPIKING 68 -0-4 1 1 • • 1 1 ' • 1 1 1 O O.O-I O.OS 0-03 o.os - C 4 - C 2 O 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 V Figure 2.32: Simulation of the zFN equations with small ks and IS < IS(SNP)- 60 current pulses of amplitude 0.1 and duration 0.5 are applied with a frequency of 100 Hz. When the drive ceases, the system exhibits 17 spontaneous secondary spikes followed by a return to rest, (a = 0.1, r = 2.5, e = 0.01, r = 10, IS = 0.090, ks = 0.001, VT = 0.3, c = 55) Chapter 2. SPONTANEOUS SECONDARY SPIKING 69 - 0 . 4 - 0 . 2 O 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 Figure 2.33: Simulation of the zFN equations with small ks and IS < IS(SNP)- 60 current pulses of amplitude 0.1 and duration 0.5 are applied with a frequency of 120 Hz. When the drive ceases, the system exhibits 30 spontaneous secondary spikes followed by a return to rest, (a = 0.1, r = 2.5, e = 0.01, r = 10, IS = 0.090, ks = 0.001, VT = 0.3, c = 55) Chapter 2. SPONTANEO US SECONDARY SPIKING 70 quiescence. If IS > 0.097, continuous spontaneous secondary spiking is possible. An example of this is shown in Figure 2.28. If we set Js=0.090, which is below IS = IS<SNP) = 0.097, then z would need to achieve a minimum value of Iz(miN)/IS = 0.3 before spontaneous secondary spiking could occur. This corresponds to a minimum frequency of 88.2 Hz, since the duration of a driven action potential is about 3.4 ms. Simulations in which we paced the system at frequencies above and below 88.2 Hz confirmed that 88.2 Hz is the (approximate) minimum frequency required for spontaneous secondary spiking. Thus, if we drove the zFN equations with a duty cycle less than 0.3 or equivalently, with a frequency of less than 88.2 Hz, IZ would never reach Iz(min)- Even very long drives would be followed directly by a return to quiescence, with no spontaneous secondary spikes, as shown in Figure 2.29a. Figure 2.29b shows that after a very long drive, the oscillation in z stabilizes so that IZ remains in the region below the limit cycles in the bifurcation diagram for the two-dimensional subsystem. If the system is driven with a duty cycle greater than 0.3, but the drive is terminated before z reaches its maximum value, we may again see a direct return to quiescence, as shown in Figure 2.30. If the drive is sufficiently long, however, we may see some spontaneous secondary spikes, as shown in Figures 2.31. Spontaneous secondary spikes will tend to occur as long as IZ remains above Iz(min). Thus, the number of spikes is proportional to the time it takes IZ to decay to Iz[min) • Longer drives can increase z until it reaches its maximum, resulting in more spontaneous secondary spikes, as shown in Figure 2.32. Increasing the duty cycle of drive increases the maximum possible value of z and, hence, can result in even more spontaneous secondary spiking, as shown in Figure 2.33. While low ks means that z builds up very slowly making spontaneous secondary spiking more difficult to achieve, it also means that z decays very slowly. Thus, long drives can result in very long trains of continuous spontaneous secondary spikes when ks Chapter 2. SPONTANEO US SECONDARY SPIKING 71 is low. When ks is higher, the length of the drive does not have as profound an influence on the length of the resulting spontaneous secondary spike train. The Effect of ks on the Bifurcat ion Structure of the K M and M E P Equat ions The effects of increasing ks in the K M and M E P equations are qualitatively the same as in the zFN equations. The bifurcation diagrams of Figures 2.34a and 2.35a show that when ks is large, the branch of periodic orbits terminates in a homoclinic connection instead of giving rise to stable large amplitude oscillations. Thus, the K M and M E P equations do not exhibit continuous spontaneous secondary spiking for large ks. As ks becomes small, we also see the same kind of qualitative changes in the K M and M E P equations seen in the zFN equations. For instance, comparing the bifurcation diagrams of Figures 2.18a and 2.18b with those of Figures 2.34b and 2.35b, respectively, we see that the saddle-node of periodics (SNP) occurs at a lower value of Is for small ks. Thus continuous spontaneous secondary spiking is possible at lower Is values when ks is small. As we did for the zFN equations, we can predict the behaviour of the K M and M E P equations in the limit of very small ks. Figure 2.36 shows the period and duty cycle of tonic firing in the fast subsystems of the K M and M E P equations as a function of applied current Iz. It also shows the times ton and t0ff for which the membrane potential is above and below Vr- Comparing Figures 2.36 and 2.25, we see that while the period T is monotonically decreasing in the K M and M E P equations, the period in the zFN equations decreases and then increases again with increasing Iz. The duration of the action potentials (ton) in the K M and M E P equations is roughly constant, while in the zFN equations it is monotonically increasing. Despite these differences, the net result is that the duty cycle DC in all three systems is monotonically increasing. Using the relationship Is = J z/(duty cycle), we can predict the dependence of z on Is during tonic Chapter 2. SPONTANEO US SECONDARY SPIKING 72 1000 (a) Large kx 0 1 2 3 4 5 6 7 Ia/55 (uA) 1 2 3 4 5 6 7 Is/55 ( M ) 0.5 v/55 o {mV) -0.5 -1.5 (b) Small ks SNP o 1 Is/55 (uA) -0.2 0 1 2 3 4 5 6 7 Is/55 (uA) 0 1 2 3 4 5 6 7 Is/55 {uA) 0 1 2 3 4 5 6 7 Ia/55 (/iA) Figure 2.34: Bifurcation diagrams for the original 5D K M model for (a) ks = 0.2 and (b) ks = 0.01 using Is as the bifurcation parameter. The projections of the steady states and hmit cycles onto v and z and the period T of the hmit cycles are shown. There is a homo-clinic connection (HC) near Is = 5.617 for large ks when the limit cycles run into the fixed point corresponding to v = VT- For small ks, there is a saddle-node of periodics (SNP) at J,=1.069*55.0=58.795 M -Chapter 2. SPONTANEOUS SECONDARY SPIKING 73 (a) Large ks , (b) Small k 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 J , / 5 5 {uA) J s / 5 5 (uA) Figure 2.35: Bifurcation diagrams for the 3D MEP model for (a) ks = 0.4 and (b) ks = 0.01 using Is as the bifurcation parameter. The projections of the steady states and limit cycles onto v and z and the period T of the limit cycles are shown. There is a homoclinic connection (HC) near Is = 6.962 * 55.0 = 382.91 for large ks (ks=0A) when the limit cycles run into the fixed point corresponding to v = VT- For small ks (rc s=0.01), there is a saddle-node of periodics (SNP) at Is=l. 187*55.0=65.285 uA. Chapter 2. SPONTANEOUS SECONDARY SPIKING 74 F i gure 2.36: The duty cycle (DC=ton/(ton + £<>//)) of tonic firing increases monotonically as a function of applied current Iz in the fast subsystem of (a) the K M and (b) the MEP equations. The period T and the times ton and t0jf for which the voltage is above and below VT=0.3, respectively, are also shown. Figure 2.37: Value of z as a function of Is during continuous spontaneous secondary spiking in (a) the K M and (b) the MEP equations predicted from the duty cycles in Figure 2.36 in the hmit of smaU ks. Chapter 2. SPONTANEO US SECONDARY SPIKING 75 firing from the duty cycles in Figure 2.36. The results, shown in Figure 2.37, indicate that there is a stable branch of tonic firing in which z is monotonically increasing for IS(SNPI) < IS < IS(SNP2)- Thus, despite the many quantitative and some qualitative differences between the zFN equations and the K M and M E P equations, the phenomenon of spontaneous secondary spiking is qualitatively the same in all three systems because of the way the duty cycles of the fast subsystems depend on Iz. 2.7 Conclusions In this chapter, we examined Kepler and Marder's model [36] of spontaneous secondary spiking in the lateral gastric axon in the stomatogastric nervous system of the crab Cancer borealis. Kepler and Marder demonstrated numerically that their model exhibited some fascinating spatial and temporal behaviour in which action potentials seemed to arise "spontaneously". One of their key predictions was that the space-clamped model has a quiescent state, as well as a tonic firing mode for sufficiently large Is. They demonstrated numerically why this tonic firing mode might occur by plotting the dependence of the firing frequency of the Hodgkin-Huxley equations on the amplitude of an applied inward current. They superimposed this on a plot of the dependence of the slow inward current on firing frequency for different values of Is. The stable tonic firing mode for large Is occurred when these two curves intersected at a high frequency. We used Kepler and Marder's results as a starting point from which we could extend our understanding of spontaneous secondary spiking. By calculating the bifurcation dia-gram using Is as the bifurcation parameter for their five-dimensional model, we confirmed their prediction and showed that the tonic firing mode arose via a saddle-node of peri-odics bifurcation. It is near this bifurcation that the model exhibits its most interesting behaviour, namely transient and continuous spontaneous secondary spiking. Chapter 2. SPONTANEO US SECONDARY SPIKING 76 In order to gain an intuitive understanding of the fundamental mechanism underlying spontaneous spiking, we reduced the five-dimensional model to three dimensions in two ways. First we replaced the four-dimensional Hodgkin-Huxley equations by the twoT dimensional FitzHugh-Nagumo equations, yielding the zFN model. These well-known equations are a caricature of the Hodgkin-Huxley equations and do not accurately reflect the quantitative and even some of the qualitative aspects of the original system. To gain a more accurate approximation to the original system, which we called the M E P model, we also reduced the four-dimensional Hodgkin-Huxley equations using a simplified version of Kepler and Abbott's method of equivalent potentials [35]. Both of these reduced models exhibit qualitatively the same behaviour as the original five-dimensional model. We advanced a qualitative explanation for spontaneous spiking by showing how the slow inward current modulates the behaviour of the two-dimensional excitable subsystem. To quantitatively predict when continuous spontaneous secondary spiking can occur, we performed a bifurcation analysis using the amplitude parameter Is of the slow inward current as the bifurcation parameter. For the z F N model, we calculated analytically the position and stability of the branches of fixed points including local (Hopf and saddle-node) bifurcations. For a more global picture, namely to calculate the branches of limit cycles, we used the numerical bifurcation analysis package AUT086 to calculate the bifurcation diagrams for each of the models. Our analysis showed that the bifurcation diagrams of the original 5D model and the reduced 3D models shared certain important features. Firstly, all three models had a stable low resting potential for all values of Is. Secondly, a branch of high amplitude limit cycles, corresponding to a stable tonic firing mode, arose via a saddle-node of periodics bifurcation as Is increased. The occurrence of spontaneous secondary spiking depends on the location of Is relative to this saddle-node of periodics. Our bifurcation analysis of the models shows that for spontaneous secondary spiking Chapter 2. SPONTANEOUS SECONDARY SPIKING 77 to occur, the time constant of the slow inward current must be slower than the time constant of the fast excitable subsystem. In particular, this analysis shows that for the zFN model continuous spontaneous secondary spiking is not possible when ks is too large. As ks increases towards er, instead of giving rise to a branch of stable large amplitude limit cycles, the unstable branch of limit cycles terminates in a homoclinic connection, and at ks = er, the three-dimensional zFN model collapses to two dimensions. Thus, when ks is large, the zFN equations do not exhibit continuous spontaneous secondary spiking for any values of Is. We showed that the branches of limit cycles in the bifurcation diagrams for the K M and M E P models also terminate in a homoclinic connection rather than giving rise to stable large amplitude limit cycles when ks is large. Kepler and Marder [36] stated that, in the limit of small ks, the asymptotic value of the gating variable z is equal to the duty cycle of tonic firing of the excitable subsystem. We employed Kepler and Marder's assertion to predict the range of Is for which continuous spontaneous spiking would occur, thus finding the critical value of Is at the saddle-node of periodics bifurcation. Also we showed how to calculate numerically the minimum frequency of stimulation which can lead to spontaneous spiking when ks is small and calculated the smallest value of Iz that must be achieved in order to see spontaneous spiking. These results give us a better understanding of why certain types of stimulation lead to spontaneous spiking while others do not. When the time constant for the slow inward current is much larger than that for the fast subsystem, we can analyze the two systems separately and use the results to predict the behaviour of the full model. Our analysis shows that the slow inward current must reach a sufficient level to induce tonic firing in the excitable subsystem, and thus induce spontaneous secondary spikes. The number of spontaneous secondary spikes is propor-tional to the amount of time it takes the inward current to decay below the minimum level for tonic firing. Thus, the number of spontaneous secondary spikes depends on the Chapter 2. SPONTANEO US SECONDARY SPIKING 78 length of the original drive, the maximum attainable amplitude of the slow inward cur-rent, and how fast the slow inward current decays. As a result, very slow inward currents can lead to long trains of spontaneous secondary spikes following lengthy initial drives. A faster slow inward current can lead to a few spontaneous secondary spikes after only a very short initial drive. Based on our analysis in this chapter, we conclude that the phenomenon of sponta-neous secondary spiking is, in a certain sense, generic. A model must have a minimum of three dimensions, namely a two-dimensional excitable subsystem plus a one-dimensional slow inward current. The excitable subsystem must exhibit either a steady state or tonic firing depending on the amplitude of the applied inward current. The slow inward cur-rent must build up during the action potentials and decay in between action potentials. The exact shape of the action potential is not important. The feedback between the slow inward current and the excitable subsystem is bidirectional. On the one hand, the amplitude of the slow inward current dictates the firing mode of the excitable subsystem, including the duty cycle of firing. On the other hand, the duty cycle of the excitable subsystem dictates the asymptotic value of the slow inward current. Because the duty cycle of the excitable subsystem is an increasing function of the slow inward current, a stable oscillation, known as continuous spontaneous secondary spiking, can be achieved for certain Is values. Spontaneous secondary spiking is of particular interest from a signal processing point of view in neurophysiology because the system responds differently to incoming signals of different frequencies and to trains of different lengths. In addition, the response of the system can be modulated, e.g., by regulating factors such as hormones, by changing either the maximum amplitude or the rate of change of the slow inward current. In a crab, the spontaneous spikes are important because they influence the duration of contraction of certain muscles in the gastric mill . Furthermore, the number of spontaneous spikes and, Chapter 2. SPONTANEO US SECONDARY SPIKING 79 hence, the contraction of the muscles can be regulated by serotonin. The simplicity of the mechanism of spontaneous secondary spiking suggests that it may occur elsewhere and, therefore, play an important physiological role in many other biological systems. Indeed, as we shall see in the next chapter, a qualitatively similar phenomenon called triggered activity occurs in cardiac cells and may have important physiological implications for pathological functioning of the heart. Chapter 3 C A R D I A C T R I G G E R E D A C T I V I T Y A N D T H E D I F R A N C E S C O - N O B L E E Q U A T I O N S Kepler and Marder's spontaneous secondary spiking bears a striking resemblance to a phenomenon in cardiac electrophysiology in which some normally quiescent cardiac cells can exhibit "spontaneous" action potentials following high frequency or lengthy stimula-tion. These "spontaneous" action potentials are referred to as cardiac triggered activity in the cardiac literature. In this chapter, we will introduce some basic concepts in cardiac electrophysiology and give the necessary background required for understanding cardiac triggered activity and its clinical relevance. In Section 3.1, we briefly review the electrophysiology of the heart and cardiac cells, including some relevant medical terminology. Then we discuss the sequence of events involved in a cardiac action potential. Using the definitions of the phases of an action potential, we differentiate between delayed afterdepolarizations, which give rise to the type of triggered activity we are studying here, and early afterdepolarizations. In Section 3.2, we discuss the normal and arrhythmogenic responses of a cell to being paced or driven at high frequency, namely overdrive suppression or overdrive excitation (triggered activity). We discuss the theoretical mechanism thought to be responsible for triggered activity; then, in Section 3.3, we review some experimental methods of promoting or suppressing triggered activity in view of this mechanism. Some clinical implications of triggered activity are discussed. In Section 3.4, we review the existing mathematical models of cardiac cells and the 80 Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 81 modeling work relating to triggered activity that may be relevant to our work. Then, in Section 3.5, we discuss a specific model, known as the DiFrancesco-Noble model, and explain what the model consists of and how the ionic currents interact to give a resting potential or action potentials. Then, in Section 3.6, we introduce a few simplifications which reduce both the dimension (the number of differential equations) and the com-plexity of the model, without significantly altering its behaviour. In Section 3.7, we review some previous work done by Varghese and Winslow in which they examined the C a 2 + subsystem [68] and the full system [69] of equations. Later, we extend some of their results and use them as a basis for the remainder of our work. Finally, in Section 3.8, we discuss the model in the context of coupled oscillators and try to define the difference between normal and calcium-driven firing. 3.1 Introduction to Cardiac Electrophysiology In the following sections, we briefly review the basic electrophysiology of the heart and cardiac cells. Then we introduce some definitions relevant to cardiac action potentials which are needed for future discussion of cardiac triggered activity. 3.1.1 Introduction to the Biology of the Heart The heart has four main compartments, the left and right atria and the left and right ventricles, as shown in Figure 3.1. In order to pump blood efficiently throughout the body, the sequence and timing of contraction in these compartments must be carefully regulated. The heart has a preferential electrical conduction system (His-Purkinje) to help coordinate its activities. The sinoatrial (SA or Sinus) node, located in the right atrium, is the heart's natural pacemaker. It normally sends out electrical impulses at a rate of 60 to 100 beats per Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 82 Figure 3.1: Electrical conduction system of the heart. (Adapted from Fig. 2-3, Mohrman and Heller [46].) minute in humans [14]. The electrical impulse spreads outward from the SA node through the atrial myocardium (muscle in the atria), causing the atria to contract. This wave of depolarization reaches the atrioventricular (AV) node, at which point the tricuspid and mitral valves open and blood enters the ventricles from the atria. The impulse then is conducted rapidly through the bundle of His into the left and right bundle branches and then into the Purkinje fibers. The Purkinje fibers stimulate the ventricular myocardium (muscle) causing the ventricles to contract. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 83 Under normal conditions, the property of beating is confined to certain cells, partic-ularly those in the SA node, AV node, and His-Purkinje system. While the SA node is the primary pacemaker for the heart, other areas of the heart also are capable of pacing or beating at their own intrinsic rate in emergency or pathological situations. Each stage in the electrical conduction system has a slower intrinsic rate than its predecessor. In humans, the atria, A V junction (where the A V node meets the bundle of His), and the ventricles can pace at their inherent rates of 60 to 80, 40 to 60, and 20 to 40 beats per minute, respectively [14]. Normally, the SA node entrains the slower pacemakers, so they beat at the faster rate of the SA node. However, under certain pathological conditions, the other pacemakers may fail to be entrained and start to beat at their own intrinsic rates. These new pacemakers are referred to as ectopic foci. Besides beating at their in-trinsically slow rate, each of these ectopic foci also may beat at a very rapid rate (150 to 250 beats per minute) under an emergency or pathological condition. If the beat rate is unusually high, it is referred to as a tachycardia, and if it is unusually slow, it is referred to as a bradycardia. As will be discussed later, cardiac triggered activity may play a role in promoting various types of tachyarrhythmias, such as ventricular tachycardia. While our study of cardiac triggered activity will focus on single cell electrophysiology, many other issues must be addressed when relating the behaviour of an individual cell to cardiac arrhythmias (abnormal rhythms). For instance, it is difficult to predict the behaviour of a collection of cells based on the behaviour of a single cell. Sometimes a lack of entrainment may lead to many different ectopic pacemakers, so that rather than beating synchronously, different segments of tissue beat out of phase. Geometrical and conduction issues become important when an artery becomes occluded (blocked), cutting off the blood supply to a particular area of the heart. This area of myocardium becomes electrically dead, changing the conduction properties of the tissue. Such a change to the geometry may result in an electrical impulse moving in a "circular" pathway outside Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 84 of the patch, a phenomenon known as re-entry. Reduced blood supply or ischemia also can promote triggered activity in Purkinje fibers [56]. Sometimes experimentalists have difficulty distinguishing between triggered activity and re-entry due to the limitations of experimental procedures. 3.1.2 Electrophysiology of a Cardiac C e l l Cardiac cells are typical excitable cells, but the dynamics are very complex. The currents are typically due to four types of ions (Na + , K + , C a 2 + , C l - ) flowing through at least nine distinct voltage or ligand operated ion channels [75]. The shape of the cardiac action potentials varies among the different types of cells. These differences are due to different proportions and types of ionic currents. Figure 3.2 shows the directions of the various depolarizing and repolarizing currents flowing through the membrane due to transmembrane ion gradients and electrical poten-tial differences. These ion gradients are maintained via the Na-K pump and the Na-Ca exchanger. The ATP-dependent Na-K pump extrudes three intracellular N a + in exchange for the uptake of two K + , resulting in a net outward current. The Na-Ca exchanger, on the other hand, uses the energy of the N a + gradient to pump out one C a 2 + for every three N a + that enter the cell. This results in a net inward current. The gradients result in inward or depolarizing N a + currents, outward or repolarizing K + currents, depolarizing C a 2 + currents, and repolarizing chloride currents. Besides the transmembrane currents, another crucial component of single cell cardiac electrophysiology is the intracellular C a 2 + dynamics. An organelle inside the cell called the sarcoplasmic reticulum, or SR for short, regulates intracellular C a 2 + concentration. Ordinarily it takes up C a 2 + in order to keep the intracellular C a 2 + concentration low. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 85 [Na ] = 140 mM [Or ] = 120 mM Figure 3.2: Typical ionic concentrations and currents found in a cardiac ventricular cell (resting potential = -85 mV). Gated membrane currents include inward N a + and C a 2 + currents and outward K+ and C l - currents. There is also a net inward current due to the Na-Ca exchanger and a net outward current due to the Na-K pump. In addition, the sarcoplasmic reticulum (SR) exchanges C a 2 + with the cytoplasm. Large, negatively charged impermeant proteins help to balance the intracellular charge. (Adapted from Fig. 7, Whalley et al. [75].) However, as C a 2 + flows into the cell during the action potential, it exhibits calcium-induced calcium release (CICR), causing a temporarily high intracellular C a 2 + concen-tration. This high intracellular C a 2 + concentration is responsible for muscle contraction [68]. The process by which an action potential initiates a cellular contraction via C a 2 + efflux from the SR is called excitation/contraction coupling. 3.1.3 Electrophysiology and Phases of the Cardiac Action Potential Figure 3.3 illustrates the phases of a typical cardiac Purkinje cell action potential, Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 86 (a) (b) time time Figure 3.3: The phases of (a) a typical cardiac action potential starting from rest and (b) a series of action potentials exhibiting phase 4 depolarization. which either starts from rest (panel (a)) or is continuously firing (panel (b)). Initially the membrane is at its resting potential or experiencing slow phase 4 depolarization, which starts at the most negative membrane potential, known as the maximum diastolic poten-tial or MDP. The mechanism of phase 4 depolarization in SA nodal cells is still subject to controversy, but is thought to involve the inactivation of an outward K + current, a time independent background current carried predominantly by N a + , and a hyperpolarization activated inward current (If) [75]. In Purkinje cells, the hyperpolarization activated in-ward current, If, is the major current responsible for phase 4 depolarization [75]. Either phase 4 depolarization or a stimulus brings the membrane potential to a threshold. In Purkinje, atrial, and ventricular cells, the membrane potential reaches the N a + thresh-old, at which the N a + current turns on rapidly. The resultant sharp upstroke of the membrane potential is called phase 0 depolarization. In SA and A V nodal cells, however, this upstroke is generated by a C a 2 + current. In these cells, the M D P is at a significantly more depolarized level, so that the N a + current is largely inactivated. The next phase, Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 87 phase 1 repolarization, is a short rapid repolarization due to voltage dependent inactiva-tion of the N a + current, combined with activation of a transient outward current. During the plateau phase (phase 2), the inward and outward currents are roughly balanced. An inward C a 2 + current, which only activates at membrane potentials above -40 mV, causes an influx of C a 2 + into the cell. The resulting high intracellular C a 2 + causes the SR to release more C a 2 + , which is pumped out by the Na-Ca exchanger or pumped back into the SR. Finally, phase 3 repolarization, which returns the membrane to rest or to its MDP, results from the dominance of outward K + currents. Some other terms commonly found in the literature are systole and diastole. Sys-tole refers to the portion of the cycle during which muscle excitation and contraction occur. Diastole refers to the period during which the cell recovers from muscle excitation and contraction. Thus phase 4 depolarization also is referred to as slow diastolic depo-larization. A cell which can spontaneously depolarize during phase 4, attain threshold potential, and initiate an action potential is said to exhibit (normal) automaticity [5]. 3.1.4 Afterdepolarizations and Triggered Activity We now can use our definitions of the phases of the action potential to define afterdepo-larizations. Afterdepolarizations come in two forms, early and delayed, as illustrated in Figure 3.4. Early afterdepolarizations, which will be referred to as EADs, occur before phase 3 repolarization is complete, interrupting or retarding membrane repolarization. EADs can lead to secondary upstrokes from a depolarized membrane potential, considered to be a form of cardiac triggered activity. While this form is clinically very important because it is thought to be associated with a fatal condition known as Torsades de Pointes, it is mechanistically very different from the other type of cardiac triggered activity which we are about to discuss. Therefore, we will not be discussing triggered activity involving Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 88 (a) 30 nrVf OmV} -90 mV early afterdepolarization (b) 30mVr OmW E E -90 mV delayed afterdepolarization / early afterhyperpolarization time Figure 3.4: Sketches of (a) early and (b) delayed afterdepolarizations. EADs here, but leave it for future work. Delayed afterdepolarizations or DADs, unlike EADs, occur after phase 3 repolar-ization is complete. These small depolarizations in the membrane potential can reach threshold (e.g., N a + threshold) and hence lead to the upstroke of an action potential. The resulting action potentials are the second form of triggered activity, which is the focus of our work. This form of triggered activity will be referred to as DAD-induced triggered activity. A key difference between these two types of triggered activity is how they are initiated. DADs and their resultant triggered activity are most likely to occur when the cell is "overdriven" (paced at a very fast rate, higher than the intrinsic pacemaking rate). They are generally associated with tachycardias. EADs and their resultant triggered activity, on the other hand, are more likely to occur when the cell is paced at a very slow rate and are associated with bradycardias. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 89 3.2 Theoretical Effects of Overdrive and C a 2 + Loading Normally, driving a cell at a higher rate than its intrinsic rate of pacing leads to a slowing down or suppression of the cells natural pacemaking activity called overdrive suppression. However, under certain circumstances, known as calcium overload, the cell may respond arrhythmogenically by pacing at an even higher rate. This behaviour is referred to as triggered activity or overdrive excitation. In the following sections, we will review the theoretical mechanisms that are thought to account for this normal and abnormal behaviour. 3.2.1 Normal Effects of Overdrive (Overdrive Suppression) As mentioned earlier, the cell maintains transmembrane ion gradients via the Na-K pump and Na-Ca exchanger. While all the ion concentrations affect the magnitudes of the transmembrane currents, the changes in some of the ion concentrations over the course of a few action potentials are insignificant. The intracellular K + and intracellular N a + are two examples of slowly varying ion concentrations. Under normal operating conditions, such as quiescence or normal pacing, these two ions reach approximate equilibrium values and can be viewed as fixed, since their changes in concentration are insignificant over the course of an action potential. However, if the cell is overdriven, or driven at a rate faster than the cell's intrinsic rate, the cell has less time to recover from each action potential. Initially the Na-K pump cannot cope with the increased ion load. As a result, since N a + flows into the cell during each action potential and K + flows out of the cell, the intracellular N a + initially builds up while the intracellular K + is depleted. At the same time, K + builds up in the small space between cells known as the extracellular cleft. As intracellular N a + and cleft K + build up, the Na-K pump responds by increasing its activity to try to restore Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 90 the ions back to their original concentrations. The increased pump activity restores the cleft K + back to its original level. However, the intracellular N a + reaches a new elevated concentration. When overdrive ceases, the intracellular N a + concentration is still high, so the Na-K pump activity is still high. As a result, cleft K + can decrease transiently. Intracellular N a + will decrease slowly back to its original level, and the Na-K pump will slow down accordingly. Vassalle discusses overdrive suppression (and excitation) in some detail [71]. He cites an illustrative experiment [70] performed on a cardiac Purkinje fiber. In this experiment, the fiber is spontaneously active at a rate of 14 beats/min. It then is overdriven at a rate of 120 beats/min for 2 minutes. Initially the maximum diastolic potential decreases in amplitude (depolarizes), probably due to the initial transient increase in cleft K + . The cleft K + is restored to its original value while the Na-K pump activity is still increasing. Because the Na-K pump is actually a net outward current, it serves to hyperpolarize the membrane so that the amplitude of the maximum diastolic potential increases. When overdrive ceases, the outward Na-K pump current is still high, so that it takes longer (about 50 seconds) for the membrane potential to reach threshold. This temporary sup-pression of the cell's natural pacemaking activity is referred to as overdrive suppression. As the Na-K pump current decreases with intracellular N a + , the cell begins to pace again slowly. The cell's pacemaking then speeds back up to its original rate. Like N a + , C a 2 + also flows into the cell during an action potential. Thus, intracellular C a 2 + also can build up during overdrive. Increased intracellular C a 2 + concentration is thought to play a minor role in overdrive suppression by increasing K + conductance and hence increasing the amplitude of the maximum diastolic potential. We will see in the next section that intracellular C a 2 + concentration plays a more important role in abnormal behaviour resulting from overdrive. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 91 Overdrive suppression plays an important role in cardiac electrophysiology. As men-tioned earlier, each stage in the electrical conduction path is overdriven, and hence en-trained, by the preceding stage. If the overdriving entrainment signal ceases temporarily, due to a skipped beat, for example, then the overdriven cells experience temporary over-drive suppression. Thus they do not immediately "escape" entrainment and start to beat at their own intrinsic rate every time there is some kind of lag in the pacemaking rhythm. Thus overdrive suppression ensures that the cell waits for its entrainment signal so that it remains synchronized with the other cells. As Vassalle [71] puts it, overdrive suppression keeps "latent pacemakers in check when not needed." 3.2.2 Calcium Overload and Triggered Activity Because high cytoplasmic C a 2 + concentrations signal myocardial cells to contract, it is important that intracellular C a 2 + concentrations increase during systole and remain low during diastole. Under normal operating conditions, cytosolic C a 2 + levels are kept low by the Na-Ca exchanger which pumps C a 2 + out of the cell and an uptake pump which pumps it into the sarcoplasmic reticulum inside of the cell. As C a 2 + flows into the cell during an action potential, the small increase in intracellular C a 2 + triggers release of C a 2 + from the SR via calcium-induced calcium release. Intracellular C a 2 + increases substantially, causing the cell to contract, and then it returns to its normally low level as it is taken up by the SR again or pumped out of the cell. In certain unusual circumstances, the cell may develop what is called calcium over-load. If too much C a 2 + enters the cell, the storage capabilities of the SR may not be able to handle it. As C a 2 + builds up in the SR in its attempt to reduce the calcium overload, it may start to release C a 2 + back into the cytoplasm in an oscillatory manner at the beginning of diastole. This oscillatory release of C a 2 + is an arrhythmogenic behaviour. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 92 As Vassalle [71] points out, it may either be a simple failure of the mechanism for main-taining stable low C a 2 + concentrations ( C a 2 + homeostasis) or "a last resort attempt at diminishing C a 2 + overload through the exchange with N a + . " This spontaneous release of C a 2 + by the SR during diastole increases the intracellular C a 2 + concentration substantially. The resulting high intracellular C a 2 + concentration turns on at least one of two possible inward currents in the cell membrane. One current that it activates is the Na-Ca exchanger current, which tries to pump the excess C a 2 + out of the cell. This current is referred to as an electrogenic current because more charge enters the cell (3 Na + ) than leaves (1 C a 2 + ) . As a result, the net inward current serves to depolarize the membrane. Another current, a nonspecific Ca 2 +-activated current carried largely by N a + ions, also may help depolarize the membrane. If the membrane is depolarized past the N a + current threshold, a N a + upstroke can occur. In this case, a full action potential results. The process will repeat itself until the cell is no longer overloaded with C a 2 + . If the depolarization is not sufficient to induce a full action potential, a small depolarization hump is seen, known as a delayed afterdepolarization or oscillatory afterpotential. As mentioned earlier, overdrive causes the buildup of C a 2 + inside the cell by reducing the diastolic interval, so that the Na-Ca exchanger cannot keep up with the C a 2 + influx (high C a 2 + load). Thus, experimentalists use high frequency or prolonged drive to induce triggered activity. Because of this, the action potentials following a period of overdrive are sometimes referred to as overdrive excitation. They also are referred to as triggered activity since one or more action potentials are required to trigger them. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 93 3.3 Expe r imen ta l Interventions for P romot ing and Suppressing Delayed Af-terdepolarizations and Triggered A c t i v i t y In view of the proposed mechanism for triggered activity outlined above, we see that two classes of interventions can promote or suppress it. First of all, the C a 2 + dynamics can be modified to promote the occurrence of calcium overload. Secondly, the membrane currents can be modified so that the cytosolic C a 2 + oscillations are able to depolarize the membrane potential past threshold. Of the membrane dynamics, the Ca 2 +-activated inward currents are the most impor-tant currents involved in depolarization of the membrane potential. Enhancement of the Na-Ca exchanger current, for instance, would tend to result in a larger depolarization when the cytosolic C a 2 + concentration goes high. Modification of other currents that set the resting membrane potential also can promote triggered activity by depolarizing the resting potential so that it is closer to threshold. A number of factors individually or in combination can contribute to the development of calcium overload and the resulting depolarizations of the membrane. Calcium overload occurs for one of three reasons: increased C a 2 + loading (influx), decreased C a 2 + efflux, or changes in the SR dynamics. Overdriving the cell, as mentioned earlier, is one of the most obvious ways of increasing C a 2 + loading. However, on its own, overdrive is not necessarily sufficient to result in calcium overload. Recall that triggered activity is considered to be arrhythmogenic behaviour, so, in general, certain other experimental interventions are necessary. 3.3.1 H i g h Ext race l lu la r C a 2 + Concentrations Another way of enhancing C a 2 + influx is to increase the extracellular C a 2 + concentration. This ensures that more C a 2 + flows into the cell via the slow inward C a 2 + current during Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 94 each action potential and also enhances the C a 2 + leak current somewhat. Ferrier and Moe [22] demonstrated that increased extracellular C a 2 + concentration enhances the magnitude of DADs following overdrive in Purkinje tissue in vitro; see [22], Figure 2. They demonstrated that an untreated Purkinje fiber does not exhibit DADs in response to pacing at basic cycle lengths of 400, 300, and 250 msec when the extracellular C a 2 + concentration is 2.5 m M . However, at an extracellular C a 2 + concentration of 10.0 m M , DADs arise and grow with decreasing basic cycle length. 3.3.2 Application of Catecholamines Application of catecholamines, such as epinephrine and norepinephrine (adrenaline and noradrenaline), also enhances Ca 2 +-influx by directly increasing the slow inward C a 2 + -current during an action potential. While catecholamines promote triggered activity, C a 2 + channel blockers such as verapamil, D600, nifedipine, and manganese, can suppress it [1]. Wit and Cranefield published several papers [78, 79, 80] which demonstrate the pro-motion of DADs and triggered activity in the simian mitral valve and canine coronary sinus by catecholamines and the effects of various pacing protocols. A n example of the effects of epinephrine on the simian mitral valve is given in [80], Figure 10. The un-treated membrane normally repolarized smoothly following an action potential. After the addition of epinephrine, however, DADs clearly could be seen following each action potential. At higher frequencies of pacing, the amplitudes of the DADs grew until they reached threshold and resulted in an action potential. In some cases, sustained triggered activity resulted. The rate of triggered activity could be significantly higher than the rate of pacing, as seen in [80], Figure 12. Further experimental evidence of catecholamines promoting triggered activity was supplied by Valenzuela and Vassalle [66] who applied Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 95 noradrenaline and high extracellular C a 2 + concentrations to canine Purkinje fibers sep-arately and in combination. 3.3.3 Inhib i t ion of the N a - K P u m p Another factor that can contribute to calcium overload is reduced Na-K pump activity. Reduced extrusion of intracellular N a + by the Na-K pump results in a buildup of N a + inside the cell. Increased intracellular N a + reduces the N a + gradient across the. cell membrane, and hence reduces the force driving the Na-Ca exchanger. This in turn reduces the ability of the Na-Ca exchanger to extrude C a 2 + , resulting in a buildup of C a 2 + in the cell. Cardiac glycosides, also referred to as digitalis glycosides, are known to reduce the Na-K pump activity [34, 62] and promote triggered activity in vitro. This type of triggered activity is generically referred to as digitalis-induced triggered activity. The effect of digitalis in inducing DADs has been studied most extensively in canine Purkinje fibers [82]. A nice example is given in Figure 3.5, where Ferrier, Saunders, and Mendez [23] induce triggered activity in a canine Purkinje fiber in the presence of digitalis by pacing it at various basic cycle lengths. While all of the examples discussed so far have only had a single D A D following the action potentials, multiple DADs can occur. Rosen and Danilo paced a canine cardiac Purkinje fiber superfused with digitalis. In Figure 19-1 of their paper [57], they demon-strated that the first three driven action potentials were followed by not one, but three DADs with decaying amplitudes. 3.3.4 A l t e r i n g the Sarcoplasmic R e t i c u l u m Dynamics Yet another way of promoting or suppressing triggered activity is to modify the uptake and release of C a 2 + by the sarcoplasmic reticulum. For instance, application of small amounts of caffeine can facilitate the release of C a 2 + from the SR and hence amplify Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 96 BCl = 800 BCL = 700 BCL = 600 BCL = 5O0 Figure 3.5: DADs and triggered activity in a canine cardiac Purkinje fiber exposed to acetyl-strophanthidin (a digitalis glycoside). Sequences of six driven beats (denoted by dots) were produced at basic cycle lengths (BCL) of 800, 700, 600, and 500 msec, resulting in zero, one, two, and three triggered spikes, respectively. (Reproduced by permission of Lippincott Williams k Wilkins from Fig. 6, Ferrier et al. [23].) D A D s . O n the other hand, large amounts of caffeine suppress the uptake of C a 2 + by the S R and hence suppress the occurrence of D A D s . Ryanodine also can suppress the occurrence of D A D s by blocking the release of C a 2 + from the sarcoplasmic ret iculum [1]. 3.3.5 Delayed Afterdepolarizations and Cardiac Arrhythmias Delayed afterdepolarization-induced triggered activity is thought to play an important role in the genesis of a number of cardiac arrhythmias. However, to date, no definitive recordings of D A D s have been made in vivo [1]. The evidence for D A D related car-diac arrhythmias is based on indirect evidence. Triggered activity and D A D s have been recorded in many in vitro experiments under conditions which are known to promote certain cardiac arrhythmias in vivo. For instance, digitalis clearly has been shown to promote D A D s and triggered activity in vitro. Digitalis toxicity also has been associated Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 97 with bigeminal rhythms and tachyarrhythmias in vivo. Many characteristics of D A D -induced triggered activity, such as their response to various pacing protocols and drugs, support their supposed role in some types of cardiac arrhythmias [1]. However, often it is impossible to rule out other mechanisms such as reentry or enhanced automaticity. There are many experimental interventions, including drugs, modification of the ionic solutions surrounding the cell, and pacing protocols, which can promote or, alternatively, suppress triggered activity. We have only reviewed some of the more popular methods for inducing triggered activity. For a more complete review of interventions which suppress and enhance DADs and triggered activity and their clinical relevance, refer to the review articles [1, 32] or the textbooks [10, 58]. 3.4 Modeling Studies of Cardiac Cells and Triggered Activity Finding cardiac mathematical models suitable for a study of triggered activity is not a trivial task. On the one hand, simple models are easier to analyze. On the other hand, the model must have sufficient details so that it at least qualitatively represents the dynamics being studied. Early models such as the Beeler-Reuter model [4] are inappropriate for the study of triggered activity because they are missing some of the most important processes, namely the Ca 2 +-activated inward current responsible for depolarizing the membrane and the dynamics of the sarcoplasmic reticulum. Traditional Hodgkin-Huxley-style models take account of currents flowing through voltage-gated ion channels in the membrane. These currents depend on electrochemical gradients established by a difference in the concentrations of ions inside and outside the membrane. In some cases, these concentration gradients are assumed to be static. However, changes in the concentrations of ions on either side of the membrane can have a substantial effect on the dynamics of the membrane potential. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 98 3.4.1 The DiFrancesco-Noble M o d e l and Its Successors One of the first cardiac models to try to take into account the important changes in intracellular ionic concentrations was the DiFrancesco-Noble model of the Purkinje fiber [12]. This comprehensive model took into account not only ion channels, but also the sodium/potassium (Na-K) pump, the sodium/calcium (Na-Ca) exchanger, and the sar-coplasmic reticulum C a 2 + pump and release mechanisms [49]. This model, which essen-tially made the earlier Purkinje fiber model by McAllister et al. [44] obsolete, seems to be the most up to date model of the Purkinje fiber. It has been adapted to describe other cells, such as the mammalian sinoatrial node (Noble and Noble [51]). The sinoatrial node model was in turn modified for a single sinoatrial cell by Noble et al. [50]. Later, Wilders et al. [76] presented another model of a rabbit sinoatrial cell. In trying to describe the mammalian atrium, Hilgemann and Noble improved the Ca 2 +-related equations [28]. These equations were used by Earm and Noble to develop a single-cell atrial model [15]. Later Noble et al. developed a model of the guinea pig ventricular cell [52]. Another model of a guinea pig ventricular myocyte was developed from the DiFrancesco-Noble equations by Nordin [53]. 3.4.2 The L u o - R u d y Mode l s At about the same time, Luo and Rudy also developed a model of the membrane potential of a mammalian ventricular cell [41]. Their model attempted to improve upon a much earlier model by Beeler and Reuter [4] by using more up to date experimental data. A recent, improved version of their model of the cardiac ventricular action potential [42, 43] contains extensive information relating to processes that regulate or depend upon intracellular C a 2 + concentration. This model exhibits triggered activity involving both early and delayed afterdepolarizations. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 99 A number of simulation studies have been performed on Luo and Rudy's models [41, 42, 43, 60, 74, 81]. In addition, Gibb et al. [25] conducted both simulation and bifurcation studies of the Luo-Rudy model [41] to understand the effects of K + block-ade on ventricular tachyarrhythmias, with a particular emphasis on the role of early afterdepolarizations. The emphasis of most of these studies has been on early afterdepolarizations. How-ever, Luo and Rudy also have conducted some fairly detailed simulation studies of delayed afterdepolarizations and DAD-induced triggered activity [43, 59] using their most recent model [42]. Their focus was on which Ca 2 +-activated inward current was largely respon-sible for depolarization at different levels of calcium overload. 3.4.3 Analys i s and Simulations of the Mode l s While many researchers have performed simulations of the above models, few have con-ducted more analytical studies. Bifurcation studies of some of the older or simplified cardiac models have been carried out by Landau et al. [38, 39], Vinet et al. [72, 73], and Chay and Lee [6, 7, 8]. Vinet and Roberge [73] performed an extensive bifurcation anal-ysis of the modified Beeler-Reuter model and several lower dimensional approximations to it. None of these studies, however, related to DADs or triggered activity. The modeling studies most relevant to the present work were conducted by Varghese and Winslow. Varghese and Winslow studied the DiFrancesco-Noble model of the cardiac Purkinje cell. They performed fairly detailed bifurcation and simulation studies of the effects of inhibition of the Na-K pump on the dynamics of the C a 2 + subsystem [68] and the membrane potential [69]. Their work set the stage for the work in this report and will be discussed in detail later. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 100 3.5 Introduction to the DiFrancesco-Noble Equations In the previous sections, we introduced the background necessary for understanding car-diac triggered activity. We now will focus on the analysis of one specific cardiac model, namely the DiFrancesco-Noble equations for a mammalian cardiac Purkinje fiber [12]. The DiFrancesco-Noble equations were originally developed in 1985 to describe cardiac action potentials and pacemaker activity [12]. While our understanding of the dynamics underlying the action potentials has improved since then, the model has the necessary features required for triggered activity, and its relatively simple form makes it easier to analyze than some other cardiac models. This particular model was chosen for a number of reasons. The most recent Luo-Rudy model [42] of a ventricular cell could have been chosen since it has been shown to exhibit triggered activity involving DADs. However, the complex C a 2 + dynamics in this model would tend to obscure the essential features of triggered activity. The DiFrancesco-Noble model has all of the components theoretically necessary for describing triggered activity. With it we can study the effects of overdriv-ing quiescent or pacemaking cardiac cells, since the parameters are readily adjusted for the cell to exhibit quiescence or tonic firing. Lastly, a considerable amount of experi-mental work has been conducted on the Purkinje fiber to examine the effects of various interventions in producing triggered activity [2, 20, 22, 23, 56, 66, 80]. In this section, we describe the various features of the model and show how these features define its behaviour. We show which features are important in determining resting potential or equilibrium values of ion concentrations. We then examine the events during an action potential and during tonic firing. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 101 3.5.1 What the DiFrancesco-Noble Equations Include The DiFrancesco-Noble equations are a system of sixteen ordinary differential equations representing the electrical activity of an equipotential mammalian cardiac Purkinje fiber. A schematic view of the model is given in Figure 3.6, while the full system of equations is listed in Appendix C (taken from the Appendix of Varghese and Winslow's paper [69]). The external ion concentrations have been set to agree with those used in the standard bathing solution in Ferrier and Moe's experiments on canine cardiac Purkinje fibers [22, 23]. Unless stated otherwise, all of our calculations will use the standard equations and standard parameter set listed in Appendix C, which lead to tonic firing. [K]„ Figure 3.6: Schematic view of the various compartments and the currents between these com-partments in the DiFrancesco-Noble equations. The sarcoplasmic reticulum consists of two compartments: C a u p and Ca r e /- (Adapted from Fig. 1, Varghese and Winslow [80].) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 102 The D N equations contain a simple model for the sarcoplasmic reticulum, which helps regulate the intracellular C a 2 + dynamics. The SR in the D N equations consists of two compartments, the uptake and release compartments, sometimes referred to in the modern literature as the network and junctional SR, respectively. The extracellular K + , N a + , and C a 2 + concentrations ( K c , Na 0 , Ca 0) are assumed to be fixed. The model keeps track of the cytosolic K + , N a + , and C a 2 + concentrations (K;, Na;, Ca;) and the C a 2 + concentrations in the SR (Ca u p , Ca r e ;) . In addition, it also keeps track of the K + concentration, K c , in the restricted subspace surrounding the cell, known as the cleft space. K c remains near the bulk extracellular K + concentration, K Q , but also is influenced by the flow of K + ions across the cell membrane. The sarcoplasmic reticulum regulates Cat- by pumping C a 2 + into the uptake compartment via the current Iup and releasing it from the release compartment via the current Ire[. The current Itr transfers C a 2 + from the uptake compartment to the release compartment. 3.5.2 H o w the Res t ing Membrane Potent ia l and Concentrations A r e Set The DiFrancesco-Noble (DN) equations normally exhibit two distinct modes of behaviour, namely quiescence (stable steady state) or tonic firing (pacing). We will begin by dis-cussing how to adjust the parameters to make the system quiescent. In principle, the steady states of a quiescent fiber can be found simply by setting the right-hand sides of all the D N equations equal to zero. Unfortunately, the resulting algebraic equations are highly nonlinear and have to be solved numerically. Nevertheless, an examination of the steady state equations can be insightful. A comparison of the equations for K / = 0 and Kc' = 0 reveals that K c must equal K 0 . In addition, all of the equations for the gating variables of the form x'= cxx(v)(l - x) - (3x{v)x = 0 (3.1) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 103 have explicit solutions in terms of u, namely, x = x^v) where x^v) = — f , , Q , (3-2) Similarly, / V — 0 has an explicit solution in terms of Cat-: h = / 2 oo (Ca t ) = kml2 (3.3) («m/2 + Ca») Taken together, the conditions C a „ / = 0 and C a r e / = 0 imply that 7up(Ca,-, Ca u p ) = J i r (Ca„ p , Ca r e / , u) = 7re/(Ca,-, Ca r ef). (3-4) Explicit formulas can be found for C a u p and C a r e ; in terms of Cat- and v: p CajCciUp . . ua r e ; — ^ 2 — — — T — , ^o.oaj p a - C a t C a M p ( l + r J p ^ i c a t l X J ) Using the above results, we see that rather than 16 equations in 16 unknowns, we now have 4 equations in terms of the 4 unknowns u, Na,-, K,-, and Ca;. The remaining equations are v' = 0 —• IfK + IfNa + ^A' + -^ A'1 + ho + A / V a + iNaK +lNaCa + J-Na + hca + hiCa + -^stA" = 0, (3.6a) N a / = 0 -> IfNa + IbNa + 3INaK + <SINaCa + INO. = 0, (3.6b) 7^ = 0 -» 7 / A- + IK + IK1 + Ito - 2INaK + IsiK = 0, (3.6c) CV = 0 ~+ IbCa - 2INaCa + LiCa = 0. (3.6d) The resting membrane potential is close to the Nernst potential for K + since the sarcolemmal K + permeability is about fifty times larger than the permeabilities of the Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 104 other ions [75]. However, it is slightly positive to this value because of some inward leak currents (particularly IbNa) and IjNa- Because the resting potential is at a hyperpolarized voltage, much below the N a + threshold, only a subset of the currents are active in a quiescent fiber. As shown in Table 3.1, Ito, IsiCai and IsiK are essentially completely turned off, while IR- and I]va only make a small contribution towards the steady state. Because IsiCa is negligible, C a / = 0 is determined by the relatively simple relationship hc'a{v->G&i) = 2lNaCa(v, Ca,-, Na,-). Though we still cannot solve this equation for Ca,-because of the other unknowns Na,- and v, we will find this relationship useful in our analysis later. IKI 33.4 IsiC a -0.2e-3 INO.K 17.6 INa -0.2 IfK 1.3 iNaCa -2.3 IK 0.4 IbCa -4.6 Ito 0.2e-4 IfNa -16.9 IsiK 0.4e-6 IbNa -28.9 Table 3.1: Magnitude of membrane currents at steady state in the DiFrancesco-Noble equations (standard parameters; quiescence via ^,-^=20 mV). Another consequence of the hyperpolarized resting potential is that Pco(v) is very close to 1 (see the activation curves for the gating variables in Appendix C). This means that our equations for C a u p and Ca r e / no longer depend on u, but are only functions of Ca,-. Figure 3.7 illustrates their Ca,- dependence. One would expect to be able to solve equations 3.6a-d for the steady state values of Na,-, K,-, and Ca,- numerically. However, it turns out that this set of equations is underdetermined. Notice that adding equations 3.6c-d together gives equation 3.6a. Therefore, there are really only three equations for four unknowns. This degeneracy is the result of a conservation equation, inherent to the original D N equations [26, 67]. We Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 105 Ca,-(/tM) Figure 3.7: Steady state values of C a u p and Ca r ej as functions of Ca,-. will discuss this in more detail in section 3.6.1. In the meantime, the degeneracy may be overcome by fixing the value of one of the variables such as K,-. 3.5.3 Ionic Events During an Action Potential Figure C.5 in Appendix C shows the sequence of events that occur during tonic firing. It illustrates the changes in membrane potential and all of the other variables, as well as the time course of all the currents. We will briefly summarize the main features here. During phase 4 depolarization (denoted P4 in Figure C.5), the membrane potential slowly depolarizes toward the N a + threshold. At such subthreshold voltages, four of the currents, 7 4 o, IsiK, hiCa, and 7jva, are turned off. The Na-K pump provides a fairly large outward current, IN<IK, which does not change over the course of the action potential. The inward Na-Ca exchanger current, iNaGai on the other hand, is generally small because Ca8- is relatively low. As the membrane depolarizes, the inward background currents, 1 Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 106 the large IbNa and the smaller hca, decay somewhat. The change in the time-dependent delayed K + current, IK, is relatively small. A more substantial change is seen in the large background K + current IKI- The most important current, however, is the hyperpolar-ization activated current / / . This current has a small outward component, I/K, carried by K + and a larger inward component, IfNa, carried by N a + . The magnitude of the If current increases substantially during phase 4 depolarization, resulting in a net inward current which depolarizes the membrane. Also, the current Itr transfers C a 2 + from the uptake compartment of the sarcoplasmic reticulum to the release compartment, depleting C a u p and charging up Ca r e / . Once the membrane potential reaches the N a + threshold, around -65 mV, the fast inward N a + current, Ipta, turns on rapidly, causing a sharp upstroke (phase 0) in the membrane potential. Rapid inactivation of I^a combined with activation of the transient outward K + current, Ito, is responsible for phase 1 repolarization (PI). When the mem-brane potential reaches about -40 mV, the inward IsiCa and outward IsiK turn on. Isica causes C a 2 + to flow into the cell. The resulting increase in Ca; causes release of C a 2 + via Irei from the sarcoplasmic reticulum with a corresponding drop in Ca r e ; . This positive feedback mechanism Ca,llS6S CcL^ to increase even further. The cell attempts to reduce Ca; by pumping C a 2 + back into the sarcoplasmic reticulum via Iup (increasing Ca u p ) and out of the cell via J/vaCa- The enhanced 7/vaCa provides a net inward current which, along with IbNa, competes with the outward currents. The dominance of IK results in early phase 3 repolarization (P3). As the membrane repolarizes, IKI turns on and repolarizes the membrane back to its maximum diastolic potential with the help of I^aK and IK-This leads to phase 4 depolarization, and the cycle repeats itself. The sequence of events during an action potential in a normally quiescent fiber is the same as in a pacing fiber, except that a stimulus depolarizes the membrane to the N a + threshold instead of IfNa-While Purkinje fibers can be found in the pacemaking mode, they are often quiescent Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 107 in vitro. Thus, in order to simulate a number of triggered activity experiments, we must make the D N equations quiescent. Since the the If currents, IfNa and IfK, strongly influence the rate at which the D N equations beat, quiescence can be established by adjusting their dynamics. One method is to shift the activation curve y ^ v ) for the If currents' gating variable y to a more negative potential. DiFrancesco and Noble, for instance, make their equations quiescent by shifting yoo{v) to the left by 10 mV [12]. A shift of 11 mV establishes quiescence in our model since the parameters in our model are only slightly different from those published in the original paper [12]. A further shift of 9 mV (vshift=20 mV) ensures a stable resting potential at a more hyperpolarized voltage (u = -84 mV). The parameter choice of us/ l i/*=20 mV will be convenient for separating normal and arrhythmogenic forms of tonic firing. Another way to achieve quiescence is to eliminate the current altogether or to scale it by a factor of A/=0.2. While this method is not biologically motivated, it does serve to further separate the two different mechanisms responsible for initiating action potentials. We get qualitatively similar results by setting A/=0.2 or vshift=20 mV. 3.6 Reduc ing the Dimens ion of the D N Equations The full 16-dimensional DiFrancesco-Noble equations, shown in Figure 3.8, are far too complicated to yield much insight into triggered activity. In this section, we will simplify the model significantly, by reducing the number of ODEs and the complexity of the interactions. We fix the values of several of the variables because they change relatively slowly compared to the other dynamics, or because they are not directly involved in triggered activity and do not qualitatively alter the observed behaviours. Finally, we will demonstrate how the DiFrancesco-Noble equations can be divided into two subsystems, the membrane oscillator and the subcellular oscillator. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 108 3.6.1 Treating Slow Variables as Parameters Triggered activity typically involves time scales of the order of several action potentials. Therefore, we are only interested in variables which vary significantly over the course of a few action potentials. Variables which do not vary significantly over this time frame may be set to appropriate values and treated as parameters. However, treating them as parameters may not be appropriate for longer simulations. If the cell is quiescent, we expect the intracellular N a + and K + to reach stable values dictated by the two equations N a / = 0 and K / = 0. Simulations also show that Na; and K ; do not vary significantly over the course of one (see Appendix C) or even several Figure 3.8: Schematic diagram illustrating the complexity of interactions in the full 16-di-mensional DiFrancesco-Noble equations, as outlined in Appendix C. (Circled quantities are variables; boxed quantities are currents; arrows point in the direction of influence.) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 109 K,-147 146 145 144 143 142 141 140 " i 1 1 1 1 r 1.0 .1.0 pacing o 0.9 +0.9 0.8 0.8 _j i i i _ 8 8.5 9 9.5 10 10.5 11 11.5 12 Na; Figure 3.9: Positions of "steady state" values of Na; and K; in quiescent and pacing fibers at different Na-K pump rates (AyvaA"=l-0, 0.9, 0.8). (DiFrancesco-Noble equations; standard parameters) action potentials. The cell attempts to maintain constant gradients across the membrane by pumping N a + out of the cell and K + into the cell via the Na-K pump. Under normal circumstances, we can treat Na; and K; as constants. However, during prolonged over-drive, Na; can build up while K; is depleted, and we must allow these variables to vary slowly. If we are to treat Na; and K; as constants, choosing the correct parameter values is essential, because they strongly modulate the behaviour of the system. The Na-K pump rate has a significant effect on the values of Na; and K; , as illustrated in Figure 3.9 by changing the Na-K pump parameter A^aK- The mode, firing or quiescence, also influences their values. Proportionately, the change in K; is small compared with its magnitude. Figure 3.10 illustrates the effects of prolonged overdrive. At a relatively low rate of overdrive (e.g., CL=800 msec), Na; does not change much. However, as the rate increases, Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 110 Na2- increases substantially as overdrive continues (e.g., CL=300 msec), plateaus, and then decays extremely slowly. K; , on the other hand, decreases monotonically throughout the overdrive. It does not appear to reach a new equilibrium value, but is continuously depleted while overdrive continues. This may reflect a flaw in the Na-K pump dynamics in the D N equations. time (sec) time (sec) Figure 3.10: Response of Na; and K,- to overdrive. The ceU is initially pacing at its own intrinsic rate, with a period of approximately 2 seconds. The system is then overdriven at cycle lengths (CL) of 300, 500, and 800 msec for 3000 seconds. (DiFrancesco-Noble equations; standard parameters) While fixing either K; or Na, or both is useful for simplifying the dynamics of the D N equations, fixing at least one variable actually is necessary to prevent difficulties in performing certain stability calculations. Varghese and Sell [67] showed that the 16 vari-ables in the D N equations are mathematically dependent due to an inherent conservation equation: v = ' ^ ( N a , - + K i + 2Ca l + ^ C a u p + ^ C a r e / ) + C 0 , (3.7) where C0 is a constant of integration. Guan et al. [26] demonstrated that, as a result of this degeneracy, the fixed points and limit cycles are not isolated. Thus, if a stable limit cycle has been obtained, a new stable limit cycle can be obtained by perturbing either Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 111 Na; or K; . This would explain difficulties experienced computing branches of limit cycles using AUT086 for the full 16-dimensional DN equations. These difficulties are overcome by fixing the value of K , , so that the Jacobian of the DN equations is no longer singular. 3.6.2 F i x i n g Variables That Do No t Qual i ta t ively A l t e r the Dynamics of Interest In this section, we will fix the values of two variables (K c , p) which have been shown to be unimportant in the study of triggered activity. The first of these two variables is the cleft K + concentration K c , which, during normal pacing, fluctuates by less than one percent (see Appendix C). When the cell is being overdriven, K + can build up in the cleft so that fluctuations of K c are more significant. This phenomenon can be important in studies of overdrive suppression. These fluctu-ations may modulate the occurrence of triggered activity somewhat, but their effects are much less important than other factors causing triggered activity. Experience has shown that setting K C = K 0 does not alter the qualitative results. Therefore, in all of the following work, we will assume that K C = K 0 . Before stating our next approximation, we first review some of the history of the C a 2 + dynamics in the D N equations. The DN model was one of the first models to try to keep track of the intracellular ion concentrations. In particular, it attempted to model the Ca; dynamics, including C a 2 + sequestering by the sarcoplasmic reticulum. DiFrancesco and Noble acknowledged that their model was, in some sense, preliminary with their statement: "It is extremely unlikely that our representation of the Ca-sequestering pro-cesses or of the Na-Ca exchange mechanism or of other Ca-activated currents (such as Iio) will remain among the best available for very long" [12]. In the ten years since then, there still is no definitive model of all the C a 2 + dynamics. Later, we will review some of the progress made and discuss some of the controversial issues in C a 2 + modeling. Here, Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 112 we will make one small modification to the D N equations for the sake of simplicity and consistency with modern views. In discussing their modeling of the sarcoplasmic reticulum C a 2 + dynamics, DiFran-cesco and Noble stated that "this part of the modelling is not thought to be very se-cure...the major issue of whether Ca2+ release is Ca2+ -induced or voltage-induced (or, perhaps, both) is still controversial" [12]. Recent models favour Ca 2 +-induced dynamics. Over the course of a single action potential, the variable p fluctuates dramatically (see Appendix C). It gates the current ITR which transfers C a 2 + from the uptake compartment to the release compartment of the sarcoplasmic reticulum. In all of the following work, we set p=l, unless otherwise stated, because the results with and without p dependence on voltage differ quantitatively but not qualitatively. This approximation simplifies the model and also brings it into agreement with more recent models of C a 2 + dynamics. By setting K C = K 0 and p=l and fixing the values of Na; and K; as discussed above, the 16-dimensional D N system has been reduced to 12 dimensions; see Figure 3.11. The complexity of the system has also been reduced considerably. Theoretically, the steady states then can be found by solving the equations v' = 0 and C a / = 0 for v and Ca;, since the remainder of the variables can be determined directly from these two quantities. 3.6.3 V i e w i n g t h e D i F r a n c e s c o - N o b l e E q u a t i o n s as T w o C o u p l e d O s c i l l a t o r s The simplifications made above make it possible to divide the system into two interact-ing subsystems, the membrane oscillator and the subcellular oscillator (terminology of Varghese and Winslow [69]). As shown in Figure 3.11, the membrane oscillator consists of all the membrane currents which directly modify the membrane potential v and their associated gating variables. The subcellular oscillator is comprised of the C a 2 + subsys-tem, including Ca;, C a u p , Ca r e ; , the gating variable f2, and all of the C a 2 + currents. The two subsystems are coupled together by several shared currents, INO-CO,, he a, and hiCa-Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 113 Membane Oscillator Subcellular Oscillator Figure 3.11: Schematic diagram illustrating the reduced complexity of the 12-dimensional approximation to the DiFrancesco-Noble equations (K c =Ko, p=l, K / = 0, Na/ = 0). Dashed boxes represent the membrane and subcellular oscillators and their interaction in the overlap region. (Circled quantities are variables; boxed quantities are currents; arrows point in the direction of influence.) The membrane oscillator additionally is influenced by Ca; through the Ca^-dependent It0 and the /Vdependent IsiK-At subthreshold voltages, the division between the two oscillators is even cleaner since It0, IsiK, and IsiCa are all turned off. Then the two subsystems share only the two currents J/vaCa and IbCa. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 114 3.7 Varghese and Winslow's Results Varghese and Winslow laid the foundation for our work in their studies of the dynamics of the C a 2 + subsystem (subcellular oscillator) in the DN equations [68] and of the interaction of the subcellular and membrane oscillators [69]. In this section, we summarize the key results which are relevant to our work. 3.7.1 Subcellular Oscillator Experimentalists have shown that cardiac cells exposed to the drug digitalis, which in-hibits the Na-K pump, can exhibit oscillations in some of its membrane currents even when the membrane potential is clamped at a fixed voltage [40]. Varghese and Winslow [68] mimicked the effect of Na-K pump inhibition under voltage-clamp conditions by fix-ing the voltage v at -80 mV (a common value used in experiments) and changing the parameter Na; to reflect different degrees of digitalis poisoning. They point out that changes in Na; caused by Na-K pump blockade occur over periods of tens of minutes, which is a few orders of magnitude longer than the time scale of the C a 2 + subsystem. Fixing v isolates the C a 2 + dynamics, yielding the simpler subsystem shown in Figure 3.12. This four-dimensional subsystem could have been reduced to three dimensions by fixing fi at its steady state value without significantly changing their results. However, they did not see any particular advantage to doing this. Theory predicts that inhibition of the Na-K pump should cause Na, to increase, which in turn should decrease C a 2 + extrusion by the Na-Ca exchanger. The resulting C a 2 + over-load can result in oscillatory or transient release of C a 2 + from the sarcoplasmic reticulum. One of Varghese and Winslow's most important contributions was to demonstrate that as Na; increases in the voltage-clamped C a 2 + subsystem, the stable steady state of the Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 115 Voltage-Clamped Subcellular Ocillator Figure 3.12: Varghese and Winslow's subcellular oscillator: C a 2 + subsystem resulting from voltage-clamped DiFrancesco-Noble equations with K / = 0 and Na/ = 0. (Circled quantities are variables; boxed quantities are currents; arrows point in the direction of influence.) system becomes unstable and stable C a 2 + oscillations arise. Figure 3.13 shows the bifur-cation diagram for the voltage-clamped subcellular oscillator in terms of the bifurcation parameter Na;. This figure is similar to Figure 4 in [68] except here we use Ca 0=2.5 m M and Nao=150 m M to be consistent with our other results. Normally, the system has a stable steady state to the left of the left-most Hopf bifurcation point. As Na; increases (with increasing digitalis poisoning), the system undergoes a Hopf bifurcation followed by a very complicated sequence of period doublings (not shown), and large amplitude stable orbits arise. Thus we see that C a 2 + oscillations occur in the D N C a 2 + subsystem. The stable C a 2 + oscillations decrease in amplitude with increasing Na, and eventually disappear through another Hopf bifurcation. Varghese and Winslow computed the same bifurcation diagram for a number of differ-ent voltage clamp values and showed that the amplitudes of the oscillations diminished with increasing v. They also computed the two-parameter curve of Hopf bifurcations in the u-Na,- parameter space. Results similar to theirs are shown in Figure 3.14, from Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 116 Figure 3.13: Bifurcation diagrams for Varghese and Winslow's subcellular oscillator (v=-90 mV, Ca0=2.5 mM, Nao=150 mM) using Na, as the bifurcation parameter: (a) Ca, and (b) Period. (Similar to Fig. 4 in [68]) Figure 3.14: Two-parameter curve of Hopf bifurcations in the t>-Na,- parameter space for Varghese and Winslow's subceUular osciUator (Ca0=2.5 mM, Nao=150 mM). (Similar to Fig. 10 in [68]) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 117 which we see that the position of the Hopf bifurcation points shifts towards smaller Na; values with increasing v, while the range of Na; decreases until the oscillations disappear altogether. 3.7.2 Coupled System Varghese and Winslow [69] examined the full DiFrancesco-Noble equations under various degrees of digitalis poisoning (Na-K pump inhibition). The finding most relevant to our work was that the D N equations have two different modes of tonic firing depending on their positions in Na,-K; parameter space. When they set Na;=8 m M and K;=140 m M , they obtained normal tonic firing (see Figure 7 in [69]). In this mode, the initial voltage upswing of the action potential is followed by a C a 2 + upswing resulting from I s ;c a and then calcium-induced calcium release from the sarcoplasmic reticulum. However, at Na;=12 m M and K;=134 m M , a C a 2 + upswing occurs before the voltage upswing (see Figure 10 in [69]). Rather than the normal pacemaking mechanism in which If brings the membrane potential to threshold, the action potential becomes entrained to the faster subcellular oscillator. Later, we will study the interaction of the C a 2 + and membrane oscillators to try to understand these two types of behaviour and their relationship with triggered activity. They also showed (see Figure 12 in [69]) that when Na;=16 m M and K;=128 m M , Ca; can exhibit a "double humped" behaviour in which the initial C a 2 + upstroke decays before the voltage upstroke. This represents a complete reversal of the normal sequence of events. With this parameter set and the appropriate initial conditions, they showed that the system can transiently exhibit action potentials alternating with DADs. Eventually "the DADs disappear as the calcium oscillations appear to dominate the usual pacemaking mechanism" [69]. Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 118 3 . 8 N o r m a l and Ca l c ium-Dr iv e n F i r i n g : W h i c h Osci l la tor Drives W h i c h ? Varghese and Winslow's work suggests that triggered activity involving DADs results from the dominance of the subcellular oscillator. In order to study this, we first de-fine some terminology. In this section, we will examine the behaviour of the simplified DiFrancesco-Noble equations with K;=145 m M , K C = K 0 , p=l , and Na, a bifurcation pa-rameter. While large changes in K; affect the behaviour, small changes do not alter it qualitatively, so we will neglect them for the time being. Na;, on the other hand, has a large effect on the qualitative behaviour. The bifurcation diagram in Figure 3.15, which uses Na; as the bifurcation parameter, shows that different ranges of Na; correspond to qualitatively different behaviours (also see the simulations in Figures 3.16 and 3.17). The system exhibits normal tonic firing or automatic i ty for low values of Na; up to and including approximately Na; = 10 m M . In this region, v leads Ca;, as shown in Figure 3.16a. Thus, the membrane oscillator is driving the C a 2 + oscillations. As Na; increases in this region, the period of oscillation increases, largely due to an increase in iNaK- This increase in period with increasing Na, is related to overdrive suppression. As we see in Figure 3.16b, a small C a 2 + upstroke precedes the voltage upstroke when Na;=10.6 m M . Between Na;=10 m M and Na t = l l m M , we have a transition region in which the two oscillators compete to initiate the action potential. Neither mechanism dominates, so both the subcellular and membrane oscillator properties are important in establishing the rhythm. For Na; between 11 m M and 16 m M , Ca; leads v (Figure 3.16c). Here the subcel-lular oscillator is entraining the membrane oscillator. We will refer to these kinds of oscillations as calcium-driven. We will see later that overdrive-excitation and triggered activity consist of calcium-driven action potentials. Notice that the period decreases with increasing Na;, the result of enhanced calcium overload. As the subcellular oscillations Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 119 8 0 0 5 l O 1 5 2 0 2 5 3 0 Na, (mM) Na; (mM) 0 5 1 0 15 2 0 2 5 3 0 Na, (mM) Figure 3.15: Bifurcation diagrams showing the voltage v, the intracellular C a 2 + concentration Ca; , and the period T as functions of the bifurcation parameter Na;. The diagram can be split up into six behavioural regions of interest: (a) normal tonic firing (1.9 < Na; < 10), (b) transition between normal and calcium-driven tonic firing (10 < Na; < 11), (c) calcium-driven tonic firing (11 < Na,- < 16), (d) transition between suprathreshold and subthreshold oscillations (16.0 < Na; < 16.9), (e) subthreshold Ca;-driven osciUations (16.9 < Na; < 24.4), and (f) quiescence (Na; > 24.4). (DiFrancesco-Noble equations; standard parameters with K;=145 m M , K C = K 0 , and p=l) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 120 Figure 3.16: Simulations of the DiFrancesco-Noble equations in (a) normal tonic firing mode (Na;=10.0 mM), (b) the transition region between normal and calcium-driven tonic fir-ing (Na;=10.6 mM), and (c) calcium-driven tonic firing mode (Na,=12.0mM). Simulations of voltage (v) and intracellular Ca 2 +(Ca;) as functions of time are shown, along with the two-dimensional projection of the trajectories on v and Ca;. (DiFrancesco-Noble equations; standard parameters with K;=145 mM, K C = K 0 , and p=l) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 121 v (mV) (a) Na,-=16.0 4 0 Na,-=16.1 (c) Na;=16.8 (d) Na;=16.9 4 0 2 4 6 8 10 time (sec) time (sec) 100 -80 -60 -40 -20 0 20 v (mV) Figure 3.17: Simulations of the DiFrancesco-Noble equations in the transition region be-tween suprathreshold and subthreshold oscillations: (a) calcium-driven tonic firing with "double-humped" C a 2 + spikes (Na;=16.0 mM), (b) action potentials alternating with DADs (Na,-=16.1 mM), (c) action potentials alternating with two DADs (Na;=16.8 mM), and (d) subthreshold calcium-driven oscillations (Na,=16.9 mM). Simulations of voltage (v) and intra-cellular C a 2 + (Ca;) as functions of time are shown, along with the two-dimensional projection of the trajectories on v and Ca;. (DiFrancesco-Noble equations; standard parameters with K t = 145 mM, K C = K 0 , and p=l) Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 122 speed up, we find that Ca; starts to decay before the action potential begins, and we see Varghese and Winslow's "double-humped" Ca; oscillations (Figure 3.17a). The region 16.0 m M < Na; < 16.9 m M is the transition region between suprathresh-old and subthreshold calcium-driven oscillations. In this region, there appears to be a "bifurcation of subharmonics" (Figure 3.17). At Na;=16.0 m M (Figure 3.17a), we see suprathreshold calcium-driven oscillations with the "double-humped" C a 2 + spikes. At Na,=16.1 m M (Figure 3.17b), every second D A D misses threshold so that action po-tentials alternate with DADs. The C a 2 + spikes associated with the DADs only have a single hump. This is because the voltage never goes high enough for Isica to turn on and cause a secondary release of C a 2 + from the sarcoplasmic reticulum. At Na,=16.8 m M (Figure 3.17c), only one of every three DADs is reaching threshold. For Na;=16.9 m M (Figure 3.17d), only subthreshold oscillations occur. The amplitude of these subthreshold oscillations decays with Na;, until finally at Na;=24.4 m M , they disappear altogether, to be replaced by a stable steady state. The subthreshold oscillations are due almost entirely to the subcellular oscillator, which drives the voltage via the Ca2 +-dependent iNaCa- There is relatively little feedback from the membrane oscillator to the subcellular oscillator in this region since Isica re-mains off. Figure 3.18 compares the subthreshold oscillations in the isolated subcellular oscillator and the coupled system. The voltage in the subcellular oscillator is fixed at -90 mV, the resting potential seen just to the right of the subcellular oscillations in the bifurcation diagram in Figure 3.15a. In the coupled system, v actually oscillates by as much as 20 mV. Figure 3.18 shows that the magnitudes of the Ca; oscillations are very similar in the isolated and coupled systems, especially for large Na;. The periods of the oscillations are also similar at large Na;. As Na; decreases, however, the voltage oscil-lations resulting from the C a 2 + oscillations increase in magnitude until they are close to threshold. At this point, the membrane dynamics start to become important. In Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 123 (a) 1.6 1.4 1.2 Ca; | (/xM) u 0.6 V \ _ i 0.4 0.2 " " 0 (b) Ca; (MM) 16 17 18 19 20 21 22 Na; (mM) 23 24 •' v \ i -¥ 16 17 18 19 20 21 22 23 24 Na; (mM) 16 17 18 19 20 21 22 23 24 Na; (mM) 16 17 18 19 20 21 22 23 24 Na8- (mM) Figure 3.18: Comparison of subthreshold Ca; oscillations in (a) the isolated subcellular oscilla-tor (v=-90 mV; IsiCa=Q] P— 1) and (b) the coupled system. The bifurcation diagrams show Ca; and the period T as functions of the bifurcation parameter Na;. (DiFrancesco-Noble equations; standard parameters with K;=145 mM, K C = K 0 , and p=l) fact, near Na;=17 m M , the oscillations become unstable as the membrane oscillator and subcellular oscillator begin to interact. The period of oscillation is faster in the coupled system because the membrane currents, like / / , assist depolarization. During suprathreshold oscillations, the feedback between the membrane and subcellu-lar oscillators goes both ways. While the subcellular oscillator may be largely responsible for initiating the action potential, the resulting voltage upswing turns on IsiCa, which in Chapter 3. CARDIAC TRIGGERED ACTIVITY AND THE DN EQNS 124 turn increases Ca;, causing calcium-induced calcium release from the sarcoplasmic retic-ulum. The suprathreshold Cat- oscillations are much larger than those associated with DADs. Chapter 4 R E S U L T S F O R T H E D I F R A N C E S C O - N O B L E E Q U A T I O N S In the previous chapter, we introduced the background material necessary for understand-ing triggered activity in the context of the DiFrancesco-Noble model of a mammalian cardiac Purkinje fiber. In this chapter, we use a combination of bifurcation analysis and simulation of the DiFrancesco-Noble model to explain the roles of various experi-mental interventions in promoting or suppressing triggered activity. We begin with an analysis of the subcellular oscillator. Then we simulate experimental interventions such as high extracellular C a 2 + concentration and application of digitalis in the pacemaking and quiescent versions1 of the model. We examine the effects of changes in intracellular K + concentration, which we assume to be constant throughout most of our work. To conclude, we discuss some of the drawbacks and inadequacies of the DiFrancesco-Noble equations that we have discovered in our study of triggered activity. 4.1 Subcel lular Osci l la tor In this section, we extend some of Varghese and Winslow's results for the subcellular oscillator. We begin by simplifying the equations. We then simulate certain experimental interventions by adjusting appropriate parameters. Lastly we examine the mechanisms responsible for subcellular oscillations and calcium-induced calcium release. 1The pacemaking and quiescent versions of the DN equations differ only in the values of the parameters (vghift or Af) describing the If currents. 125 Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 126 4.1.1 Simplifying the Equations The voltage-clamped subcellular oscillator with Na; fixed consists of the following equa-tions: dC&i —1 dt 2VtF dt 2V,mFKup {IbCa — 21 /VaCa + IsiCa + hp ~ hel), (4.1a) (hp-hr), (4.1b) 1 upJ dt 2Vre,F {hr-hel), {4 Ac) ~df = 6 ( 1 - / 2 ) - ^ / 2 ' <4"ld» where IbCa{CcLi; v, Ca 0) = Gbca{v - Eca), T ( r a - „ r * Na e ^ A ^ / ^ N a / C a ^ - e - C ^ ^ / C ^ / ^ N a / C a ; JNaCa[Cai, V, ba 0 , INa;, INa0j = fc/VaCa ,T\T 3 p -Nj 3 n a L , i+d^oCo(Na 0 Ca;+Na; Ca 0 ) W C a ^ C a . ) = 4 f C t t d 0 0 ( t ; ) / 0 0 ( t ; ) / 2 , ( C a i ) ( 1 J e ^ K ) J . ) ) *(Ca;e 5 0 /( f i r / 2 i ? ) - Ca o e-^- 5 0 ) / ( f i T / 2 F ) ) , T fPa- Pa ^ — 2 V;Fp- ,Caup-C&Up ^ 7j r(Ca;, C a u p , Ca r e / ; v) = — p c o ( i > ) ( C a u p — Ca r e / ) , hel{C&i, C&rel) = 2 ^ r " ' / ? C a r e / ( „ ^ * )• 1 ' mCa The voltage-clamped subcellular oscillator has four adjustable parameters, v, Ca 0 , Na 0 , and Na;, and four variables, Ca,, C a u p , Ca r e / , and f2. We are only interested in the subthreshold behaviour of the isolated system. Above threshold, the interaction between v and Ca; is too large for volt age-clamped results to be useful. At subthreshold voltages, hiCa is virtually turned off, so the effects of f2 are negligible. Therefore, the subthreshold subcellular oscillator really only consists of three variables, as shown in Figure 4.1. In Figure 4.2, we recompute the two-parameter curve of Hopf bifurcations in u-Na, space with and without the approximations l s;ca=0 and P o o (v )= l . We see that the approximation hiCa=0 has a small effect at high voltages, and a negligible effect at Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 127 StaCa b C a rel up A V A V Figure 4.1: Subthreshold subcellular osciUator: C a 2 + subsystem resulting from DiFran-cesco-Noble equations with K / = 0, Na/ = 0, and membrane potential clamped at a sub-threshold voltage. (Circled quantities are variables; boxed quantities are currents; arrows point in the direction of influence.) subthreshold voltages. The variable p normally turns off oscillations of the subcellular oscillator at high voltages; however, setting p=l has little effect on the subthreshold behaviour. Given these two approximations, we can solve explicitly for C a u p and Ca r e / in terms of Ca,-, as shown earlier. Ca, is defined implicitly by the equation 7(,co(Ca,-; v, Ca 0)=2 lNaCa(Ca.i] v, C a 0 , Na i 5 Na 0 ). 4.1.2 Adjus t ing the Parameters to Simulate Exper imenta l Interventions The behaviour of the subthreshold voltage-clamped subcellular oscillator may be modified by adjusting any one of the four observables v, Ca 0 , Na;, and Na 0 . These four parameters modulate the membrane currents, but not the currents in the sarcoplasmic reticulum. Na,- and Na„ only affect the behaviour through modulation of -Zyvaca, while v and Ca 0 modulate 7yvaCa and IBCA-In the triggered activity experiments, Na 0 is typically fixed. On the other hand, Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 128 Figure 4.2: Two-parameter curves of Hopf bifurcations for the voltage-clamped subcellular osciUator in the w-Na; parameter space: (a) full four-dimensional system, (b) IsiCa=0, (c) p—1, and (d) lsica=0 and p=l. The curve of Hopf bifurcation points separates regions of stable and unstable fixed points, (standard parameters; CaQ=2.5 mM) Na; reflects the effects of experimental interventions such as fast pacing or application of digitalis and strongly influences the behaviour of the subcellular oscillator. In most of the analysis to follow, it will be treated as the primary bifurcation parameter. A second important parameter is Ca 0 , which experimentalists have shown to promote the occurrence of DADs and triggered activity when increased. From Figure 4.3, we see that Ca 0 only has a minor effect on hca, while Na; does not affect it at all. The major effects of Ca Q and Na; on the subcellular oscillator are Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 129 Ca 0=2.5, 3.5, 8.0 (a) < c 0 \ i\ i \ i i i r i -2 \2 J / V a C a --4 IbCa \ , i 2.5 8.0 -6 \ \ --8 - 2.5\ \3.5 8.6., --10 i i \ i i i \ i 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Ca;(^M) (b) < c 0 -2 -4 -6 -8 -10 Na;=8, 10, 12, 14, 16 yr-\—F; r r — i 1 1 r \ \ \ \ 2 iNaCa 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Figure 4.3: The intersection (dot) of I^Ca (solid line) with 2Ij^aca (dashed line) determines the steady state value of Ca;. (a) As Ca c increases from 2.5 to 8.0 mM, the magnitude of IhCa increases slightly, 2I^aca moves to the right, and the steady state value of Ca; increases, (b) As Na; increases from 8 to 16 mM, I\,ca remains fixed, while 2/jVaCa moves to the right, and the steady state value of Ca; increases. (t>=-80 mV, Na o = 150 mM, Na; = 10 mM, CaD=2.5 mM) through modulation of Ii^aCa- As Na; increases, the N a + gradient, which drives the Na-Ca exchanger, decreases, resulting in reduced I^aCa- As a result, C a 2 + builds up inside the cell. Increased C a 0 also, reduces IjvaCa, causing an increase in the steady state value of Ca;. If we neglect the relatively minor effect of Ca 0 on hca, then we see that Ca 0 and Na; only affect the subcellular oscillator's behaviour through the term Na; 3 Ca 0 in JyvaCa-Because Na; and Ca Q are coupled, any behaviour observed for a pair (7Va;,Ca0) will also occur for other pairs (Na;,Ca 0) defined according to the equation Na ; 3 Ca 0 = NalCa0. Figure 4.4 shows the magnitude of the Ca; oscillations and their corresponding periods as a function of the bifurcation parameter Na; at two values of Ca D . An increase in Ca Q results in compression of the Na; scale without significantly altering the dynamics, and the steady state value of Ca; increases more steeply with Na.;. The magnitudes and periods of the oscillations are not significantly affected but the region of oscillations shrinks and shifts to the left. Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 130 Na, (mM) Na,- (mM) Figure 4.4: Bifurcation diagram showing Ca; and the period T for the voltage-clamped sub-cellular o'sciUator as a function of the bifurcation parameter Na; for (a) Ca0=2.5 mM and (b) Cao=8.0 mM. (standard parameters; p=l, I S j C a = 0 , v=-90 mV) We determine the values of Na; at the bifurcation points when Ca 0=2.5 m M for the exact system. Using these quantities from Figure 4.4a for 7Va; and C a 0 , we use the relation Na , 3 Ca 0 = Na{Ca0 to calculate the 2-parameter curve of Hopf bifurcations for the voltage-clamped subcellular oscillator in Ca 0-Na; parameter space (Figure 4.5b). A comparison of this curve with that in Figure 4.5a calculated using X P P A U T confirms that neglecting the effect of Ca 0 on I^ca is valid here. An important result of this analysis is that the region of subcellular oscillations shrinks and shifts to lower values of Na, as Ca 0 is increased. These results are used later when we study the coupled system to Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 131 explain the effects of pacing, digitalis, and high extracellular C a 2 + concentration. Na, (mM) Na; (mM) Figure 4.5: Two-parameter curve of Hopf bifurcations for the volt age-clamped subcellular oscillator in the Ca 0-Na; parameter space (v=-90 mV) calculated (a) using XPPAUT on the exact equations and (b) using the relationship Na; 3 Ca 0 = Na{ Ca0 where Ca0=2.5 mM and JVa; is the corresponding bifurcation point found in Figure 4.4a. The curve of Hopf bifurcation points separates regions of stable and unstable fixed points, (standard parameters; p=l; IsiCa—0) 4.1.3 Subcel lular Oscillations and Calc ium-Induced C a l c i u m Release The sequence of events involved during subcellular oscillations is illustrated in Figure 4.6. When Ca,- is low, C a 2 + is transferred from the uptake compartment to the release compartment of the sarcoplasmic reticulum. Ca r e / continues to build up until the release current Irei, which is proportional to Ca r e ; , becomes larger than the uptake current Iup, which pumps C a 2 + back into the SR. A strong positive feedback between Ca; and Irei causes further release known as calcium-induced calcium release or CICR. The release compartment is quickly depleted. While Ca,- is high, the uptake compartment attempts to pump the C a 2 + back in, restoring C a u p to its high value. At the same time, the Na-Ca exchanger removes C a 2 + from the cell, resulting in a large inward Ij^aCa capable of depolarizing the membrane in the full system. Ca, is restored to its normally low value, Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 132 and the process is ready to repeat itself. Normally, the resting value of Ca; lies below the threshold at which Ca; turns on Irei, and small increases in Ca; are insufficient to trigger Ire\ significantly. At very high steady state values of Ca;, caused by high Na;, Ca; is so large that Ire\ is mostly turned on. Further increases in Ca; cannot initiate a large increase in Ire\. However, if Ca; is somewhere in between, a small increase in Ca; can initiate a large CICR, and oscillations are possible. Whether this positive feedback is sufficient to cause an avalanche effect resulting in a large increase in Ca; depends on many factors, including the rate of removal of C a 2 + from the cytoplasm by J / V a C a and Iup and the rate of transfer of Ca 2 +between the two SR compartments. At values of Na; just to the left of the oscillatory range, spontaneous oscillatory behaviour cannot occur, but CICR is possible with the right initial conditions (calcium overload). Figure 4.7 illustrates what happens if C a u p starts at an unusually high value, (a) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -i i 1000 Ca, i i i 1 -^^Caup C a r e ; -• i L i 1 2 3 4 5 time (sec) (b) < c 250 200 150 100 50 0 -50 -100 -150 1 1 - \lrel _ ii 1 ii ii ii ii ii 1 1 J Ii -ii !i ji i i ";!.. hr i i i i i i / i -JbC a !i / J \ -- 1?2 iNaCa "V - \l~Iup i i V 1 2 3 time (sec) Figure 4.6: Simulation of stable oscillations in the subcellular oscillator: (a) C a 2 + concen-trations in the cytoplasm (Ca;), the uptake compartment of the SR (Ca„ p), and the release compartment of the SR (Ca r e;), and (b) currents flowing into the cytoplasm (Irel, -hea), °ut of the cytoplasm (-J u p, 2 / j V a C a ) , and from the uptake to release compartment of the SR (Lr)-(subceUular oscillator: standard parameters; p=l, 7s;ca=0, Na;=16.0 mM, Ca0=2.5 mM) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 133 which could occur in the coupled system if the cell was paced at a high rate to "charge up" the uptake compartment of the SR. Because C a u p is above its equilibrium value, the uptake compartment keeps charging up the release compartment with C a 2 + until CICR occurs. The process continues to occur while C a u p remains high. Then C a u p gradually decreases to its equilibrium value as C a 2 + is removed from the cell by INUCU- Eventually the system returns to rest. o 1 " 1000 Ca, ~~-...Caup 1 1 1 Ca r e ; 0 2 4 6 8 10 time (sec) Figure 4.7: Simulation of the transient response of the subcellular osciUator to an initially high C a u p : C a 2 + concentrations in the cytoplasm (Ca,), the uptake compartment of the SR ( C a u p ) , and the release compartment of the SR (Ca r e ; ) . The system is initially at steady state except C a u p which is set to 1.3 m M instead of its steady state value of 1.1 m M . (subcellular oscillator: standard parameters; p=l, IsiCa=0, Na;=13.0 m M , Ca 0=2.5 mM) 4.2 Simula t ing Exper imenta l Interventions in the Pacemaking System Here, we examine the effects of various experimental interventions on the pacemaker version of the D N equations. Our main purpose is to uncover the primary mechanism for triggered activity. We use the simplifications discussed earlier to make the most important events more visible. We set p—1 and K C = K 0 , and in addition, K , = 145 m M , the approximate value seen in the pacemaking fiber under normal conditions. Numerical Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 134 simulations of the full and reduced D N equations indicate that these approximations do not significantly alter the behaviour. We compare the normal behaviour with the behaviour under the influence of digitalis or in the presence of high extracellular C a 2 + concentrations. Then we demonstrate the effects of pacing the system. 4.2.1 N o r m a l Behaviour The normal pacemaking version of the D N equations exhibits tonic firing with a period of approximately 1.8 seconds. The intracellular N a + concentration remains within 0.04 m M of Na;=9.10 m M , i.e., the fluctuation in Na,- is less than 0.5 percent. In Figure 4.8, we zoom in on the region of interest in the bifurcation diagram of Figure 3.15, namely the normal and calcium-driven tonic firing. We also include the subthreshold oscillations to show how the suprathreshold oscillations arise. Comparing the pseudo-steady state value of Na; with the bifurcation diagram, we see that the D N equations undergo normal tonic firing with Ca0=2.5 m M and no alterations to the If currents. A simulation of this behaviour is given in Figure 4.9. 4.2.2 The Effects of Digi ta l is Poisoning and H i g h C a 0 Earlier we showed that normal and calcium-driven tonic firing correspond to different Na; ranges in the bifurcation diagram (Figure 3.15 or 4.8). However, Na, is not, in fact, a parameter of the system, but tends to stay within 0.5 percent of some limiting value. The result is that Na; remains in the range in which normal tonic firing is seen. How then is calcium-driven tonic firing achieved? To answer this question, we explore the effects of changing the parameters Aj^aK and C a 0 on the behaviour of the system. The bifurcation diagrams showing v as a function of the bifurcation parameter Na,- are compared for four (A/Vajf , Ca 0) pairs in Figure 4.10. Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 135 v (mV) 4 ( ) 2 0 o - 2 0 -4-0 - 6 0 - 8 O I O O - 1 — . - V -. 1 -Ca; (uM) T (sec) 1 6 1 8 2 0 2 2 2 - * Na8- (mM) Na,(tf)=9.1 1 6 1 8 2 0 2 2 2 4 Na, (mM) Figure 4.8: Pacemaking DiFrancesco-Noble equations. Bifurcation diagram shows the voltage v, the intracellular C a 2 + concentration Ca;, and the period T as functions of the bifurcation parameter Na;. Under these conditions, Na; naturally tends towards a pseudo-steady state value of Na;(tf)=9.1 m M so that the system exhibits normal tonic firing. (DN equations; standard parameters with K;=145 m M , K C = K 0 , p=l; vshift=0 mV, A/=1.0; Ca 0=2.5 m M , A/vaA' = l-0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 136 0 2 4 6 8 10 -100 -80 -60 -40 -20 0 20 time (sec) v (mV) Figure 4.9: Simulation of the pacemaking DiFrancesco-Noble equations. The membrane po-tential v is plotted as a function of time, and intracellular C a 2 + concentration Ca; is plotted against v. The system is exhibiting normal tonic firing. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=0 mV, A / = 1.0; Ca0=2.5 mM, ANaK=1.0) To see what happens to Ca; and the period of the limit cycles for cases (b), (c), and (d), see Figures D . l , D.2, and D.3 in Appendix D. Digi ta l i s Poisoning One method of achieving calcium-driven tonic firing is to shift the pseudo-steady state value of Na; to a higher value. This is the major effect of digitalis, which inhibits the Na-K pump. We mimic this effect by scaling the outward current INO.K by a factor ANaK < 1 to reflect graded inhibition of the Na-K pump. In this thesis, we refer to small amounts of inhibition as "weak digitalis poisoning" and larger amounts of inhibition as "strong digitalis poisoning." The immediate effect of Na-K pump inhibition is to reduce the net outward current, making it easier for the membrane to depolarize. This results in tonic firing at higher frequencies. Reduction of INO.K also means that less N a + is pumped Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 137 (a) ANaK=l.O, Ca0=2.5 m M (b) ANaK=0.S5, Ca 0=2.5 m M I—i 1 T ' 1 1 1 ' ' 1—I I—r 1 1—i—i 1 1 1 1 1 Nai(tf)=9.3 Na t (mM) Na;(tf)=10.2 N a w ( m M ) Figure 4.10: Pacemaking version of the DN equations at various ANaK and Ca 0 . Bifurcation diagrams show the voltage v as a function of the bifurcation parameter Na;. The approximate value of Na; during tonic firing (Na,(tf)) is indicated in each figure. Note: v, Ca;, and the period of the limit cycles are shown as functions of the bifurcation parameters Na; in Appendix D for (b), (c), and (d). (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; Ca0=2.5, 3.5, or 8.0 mM, AivaA-=1.0 or 0.85) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 138 out of the cell, resulting in an accumulation of N a + in the cell, and hence high Na,-. Figures 4.10b and D . l show the new bifurcation diagram with Na; as a bifurcation parameter under weak digitalis poisoning (A/Va/i"=0.85). A comparison with Figures 4.10a or 4.8 shows that the bifurcation diagram has not changed significantly, except for a decrease in the period of tonic firing (Figure D. l ) . In particular, the transition from normal to calcium-driven tonic firing has not moved much. The pseudo-steady state value of Na;, on the other hand, has increased from 9.1 m M to 11.0 m M . As a result, the system now sits in the Na; range where it exhibits tonic firing in the transition region between normal and calcium-driven tonic firing. A simulation of this behaviour is given in Figure 4.11a. Stronger digitalis poisoning (A/vaA-=0.7) causes a further increase in Na; to 13.5 m M . As a result, the system exhibits calcium-driven tonic firing, as shown in Figure 4.11b. High Extracellular C a 2 + Concentration Another method to achieve calcium-driven tonic firing is to increase Ca 0 . This increases the C a 2 + gradient across the cell membrane, resulting in enhanced C a 2 + influx and re-duced C a 2 + extrusion by 7/VaCa- Consequently, the steady state value of Ca; increases, resulting in calcium overload. We showed in Figure 4.4 that an increase in Ca 0 shifts the oscillations in the bifurcation diagram for the voltage-clamped subcellular oscillator to lower values of Na;. In the coupled system, it also shifts the calcium-driven behaviour to lower values of Na,, cf. Figures 4.10c and 4.10d. At the same time, Na; increases slightly. When Ca 0=3.5 m M , the system exhibits tonic firing in the transition region be-tween normal and calcium-driven firing (Figure 4.12a). When Ca o=8.0 m M , the system exhibits calcium-driven tonic firing (Figure 4.12b). The bifurcations diagrams for Ca„=2.5 m M (Figure 4.10a or 4.8) and Ca 0=3.5 m M (Figure 4.10c or D.2) look qualitatively quite similar. The important difference is that for Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 139 Figure 4.11: Simulation of the pacemaking DiFrancesco-Noble equations under (a) weak (A/VaA"=0.85) and (b) strong (AjvaA"=0.7) digitalis poisoning. The membrane potential v is plotted as a function of time, and intracellular C a 2 + concentration Ca; is plotted against v. For weak digitalis poisoning, the system exhibits tonic firing in the transition region between normal and calcium-driven firing. For strong digitalis poisoning, the system exhibits calcium-driven tonic firing. (DN equations; standard parameters with K4=145 mM, K C = K 0 , p=l; vshift=0 mV, A / = 1.0; Ca0=2.5 mM, ANaK=0M, 0.7) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 140 Figure 4.12: Simulation of the pacemaking DiFrancesco-Noble equations with (a) an interme-diate Ca 0 value (3.5 mM) and (b) a high Ca 0 value (8.0 mM). The membrane potential v is plotted as a function of time, and intracellular C a 2 + concentration Ca; is plotted against v. For intermediate Ca 0 , the system exhibits tonic firing in the transition region between normal and calcium-driven firing. For high Ca 0 , the system exhibits calcium-driven tonic firing. (DN equa-tions; standard parameters with K; = 145 mM, K C = K 0 , p=l; vs^ft=0 mV, A / = 1.0; Ca0=3.5, 8.0 mM, ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 141 Ca 0=2.5 m M , the transition between normal and calcium-driven tonic firing is centered around Na;= 10.5 m M , while for Ca 0=3.5 m M , it is centered around a lower value of Na;= 9.5 m M . As in the case for Ca0=2.5 m M , the subthreshold C a 2 + oscillations in the isolated voltage-clamped subcellular oscillator resemble those of the coupled system when CaG=3.5 m M . For Ca o=8.0 m M , the bifurcation diagram (Figure 4.10d) looks qualitatively different than those for the lower values of Ca 0 (Figures 4.10a and 4.10c). We see that rather than a smooth transition from subthreshold oscillations to calcium-driven tonic firing, there are actually two distinct branches of periodic orbits. Comparing the subthreshold Ca; oscillations of the isolated (voltage-clamped) and coupled system in Figure 4.13, we see that the subthreshold oscillations for high Na; are relatively undisturbed by the coupling. However, as Na; decreases, the membrane depolarizes and the interactions between the two oscillators begin to become more important. The behaviours of the isolated and coupled systems diverge. The subthreshold oscillations of the coupled system become unstable and eventually disappear. Calcium-driven tonic firing arises, but not directly out of the subthreshold oscillations of the subcellular oscillator. This diagram helps to illustrate a very important point. The calcium-driven tonic firing is not merely an extension of the subthreshold oscillations. In the case of the sub-threshold oscillations, the subcellular oscillations tend to drive the membrane potential with very little feedback. Thus the subcellular oscillator alone sets the rhythm. The calcium-driven tonic firing, on the other hand, does not rely solely on the intrinsic oscil-lations of the subcellular oscillator. It also requires the influx of C a 2 + during each action potential via Isica-Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 142 Figure 4.13: Comparison of subthreshold Ca; oscillations in (a) the isolated subcellular os-cillator (n=-80 mV; IsiCa=Q', P=l) a n d (b) the coupled system for pacemaking DN equations with Cao=8.0 mM. The bifurcation diagram shows Ca; and the period T as a function of the bifurcation parameter Na;. (DN equations; standard parameters with K; = 145 mM, K C = K 0 , P=l; v8hift=0 mV, Af = 1.0; Cao=8.0 mM, ANaK=1.0) Digitalis Poisoning and High Ca 0 Together Digitalis and high Ca 0 may be applied separately or together. Table 4.1 summarizes some of the results. The first column (Ca0=2.5 mM) illustrates the rapid increase in the pseudo-steady state value of Na; with increasing digitalis poisoning and the transition from normal to calcium-driven tonic firing. The first row (A/vaA"=l-0) illustrates the slower increase of Na; with increasing Ca 0 . The transition from normal to calcium-driven Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 143 tonic firing with increasing Ca 0 relies more heavily on the shift of the C a 2 + oscillations to lower Na, values. The remaining entries in the table show that the two interventions act in concert to promote calcium-driven tonic firing. Ca0—2.5 Cao=3.0 Ca0—3.5 Ca o=4.0 Ca o=8.0 normal Na,=9.1 normal Na,=9.2 transition Na,=9.3 Ca;-driven Na,-=9.4 Ca,-driven Na,=10.1 ANaK=0.9 normal Na,=10.2 transition Na i = 10.4 C a,-driven Na,=10.5 ANaK=0.85 transition Na; = 11.0 Ca,-driven Na, = l l . l ANO.K=0-8 C a,-driven Na,-=11.8 ANaK=0.7 C a,- driven Na,=13.5 ANaK=0.6 Ca;-driven Na,=15.8 Table 4.1: The type of tonic firing and pseudo-steady state Na,- value (in mM) exhibited by the pacemaking version of the DN equations depends on the amount of digitalis poisoning (A/vaA") and the extracellular C a 2 + concentration Ca 0 (in mM). The bifurcation diagrams using Na, as a bifurcation parameter are given in Appendix D for the elements of the table shown in bold. (DN equations; standard parameters with K,-=145 mM, K C = K 0 , p=l; v s^/t=0 mV, Ay = 1.0) (normal = normal tonic firing; transition = tonic firing in transition region between normal and calcium-driven; Cai-driven = calcium-driven tonic firing) 4.2.3 The Effects of Overdrive Digitalis poisoning and application of high extracellular C a 2 + lead to tonic firing which is calcium-driven. It is also possible to induce a transient response consisting of a train of calcium-driven action potentials and then a return to normal tonic firing by overdriving the pacemaker using a train of relatively high frequency current pulses. We examine the effects of short and long durations of overdrive on the occurrence of Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 144 calcium-driven firing. For the sake of illustration, we will use the pacemaking version of the D N equations with Ca 0=3.5 m M . This system is just barely at the edge of the transition region between normal and calcium-driven tonic firing. A slight perturbation of the dynamics can lead to calcium-driven tonic firing via two mechanisms. One mecha-nism, namely the buildup of Na; requires long durations of overdrive. Shorter durations of overdrive result in a buildup of Ca„ p without significant changes of Na;. Prolonged Overdrive Figure 4.14 illustrates the effect of prolonged overdrive on the Na; level. During overdrive at a basic cycle length of 500 msec, Na; slowly builds up and then eventually plateaus at an abnormally high value (Na;=10.58 mM). This temporarily high Na; level corresponds with calcium-driven tonic firing with a period corresponding approximately to the period Figure 4.14: Simulation of the effect of prolonged overdrive on the Na; level in the pacemaking DN equations. The intracellular N a + concentration Na; is plotted as a function of time. The system is overdriven with rectangular current pulses of amplitude 900 nA, duration 3 msec, and basic cycle length of 500 msec for 3000 sec. Na; builds up during the overdrive, eventually reaching a plateau. When overdrive ceases, Na; immediately begins to decay back to its original level. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p—l\ vshift=0 rnV, A/ = 1.0; Ca0=3.5 mM, ANaK=1.0) -i o . s Na, (mM) s o o o Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 145 associated with its current Na; value. Thus we can view the major effect of prolonged overdrive as a "parameter change" (the change in Na;) which puts the system in a different firing regime. When overdrive ceases, Na; slowly decays back to its original level. Figure 4.15: Simulation of the effect of prolonged overdrive on the pacemaking DN equations with Ca0=3.5 mM. The system is overdriven with rectangular current pulses of amplitude 900 nA, duration 3 msec, and basic cycle length of 500 msec for 600 cycles (300 sec). The intra-cellular N a + concentration Na; is plotted as a function of time in panel (a). The intracellular C a 2 + concentration Ca; is plotted as a function of the voltage v for a period of ten seconds starting at times 400, 600, and 1500 seconds in panels (b), (c), and (d), respectively. While Na; is allowed to vary in these simulations, it varies so slowly that its effects are barely noticeable over the course of several action potentials. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=0 mV, A/=1.0; Ca0=3.5 mM, ANaK=1.0) In Figure 4.15, we give an example of this kind of effect. The system is driven at a basic cycle length of 500 msec for 300 seconds, long enough to cause a substantial increase Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 146 in Na,-. We show the projections of the action potentials following overdrive onto Ca, and v at times 400, 600, and 1500 seconds in panels (b), (c), and (d), respectively. Each plot shows a simulation 10 seconds long, which corresponds to approximately five action potentials. As we can see, the dynamics of the action potentials do not change significantly over the course of a few action potentials since Na,- is decaying very slowly in comparison with the period of the action potentials (of the order of 2 seconds). Thus, under these circumstances, we can view Na,- as a slowly varying parameter. At the end of overdrive, the Na; level is much further into the calcium-driven tonic firing region than it was before overdrive (see Figure 4.10c). Thus, we would expect that the system would exhibit action potentials that rely more heavily on the initial Ca,-upstroke. An examination of panel (b) confirms this, since there is a large increase in Ca,- before the voltage upstroke. As Na,- decreases, this effect become less pronounced (panel (c)) until Na,- decreases so much that the Ca,- upstroke is barely noticeable (panel (d)) before the voltage upstroke. Short Overdrive Calcium-driven tonic firing also can result from short instances of overdrive, even when Na,- is fixed. To demonstrate this, we overdrive the pacemaking version of the D N equa-tions with Ca 0=3.5 m M for 15 seconds at a basic cycle length of 500 msec while fixing Na; at its pseudo-steady state value of 9.3 m M . The results are shown in Figure 4.16. We see that Ca„ p builds up slowly during overdrive, and when overdrive ceases, the uptake compartment of the sarcoplasmic reticulum releases its excess C a 2 + into the re-lease compartment. This results in calcium-driven tonic firing, as indicated in panel (b). As C a u p decays back to normal levels, the magnitude of the Ca,- upstroke preceding the voltage upstroke decreases. Eventually, the system resumes its regular firing mode at the edge of the transition region between normal and calcium-driven tonic firing. Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 147 - 1 OO 1 1 1 1 " 1 • 1 1 O S 1 O 1 S SO S S 3 0 3 5 >^ o time (sec) -100 -80 -60 -40 -20 0 2 0 4 0 0 5 10 15 2 0 2 5 3 0 3 5 4 0 v (mV) time (sec) 0 5 10 15 2 0 25 3 0 3 5 4 0 0 5 10 15 2 0 2 5 3 0 3 5 4 0 time (sec) time (sec) Figure 4.16: Simulation of the pacemaking version of the DN equations with Ca0=3.5 mM in response to short overdrive with Na; fixed at 9.3 mM. Thirty stimuli induce thirty action potentials, foUowed directly by calcium-driven tonic firing which becomes less calcium-driven with time. The stimuli are current pulses of amplitude 900 nA, duration 3 msec, and a basic cycle length of 500 msec. Following overdrive, the system exhibits calcium-driven tonic firing which become less dependent on the Ca; upstroke as C a u p decays. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=0 mV, A/=1.0; Ca0=3.5 mM, A/va/<- = 1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 148 Thus, we have seen that overdrive has two effects on the system. It results in Na; buildup since the Na-K pump cannot keep up with the accelerated influx of N a + during fast pacing. Since the buildup of Na; is relatively slow, its effects are generally only noticeable after relatively long instances of overdrive. We can view Na; as a slowly changing parameter. The buildup of C a 2 + in the uptake compartment of the sarcoplas-mic reticulum, on the other hand, happens relatively quickly. Therefore, it can lead to calcium-driven firing after short instances of overdrive. The rate of calcium-driven firing is not sufficient to maintain the high C a u p level, so this behaviour does not persist. 4.3 S imula t ing Exper imenta l Interventions in the Quiescent System In this section, we examine the effects of various experimental interventions on the qui-escent version of the D N equations with the same simplifications (p=l, K C = K 0 , K;=145 mM) used in the pacemaking version in the previous section. We begin with a discussion of the normal behaviour of the quiescent system. We then demonstrate the effects of digitalis poisoning and high extracellular C a 2 + . We show that both of these two experi-mental interventions lead to triggered activity. We also show that prolonged pacing can result in triggered activity. 4.3.1 N o r m a l Behaviour To make the system quiescent, we shift the activation curve, ?/co( v), which gates the Ij currents, by setting the parameter vshift equal to 20 mV. The resulting bifurcation diagram, where Na; is the bifurcation parameter, is shown in Figure 4.17. Comparing this bifurcation diagram with the equivalent bifurcation diagram for the pacemaking system (Figure 4.8), we see that the normal tonic firing persists at very low Na; when vshift is set to 20 mV. Much of the calcium-driven tonic firing also persists. However, normal tonic Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 149 T (sec) 1 0 2 0 2 5 3 0 Na; (mM) 2 0 3 0 Na; (mM) Figure 4.17: Quiescent version of the DN equations under normal conditions. Bifurcation diagram shows the voltage v, the intraceUular C a 2 + concentration Ca;, and the period T as functions of the bifurcation parameter Na;. Under these conditions, the steady state value of Na; is 8.5 mM and the system is quiescent. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=20 mV, A / = 1.0; Ca0=2.5 mM, ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 150 firing disappears at intermediate values of Na;, to be replaced by a stable steady state. Therefore, under normal conditions, the system is quiescent with Na; resting at 8.5 m M . The resting value of Na; is lower than in the pacemaking version because there are no action potentials and, hence, no rapid N a + influx. Simulations of this system (Figure 4.18) show that it does not yield triggered activity in response to pacing. v (mV) time (sec) Figure 4.18: Simulation of the quiescent version of the DN equations under normal conditions in response to a train of stimuli. Twenty stimuli induce twenty action potentials, followed directly by a return to quiescence. The stimuli are current pulses of amplitude 900 nA, duration 3 msec, and a basic cycle length of 800 msec. The membrane potential v is plotted as a function of time. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=20 mV, A / = 1.0; Cac=2.5 mM, ANaK=l.0) 4.3.2 The Effects of Digi ta l i s Poisoning and H i g h C a 0 We simulate the effects of applying digitalis and high extracellular C a 2 + concentration to the quiescent system. We present a table of our results (Table 4.2), and then discuss each of the interventions in turn, using the relevant bifurcation diagrams to make sense of our results. Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 151 (J>£L0—2.5 CJ&O—3.5 Ca 0=4.5 Cao=6.0 Ca o=7.0 ANaK=l.O no T . A . Na t=8.5 no T .A. N& 1=8.5 short T . A . Na,—8.5 long T . A . Na,-=8.5 bistable Na t=8.5/8.2 ANAK=0.9 no T .A. Na;=9.6 short T .A. Na;=9.6 long T .A . Na , -9.6 bistable Na,-=9.6/9.1 ANai<=0.8 short T . A . Na t = l l . l long T .A. N a ; = l l . l tonic firing Nat=10.1 ANaK=0.7 long T .A . Na;=13.0 tonic firing Na,-=11.6 ANaK = 0-& tonic firing Na 8 = 13.5 Table 4.2: The type of response to pacing and steady state Na; value (in mM) exhibited by the quiescent version of the DN equations depends on the amount of digitalis poisoning (AyvaA") and the extraceUular C a 2 + concentration Ca 0 (in mM). The bifurcation diagrams using Na8- as a bifurcation parameter are given in Appendix E for the elements of the table shown in bold. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshi/t=20 mV, Af=1.0) (no T.A. = no triggered activity; short T.A. = triggered activity that lasts less than a minute; long T.A. = triggered activity that lasts for minutes; bistable = system has a stable steady state (ss) and stable tonic firing (tf) at the values of Nat- indicated (ss/tf); tonic firing = calcium-driven tonic firing) High Extracellular C a 2 + Concentration First, we look at the effects of changing the extracellular C a 2 + concentration Ca Q in the absence of digitalis poisoning. The system has a stable steady state with a resting Na; value of 8.5 m M . Referring to the first row of Table 4.2 (A/v aA"=l-0), we see that the value of Na; is virtually unaffected by a change in Ca 0 . However, the system's response to stimuli changes considerably as Ca 0 increases. An examination of the bifurcation diagrams shown in Figure 4.19 for several values of Ca 0 will make the reason clear. The behaviour of the quiescent system parallels the behaviour of the pacemaking system as Ca 0 increases. The only major difference is that the region of normal tonic firing has been replaced by a stable steady state. Thus, as Ca 0 increases, the subthreshold oscillations Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 152 40 20 0 v -20 (mV) -40 -60 -100 (a) Ca 0=2.5 m M 40 20 0 v -20 (mV) -40 -60 100 1 1—1 ISNP^ i 1 • 1 c 0 0 0 8 (b) Ca 0=4.5 m M 10 15 20 25 30 N a i(ss)=8.5 N a . ( m M ) (c) Cao=6.0 m M , , 1—1— - to 1 1 1 10 Na,-(ss)=8.5 15 20 25 30 Na; (mM) 40 20 0 v -20 (mV) -40 -60 -100 N a i(ss)=8.5 Na,- (mM) (d) Cao=7.0 m M j 8 5 10 15 Na,(tf ,ss)=8.2,8.5 20 25 30 Na; (mM) Figure 4.19: Quiescent version of the DN equations at various Ca 0 . Bifurcation diagrams show the voltage v as a function of the bifurcation parameter Na,-. The steady state value (Na,-(ss)) or approximate value during calcium-driven tonic firing (Na,(tf)) for Na,- is indicated in each figure. The positions of the Hopf bifurcations (HB) and the saddle-node of periodics (SNP) are indicated in panel (a). (DN equations; standard parameters with K, = 145 mM, K C = K 0 , p=l; vshift=20 mV, A/=1.0; Ca0=2.5, 4.5, 6.0, or 7.0 mM, ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 153 and the region of calcium-driven tonic firing move to lower values of Na;. Since the steady state value of Na; does not move, the net result is that the calcium-driven tonic firing region approaches the resting value of Na,-. Let us examine each of the bifurcation diagrams of Figure 4.19 in turn. In each of the bifurcation diagrams of Figure 4.19, there is a region of normal tonic firing at very low Na,- which has been omitted since it is not relevant to our study. Around Na,(ss) is a region of stable steady states, the normal behaviour expected for the quiescent fiber. The right-most branch of limit cycles, which contains the region of stable calcium-driven tonic firing, arises from two Hopf bifurcation points. At the left end of this region, the stable and unstable limit cycles coalesce and annihilate each other. This phenomenon is called a saddle-node bifurcation of limit cycles or saddle-node of periodics (SNP). This point is important since it marks the lowest value of Na; at which we can expect stable calcium-driven tonic firing for fixed Na;. If Na,- is sufficiently close to this point, we can expect a transient response resembling tonic firing. From Figure 4.19a, where Ca 0=2.5 m M , we see that the resting value of Na,- is far away from any of the tonic firing regions and in particular far away from the saddle-node of periodics (SNP). As a result, short durations of overdrive, which do not substantially affect Na;, are quickly followed by a return to rest, as shown in the simulation in Figure 4.18. From Figure 4.19b, where Ca0=4.5 m M , we see that the resting value of Na; is much closer to the SNP, and we might expect the system to exhibit some transient behaviour that closely resembles the nearby oscillations. In fact, two "spontaneous" spikes follow a train of directly induced action potentials, as shown in Figure 4.20a, and are referred to as triggered activity. From Figure 4.20b, we see that the triggered activity is a transient form of calcium-driven firing. It results from the buildup of C a 2 + in the uptake compartment of the sarcoplasmic reticulum during overdrive (Figure 4.20d) and the ensuing release from Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 154 Figure 4.20: Simulation of the quiescent version of the DN equations with Ca0=4.5 mM in response to overdrive. Twenty stimuli induce twenty action potentials, foUowed directly by two "spontaneous" action potentials (triggered activity). The stimuli are current pulses of amplitude 900 nA, duration 3 msec, and a basic cycle length of 800 msec, v, Ca t , Ca,up, and Ca r e ; are plotted as functions of time during overdrive (OD) and triggered activity (TA). The plot of Ca; vs v shows that triggered activity consists of calcium-driven action potentials. (DN equations; standard parameters with K; = 145 mM, K C =K D , p=l; t;s/l;/i=20 mV, A/=1.0; Ca0=4.5 mM, ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 155 the release compartment of the SR (Figure 4.20e). The resulting upstroke of Ca; (Figure 4.20c) enhances iNaCa (not shown), which in turn depolarizes the membrane. The changes in Na,- during this whole episode are very small and do not contribute substantially to the behaviour. In fact, if we fix Na,- at its initial value rather than letting it vary, we still get two spontaneous spikes in response to the same pacing protocol. Figure 4.21: Simulation of the quiescent version of the D N equations with Ca o=6.0 m M in response to overdrive. Three stimuli induce three action potentials, followed directly by about two minutes of triggered activity (TA) and then a return to quiescence (Q). The stimuli are current pulses of amplitude 900 nA, duration 3 msec, and a basic cycle length of 800 msec, v and Na,- are plotted as functions of time during overdrive (OD) and triggered activity (TA). The plot of Ca; vs v shows that triggered activity consists of calcium-driven action potentials. (DN equations; standard parameters with K;=145 m M , K C = K 0 , p=l; vshift=20 mV, Ay = 1.0; Ca o=6.0 m M , ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 156 When Ca 0 increases to 6.0 m M , the calcium-driven behaviour shifts even further to the left in the bifurcation diagram of Figure 4.19c, and Na; rests just inside of the bistable region of the bifurcation diagram. Only a few stimuli cause the formerly quiescent system to exhibit calcium-driven tonic firing. This firing persists for a couple of minutes, as shown in Figure 4.21a. However, as the firing continues, Na; decays slowly (Figure 4.21c), until finally it overshoots the saddle-node of periodics at Na;=8.44 m M and "falls off" the branch of periodic orbits. The system becomes quiescent and Na; is eventually restored to its prior resting value of 8.53 m M . A further increase of Ca 0 to 7.0 m M shifts the calcium-driven behaviour even further to the left, as shown in Figure 4.19d. The system still has a stable steady state at Na;=8.5 m M and can exhibit calcium-driven tonic firing (Figure 4.22a,b). Only this time, it does not "fall off" the branch of periodic orbits. Instead Na; decreases slowly, until at Na;=8.2 m M , it achieves a stable oscillation (Figure 4.22c). Thus, at Cao=7.0 m M , we have a truly bistable system capable of quiescence or stable calcium-driven tonic firing. Compar ing the Effects of H i g h C a 0 and Digi ta l i s Poisoning From the first column of Table 4.2 (Ca0=2.5 mM), we see that inhibition of the Na-K pump by digitalis has the major effect of increasing the steady state value of Na; in the quiescent system. Figure 4.23c shows the bifurcation diagram of the quiescent system as a function of the bifurcation parameter Na, under digitalis poisoning (ANO,K=0.8). A comparison with the equivalent diagram for the unpoisoned system (Figure 4.23a) shows that the bifurcation diagram itself is relatively unchanged. Thus, while increasing Ca D causes the saddle-node of periodics to approach the resting value of Na;, digitalis poisoning causes the resting value of Na; to approach the saddle-node of periodics. Regardless of whether the saddle-node of periodics approached Na;(ss) or vice versa, the effect on the response to overdrive is the same. Comparing the bifurcation diagrams Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 157 Figure 4.22: Simulation of the quiescent version of the DN equations with Cao=7.0 mM in response to overdrive. Only one stimulus (amplitude 900 nA, duration 3 msec) is required to switch this bistable system from quiescence with Na;=8.5 mM to calcium-driven tonic firing with Na;=8.2 mM. v and Na; are plotted as functions of time during the first driven action potential (OD) and triggered activity (TA). The plot of Ca; vs v shows that continuous triggered activity consists of calcium-driven action potentials. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=20 mV, A/=1.0; Cao=7.0 mM, ANaK=1.0) of Figures 4.23b and 4.23c, we would expect the same type of response whether we increased Ca 0 to 4.5 m M or decreased AjvaA" 1° 0.8. If we apply the same pacing protocol for AjvaA-=0.8 as we did for Ca0=4.5 m M , we get one "spontaneous" action potential, shown in Figure 4.24a. Thus, in both cases (see Figure 4.20a for comparison), we get a small amount of triggered activity following a train of twenty stimuli. If we increase the number of stimuli, we get more triggered activity (shown in Figure 4.24b) as the cell Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 158 a) ANaK=1.0, Ca 0=2.5 m M (b) ANaK=l.O, Ca 0=4.5 m M 0 5 J 10 15 20 25 30 N a i(ss)=8.5 Na, (mM) (c) ANaK=0.8, CaQ=2.5 m M Na, ( s s )= l l . l 20 25 30 Na,- (mM) 20 25 30 Na,(ss)=8.5 Na, (mM) (d) ANaK=0.8, Ca 0=4.5 m M 40 20 0 v -20 (mV) -40 -60 0 1 I 0 5 lj) 15 Na,( t f )=10.1 20 25 30 Na,- (mM) Figure 4.23: Quiescent version of the DN equations at various A^aK and Ca 0 . Bifurcation diagrams show the voltage v as a function of the bifurcation parameter Na,-. The steady state value (Na,(ss)) or approximate value during calcium-driven tonic firing (Na,(tf)) for Na,- is indicated in each figure. (DN equations; standard parameters with K; = 145 mM, K C = K 0 , p=l; vshift=20 mV, A/=1.0; Ca0=2.5 or 4.5 mM, ANaK=l.O or 0.8) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 159 (a) v (mV) (b) v (mV) time (sec) 1 S 20 time (sec) Figure 4.24: Simulation of the quiescent version of the DN equations with digitalis poisoning (A/VaA"=0.8) in response to a train of stimuli, (a) 20 stimuli induce 20 action potentials, followed by one "spontaneous" action potential (triggered activity), (b) 23 stimuli induce 23 action potentials, followed by two "spontaneous" action potentials. The stimuli are current pulses of amplitude 900 nA, duration 3 msec, and a basic cycle length of 800 msec. The membrane potential v is plotted as a function of time. (DN equations; standard parameters with K; = 145 mM, K C = K 0 , p=l; vshtft=20 mV, A/=1.0; Ca0=2.5 mM, ANaK=0.8) becomes more overloaded with C a 2 + . A combination of high Ca 0 and digitalis poisoning promote triggered activity even more by moving Na;(ss) and the saddle-node of periodics together simultaneously. We see from Table 4.2, for instance, that while setting either AjvOA"=0.9 or Ca 0=3.5 m M does not result in triggered activity, the combination of the two produces short triggered activity following overdrive. If we set ANaK—0.8 and Ca 0=4.5 m M simultaneously, we get a new Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 160 behaviour. For these parameters, the system only exhibits stable calcium-driven tonic firing. It does not have a stable steady state. We can explain this by referring to Table 4.2 and the bifurcation diagram for this parameter set (Figure 4.23d). For Ajva/<=0.8 in Table 4.2, we see that Na;(ss) remains at 11.1 m M for Ca 0=2.5 m M and 3.5 m M . In Figure 4.23d, this value of Na; is just slightly to the right of the Hopf bifurcation point at Na;=10.9. Thus, the anticipated steady state is unstable when Ca 0=4.5 m M , and only stable calcium-driven tonic firing is observed (Figure 4.25). 0 5 1 0 1 5 2 0 - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 t ime (sec) v (mV) Figure 4.25: The "quiescent" version of the D N equations exhibits only calcium-driven tonic firing when Ca 0=4.5 m M and A7vaA"=0.8. (a) The membrane potential v is plotted as a function of time, (b) Plotting Ca; as a function of v shows that this stable behaviour is calcium-driven tonic firing. (DN equations; standard parameters with K;=145 m M , K C = K 0 , p=l; v s/ l;/t=20 mV, A / = 1.0; CaG=4.5 m M , AAr a A-=0.8) 4.3.3 The Effects of Prolonged Overdrive If we overdrive the quiescent version of the D N equations for a long time, then Na; can build up significantly. This effect is added to the transient effect due to C a u p buildup. The system may or may not exhibit triggered activity, depending on where the final Na; level is with respect to the calcium-driven tonic firing region. If we overdrive the quiescent system with ApfAK=1.0 and Ca 0=2.5 m M at a basic cycle Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 161 length of 500 msec for about 1000 seconds, we find that Na; plateaus at about 9.72 m M . This Na; level is still very far away from the saddle-node of periodics at the beginning of the calcium-driven tonic firing region (Figure 4.17). Therefore, under normal conditions in the quiescent system, no triggered activity is seen. (a) Na; (mM) overdrive -OD ceases k triggered activity \< TA ceases quiescence G O O 1 o o o i s o o s o o o '.' r_>oo 3 0 0 0 3 S O O - * o o o time (sec) time (sec) Figure 4.26: Simulation of the effect of prolonged overdrive on the quiescent DN equations with Ca0=4.5 mM. The system is overdriven with rectangular current pulses of amplitude 900 nA, duration 3 msec, and basic cycle length of 500 msec for 2000 sec. The intracellular N a + concentration Na; and voltage v are plotted as functions of time, (a) Na; builds up during the overdrive, eventually reaching a plateau. When overdrive ceases, Na; immediately begins to decay back to its original level, (b) The last ten seconds of overdrive are shown, followed by triggered activity, and then a return to quiescence. (DN equations; standard parameters with K;=145 mM, K C = K 0 , p=l; vshift=20 mV, A/=1.0; Ca0=4.5 mM, ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 162 When Ca 0=4.5 m M , triggered activity appears after only a short overdrive, largely due to the buildup of Ca„ p . Prolonged overdrive results in a Na; plateau at approximately 10.8 m M (see Figure 4.26a), which is quite far into the calcium-driven tonic firing region (Figure 4.19b or E . l ) . Triggered activity begins after the overdrive, as shown in Figure 4.26b. The frequency of the triggered activity decreases as Na; decreases, qualitatively in agreement with the period of oscillation predicted by the bifurcation diagram in Figure E . l . The magnitude of the envelope of the Ca; oscillations increases. Although this triggered activity is largely due to a change in the Na; level, viewing it as a mere parameter change is not accurate for two reasons. First, Na; is decaying quickly, so the changes are noticeable on the time scale of a few action potentials. Secondly, the buildup of Caup has a fairly substantial effect on the system, especially just after overdrive ceases. Eventually, the Na; level overshoots the Na; value at the saddle-node of periodics and the system becomes quiescent. 4.4 Examina t i on of the Effects of Changes in K ; Here we comment on some of the simplifications we made to the D N equations. The ap-proximations K C = K 0 and p=l have a relatively small effect in that they alter the number of "spontaneous" spikes following a train of stimuli but not the qualitative behaviour. Setting K;=145 m M for all of the calculations influences the behaviour a bit more. As discussed earlier, driving the cell at a high frequency and poisoning the system with digi-talis, thereby inhibiting the Na-K pump, decreases the resting value of K; . The effects are only significant under heavy digitalis poisoning or prolonged overdrive. Since K ; varies slowly, we treat it as a parameter. We first examine the effect of changes in K; on the fixed points of the system. Figure 4.27a is a two-parameter continuation of the critical fixed points, including the Hopf Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 163 Figure 4.27: Two-parameter continuation in (Na;, K;) of the critical fixed points of the qui-escent version of the D N equations. The Hopf bifurcation points (HB) and saddle-node points (SN) are traced out as Na; and K ; vary. Panel (b) is a zoomed in view of panel (a) over a more restricted range of K ; . (DN equations; standard parameters with K C = K 0 , p=\\ -y s/ l;/i=20 mV, A/=1.0; Ca 0=2.5 m M , ANaK=1.0) Chapter 4. RESULTS FOR THE.DIFRANCESCO-NOBLE EQUATIONS 164 bifurcation points and the saddle-node points. The range of K; shown here is not meant to be biologically realistic, but rather is shown so that we can see how the various points arise and disappear with increasing K; . Clearly K,- substantially affects the dynamics of the D N equations. If we zoom in on a more biological range of values, as shown in Figure 4.27b, we see that the positions of the fixed points still depend on K,-, but not as drastically. We note here that a change from K, = 145 m M to K;=100 m M could only result from a very long overdrive. The bifurcation diagrams corresponding to these two K,- values are given in Figure 4.28 where Na,- is the bifurcation parameter. The general shape of the curve of fixed points remains the same as K; decreases, although the resting potential for K,=100 m M is somewhat depolarized compared with K;=145 m M . The shape of the leftmost branch of periodic orbits changes substantially, although this range of Na,- is not particularly relevant to our study. The most important change is in the position of the branch of calcium-driven oscillations. For low K, , the branch corresponds to a lower range of Na,. This means that triggered activity will develop more easily at low K; . Considering that the difference in K; represents a change nearing 50 m M , the two bifurcation diagrams do not really vary that significantly. One obvious difference is that for K;=100 m M , the subthreshold and suprathreshold calcium-driven oscillation regions are not connected. Instead, the suprathreshold oscillations appear as an isola 2. This difference is not impor-tant in the context of triggered activity, except that it illustrates that the subthreshold and suprathreshold oscillations are two distinct phenomena. A careful examination of the bifurcation diagrams of Figure 4.28 shows that the saddle node point moves from Na,=12.79 m M at K,=145 m M to Na;=12.29 m M at K;=100 m M , 2This branch of periodic orbits is not accessible via the continuation method employed by AUTO from any other branch on this diagram. Rather, one must start with a known branch using a different parameter value (for example, K; = 145 mM) and then continue the calculation using that parameter as the bifurcation parameter, say for Na,-=13.5 mM, until the correct parameter value (K;=;100 mM) is reached. Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 165 (a) K;=145 m M (b) K t=100 m M Na; (mM) Na; (mM) Figure 4.28: Quiescent version of the DN equations at normal and low K;. Bifurcation diagrams show the voltage v and the intracellular C a 2 + concentration Ca; as functions of the bifurcation parameter Na; for (a) K;=145 mM and (b) K; = 100 mM. (DN equations; standard parameters with K;=145 or 100 mM, K C = K 0 , p=l; vshift=20 mV, A/=1.0; Ca0=2.5 mM, ANaK=1.0) Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 166 a change in Na; of 0.5 m M . Since the position of the saddle-node of periodics relative to the current value of Na; is the crucial determinant of triggered activity, we might expect K ; variation to have some effects on our results. However, simulations show that Kt does not vary fast enough during a short overdrive to noticeably influence the results. In fact, a simulation in which the cell was overdriven for 300 seconds at a cycle length of 500 msec (i.e., 600 cycles) resulted in the same amount of triggered activity (when CaG=3.5 mM) regardless of whether K; varied or not. In this case, K ; only dropped from 145 m M to 142.6 m M . Thus, we would need a much longer overdrive in order for K ; to have a major effect. 4.5 Conclusions The work in this chapter was based on two very important observations by Varghese and Winslow. First, they pointed out that the D N equations consist of two subsystems, a membrane oscillator and a subcellular oscillator. They showed that in a particular range of Na I 5 the subcellular oscillator is capable of oscillating even when the voltage is clamped [68]. Their second key observation was that the system can exhibit two distinct types of behaviour: normal tonic firing and calcium-driven tonic firing. They suggested that the latter form of firing occurs when the subcellular oscillator dominates the pacemaking mechanism. They showed that DADs occurred transiently just before the membrane oscillator became entrained by the faster subcellular oscillator [69]. Their observations provided a framework for understanding overdrive excitation or triggered activity in the context of the interactions of the two oscillators. Triggered activity is the result of a transition between normal and calcium-driven tonic firing. In this work, we have given a mathematical explanation of the effects of a number of experimental interventions in promoting triggered activity. We saw that Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 167 different regions of biological parameter space corresponded with quiescence, normal tonic firing, or calcium-driven tonic firing. While each experimental intervention affected the system in a different way, those that led to triggered activity shifted the system into or near the region of parameter space corresponding to calcium-driven tonic firing. To see how such a transition between normal and calcium-driven tonic firing might occur, we examined the stability of the D N equations with respect to the bifurcation parameter Na;, which varies sufficiently slowly compared to the time course of a few action potentials that we can treat it as a constant. Using AUT086 , we first computed the bifurcation diagram for the pacemaking version of the equations. It showed a smooth transition between normal and calcium-driven tonic firing as Na; increased. By plotting v versus Ca;, we ascertained which ranges of Na; corresponded with normal and calcium-driven tonic firing by determining whether the v or Ca; upstroke came first. For normal values of Na;, the system exhibits normal tonic firing. As Na; increases, the subcellular oscillator starts to dominate in the transition region. Eventually, the system is entrained by the subcellular oscillator, resulting in calcium-driven tonic firing. At higher Na; values, the C a 2 + oscillator continues to drive the system, but the tonic firing eventually gives way to subthreshold calcium-driven membrane oscillations. By comparing the bifurcation diagrams for the full system and the voltage-clamped sub-cellular oscillator, we saw that these subthreshold oscillations are due to the intrinsic oscillations of the voltage-clamped subsystem. The calcium-driven tonic firing, how-ever, is not due to the intrinsic oscillations of the subcellular oscillator alone, but rather to the interaction of the two subsystems. The feedback between the two subsystems is bidirectional. The C a 2 + oscillations are driven by the influx of C a 2 + during the action po-tentials, while the action potentials are driven by intracellular C a 2 + oscillations resulting from calcium-induced calcium release from the sarcoplasmic reticulum. This bidirection-ality is a key feature of triggered activity (and the related phenomenon of spontaneous Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 168 secondary spiking) since it allows the system to respond differently to different types of overdrive. We also computed the bifurcation diagram for the quiescent version of the D N equa-tions, which looks similar to that for the pacemaking system except that the region of normal tonic firing is largely absent. Thus, the calcium-driven tonic firing arises suddenly from normal tonic firing via a saddle-node of periodics bifurcation. To the right of the saddle-node of periodics, the system is actually bistable, exhibiting either calcium-driven tonic firing or quiescence. The appearance of calcium-driven action potentials depends on how close the current Na; value is to the transition region in the pacemaking version or the saddle-node of periodics in the quiescent version of the D N equations. If Na; is to the right of the transition region or saddle-node of periodics, we expect to see calcium-driven action potentials. If Na; is just to the left of these regions, transient calcium-driven action potentials may occur in response to short periods of overdrive which can lead to calcium overload. Thus, interventions that place Na; close to or in the region of calcium-driven tonic firing will promote triggered activity. Under normal circumstances, Na/ is sufficiently small that only normal action potentials occur. Certain interventions, such as very long periods of overdrive, lead to either an increase in Na, or a shift in the calcium-driven tonic firing region to lower values of Na;. If the buildup of Na; during overdrive places Na; in the calcium-driven tonic firing region, then we expect to see a relatively long train of calcium-driven action potentials when the overdrive ceases as Na; slowly decays back to its original value. In the pacemaking system, we refer to this calcium-driven tonic firing as "overdrive excitation" since the natural rhythms of the system are "overdriven". In the quiescent system, we refer to it as "triggered activity". If the initial drive fails to place Na; in the calcium-driven firing region, we still may see a few transient calcium-driven Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQ UATIONS 169 spikes due to temporary calcium overload. Another way to increase Na; to a higher value is by poisoning the Na-K pump with digitalis. We showed that inhibition of the Na-K pump did not significantly alter the bifurcation diagram for either the quiescent or pacemaking D N equations. However, simulation of the systems showed that Na; rested at a much higher pseudo-steady state value. The net result was that Na; ended up resting much closer to (or inside of) the calcium-driven tonic firing region. Our simulations showed that calcium-driven firing was more likely to result in response to pacing when the Na-K pump was inhibited. Thus inhibition of the Na-K pump promotes overdrive excitation and triggered activity by increasing Na;. High Ca 0 promotes overdrive excitation and triggered activity in a different way. By enhancing C a 2 + influx and reducing C a 2 + extrusion by the Na-Ca exchanger, it results in calcium overload at lower values of Na;. Thus, rather than shifting Na; to higher values, it shifted the calcium-driven tonic firing region to the left in the bifurcation diagrams for both the quiescent and pacemaking systems. The net result, nevertheless, was that Na; ended up resting closer to the calcium-driven tonic firing region. Again our simulations showed that calcium-driven tonic firing was more likely to occur in response to pacing when Ca 0 was high. A combination of overdrive, inhibition of the Na-K pump, and high Ca 0 will promote the occurrence of calcium-driven action potentials more than any one of these mecha-nisms alone. They all enhance calcium overload. Many other interventions, such as the application of catecholamines or caffeine, also can result in calcium overload. They may do this by enhancing the influx of C a 2 + during each action potential, decreasing the cell's ability to extrude C a 2 + , changing the dynamics of the sarcoplasmic reticulum so it becomes overloaded more easily, or, as in the case of high frequency overdrive, simply by causing more C a 2 + to enter the cell by increasing the rate of the action potentials. Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 170 We focussed on digitalis-induced triggered activity in particular because it is thought to play a role in the generation of certain cardiac arrhythmias. While digitalis glycosides are used to treat cardiac disorders such as congestive heart failure, too high a dose of digitalis can lead to digitalis toxicity which can result in any one of a number of rhythm disturbances. Hypercalcemia, which we can interpret as high Ca D , has also been known to contribute to the occurrence of digitalis toxicity. Perhaps a better understanding of the mechanisms responsible for triggered activity will eventually help lead to prevention of some of the dangerous symptoms associated with digitalis toxicity. Finally, we comment on some of the inadequacies of the D N equations. As mentioned earlier, a latent conservation principle in the D N equations results in a degeneracy, which, unless corrected, can cause difficulty with certain numerical schemes [26, 67]. Fixing one of the slowly varying variables, namely K, , removes this problem and simplifies the D N equations at the same time. The more serious problems with the DN equations involve the C a 2 + dynamics. A comparison of the simulations of triggered activity in this report (e.g., Figure 4.24b) with experimental results such as the one presented in Figure 3.5 reveals two major shortcomings of the D N equations. First, the frequency of triggered activity in the D N equations is far too low. In Figure 4.24b, triggered activity occurs at a much slower rate than the rate of pacing during overdrive. However, in real experiments, triggered activity tends to occur at a rate as fast or faster than the overdrive rate. The rate of triggered activity in the D N equations reflects the rate of oscillation in the subcellular oscillator, which is much too slow. Another difference between the behaviour seen in Figures 4.24b and 3.5 is the lack of DADs in the simulations. The Ca; spikes are too large in the D N equations and generally result in all-or-none action potentials. The CICR mechanism in the SR does not exhibit graded C a 2 + release. In the next chapter, we will discuss some specific inadequacies of Chapter 4. RESULTS FOR THE DIFRANCESCO-NOBLE EQUATIONS 171 the C a 2 + dynamics in the D N equations and the current efforts towards improving the C a 2 + dynamics in cardiac models. Chapter 5 C O N C L U S I O N S A N D D I S C U S S I O N In this thesis, we examined two biological models, both of which are capable of generating "spontaneous" action potentials following high frequency or lengthy durations of pacing. The first was the Kepler and Marder (KM) model [36] of spontaneous secondary spiking in the lateral gastric neuron of the stomatogastric ganglion of a crab. The second was the quiescent version of the DiFrancesco-Noble (DN) equations [12], which can exhibit cardiac triggered activity under certain conditions. Here, we compare the two models and discuss the common features which are essential to producing the "spontaneous" action potentials. Also, we discuss some problems with the C a 2 + dynamics of the DiFrancesco-Noble equations and what is being done to rectify them. 5.1 Compar i son of the K M M o d e l and the D N Equations On the surface, the rather simple K M model and the extremely complex D N equations do not appear to have much in common. However, if we treat the slowly varying variable Na; as a bifurcation parameter in the quiescent version of the D N equations and the parameter Is as a bifurcation parameter in the K M model, we see that their bifurcation diagrams have a few common features. While the D N equations only have a single branch of steady states and the K M model has two such branches, both models only have a single stable steady state for low to intermediate values of their bifurcation parameter. As the bifurcation parameter increases in each model, two periodic orbits, one stable and one unstable, arise via a 172 Chapter 5. CONCLUSIONS AND DISCUSSION 173 saddle-node of periodics (SNP) bifurcation. To the right of the SNP, the stable steady state coexists with the stable limit cycle. If the bifurcation parameter is in this bistable region, the system is capable of either quiescence or stable tonic firing. To the left of the SNP, the system always must return to the resting potential since, based on numerical evidence, the steady state seems to be the only stable behaviour there. Each model will respond to pacing in one of three ways, depending on where the bifurcation parameter sits with respect to the SNP. For very low values of the bifurcation parameter far to the left of the SNP, they will return directly to steady state following pacing. In the bistable region to the right of the SNP, minimal pacing will likely switch the behaviour from quiescence to stable tonic firing. Just to the left of the SNP, the system is "on the verge" of oscillating and, therefore, is capable of transiently exhibiting a few action potentials following appropriate stimulation. These action potentials, which typically follow high frequency or lengthy durations of pacing, are called "spontaneous secondary spikes" in the K M model and "cardiac triggered activity" in the D N equations. Depending on how close the bifurcation parameter is to the SNP and on the type of pacing, these trains of "spontaneous" action potentials can persist for a long time, but will eventually be followed by quiescence. Both models can be divided into two subsystems: an excitable subsystem representing the fast dynamics of the action potentials and a second subsystem capable of some kind of slow buildup in response to pacing. The feedback between the two subsystems is bidirectional. Fast action potentials must result in the requisite buildup in the slower subsystem, so that the slower subsystem can respond appropriately to different types of pacing. The slower subsystem must be able to trigger an inward current which depolarizes the excitable subsystem to produce a "spontaneous" action potential. We showed that the 4D Hodgkin-Huxley type excitable subsystem in the K M model can be reduced to a couple of different 2D subsystems without affecting the ability of Chapter 5. CONCLUSIONS AND DISCUSSION 174 the resulting 3D zFN or M E P models to produce spontaneous secondary spiking. Thus, the exact form of the action potentials is not a crucial element of the model, although their duration is important for spontaneous secondary spiking. This suggests that the full complexity of the action potential dynamics in the D N equations is not necessary for triggered activity. The slow subsystem of the K M model simply consists of a single gating variable z which directly modulates the inward current responsible for triggering "spontaneous" action potentials. It remembers high frequency or lengthy pacing by building up during action potentials and decaying in between action potentials. The result is that the inward current Iz tends to be at least partly turned on most of the time, except when the system is quiescent. The biophysical interpretation of this current is unclear, although it has been suggested that it may be some kind of inward C a 2 + current or C a 2 + -dependent inward current. The latter interpretation would indicate that the intracellular C a 2 + concentration remains high for extended periods of time. However, a continuously high intracellular C a 2 + concentration is not appropriate in a model for cardiac triggered activity. In a cardiac model, Ca; is maintained at low levels except for temporary elevations due to calcium-induced calcium release. Therefore, the slower subsystem in the DN equations is necessarily more complicated than in the K M model since it must allow for a slow buildup separate from the inward current mechanism which triggers the "spontaneous" action potentials. The slower subsystem, referred to as the subcellular oscillator, regulates Ca; by pumping it out of the cell or taking it up into the sarcoplasmic reticulum (SR). Driving the cell can result in calcium overload and the subsequent spontaneous calcium-induced calcium release from the SR which activates the net inward iNaCa responsible for triggered activity. While the subcellular oscillator in the D N equations can oscillate on its own in a certain range of Na,-, triggered activity is not due to these intrinsic oscillations, but to Chapter 5. CONCLUSIONS AND DISCUSSION 175 the interaction of the subcellular oscillator and the membrane oscillator in a different range of Na;. Similarly, in the K M model, spontaneous secondary spiking results from the interaction of the excitable subsystem and the slower subsystem, neither of which would be oscillating on their own. The synchronization that results from this mutual interaction may prevent cardiac cells from escaping entrainment during high frequency pacing. • The appearance of spontaneous secondary spikes or triggered activity depends on how close the bifurcation parameter is to the SNP. Serotonin may promote the occurrence of spontaneous secondary spiking by shifting the bifurcation parameter Is closer to the SNP or by increasing the magnitude of the parameter ks, which controls the position of the SNP and the rate at which Iz builds up. Similarly, in the D N equations, triggered activity may be promoted by increasing Ca 0 , which shifts the SNP to a lower Na; value, or by digitalis poisoning, which increases the resting value of Na;. We should note that Na; is not a true parameter, but can be increased by lengthy pacing so that it ends up in the bistable region of the bifurcation diagram. In this case, triggered activity will result partly due to temporary calcium overload as it does when Na; is to the left of the SNP, but mostly due to the fact that Na; transiently sits in a region of stable tonic firing. When Na; eventually decays past the SNP, the system will return to resting potential. The behaviour seen for the quiescent DN equations persists in the pacemaking ver-sion. Instead of an SNP, the bifurcation diagram for the pacemaking version contains a transition region between normal and calcium-driven tonic firing. The same interventions which result in triggered activity in the quiescent version can lead to transient or tonic calcium-driven firing in the pacemaking version, which can be referred to as triggered activity or overdrive excitation. Chapter 5. CONCLUSIONS AND DISCUSSION 176 5.2 Bio log ica l Implicat ions While the phenomena of spontaneous secondary spiking and triggered activity are fas-cinating in their own right, their occurrence also has important biological implications. For instance, the number of spontaneous secondary spikes generated in the peripheral spike initiation zone in the lateral gastric axon of a crab depends on the duration and frequency of the incoming spike train initiated in the soma. The number of spontaneous secondary spikes also is controlled by the level of serotonin present. Since the sponta-neous secondary spikes prolong the contraction of some of the muscles controlling the lateral teeth of the gastric mill , these extra spikes may be an important signal processing mechanism whereby the crab controls the movement of its teeth. Since several different models exhibit spontaneous secondary spiking despite the fact that the shape of their action potentials are different, this phenomenon is generic. We demonstrated that models in as few as three dimensions, consisting of a two-dimensional excitable subsystem plus a one-dimensional slow inward current, can exhibit spontaneous secondary spiking. We also showed that the relative time constants of the fast and slow subsystems were crucial determinants of this kind of model's behaviour. The simplicity of this ionic mechanism suggests that the phenomenon may be found in many other types of cells and play important roles in different types of biological systems. In fact, there is a qualitatively similar phenomenon called triggered activity in cardiac cells. Our understanding of spontaneous secondary spiking suggests that although the cardiac model is extremely complicated, all of the extra details are not required to understand the essential nature of triggered activity. Instead, we should look for interaction of an excitable subsystem (the membrane oscillator) and another subsystem containing a slow variable (the subcellular oscillator). By looking at the dynamics of the subcellular oscillator and comparing them with the full system, we saw that the Chapter 5. CONCLUSIONS AND DISCUSSION 177 calcium-driven firing or triggered activity resulted from the mutual interaction of the two oscillators, rather than being driven by the subcellular oscillator alone. This mutual interaction appears to be a crucial factor in the occurrence of this type of rate-dependent activity, since it allows the system to respond to different types of stimuli to different degrees. An understanding of delayed afterdepolarization-induced triggered activity is impor-tant from a clinical perspective because it is thought to play a role in the generation of a number of cardiac arrhythmias. In this work, we focussed on the effects of digitalis in promoting triggered activity, since it clearly has been shown to promote DADs and triggered activity in vitro. Digitalis glycosides have been used for decades to treat cardiac disorders such as congestive heart failure. However, too high a dose of digitalis can lead to digitalis toxicity, which can result in rhythm disturbances such as atrial and ventricular premature complexes, ventricular bigeminy, and ventricular tachycardia. Digitalis toxic-ity is more frequent in patients with hypokalemia (low [K + ]) , hypomagnesia (low [Mg 2 + ]), or hypercalcemia (high [Ca 2 +]). While triggered activity has not been identified in vivo, many characteristics of DAD-induced triggered activity, such as their response to various pacing protocols and drugs, support their role in some types of cardiac arrhythmias. We have examined the mathematical effects of a number of experimental interven-tions known to promote triggered activity in vitro. Our analysis suggests that transient triggered activity is likely to occur when the biological parameters lie close to a critical bifurcation point (saddle-node of periodics for the quiescent D N equations) in biological parameter space. If the parameters are pushed past this bifurcation point, we may see calcium-driven firing for an extended period of time. This implies that experimental interventions that are known to lead to triggered activity in vitro may be altering the relative positions of a particular parameter value and an important saddle-node of pe-riodics bifurcation. We have shown that while application of high extracellular calcium Chapter 5. CONCLUSIONS AND DISCUSSION 178 concentration and digitalis have different effects on the system, they both can lead to triggered activity by bringing the parameter value and the saddle-node of periodics closer together. Similarly, lengthy overdrive results in a transient increase in Na; which may place Na4- close enough to the saddle-node of periodics to result in triggered activity. Since the system can respond with a few transient calcium-driven action potentials when it is near the saddle-node of periodics, we can see why short drives are sometimes ad-equate for producing triggered activity while at other times long drives are necessary. Similar results hold for the pacemaking system except that the system must be close to the transition region between normal and calcium-driven tonic firing rather than near a saddle-node of periodics for triggered activity or overdrive excitation to occur. Thus, to understand when triggered activity might occur, it is important to find the point in parameter space where calcium-driven firing first occurs and to see how various experi-mental interventions change the biological parameters relative to this point. Hopefully this work will provide experimentalists with a framework within which they can interpret their experimental results. 5.3 Cardiac C a 2 + Dynamics and the D N Equations Some aspects of our simulations of triggered activity in the D N equations did not match experimental observations. First, the rate of triggered activity in the simulations was too low. Also, the amplitudes of the Ca; oscillations were too high, resulting in very sharp upstrokes of the "spontaneous" action potentials. Furthermore, we did not observe graded C a 2 + release. Instead, the Ca; spikes were "all or nothing" so that they only resulted in full action potentials, not delayed afterdepolarizations of various sizes. These discrepancies are the result of shortcomings in the C a 2 + modeling in the D N equations. The C a 2 + subsystem in the D N equations is a phenomenological model, and, hence, Chapter 5. CONCLUSIONS AND DISCUSSION 179 fails to capture the biophysical details of the mechanisms involved [31]. A number of aspects of the C a 2 + subsystem should be updated, especially the dynamics of the currents in the SR. We already have removed the voltage-dependent gating variable p from Itr, the current flowing between the uptake and release compartments of the SR. The uptake current for the SR, Iup, should be replaced by a SR Ca 2 + -ATPase (SERCA pump) of the form Iup — K\Ca,i2/(K22 + Ca; 2), as in [31], [33], and [65]. The calcium-induced calcium release current, Ire[, also needs to be updated to reflect the dynamics of the ryanodine receptor (RyR), although there is no consensus yet on exactly how it should be modeled. A number of models have been developed to address the issue of "adaptation" of the RyR (e.g., [33] and [65]) in response to some experiments by Gyorke and Fille [27], where they showed that incremental steps in Ca; can result in incremental release of C a 2 + from the RyR. Besides the SR currents, the question of SR and cell geometry must be addressed. For instance, there is reason to believe that C a 2 + may be released into a restricted subspace of the cytoplasm by the SR, resulting in higher local C a 2 + concentrations [31, 53]. Another issue not addressed by the DN equations is the existence of C a 2 + buffers in the cytoplasm and SR. Some of the membrane currents, such as IsiCa and iNaCa, also need updating [31]. In addition, some of the inward current responsible for triggered activity may be carried by a nonspecific Ca 2 +-activated current [43], which should be incorporated into the model. Many of these issues have been addressed in a recent model of the ventricular myocyte by Jafri, Rice, and Winslow [31]. Even so, the model still does not exhibit graded C a 2 + release. Stern has considered this issue and has suggested that all cell models in which the L-type C a 2 + current and the ryanodine-sensitive C a 2 + release channel current empty into a common C a 2 + pool cannot exhibit graded C a 2 + release [63]. According to Winslow [77], current thinking is that release of C a 2 + from the SR is under "local control". Since each Chapter 5. CONCLUSIONS AND DISCUSSION 180 L-type C a 2 + channel in the sarcolemmal membrane is close to a small number of RyR C a 2 + release channels, there is a high probability that each of these RyR C a 2 + release channels will open when the corresponding L-type C a 2 + channel opens. Thus, if the total C a 2 + release is computed by averaging across all of the sets of RyR C a 2 + release channels, then C a 2 + release as a function of membrane potential would be graded. Winslow has developed stochastic computational models to test this hypothesis and has found it to be true [77]. Unfortunately such models can only be studied using Monte Carlo simulation techniques. How these types of models might be simplified so that they can be described by a system of differential equations is still an open question. Another question worth considering is whether the oscillatory release of C a 2 + from the SR is merely an arrhythmogenic event or whether it has a functional value. Diaz et al [11] have suggested that these C a 2 + oscillations may remove C a 2 + from the cell under conditions of C a 2 + overload in a very efficient way. In our simulations of the D N equa-tions, we have seen a noticeable buildup of C a 2 + inside the uptake compartment of the SR during overdrive as it tries to regulate the cytosolic C a 2 + concentration. Following overdrive, the oscillatory release of C a 2 + that gives rise to triggered activity is accom-panied by a fairly rapid decay in the C a 2 + concentration in the uptake compartment of the SR as C a 2 + is pumped out of the cell by the Na-Ca exchanger. Thus, it is possible that the cell is trying to alleviate the problem of C a 2 + overload by releasing C a 2 + in this oscillatory manner, thereby minimizing any increase of diastolic tension due to the calcium overload [11]. B i b l i o g r a p h y [1] C. Antzelevitch and S. Sicouri. Clinical relevance of cardiac arrhythmias generated by afterdepolarizations. 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I. Lie, and M . J . Janse, editors, The Conduction System of the Heart: Structure, Function and Clinical Implications, pages 163-181. Lea and Febiger, Philadelphia, 1976. [81] J. Zeng and Y . Rudy. Early afterdepolarizations in cardiac myocytes: mechanism and rate dependence. Biophysical Journal, 68:949-964, Mar. 1995. [82] D. P. Zipes, J . C. Bailey, and V . Elharrar. The Slow Inward Current and Cardiac Arrhythmias. Martinus Nijhoff Publishers, The Hague, 1980. Appendix A K E P L E R A N D M A R D E R ' S M O D E L OF S P O N T A N E O U S S E C O N D A R Y SPIKING Kepler and Marder's model of spontaneous secondary spiking [36] is given by + lHH{v,m,h,n) - zls dm ~dt dh ~dt dn ^ 1 - T 7 Q__ 9 — lappet), dx2 = am(v)(l - m) - 8m(v)m, = ah(v)(l - h) - 8h(v)h, = an(v)(l - n) - 8n(v)n, dt ft = ks[9(v-VT)-z] (A.la) (A.lb) (A.lc) (A.ld) (A.le) where IHH = 9Nam3h(v - vNa) + gKnA{v - vK) + gL(v - vL) (A-lf) and 0(v - VT) 1 v - V T > 0 , 0 v - VT < 0. The parameters are given by VNa = 55 mV, VK = -72 mV, VT, = -67.9 mV, gj\ra = 120 mmho/cm 2, §K = 20 mmho/cm 2, gi, — 0.3 mmho/cm 2 , C = 1 uF2, ks = 0.1 ms" 1 , VT = -30 mV. (A.lg) 188 Appendix A. KEPLER AND MARDER'S MODEL \ 189 The as and /3 s have the voltage dependencies: 1 0 ^ + 48N „ , , 1 ah(v) = 0.07exp( — - ) , fdh(v) = 20 " r " y - J exp(-^) + V a (v)- - 0 - 0 1 ^ + 45.7) ( r ) _ 0 1 2 5 c „ , ^ + 55-7 The equations for m, h, and n also can be written as dm rriooW-m am(v) dt rm(v) ' 0 0 ^ + m + d/i - h ah(v) 1 ~?r~ = /—\—•> where L B = — — n , N , 77, (t;) = — — , dt rh(v) °°K } ah(v) +/3h(vY ah(v) + dn ~di a , 1 a n ( - y ) + j3n(v)' Figure A . l : Voltage dependence of (a) the activation curves m ^ v ) , h^v), and n ^ t ; ) and (b) the time constants rm(v), r^v), and rn(v) in Kepler and Marder's model. A p p e n d i x B G U I D E T O B I F U R C A T I O N D I A G R A M S In this appendix, we briefly introduce some terminology from dynamical systems theory. We explain the concept of a bifurcation diagram and describe the main features seen in the bifurcation diagrams in this work. More information on dynamical systems theory and bifurcation diagrams can be found in [64]. The behaviour of a system is described by the time courses of a number of depen-dent variables called state variables, which describe the states of the system, such as the membrane potential, the intracellular C a 2 + concentration, and so forth. The ranges of possible values for these state variables are referred to as the state space of the system. The system of ordinary differential equations describing the behaviour of the state vari-ables may contain a number of parameters. The numerical values of these parameters influence the qualitative and quantitative behaviour of the system. For any given set of parameters, the system will exhibit a certain set of behaviours, such as steady states or limit cycles, either of which may be stable or unstable. An example of a stable limit cycle is the tonic firing mode in the Hodgkin-Huxley equations, while the resting potential is a stable steady state. Steady states also are called fixed points, equilibrium points, or singular points. A locally stable (attracting) fixed point is called a sink or, in two dimensions, a stable node or spiral. A locally unstable (repelling) fixed point is called a source or, in two dimensions, an unstable node or spiral. If the fixed point attracts in certain directions and repels in others, it is called a saddle. 190 Appendix B. GUIDE TO BIFURCATION DIAGRAMS 191 The set of all possible parameter values is called parameter space. As the set of values of the parameters moves through parameter space, the behaviour of the system changes. In many cases, the change may not be radical (i.e., a quantitative change only). For instance, the shape or amplitude of a limit cycle may change as we vary a certain parameter. In other cases, the behaviour may change qualitatively, and we say that a bifurcation has occurred. For instance, in the Hodgkin-Huxley equations, if we increase the amplitude of an applied current, at a certain amplitude, the fixed point will change from stable to unstable and a limit cycle will appear. We call the point in parameter space where such a qualitative change occurs the bifurcation point. A one-parameter bifurcation diagram is a diagram that shows the stability charac-teristics of the system, including stationary and oscillatory states (fixed points and limit cycles), over a range of one particular parameter, called the bifurcation parameter. Fig-ure B . l is an example of a typical one-parameter bifurcation diagram. In panel (a), the state variable v (membrane potential) is plotted on the y-axis and the bifurcation pa-rameter Is is plotted on the z-axis. A separate bifurcation diagram can be plotted for each state variable in the system. Thus, a second state variable z is plotted versus Is in panel (b). Panel (c) indicates the periods of the limit cycles at each value of Is. A l l of these bifurcation diagrams were produced using the numerical bifurcation pack-age AUT086 [13] via the X P P A U T interface [17]. This package computes curves or branches of fixed points and limit cycles that show how the fixed points and limit cycles change as the bifurcation parameter is varied. Stable fixed points (sinks) are indicated by solid lines, while unstable fixed points (sources or saddles) are indicated by dashed lines. The filled circles denote the maximum and minimum amplitudes of stable limit cycles, while the open circles refer to unstable limit cycles. Thus, Figure B . l indicates that when Is=l, the system of equations has a single stable fixed point. At Is=2, the system still has a stable fixed point, but also has two unstable Appendix B. GUIDE TO BIFURCATION DIAGRAMS 192 (a) 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 (c) T 1 1 1 1 1 1 1 HB SN( \ S N P V \ V \ \ \ \ H C . % o * 1 . 1 1 1 1 1 i 3 4 5 6 7 Figure B . l : Typical one-parameter bifurcation diagrams produced by AUT086 using Is as the bifurcation parameter. The projections of the fixed points and limit cycles onto v and z and the period T of the limit cycles are shown. Stable (unstable) fixed points are indicated by solid (dashed) lines. The maximum and minimum of the stable (unstable) limit cycles are indicated by filled (open) circles. (HC = homoclinic connection; HB = Hopf bifurcation; SN = saddle-node; SNP = saddle-node of periodics.) fixed points. At J s=6, the system now has two stable fixed points, one unstable fixed point, and an unstable limit cycle. The system behaves qualitatively differently for each of the parameter values considered. The transitions between different types of qualitative Appendix B. GUIDE TO BIFURCATION DIAGRAMS 193 behaviours occur at the bifurcation points, marked SN, SNP, H B , and HC in Figure B . l . The bifurcation point marked SN is called a saddle-node point or limit point. At a saddle-node point, two fixed points approach each other, collide, and then annihilate each other. In the prototypical example of this type of bifurcation, one fixed point is a saddle and the other is a node, hence the name saddle-node bifurcation. At the saddle-node point SN in Figure B . l , two unstable fixed points are created or destroyed as Is is increased or decreased, respectively. Another common bifurcation is called a Hopf bifurcation (HB). At a Hopf bifurcation, the stability of the fixed point changes and small amplitude oscillations arise. Thus, in Figure B . l , we see that as Is decreases past the HB point, the stable fixed point suddenly becomes unstable and small stable limit cycles arise. These limit cycles grow in amplitude. At the point marked SNP, the stable limit cycle coalesces with an unstable limit cycle and they both disappear. When two limit cycles coalesce and annihilate each other like this, it is called a saddle-node of periodics (SNP), a saddle-node bifurcation of cycles, or a fold. The unstable limit cycles arising from the SNP grow in amplitude as Is increases and then suddenly disappear at the bifurcation point HC. At this point, the unstable limit cycle touches the saddle point and becomes a homoclinic orbit (an orbit that begins and ends at the saddle point). This kind of bifurcation is called a homoclinic connection (HC), a homoclinic bifurcation, or a saddle-loop. We have seen that if we fix all but one of the parameters of a system, a bifurcation will occur at a particular value of the bifurcation parameter. A two-parameter bifurcation diagram allows us to see where this bifurcation occurs if two parameters are allowed to vary at once. The two-parameter bifurcation diagram in Figure B.2 shows how the Hopf bifurcation (HB) points move as two parameters, P i and P2, are varied. When P2=-70, Appendix B. GUIDE TO BIFURCATION DIAGRAMS 194 there are HB points for two different values of P x . As P2 increases, the positions of these H B points shifts with respect to P i until they coalesce and disappear near P2=-37. Pi Figure B.2: Typical two-parameter bifurcation diagram produced by AUT086, showing how the Hopf bifurcation points move in parameter space as parameters Pi and P2 are varied. Lastly, we should mention that if our system of equations describes only two variables, say x and y, we can represent their dynamics by a plot of x versus y called a phase portrait. In the phase portrait, we would plot the two curves describing dx/dt = 0 and dy/dt = 0, which are called the x and y nullclines. These curves separate regions of increasing and decreasing x and y, respectively. Their intersections describe the fixed points of the system, since dx/dt = dy/dt = 0 there. Finally, we should mention that in a three-dimensional system, the equations dx/dt = 0, dy/dt = 0, and dz/dt = 0 describe the nullsurfaces for x, y, and z. A p p e n d i x C T H E D I F R A N C E S C O - N O B L E E Q U A T I O N S The DiFrancesco-Noble (DN) equations [12] are a system of 16 ordinary differential equa-tions describing the electrical activity of a mammalian cardiac Purkinje fiber. The version listed here was taken from the Appendix of Varghese and Winslow [69]. C l The Equations -77 = ~7r~(IjK + IfNa + IK + IKI + ho + IbNa at Lim + lNaK + iNaCa + 1-Na. + IbCa + hiCa + hiR'), (C. la d y — = ay(v + vshift)(l - y) - Py(v + vshift)y, ( C l b d x — = ctx(v)(l - x) - fix(v)x, (C. lc ^ = a r ( u ) ( l - r ) ( C . l d i in — = am(v) (1 - m) - j3m(v)m, (C.le ^ = ah(v) (1 - h) - fa(v) h, (C. l f ^ = ad(v)(l-d)-/3d(v)d, (C.lg; d-£ = af(v)(l-f)-Pf(v)f, (C . lh ^ = ap(v)(l-p)-/3p(v)p, (C . l i ^ = a / 2 ( C a ? ) ( l - / 2 ) - ^ 2 ( C a 4 ) / 2 , (C.lj TT- = TTT;(IfNa + IbNa + 3IN<LK + SlNaCa + INO), (C.Ik it ViP 195 Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 196 {IfK + IK + IKI + Ito — 2INaK -f ISXK), ( C l l ) dt V{F dKc _ J _ ~df ~ VeF dC&i —1 dt ~ W~F (IfK + IK + IKI + Ito - 2INaK + IstK) - DK(KC - K0), (C.lm) (IbCa — 2lNaCa + hiCa + lup ~ Irei), (C.ln) (Iup ~ Itr), (C.lo) dt 2VupF d C^e/ 1 , . = 2 v ^ F { I t r - I r e l ) - (c-lp) The ionic currents in the above equations are defined in Table C l and the relevant reversal potentials in Table C.2. The equations for the gating variables (C.lb - C.lj) have been written in the form ^ = ax(l - x ) - Bxx (C.2) where ax and 3X are given in Table C.3. An alternative formulation for the gating equations is d x tC ("v~) %c , ^ where x^ = ax/(etx + 0X) and Tx = l/(ctx + 8X). The activation curve X Q Q and time constant TX are plotted for each of the nine gating variables in Figures C l to C.4. The parameter values listed in Table C.4 are considered to be the standard parameter set. The fixed parameters in the top part of the table were taken directly from the Appendix of Varghese and Winslow [69]. The extracellular ion concentrations were chosen to match Ferrier and Moe's experiments on canine cardiac Purkinje fibers [22, 23]. These can be adjusted to simulate experimental protocols. We also use a number of scaling factors (vshift, Aj, AfjAK) to adjust the behaviour of the model. In Table C.4, they are set to values corresponding to a pacing fiber under normal conditions. Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 197 Hyperpolarization activated K + current: Ifi< = A S K ^ k m G f K { v EK)y Hyperpolarization activated N a + current: IfNa = Af Kc+%mf GfNa(v ENa)v Time-dependent (delayed) K + current: IK = ^ ( K i - Kce-^RT^)x Time-dependent (background) K + current: Ihl — Ghl(Kc+kmK1)(1 + e(v-EK+W)/(RT/2F)) Transient outward K + current: it0 = 0.28(0.2 + , J : ; A 0 ( T ^ f e ) ( 1 _ e _ ^ + 1 o ) M e0.02v _ KcB-o.02vy Background N a + current: IbNa = GbNa(v ~ EN a) Fast N a + current: IN a. = GNa{v - Emh)m3h Background C a 2 + current: IbC a = GbCa{v — Eca) Slow inward C a 2 + current: IsiCa = ^ C a d f H j ^ i ^ k ^ C a ^ l ^ l ^ - C« 0 e - ( " - 5 ° ) / ( f i T / 2 J ? ) ) Slow inward K + current: _ IsiK = PcaKPcadff2(T^X^)(Kie^m -- Kce-(V-5°V(RTIF)) Sodium-potassium exchange pump current: iNaR - A N A R INaR max{Kc+kmK ) ( Na,+kmNa ) Sodium-calcium exchanger current: e-,v/(RT/F)Na3Cao_e-(l-t)v/(RT/F)Na3Cai iNaCa - KNaCa l + < % a C a ( / V a 3 C a ; + / V a ? C a 0 ) C a 2 + uptake from cytosol to SR uptake store: T 2ViFn„ 1 Cciup-C&up \ hp - T U PC M c~aup ) C a 2 + transfer from SR uptake store to release store: hr = ^ ^ ( C a ^ - Ca r e / ) C a 2 + release from SR release store to cytosol: Table C l : Currents in the DiFrancesco-Noble equations. Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 198 Reversal potential for IjNa and IbNa'- ENa F U i \ N a i ) Reversal potential for Tyva : Emh _ KT] (Nao+0.12Kc\ ~ F '"V Na,+0.12Ki ' K + reversal potential: EK-- = ¥HW C a 2 + reversal potential: Eca Table C.2: Reversal potentials in the DiFrancesco-Noble equations. ay(v) = 0.05e-°-o67(v+v^ft+4V ~~ 1_e-0-2(v + VshiJt+i2) . ( \ _ o.5e0 0 8 2 6 ^ + 5 0 ) a (,,\ _ 1.3e-0 0 6 C + 2 ° ) Ctx\y) — 1 ^ _ e 0 . 0 5 7 ( « + 5 0 ) Px\U) — 1_j_e_0.04(t> + 20) ar(v) = 0.033e- u/ 1 7 Pr(v) — 1 + e _ ( „ + io)/8 a ( v ) - 2 0 0 ( „ + 4 1 ) /3m(v) = 8000e-° - O 5 6 ^ + 6 6 ) ah(v) = 20e-°- 1 2 5 ^+ 7 5 ) Q (,,\ _ 2000 Hh\U) — 1 + 3 2 0 e - o . l ( v + 7 5 ) / \ 30(u+19) ad(v) = ^ ^ ( 7 + 1 9 ^ / 4 / \ 6.25(t;+34) af(v) = e(.+3«)74_i - 1 + e - ( * + 34)/4 / \ 0.625(^+34) /?p(^) — 1 + e _ ( „ + 3 4 ) / 4 a/ 2(Ca 8) = 5 Table C.3: Definitions of a and f3 for the gating variables in the DiFrancesco-Noble equations. Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 199 F i x e d Parameters Cm 0.075 nF d-Kmax 180.0 nA C QUp 5.0 m M ^NaKmax 125.0 nA DK 0.7 Pea 15.0 n A / m M d-NaCa 0.001 PcaK 0.01 GbCa 0.02 fiS 7 0.5 GbNa 0.18 0.025 sec GfK 3.0 [iS Ttr 2 sec GfNa 3.0 /iS 7~rel 0.050 sec GRI 920.0 fiS F 96485 C/mol GNa 750.0 nS R 8314.41 mJ/(mol K) kactA 0.0005 m M T 310.0 K kmCa 0.001 m M radius 0.05 mm kmf 45.0 m M length 2.0 mm kmf2 0.001 m M V Tr(radius2)(length) 1.0 m M Vecs 0.1 kmKi 210.0 m M ve vecsv kmNa 40.0 m M Vi (i-vecs)v h * ""mto 10.0 m M VUp 0.05V J^NaCa 0.02 nA Vrel 0.02V Adjustable Parameters Ca0 2.5 m M Vshift 0.0 mV Ko 4.0 m M 1.0 Na0 150.0 m M ANaK 1.0 Table C.4: Standard parameter set for the DiFrancesco-Noble equations. The parameters in the top portion of the table remain fixed throughout this work. The parameters in the bottom portion of the table may be adjusted to simulate experiments. Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 200 0 2 4 6 8 10 Ca,i(uM) Figure C l : Activation curve f2oo and time constant r/ 2 (sec) of f2 as functions of Ca; in the DN equations (standard parameters). Figure C.2: Activation curves of gating variables in the DN equations (standard parameters). Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 201 -100 -80 -60 -40 -20 0 20 40 v (mV) Figure C.3: Time constants of slow gating variables in the DN equations (standard parameters). 0.05 v (mV) Figure C.4: Time constants of faster gating variables in the DN equations (standard parame-ters). Appendix C. TEE DIFRANCESCO-NOBLE EQUATIONS 202 C .2 Simulations The D N equations exhibit tonic firing when the standard parameters of Table C.4 are used. To make the system quiescent, we can either set vshijt=20 or A/=0.2. The resting values of all of the variables in the two quiescent versions of the D N equations are given in Table C.2. A n initial condition is also given for the pacemaking version of the equations. The following four pages of figures show the sequence of events during tonic firing. The full D N equations were used with standard parameters. A l l 16 variables and 15 ionic currents are shown. Quiescent (vshift—20) Quiescent (Aj=0.2) Pacing V -83.6437 mV -87.0479 -79.6034 mV y 0.424372 0.994644 0.620223 X 0.624962e-2 0.447537e-2 0.894917e-2 r 0.999267 0.999608 0.994572 m 0.562857e-2 0.356644e-2 0.959848e-2 h 0.957275 0.979665 0.907458 d 0.239146e-6 0.102153e-6 0.652591e-6 f 0.999999 1.000000 0.999998 P 0.962639 0.975924 0.960635 h 0.999999 1.000000 0.999998 Ncn 8.48453 m M 7.18945 9.08717 m M Kt 143.592 m M 144.817 144.928 m M Kc 4.00000 m M 4.00000 3.99191 m M Ca,i 0.388106e-4 m M 0.246696e-4 0.419138e-4 m M Cawp 1.76825 m M 2.26918 1.58388 m M Ca r e / 1.66791 m M 2.21529 0.713930 m M Table C.5: Initial conditions for the quiescent and pacemaking versions of the D N equa-tions (standard parameters except for the indicated changes to vshift and Af to make the system quiescent.) Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time (sec) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time (sec) Figure C.5: Simulation of the full DN equations (standard parameters). Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 204 O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) time (sec) d 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) time (sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) time (sec) Figure C.5: Simulation of the full DN equations (standard parameters). Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time (sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time (sec) Figure C.5: Simulation of the full DN equations (standard parameters). Appendix C. THE DIFRANCESCO-NOBLE EQUATIONS INaK (nA) 20 18 F~ 16 14 12 10 8 6 4 h 2 O n l n _ l l i i i _ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) IsiK (nA) 50 45 40 35 30 25 20 15 10 5 0 _i 1 L . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 time (sec) IsiCa (nA) 0 -20 -40 -60 h -80 l--100 -120 h -140 n 1 1 1 1 1 r -J I I I L_ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) 15 10 5 O iNaCa "5 (nA) " 1 0 -15 -20 i 1 1 1 1 1 1 r _ l l i i i i _ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 time (sec) lup (nA) 450 400 I-350 |-300 250 h 200 150 (-100 50 O 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) Figure C.5: Simulation of the full DN equations (standard parameters). A p p e n d i x D B I F U R C A T I O N D I A G R A M S F O R T H E P A C E M A K I N G D I F R A N C E S C O - N O B L E E Q U A T I O N S F O R V A R I O U S C a 0 A N D ANaK The following bifurcation diagrams are for the pacemaking version of the D N equations listed in Appendix C (standard parameters) assuming K, = 145 m M , K C = K 0 , and p=l. The bifurcation diagrams show the voltage v, the intracellular C a 2 + concentration Ca;, and the period T as functions of the bifurcation parameter Na,-. The approximate value of Na; during tonic firing (Na;(tf)) is indicated on each bifurcation diagram. The bifurcation diagram for the pacemaking D N equations under normal conditions (^4jVaA'=l-0, Ca 0=2.5 mM) is given in Figure 4.8. The bifurcation diagrams in this ap-pendix are computed for various extracellular C a 2 + concentrations or degrees of digitalis poisoning as follows: • Figure D . l Weak digitalis poisoning: AyvaA"=0.85, Ca 0=2.5 m M • Figure D.2 Intermediate Ca 0 : A;vaA"=l-0, Ca 0=3.5 m M • Figure D.3 High Ca 0 : ANaK=l.O, Cao=8.0 m M 207 Appendix D. BIFURCATION DIAGRAMS FOR PACEMAKING DN EQNS 208 6 8 lO 12 14 16 18 20 22 24 Na t (mM) Na;(tf)=11.0 Figure D . l : The pacemaking version of the DN equations with Ca0=2.5 mM and AN AK—0-85. Na; naturally tends towards a pseudo-steady state value of Na;(tf)=11.0 mM so that the system exhibits tonic firing in the transition region between normal and calcium-driven tonic firing. Appendix D. BIFURCATION DIAGRAMS FOR PACEMAKING DN EQNS 209 v (mV) Ca; (/*M) r [sec) lO 12 14 16 18 20 22 24 Na; (mM) Na;(tf)=9.3 Figure D.2: The pacemaking version of the DN equations with Ca0=3.5 mM and Ajva^- = 1.0. Na; naturally tends towards a pseudo-steady state value of Na;(tf )=9.3 mM so that the system exhibits tonic firing in the transition region between normal and calcium-driven tonic firing. Appendix D. BIFURCATION DIAGRAMS FOR PACEMAKING DN EQNS 40 U Na; (mM) Na; (mM) 6 8 l(p 12 1-4 16 18 20 22 24 ! Na; (mM) Na;(tf)=10.2 Figure D.3: The pacemaking version of the DN equations with Cao=8.0 mM and A/v aA' = l -Na; naturally tends towards a pseudo-steady state value of Naj(tf)=10.2 mM so that the syste: exhibits calcium-driven tonic firing. A p p e n d i x E B I F U R C A T I O N D I A G R A M S F O R T H E Q U I E S C E N T D I F R A N C E S C O - N O B L E E Q U A T I O N S F O R V A R I O U S C a c A N D A The following bifurcation diagrams are for the quiescent version of the D N equations listed in Appendix C (standard parameters; vskift=20 mV) assuming K,-=145 m M , K C = K 0 , and p=l. The bifurcation diagrams show the voltage v, the intracellular C a 2 + concentration Ca2-, and the period T as functions of the bifurcation parameter Na,-. The approximate value of Na,- at steady state (Na,(ss)) or during calcium-driven tonic firing (Na;(tf)) is indicated on each bifurcation diagram. The bifurcation diagram for the quiescent D N equations under normal conditions (A/vaA-=1.0, Ca 0=2.5 mM) is given in Figure 4.17. Under normal conditions, the D N equations do not exhibit triggered activity in response to short trains of stimuli. The bifurcation diagrams in this appendix are computed for various extracellular C a 2 + con-centrations or degrees of digitalis poisoning as follows: • Figure E . l Intermediate Ca 0 : A/va/<=l-0, Ca0=4.5 m M • Figure E.2 High Ca 0 : ANaK=l.O, Cao=6.0 m M • Figure E.3 Very high Ca 0 : ANaK=1.0, Cao=7.0 m M • Figure E.4 Digitalis poisoning: AjvaA"=0.8, Ca0=2.5 m M • Figure E.5 Digitalis poisoning and intermediate Ca Q : AiVaA'=0.8, Ca 0=4.5 m M 211 Appendix E. BIFURCATION DIAGRAMS FOR QUIESCENT DN EQNS 212 Ca2-(^M) Na;(ss)=8.5 Na; (mM) Figure E . l : The quiescent version of the DN equations exhibits a small amount of triggered activity in response to short trains of stimuli for Ca0=4.5 mM and ANaK=l-0. Appendix E. BIFURCATION DIAGRAMS FOR QUIESCENT DN EQNS 213 30 30 Na8-(ss)=8.5 Na; (mM) Figure E.2: The quiescent version of the DN equations exhibits triggered activity that lasts several minutes in response to short trains of stimuli for Cao=6.0 mM and Ajva/f=1.0. Appendix E. BIFURCATION DIAGRAMS FOR Q UIESCENT DN EQNS 214 3 O 3 0 Na,-(tf,ss)=8.2,8.5 2 0 2 5 3 0 Na; (mM) Figure E.3: The quiescent version of the DN equations is bistable for Cao=7.0 mM and AjVaA'=l-0- It can exhibit a stable steady state or stable calcium-driven tonic firing. Appendix E. BIFURCATION DIAGRAMS FOR QUIESCENT DN EQNS 215 Figure E.4: The quiescent version of the DN equations exhibits a small amount of triggered activity in response to short trains of stimuli for Ca0=2.5 mM and A/\raA'=0.8. Appendix E. BIFURCATION DIAGRAMS FOR QUIESCENT DN EQNS 216 4 0 I I Na4- (mM) i Nat- (mM) i ! Na; (mM) Na t(tf)=10.1 Figure E.5: The "quiescent" version of the DN equations exhibits calcium-driven tonic firing for Ca0=4.5 mM and A/vaA-=0.8. 

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