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A study of phase resetting, mutual entrainment, and modified ventricular parasystole using a model of… Foster, Toni 1992

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A STUDY OF PHASE RESETTING, MUTUAL ENTRAINMENT, ANDMODIFIED VENTRICULAR PARASYSTOLE USING A MODEL OFCOUPLED HEART CELLSByToni FosterB. Sc. (Mathematics/Computer Science) University of VictoriaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MATHEMATICS, INSTITUTE OF APPLIED MATHEMATICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1992© Toni Foster, 1992In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of Mathematics, Institute of Applied MathematicsThe University of British ColumbiaVancouver, CanadaV6T 1Z2Date:/larch c23 AbstractSeveral aspects of cardiac electrophysiology including phase-resetting and entrainmentare investigated through the dynamic interactions of coupled mathematical models ofcardiac cells. Unlike previous studies of cell interaction which have used simplifiedmodels of a general cardiac cell or oscillators characterized only by their phase transitioncurves the present work incorporates the more physiologically realistic cell models ofthe Hodgkin-Huxley type. Furthermore, this caricature of the heart includes modelsrepresenting cells from each of the major components of the heart's electrical conductionsystem, the SA node, AV node, Purkinje fibre network, and ventricular myocardium.Since there does not exist a model for an AV node cell, a new model is created bymodifying the SA node cell equations.Because spontaneously firing cardiac cells exhibit a wide range of oscillation frequen-cies, certain physiologically based parameters in the model equations were altered toproduce a similar range of intrinsic frequencies in the model cells. These model cellsare coupled by assuming that direction-dependent and purely resistive coupling currentsflow between them. Furthermore, these coupling currents incorporate a time delay rep-resenting the intercell impulse propagation time which is significant between differentregions of the heart. Each of the conduction components, the SA and AV nodes, Purkinjefibre network, and ventricular myocardium, can be modelled with either one or severalof their model cells. These model components can then be coupled with appropriateconductances and propagation time delays to model the spread of waves of excitationfrom one region of the heart to another.When cells interact, their intrinsic cycles may be perturbed. These alterations oriiphase-shifts which determine the ultimate rhythm of coupled cells are studied for variouscombinations of the model cells and different strengths of interaction. In each case resultsare plotted in a phase response curve whose shape is found to depend on such thingsas coupling conductance and cell frequency. Phase-resetting is a prerequisite to theentrainment of cells in which there are m cycles of one cell for every n cycles of another.The phenomenon of entrainment is studied as part of the behavior of a healthy heartwhere it provides the mechanism by which cells synchronize to produce a single waveof excitation for each heartbeat. Mutual entrainment is also studied in a commonlyoccurring pathological situation, modulated ventricular parasystole. It is demonstratedthat the particular pattern of m:n entrainment is dependent on the differences in theintrinsic frequencies of the cells and on the strength of the interaction. Results obtainedusing different types of model cells located in different regions of the heart are comparedand contrasted to those of previous studies of adjacent cells which are identical.iiiTable of ContentsAbstract^ iiList of Tables viiList of Figures^ viiiAcknowledgements xi1 Introduction 11.1 Cardiac Physiology ^ 31.1.1^Action Potentials 31.1.2^The Conduction Path in a Healthy Heart ^ 41.1.3^Conduction Time in a Healthy Heart 61.2 Modelling the Heart as a System of Coupled Nonlinear Oscillators 71.3 The Present Work^ 82 The Model 122.1 Methods ^ 122.2 Modelling Individual Cells ^ 132.2.1^The SA Node Cell 162.2.2^The AV Node Cell ^ 162.2.3^The Purkinje Fibre Cell 172.2.4^The Ventricular Myocardial Cell ^ 172.2.5^Control of Pacemaker Periodicity 18iv2.32.42.52.6Modelling Adjacent Cell Interaction ^Modelling Nonadjacent Cell Interaction Range of Coupling Conductances ^Propagation Time Delays^252629303 Phasic Interactions of Pacemaker Cells 313.1 Phase Resetting ^ 313.2 Methods 333.3 An SA Node Cell and a Purkinje Fibre Cell ^ 353.4 An SA Node Cell and an AV Node Cell 403.5 Fast and Slow Cells ^ 433.6 Phase Response Curves and Zones of Entrainment ^ 443.7 Pulsed versus Continuous Coupling ^ 464 Behavior of a Healthy Heart 494.1 Activation of a Nonspontaneous Cell ^ 494.2 The Production of a Heartbeat 545 Modulated Ventricular Parasystole 615.1 Cardiac Arrhythmias and Ectopic Pacemakers ^ 615.2 Modulated Ventricular Parasystole ^ 625.3 The Lead Cell ^ 635.4 TP/T8 = 3 665.4.1^Regions of Synchronous and Asynchronous Behavior ^ 665.4.2^1:1 s Entrainment^ 705.4.3^1:1 p Entrainment 705.5 TpIT, = 2 76v5.5.1 Regions of Synchronous and Asynchronous Behavior ^ 775.6 Tp/T, = 0.75 ^  795.6.1 Regions of Synchronous and Asynchronous Behavior ^ 795.7 The Greater Influence of the Purkinje Fibre Cell ^ 795.8 The Lead Cell and the Entrained Oscillation Period  815.9 The Effect of Changes in Tp and T8 ^  826 Discussion^ 836.1 Summary of the Heart Model ^  836.2 Discussion of Results ^  846.3 Suggestions for Future Research ^  88Bibliography^ 90Appendix 95A The Cell Models^ 95A.1 The Sinus Node Cell ^  96A.2 The AV Node Cell  98A.3 The Purkinje Fibre Cell ^  98A.4 The Ventricular Myocardial Cell  102viList of Tables2.1 Parameter values and resulting frequencies for model SA node cells. . . 202.2 Parameter values and resulting frequencies for model AV node cells. . . 222.3 Parameter values and resulting frequencies for model Purkinje fibre cells. 254.4 Coupling conductance and delay values. ^ 58viiList of Figures1.1 Diagrammatic representation of a cardiac cell transmembrane action po-tential.  ^41.2 Cardiac cells and their action potentials.  ^52.1 Model SA node cells^  212.2 Model AV node cells  232.3 Model Purkinje fibre cells. ^  242.4 Schematic diagram depicting the flow of an action potential between twoadjacent cells across a gap junction. ^  262.5 Schematic diagram depicting the flow of an action potential from cell i tocell j.  ^273.1 Terminology used to describe the influence of an action potential of cell ion cell j ^  343.2 Phase response curves summarizing the effects of a single action potentialfrom p on the cycle length of s for two values of gp,,. ^ 373.3 Phase response curves summarizing the effects of a single action potentialfrom s on the cycle length of p for two values of gs ,p. ^ 393.4 Phase response curve summarizing the effects of a single action potentialfrom s on the cycle length of a. ^  413.5 Phase response curve summarizing the effects of a single action potentialfrom a on the cycle length of s. ^  42viii3.6 Phase response curves summarizing the effects of a single action potentialfrom s on two different Purkinje fibre cells. ^  453.7 Phase response curves summarizing the effects of p on s and s on p underconditions of continuous coupling. ^  474.1 Schematic representation of the interaction of a Purkinje fibre cell and anadjacent ventricular myocardial cell. ^  504.2 Transmembrane potentials of a Purkinje fibre cell and a ventricular my-ocardial cell for increasing values of the coupling conductance g^ 524.3 Period of oscillation of the Purkinje fibre cell as a function of couplingconductance g. ^  544.4 Schematic representation of the model used to simulate the production ofa heartbeat^  554.5 Electrical activity of SA node, AV node, Purkinje fibre, and ventricularmuscle cells during the simulation of a heartbeat. ^  574.6 Peak action potential times of the AV node, Purkinje fibre, and ventricularmyocardial cells relative to the peak action potential times of the SA nodeduring a simulation of the production of a heartbeart. ^ 595.1 Schematic diagram of the model used to simulate modulated ventricularparasystole^  635.2 Action potentials of cells i and j at 1:1 entrainment.^ 645.3 Regions of entrainment for Tp/T. — 3. ^  685.4 Changes in cycle lengths Tp and T, for gp,, = 0.001 mS/cm 2 and increasingg8,p •  ^715.5 Action potentials of cells s and p for gp,, = 0.001 mS/cm 2 and increasing98,p ^  72ix5.6 Changes in cycle lengths Tp and T8 for gs ,p = 0.04 mS/cm2 and increasinggp,8• 745.7 Action potentials of cells s and p for gs ,p = 0.04 mS/cm2 and increasinggp ,s .  ^755.8 ECG recording showing 4:3 entrainment ^  765.9 Regions of entrainment for TpIT, = 2.  78^5.10 Regions of entrainment for TpIT, = 0.75    80xAcknowledgementsThank God it's over! Thanks also to Dr. Robert Miura for his indispensible guidancein the preparation of this thesis and Drs. Leah Edelstein-Keshet and Brian Seymour forgiving it the final OK. Thanks to Rob for being here to share this whole Master's thingwith me and for being such an entirely excellent office-mate and friend. I would also liketo acknowledge the much needed financial assistance of the Bank of Foster and NSERC.Finally, special thanks to Mark for his support and encouragement and for continuallyreminding me that life is too short not to have fun.xiChapter 1IntroductionThe heart's function is to provide the energy necessary to circulate blood through thecardiovascular system. Electrical excitations originating from specialized regions spreadthroughout the heart causing atrial, and then ventricular, muscle contractions. In hu-mans, this process takes place continuously at an average rate of 60 to 100 times perminute for an entire lifetime. It is not surprising that, despite decades of research, manyaspects of the physiology of such a sophisticated organ are still not understood. Forexample, the mechanism behind one of the most fundamental properties of cardiac cells,their ability to synchronize to a common frequency, enabling the heart to beat in aregular manner, is not known [30,39]. Fundamental aspects of cardiac pathophysiologyalso are poorly understood. Although, it is known that cardiac arrhythmias (irregularrhythms) are caused by abnormalities in either the initiation or the conduction of a waveof excitation, the causes of the abnormalities themselves are, in many cases, not known.Abnormal initiation refers to the formation of waves at either abnormal locations or atrates outside the usual range. Abnormal conduction of a wave of excitation refers to ei-ther a change in the conduction velocity or in the path taken over the cardiac tissue. Theresult of these malfunctions is sometimes an arrhythmia known as ventricular fibrillationin which the ventricles can contract at rates as high as 500 beats per minute. Clearly, aknowledge of the mechanisms which trigger the onset of such life-threatening arrhyth-mias is vital to their prevention and treatment. The lack of a complete understandingof the physiology of the heart, combined with its essential role in human health, has1Chapter 1. Introduction^ 2provided the motivation for ongoing research. Much of this research has been directedat both the mechanical and electrical aspects of the heart's functioning.The mechanical properties of the myocardium (heart muscle) have been studied by,amongst others, Horowitz [19] using a microstructural model based on the structuralarrangement of cardiac muscle cells and properties of the complex connective tissue.Such studies have been aimed at obtaining a suitable mathematical description of thetime-varying properties of active muscle tissue. With such a mathematical description,finite element models [16,20,23,33,36] have been employed to describe regional mechanicsof the heart wall by continuous distributions of stress and strain. This approach wasused by McCulloch [29] who also compared his results with experimental measurements.The electrical properties of cardiac tissue, which are the focus of this thesis, have,for the most part, been studied independently of the mechanical properties. The heart'selectrical activity is caused by the opening and closing of ionic channels in the cardiaccell membrane. The resulting flow of electric current between adjacent cells has beenmodelled theoretically using systems of nonlinear ordinary differential equations. Modelsbased on these ideas will be employed in the present study. The resulting systemsof equations can be very complex and usually cannot be solved analytically. As analternative to this approach, low dimensional finite difference equations have been usedto model the responses of cardiac tissue to various inputs such as periodic stimulation[9,11,12,14,18,21,26,37]. In addition, cellular automata (finite state) models of cardiacconduction have been applied to the study of the patterns of electrical activity in theheart ([10], chapter 18).Chapter 1. Introduction^ 31.1 Cardiac Physiology1.1.1 Action PotentialsAll cells have an electrical potential (voltage) across their membranes. These membranepotentials exist because the ionic concentrations inside the cells are different from thosein the interstitial fluid outside the cells. The resulting concentration gradients cause theflow of ions across the semipermeable membranes and generate electrical gradients.Abrupt changes in the permeability of the membrane to specific ions, mainly sodiumand potassium, and to a lesser extent calcium, cause an abrupt rise (depolarization) inmembrane potential. The muscle contractions which cause the heart to pump bloodare triggered by these sudden voltage changes called action potentials in cardiac musclecells.All cardiac muscle cells are excitable [6]. That is, they can respond to a given stimulusby the production of an action potential. Some cells of the myocardium, however,have the ability to spontaneously depolarize and, thereby, initiate action potentials.These cells, known as pacemakers or spontaneous cells, are located in various regionsthroughout the heart. For the remainder of the myocardial cells, which do not possessthis inherent rhythmicity (nonpacemaker or nonspontaneous cells), an action potentialis generated only by external stimulation.Although the action potentials of different excitable cardiac cells are not identical,they share a common qualitative time sequence characterized by some or all of the fol-lowing five separate phases (Figure 1.1). Phase 4 (as it is normally denoted), the restingphase, is the transmembrane potential recorded between impulses. For a nonsponta-neous cell this potential will be constant; however, for a pacemaker cell this phase ischaracterized by a slow spontaneous depolarization. When either this depolarizationor an external stimulus causes the membrane potential to reach a critical value, called20threshold potentialE:EiOEE-100Chapter 1. Introduction^ 4resting potential phase^action potential phase^A^Figure 1.1: Diagrammatic representation of a cardiac cell transmembrane action poten-tial. Numbers indicate phases. The letters, D and R, indicate regions of membranedepolarization and repolarization, respectively.the threshold potential, rapid depolarization (phase 0) occurs. This sharp upstroke isfollowed by an early rapid repolarization or decrease in membrane potential (phase 1),a prolonged phase of slow repolarization (phase 2) constituting a plateau, and a finalphase of rapid repolarization (phase 3) to the resting level.1.1.2 The Conduction Path in a Healthy HeartAction potentials propagate within the heart by crossing the membranes between ad-jacent cells. This cell-to-cell flow of electric current is the means by which a wave ofexcitation is conducted throughout the heart. In a healthy heart there is a specializednetwork of cardiac cells specifically adapted to generate and conduct the wave of exci-tation for each heartbeat. The major components of this network are: the sinoatrial(SA or sinus) node, the atrioventricular (AV) node, the Purkinje fibre system, and theventricular myocardium (muscle cells). The different membrane electrical properties ofSA node o—100iamturiscal le Batrial^cmuscle IAV node D.__Purkinie fiber Eventricular muscleIFventricular muscle1G0.2 sChapter 1. Introduction^ 5Figure 1.2: Cardiac cells and their action potentials (taken from [17], p.22).the cells in each of these components are reflected in the variations amongst their actionpotentials (Figure 1.2).The SA node, located in the wall of the upper right atrium (Figure 1.2), is a groupof pacemaker cells which, among all cells in a healthy heart, exhibit the most rapidspontaneous depolarization during the resting phase [4,6,38]. Consequently, SA nodecells reach their threshold and generate action potentials at a rate faster than othercardiac cells which then become entrained to the SA node rhythm. For this reason, theSA node is sometimes referred to as the heart's primary pacemaker.The excitation initiated at the SA node spreads radially throughout the atria. Atthe base of the atrium the wave of excitation encounters a small mass of specializedcells called the AV node (Figure 1.2). The conduction velocity of the excitation isChapter 1. Introduction^ 6slowed through the AV node, which provides the only contiguous bridge of cardiactissue crossing the cartilaginous structure that separates the atria from the ventricles.There is a substantial delay before the wave of excitation emerges at the base of theAV node and enters the Purkinje fibre system. The Purkinje fibres are specializedlarge-diameter conducting fibres spreading throughout the ventricles in finer and finerbranches which terminate on ordinary ventricular muscle (myocardial) cells. Conductionof the excitation through the Purkinje fibre network is extremely rapid causing the leftand right ventricles to contract nearly in unison.1.1.3 Conduction Time in a Healthy HeartThe temporal displacements of the action potentials in Figure 1.2 indicates the timerequired for a wave of excitation to propagate throughout the heart. Approximately 80-120 msec are required for atrial depolarization and a further 50 msec for passage throughthe AV node. Thus, from the generation of the first action potential at the SA node tothe emergence of the wave of excitation at the lower portion of the AV node into thePurkinje fibre network requires an average of about 160 msec. The spread of the actionpotential over the ventricles requires an average of 60-100 msec and the total durationof the activation process of the entire heart has an average value of 250 msec [6). Underpathological conditions these conduction times between different regions of the heart canvary considerably. For example, a wave of excitation may require more than 200 msecto pass through the AV node and, in some situations such as when there is damagedtissue, the excitation may not be conducted at all.Chapter 1. Introduction^ 71.2 Modelling the Heart as a System of Coupled Nonlinear OscillatorsPacemaker (spontaneous) cells from different regions of the heart may have widely vary-ing intrinsic oscillation rates [4,6,17]. Even cells from a small localized region are unlikelyto have identical intrinsic frequencies [30]. Experimentally, it is known that when cellswith different frequencies are coupled their cycle lengths may be altered [9,30,39]. Incertain cases, the cells may become entrained so that there are m cycles of one for everyn cycles of another. Synchronization, a particular case of this m:n entrainment in whichthe coupled cells oscillate with a common frequency, may also occur. For example, thepacemaker cells of the SA node synchronize to produce a single wave of excitation foreach heartbeat [30]. The mechanism by which coupled cells become entrained is notcompletely understood; however, it is thought that the electric currents caused by theflow of ions between adjacent cells may mediate the process [5,30,39].Theoretical work on the entrainment of pacemaker cells was investigated as earlyas 1928 when van der Pol and van der Mark [35] simulated some typical phenomena(including synchronization) of the heartbeat using three coupled electronic relaxationoscillators. The analogy between the behavior of mathematical forced oscillators andthe heartbeat led to the idea of considering the heart as a system of coupled nonlinearoscillators where the periodic stimuli are provided by one or more pacemaker cells.Since that time, many other studies [5,9,18,30,34,39] have also used coupled oscillatorsto model the entrainment of cardiac cells.In 1962 Denis Noble modified Hodgkin and Huxley's equations describing the trans-membrane potentials of squid giant axons to model the electrical activity of Purkinjefibre cells [31]. Since then, new models for the Purkinje fibre cell, as well as models forother cardiac cells, have been developed. These Hodgkin-Huxley-type models involvevoltage- and time-dependent conductances in parallel with a capacitance and includeChapter 1. Introduction^ 8as many as nine ionic channels. These models cannot be solved analytically; therefore,their use in modelling entrainment phenomena requires extensive numerical computa-tions. For this reason, many previous studies of the propagation of the cardiac impulsethrough systems of coupled cells have chosen not to use these models but rather to usefinite difference equations and oscillators characterized only by their phase transitioncurves [9,11,12,14,18,21,26,37]. Others have used simplified Hodgkin-Huxley-type mod-els of the cardiac action potential to model the behavior of a general heart cell [5,34,39].Furthermore, many of those who have used the more physiologically realistic Hodgkin-Huxley-type models have chosen a specific type of cell (for example, an SA node cell[30]) and studied interactions among cells of only that particular type. In some of thesestudies, the propagation of action potentials from cell to cell is modelled by invokinga purely resistive coupling current between adjacent cells. Since even a small area ofcardiac tissue is comprised of hundreds of cells, solving complex Hodgkin-Huxley-typeoscillators numerically would be far too time consuming and costly; therefore, this typeof adjacent-cell-coupling has meant that studies of the propagation of the cardiac actionpotential have been restricted to small localized regions of the heart.1.3 The Present WorkIn this thesis, a simple model of the entire heart is created and used to study phenomenaobserved in cardiac electrophysiology. This study is focussed on those aspects due toimpulse initiation and conduction, since it is abnormalities in these areas which arethe cause of cardiac arrhythmias. In order to limit the number of cells and yet retaina physiologically realistic model, only a small number of cells representing the majorcomponents of the heart's conduction system are included. The models for these cellsof the SA node, AV node, Purkinje fibre system, and ventricular myocardium are of theChapter 1. Introduction^ 9Hodgkin-Huxley type.The intrinsic (free-running) oscillation frequency of a particular type of spontaneouscardiac cell varies widely under different physiological conditions. For example, in ahealthy heart, cells of the AV node oscillate between 45 and 60 cycles/min, whereas ina particular pathological state called AV junctional tachycardia, they can fire at morethan 250 beats/min [4,6]. In order to model a wide range of cardiac phenomena, a meansof altering the intrinsic frequencies of the pacemaker cells of the model (this excludesthe ventricular myocardial cell which is nonspontaneous [1,6]) was devised. The model,which is presented in detail in Chapter 2, allows the coupling of any number of eachof the four types of cells where the intrinsic frequencies of the pacemaker cells can bespecified in some range.As in previous studies, coupling in the model is achieved through the addition ofpurely resistive coupling currents which flow between interacting cells with the strengthof the interaction controlled by the choice of an appropriate value of the coupling con-ductance. The difference between this new model and those of previous studies is thathere the interacting model cells are not necessarily physically adjacent and may, in fact,be from completely different regions of the heart. A real excitation travelling from oneregion of the heart to another will likely take a path over the tissue which is differentfrom the path taken by an excitation travelling in the reverse direction. Consequently,the coupling currents flowing between the model cells also must be different in eachdirection. In addition, the propagation times of the excitation between various regionsof the heart are significant (Section 1.1.3); therefore, an excitation generated at a cellfrom one region does not immediately affect the activity of a cell from another region.These latencies are incorporated into the model by means of appropriately chosen delaysadded to the coupling currents. In previous studies of adjacent cells these propagationtime delays were not required since an action potential generated at one cell affects itsChapter 1. Introduction^ 10neighbouring cells almost immediately simply by crossing the intercell membranes.When two cells of different intrinsic oscillation periods are coupled, the action po-tentials initiated at one may cause subthreshold responses in the other. These inducedresponses in a cell can either accelerate or delay its subsequent discharge compared toits unperturbed oscillation. This phase shifting causes changes in the basic rhythms ofthe cells and, in a real heart, is the cause of both the normal healthy periodic heart-beat as well as the abnormal rhythms (arrhythmias) observed in pathological situations.Chapter 3, through the use of phase response curves (PRCs), presents an investigationof the phase shifts which determine the ultimate rhythm of coupled cells. PRCs areobtained for various pairs of coupled cells of different intrinsic periods and for differentstrengths of interaction. The effects of the coupling conductance and the type of cellsbeing coupled on the shape of PRCs are studied. The magnitude of the phase shiftsinduced in one cell by another determine whether or not the cells may be synchronizedto a common frequency. Previous studies have indicated that a slower cell (one with alonger period of oscillation) may be entrained to a faster cell. This is also demonstratedwith the present model. In addition, the PRCs obtained with the present model indicatethat a fast cell may become synchronized to a slower cell.In Chapter 4, cells are coupled to demonstrate that the present model can simulatetwo of the most basic functions of a healthy heart, namely the activation of a nonspon-taneous muscle cell and the production of a heartbeat. Each cardiac impulse spreadsthroughout the ventricles by activating nonspontaneous muscle cells causing their con-traction. Questions such as: (1) "What is the effect of the coupling conductance onthe level of activation of the muscle cell?" and, (2) "How does the coupling affect theintrinsic period of the cell providing the stimulus?" are investigated. The second basicfunction of a healthy heart is the production of a heartbeat. The spread of waves ofexcitation from the SA node throughout the rest of the heart is modelled to determineChapter 1. Introduction^ 11whether, as in a real heart, each model cell along the conduction path will be excited inturn and become entrained to the SA node rhythm.Modulated ventricular parasystole, a particular pathological situation in which im-pulses are initiated at an abnormal (nonsinus) location is studied in Chapter 5 usinga simple two-cell model. Computations are repeated for cells of different intrinsic fre-quencies and, in each case, ranges of the coupling conductances at which the cells beatsynchronously are determined. Various patterns of m:n entrainment, in which thereare m beats of one cell to every n of the other, are demonstrated. These patterns ofentrainment are also readily observable in clinical electrocardiogram (ECG) recordings.Questions such as: (1) "How does the common period of two cells exhibiting 1:1 en-trainment relate to their intrinsic periods?", (2) "How do the intrinsic periods affect thestrength of interaction required to obtain the synchronization of the cells?" and, (3)"For equal values of the direction dependent coupling conductances, do different cellsexert equal influences on each other?" are investigated.Finally, a discussion of results and suggestions for further research are given in Chap-ter 6.Chapter 2The Model2.1 MethodsThe action potentials of the cells which comprise the major components of the heart'sconduction system, the SA and AV nodes, Purkinje fibre network, and ventricular my-ocardium, were simulated using sets of differential equations of the Hodgkin-Huxleytype. Existing models were available for all but the AV node cell, for which a new setof equations was developed. These individual cells were coupled by means of resistivecoupling currents which model the flow of ionic currents across intercell membranes. Thestrength of the cell interaction is determined by a user-defined value of the membraneconductance. For cells which are separated by some distance, the propagation timefor a wave of excitation travelling between them was incorporated into the model by adelay in the resistive coupling terms. Because two coupled cells may not exert equalinfluences on each other, the coupling conductances and delays are direction dependent.Each of the four components (SA node, AV node, Purkinje fibre network, and ventricu-lar myocardium) can be modelled by one model cell, or a few model cells coupled withappropriate conductances and delays. These model components can, in turn, be coupledto model the propagation of waves of excitation from one region of the heart to another.The mathematical models were programmed in FORTRAN and run on an IBM3090/150S mainframe with Vector Facility using the routine LSODE in the ODEPACKpackage (Hindmarsh 1980) to perform the integration of the stiff systems. The solutions12Chapter 2. The Model^ 13of the sets of differential equations showed constant rhythms after approximately 3 sec-onds; therefore, the solutions at 3 seconds were used as initial values in all computations.The model allows the user to specify, not only the length of time coupled cells will in-teract, but also the exact time within a computation the interaction will begin and end.The values of all dynamic variables can be stored upon termination of a computation sothat it can be resumed at a later time. Results of all computations were stored on diskand later plotted using the Tell-a-Graf graphics package.2.2 Modelling Individual CellsThe electrical activities of individual cells have been modelled using Hodgkin-Huxley-type models. These models are based on the fact that the changes in membrane potentialresponsible for initiating muscle contraction are primarily caused by the flow of ioniccurrents through individual channels in the cell membrane. Experimentally, this flowof ions can be measured through a single channel in a small patch of membrane usingwhat is known as the patch-clamp technique. This technique involves measuring thetransmembrane potential of a cell and clamping the membrane potential at a fixedvoltage by counteracting any ionic currents with an injected current of equal magnitude.Experiments have shown that there are many different types of channels in a cardiaccell membrane. These channels differ in their permeabilities to different ions and intheir responses to changes in the transmembrane potential. The total ionic currentflowing across the cell membrane at any particular time is the sum of the currents dueto individual ions, such as Na+ and K+, flowing across all channels in the membrane.In a patch-clamp experiment the transmembrane potential of a cell is clamped at acertain potential and once a steady state current is attained, the membrane potential isclamped to a new potential value. There are three types of behavior observed as a newChapter 2. The Model^ 14steady state is achieved. The current at the new potential will (1) instantaneously reachits new steady state value, (2) take some time to be activated as appropriate channelsopen, thus gradually reach its steady state value, or (3) take some time to be activatedbut then gradually, as channels close, become inactivated. In the first case, the currentis time-independent and is often referred to as a background current. In the second case,the current is time-dependent and exhibits only activation processes while, in the thirdcase, the time-dependent current exhibits both activation and inactivation processes.In the Hodgkin-Huxley formulation for a particular cell, cell j, time-independent orbackground currents are often described by an equation of the form:^i g(Ei)^(Ei — Erev)^ (2.1)where Ei is the transmembrane potential of the cell at a particular time and Erev is thereversal potential at which the ionic current flowing across the cell membrane changesdirection. The membrane conductance, g(E;), is often assumed to be constant.For those components i which are time-dependent, the Hodgkin-Huxley formulationoften describes their kinetics by an equation of the form:^i = g(EJ) x^—Erev)^(2.2)if i is an activation current, and^i g(E;) x y^— E„,)^(2.3)if i shows both activation and inactivation processes. Here g(Ei), Ei, and E„, areas in (2.1). The activation variable x and inactivation variable y are gating variablesindicating the fraction of channels specific to a particular ion which are open at any timet. Both x and y take on values between 0 and 1 and follow the first-order equation, e.g.,for x, given by:dxax (1 — x) — ps (x).dt(2.4)Chapter 2. The Model^ 15The rate constants ax and Q1 are generally extremely complex nonlinear functions ofthe cell's membrane potential Ei. A similar equation holds for y but with a x and flxreplaced by ay and fly , respectively. The rate constants for a particular gating variabledetermine the timing of its activation and/or inactivation. For an activation variable,the rate constants cause the variable to increase as the membrane potential becomesless negative (depolarizes) while, for an inactivation variable, they cause a decrease inthe variable with increases in membrane potential.Assuming there are no external current sources and that the changes in membranepotential are the result of the ionic current components, the Hodgkin-Huxley formulationdescribes the rate of change of the transmembrane potential of a cell by a governingequation of the form:dEi(t)1dt =^Z ionic (2.5)where t is the time in msec, j denotes the cell type and, in this thesis, is one of: s, a,p, or v indicating SA node, AV node, Purkinje fibre, and ventricular myocardial cellsrespectively, Ei(t) is the membrane potential in mV (expressed as the inside potentialminus the outside potential) of cell j at time t, C is the membrane capacitance inFF/cm2 , and iion ic is the total ionic current in itA/cm2 flowing out of cell j.There are many Hodgkin-Huxley-type models for cardiac cells in existence today. Inthis thesis, the electrical activity of the SA node cell has been modelled using equationsdevised by Yanagihara et al. (1980) [38]. Because of the relative complexity of DenisNoble's 1984 model for the Purkinje fibre cell, the 1975 model devised by McAllister,Noble, and Tsien [28] is used. For the ventricular myocardial cell, the model employedis due to Beeler and Reuter (1977) [1]. Finally, due to the lack of an existing model, andmotivated by the fact that action potentials of an AV node cell are very similar to thoseof an SA node cell (Figure 1.2), a model for an AV node cell was created by modifyingChapter 2. The Model^ 16the model equations for an SA node cell.2.2.1 The SA Node CellYanagihara et al. [38] modelled the electrical activity of an SA node cell using fourdynamic currents and a time-independent leak current denoted il. The dynamic cur-rents are: a slow inward current, a sodium current, iNa , a delayed inward currentactivated by hyperpolarization, ih, and a potassium current, iK. Both i 8i and iNo in-volve activation and inactivation, while ih and iK involve only activation processes. Thegoverning equation is given by (2.5) with:ionic = isi^iNa^iK^ih^il• (2.6)The model exhibits spontaneous action potentials where the depolarization (phases 4 and0) of the cell membrane is primarily due to the transient flow of i 8i and repolarization(phases 1, 2, and 3) is caused by the combination of a decrease in i 81 and an increase iniK. A large fraction of the total ionic current is provided by i si, iK, and il, as ih and iNaare much smaller in magnitude. The equations describing each of these components ofthe ionic current are given in the Appendix, A.1.2.2.2 The AV Node CellThe action potentials of AV node cells are very similar in shape to the action potentialsof SA node cells (Figure 1.2); therefore, a model for an AV node cell was created usingthe equations for an SA node cell [38]. Minor changes were made to slow the rate ofdepolarization which is not as fast as for SA node cells. The total ionic current for theAV node cell is given by (2.6) and the individual current components are described inthe Appendix, A.2.lIn the original paper [38], the slow inward current is denoted i, rather than i31.Chapter 2. The Model^ 172.2.3 The Purkinje Fibre CellThe McAllister, Noble, and Tsien model for the Purkinje fibre cell [28] describes thetransmembrane potential by (2.5) where:iionic^iNa^i81+ iqr^iK2+^jor2^iK1+ iNab+ iClb^(2.7)and, as in the original paper, the outward (time-independent) background current, iKi ,is listed after the time-dependent pacemaker potassium current, i K2 .Just as in the previous models, this model also exhibits spontaneous action potentials.The upstroke (phase 0) is mainly due to the activation of the sodium current, iNa,in contrast to the SA and AV node cell models where iNa plays only a minor role.The initial repolarization from the peak of the action potential (phase 1) is primarilycaused by the transient chloride current, iqr. The role of the slow inward current, i 82 ,is to slow this rapid repolarization and to produce the plateau (phase 2). Phase 3repolarization is triggered by the onset of one of the plateau potassium currents, ix„which becomes activated over the plateau range of potentials. Two other potassiumcurrents, is2 and iK2 , which also activate over this range, have only a minor influence onthe repolarization. However, iK2 is responsible for the slow phase 4 depolarization. Theremaining currents, iK„ an outward potassium current component, and iNab and jo b ,time-independent background currents carried by sodium and chloride ions respectively,all play a relatively minor role. The equations describing the individual components ofthe ionic current are given in the Appendix, A.3.2.2.4 The Ventricular Myocardial CellThe model, due to Beeler and Reuter [1], for the ventricular muscle cell describes thetotal ionic current using three dynamic currents and one time-independent current, iK1 .The dynamic currents are: a sodium current, iNa , an outward current, i s1 , primarilyChapter 2. The Model^ 18carried by potassium ions, and a slow inward current, i si 2 , primarily carried by calciumions. The governing equation is given by (2.5) where:iionic = iNa 28i. (2.8)Most muscle cells of the ventricles are nonspontaneous; therefore, unlike the previousthree model cells, the ventricular myocardial cell model does not exhibit spontaneousaction potentials. When excitation does occur, due to an external stimulus, then, as forthe Purkinje fibre cell model, the sodium current, iNa , is primarily responsible for therapid phase 0 depolarization. The early phase 1 repolarization is due to the continuedactivation of i 81 . The plateau is determined by the antagonism between the outwardcurrents, iK1 and ix„ and the slow inward current, i81 . Individual current componentsare described in detail in the Appendix, A.4.2.2.5 Control of Pacemaker PeriodicityPacemaker cells from different regions of the heart have characteristically different in-trinsic frequencies. For example, cells in the AV node have an inherent firing rate of45 — 60 cycles/min while Purkinje fibre cells have an intrinsic frequency between 20 and40 cycles/min. In a healthy heart, all cells discharge at the intrinsic SA node frequencywhich is between 60 and 100 cycles/min. Furthermore, under pathological conditions,the heart rate can vary dramatically and cells can fire at frequencies as high as 650cycles/min [6]. In order for the model of the heart to be able to accommodate such awide variation, it was necessary to devise a means of altering the frequencies of the spon-taneous cells of the model. The frequency of a particular cell is defined as the numberof times per minute that its transmembrane potential crosses its threshold potential in21n the original paper [1] the slow inward current is denoted i, rather than ii ,.Chapter 2. The Model^ 19a depolarizing direction (Figure 1.1). The threshold potential for the ventricular my-ocardial and Purkinje fibre cells is -60 mV [1,28]. For the SA and AV node cells, thethresholds were estimated by analyzing the rate of change of the membrane potentialsand were set at -30 mV for both.Since most cells of the ventricular myocardium are nonspontaneous [1,6], the modelcell used for all simulations did not fire spontaneously. The Beeler and Reuter equations[1], as presented in the previous section, model such a nonpacemaker muscle cell. Theresting membrane potential is constant at approximately —84 mV and remains at thisvalue until some external stimulus of sufficient strength causes the cell's excitation. Incontrast, the SA and AV node and Purkinje fibre cells are all spontaneous; therefore,each exhibits a wide range of oscillation rates. To model the behavior of a real SA nodecell, 5 model SA node cells each having a different intrinsic frequency were created byaltering certain constants in the SA node model equations. The frequencies of the 5model cells were chosen to span a range which includes both the normal healthy rangeof oscillation rates for an SA node cell and a range of rates which commonly occur inpathological situations. Similarly, once the appropriate ranges had been determined forthe frequencies of the AV node and Purkinje fibre cells, 5 model cells of each type, withintrinsic frequencies spanning those ranges, were created by altering the appropriateconstants in the equations of both models.SA node cells in a healthy adult heart typically have an oscillation rate which isbetween 60 and 100 cycles/min. Under certain pathological conditions this rate canincrease dramatically; however, in the majority of cases the rate remains less than 200cycles/min. Model SA node cells were created with intrinsic frequencies from approx-imately 80 to 180 cycles/min, a range which includes much of the normal and patho-logical behavior of an SA node cell. There are two currents primarily responsible forthe pacemaker activity of the SA node cell: the potassium current, iK, and the slowChapter 2. The Model^ 20Figure 2.1 Frequency(cycles/min)Period(cosec)E,,,,(mV)EN,(mV)(a) -6.7 -65.0 79.65 753.25(b) -6.0 -62.0 100.08 599.53(c) -5.0 -58.0 119.42 502.44(d) -4.0 -49.7 149.63 401.00(e) -3.8 -39.0 178.26 336.58Table 2.1: Parameter values and resulting frequencies for model SA node cells.inward current, i 81 [2,3]. The slow inward current is responsible for the relatively sharpupstroke in the action potential but contributes only during the last 30% of the timecourse (the latter part of phase 4 and phase 0) of depolarization. The decay of iK,however, has a much longer time course, and is responsible for the preceding slow rise.Slight changes in the parameters, Eap and Epp of iK (A.17), produce a significant changein frequency. Figure 2.1 demonstrates the sensitivity of the cell frequency on the valuesof these parameters (Table 2.1). With E„p = —6.7 mV and Fop = —65 mV (top trace)the frequency is 79.65 cycles/min. As both parameters are shifted to less negative po-tentials, the frequency increases (lower traces) and the action potential moves slightlyupward on the voltage axis. The range of frequencies produced is typical of the rangefound in a human heart, where 80 cycles/min is a normal healthy rate for SA node cellsand 180 cycles/min is representative of an arrhythmia known as sinus tachycardia.In a normal heart, AV node cells have an intrinsic frequency which is slightly lessthan that of the SA node; therefore, to produce AV node cell models the five SA nodemodels were modified to reduce their rate of spontaneous depolarization. Using the sameadjustments in Eap and EN as for the model SA node cells, the maximum value of theslow inward current, isi , was reduced in each by multiplying it by a constant fraction,1.7—20-ESIIAA—80—00-1—100^O40-20-0 1^A (b) 100.08(c) 119.4240200—20a. —40W—80—$ 0—100—20-E—40— )4.1—GO-0——100^I(d) 149.63Chapter 2. The Model^ 2140 —20—^O^—20-ELa—80—^.100^40-20—(a) 79.651000^20'00^3000^4000^5000^6000(e) 178.26 time (msec)Figure 2.1: Model SA node cells (numbers indicate cell frequency in cycles/min).24000 ^—40-W—50——100^0h^A h^ftI VChapter 2. The Model^ 22Figure 2.2 isi Frequency(cycles/min)Period(msec)Ea p(mV)Epp(mV)(a) -6.7 -65.0 0.99 75.66 793.00(b) -6.0 -62.0 0.94 87.81 683.31(c) -5.0 -58.0 0.90 103.12 581.84(d) -4.0 -49.7 0.80 121.21 495.00(e) -3.8 -39.0 0.75 146.32 410.07Table 2.2: Parameter values and resulting frequencies for model AV node cells.z si. This is one of the methods used by Michaels et al. in their study of the interactionof SA node cells [30]. Table 2.2 lists the scaling values of i si used and the resulting cellfrequencies. Figures 2.2 a-e show the resulting action potentials which are modificationsto the action potentials of Figures 2.1a-e, respectively. In the top trace, with Eap, = -6.7mV, Epp = -65 mV, and i si = 0.99 the resulting reduction in i 8i produces a model foran AV node cell with a period of 793 msec which is slightly longer than the 753.25 msecperiod of the corresponding SA node cell (Figure 2.1a). There is also a small decrease inthe maximum membrane potential which accompanies the scaling of i 8i. Similarly, lowertraces show AV node cell models with intrinsic frequencies and maximum membranepotentials slightly less than that of the respective model SA node cells.Purkinje fibre cells in a healthy adult heart have an intrinsic frequency between 20and 40 cycles/min; however, this rate can increase significantly during many arrhyth-mias. Model cells with frequencies ranging from 40 to 120 cycles/min were producedby shifting the parameter, Ek, of the pacemaker potassium current, iK2 (A.34). Fig-ure 2.3 shows the corresponding action potentials for the various values of Ek given inTable 2.3. In the top trace, with Ek = -54.7 mV, the intrinsic period is 1506 msec.This corresponds to a firing frequency of approximately 40 cycles/min which is a typicalL40-20-0^-20-d -40-,///-60--8o-100(a) 75.6640-20--20• -40-W - 80-- 80-- 100 ^(b) 87.81(c) 103.12(d) 121.2120-^0^-20-E-60-80-- 100^Chapter 2. The Model^ 2324001E -20-Lad -40-- 80-- 80- 100 I^1^Ai^A^A^ tAAAA/l/^0/O loO o^20'00^•^30700^40'00^soIo o^6000(e) 146.32 time (msec)Figure 2.2: Model AV node cells (numbers indicate cell frequency in cycles/min).040-.20-E5,%La-20--40- PvL^AO Pt-.80-8o--100A40-20-0 E -20-P. -40-- 60-- 80-^-100 ^0 1000^2000^3000(e) 119.735000^6000time (msec)4000Chapter 2. The Model^2440- 200E -20-40▪ -60-)-e0--100 ^(a) 39.8440-20- N1NE40-20-0 - 20-- 40-- 80-- 80-0)-100^40-Ep, -40-W- 50-- 8011-100^20-(b) 60.27(c) 80.20\2\(^d) 100.71Figure 2.3: Model Purkinje fibre cells (numbers indicate cell frequency in cycles/min).Chapter 2. The Model^ 25Figure 2.3 Frequency(cycles/min)Period(cosec)Ek(mV)(a) -54.7 39.84 1506.00(b) -45.0 60.27 995.50(c) -35.0 80.20 748.13(d) -25.0 100.71 595.75(e) -15.0 119.73 501.11Table 2.3: Parameter values and resulting frequencies for model Purkinje fibre cells.value for a Purkinje fibre cell in a healthy human heart. As Ek is progressively shiftedtowards zero (lower traces) the frequency increases. As for the SA node cell, this increaseis accompanied by a slight upward shift of the action potentials along the voltage axis.The five resulting models allow simulation of the electrical activity of Purkinje fibre cellswith frequencies of approximately 40,60,80,100, and 120 cycles/min.It should be noted that similar control of pacemaker periodicity can be obtained bythe application of an external current to each of the models. A hyperpolarizing (outward)current decreases, while a depolarizing (inward) current increases, cell frequency [30].2.3 Modelling Adjacent Cell InteractionAction potentials propagate within the heart by moving from cell to cell over regionsof close membrane association called gap junctions which exist between adjacent cells(Figure 2.4). These junctions provide a low resistance pathway through which electriccurrent can easily flow [17,30,39].To describe the interaction of two adjacent cells, cell i and cell j, it is assumed thatcoupling currents flow between the cells. For two neighbouring cells, these coupling cur-rents simply describe the movement of action potentials over the gap junction betweenChapter 2. The Model^ 26gap junction (nexus►Figure 2.4: Schematic diagram depicting the flow of an action potential between twoadjacent cells across a gap junction (taken from [17], p.20).the cells. This gap junction constitutes the entire propagation path for impulses trav-elling in both directions; therefore, the influence of one cell on the other will be almostimmediate. This influence of cell i on cell j is described by a coupling current, ici , whichis a function of intercell membrane conductance, g, and voltage difference as follows:^= g [E,(t) — Ei (t)]•^ (2.9)Similarly, the influence of cell j on cell i is given by:^ici = g [Ei(t) — Ei (t)].^ (2.10)These currents, (2.9) and (2.10), are included in the models for cell j and cell i, respec-tively, and integrated along with the other ionic currents. Also, (2.9) and (2.10) have thesame magnitude but are of opposite sign implying that, for two adjacent cells, the samecurrent that flows out of one cell flows in to the other. This is the form of the couplingcurrents employed in the previous studies by both Michaels et al. [30] and Lambert andChay [5] where only adjacent cell interactions were modelled.2.4 Modelling Nonadjacent Cell InteractionCertain types of cardiac cells, e.g., an SA node cell and a Purkinje fibre cell, are neverphysically adjacent. Even two cells of the same type, such as two ventricular muscleChapter 2. The Model^ 27•••Figure 2.5: Schematic diagram depicting the flow of an action potential from cell i tocell j.cells, may be located some distance from each other. For two such nonadjacent cells,cell i and cell j, the time required for action potential propagation between them willbe significant relative to the duration of their action potentials. Furthermore, the pathtaken by an impulse from one to the other may not be the same as the path taken in thereverse direction. Consequently, in this case, the cells may not exert equal influences oneach other and the coupling currents must be direction dependent.An impulse originating at a particular cell, cell i, eventually reaches another distantcell, cell j, by travelling cell-to-cell crossing gap junctions over a path from cell i tocell j. Finally, the impulse will arrive at cell j from an adjacent cell denoted cell j-i(Figure 2.5).The coupling current flowing between the adjacent cells, cell j-i and cell j, is anal-ogous to (2.9) with subscripts changed and is described by:= gj-i,, [EE(t) —^(0]^ (2.11)Chapter 2. The Model^ 28where gj_ i ,j is a constant representing the conductance in mS/cm 2 of the gap junctionbetween cells j-i and j.In describing the influence of cell i on cell j it is the electrical activities of these twocells which are of concern in the model; therefore, the coupling current flowing betweencell i and cell j should be expressed in terms of parameters describing only these twocells. In order to accomplish this, it is assumed that the impulse generated at cell i isnot affected by its propagation to cell j or, in other words, that the membrane potentialat cell i reaches cell j through cell j-i , after a finite amount of time, without a changein magnitude. Mathematically,Ej_1(t) = Ei(t — ri,j-i)where Ti,j_ i is a constant representing the time in msec required for the conduction ofan impulse from cell i to cell j-i . With this assumption, (2.11) becomes:ici = gj- Li [Ej(t) — Ei(t —^ (2.12)To remove any reference to cell j-i in the above expression we let gi,j = gj_ i ,j andTi j = ri,j_ 1 . With this new notation, gi,j should be interpreted as the conductance ofthe gap junction between cell j and the cell adjacent to it in the conductance path fromcell i to cell j. Similarly, ri,j is the time required for an impulse to travel from cell i tothe cell immediately preceding cell j along a path to cell j. Consequently, the couplingcurrent which represents the influence of cell i on cell j can be written:ici = gi,j {Ei (t)— Ei (t — rid )].^ (2.13)This coupling current is included in the model for cell j and integrated along with theionic currents. The governing equation for cell j, (2.5), becomes:dEi (t) 1dt =^(zionic^ic; )(2.14)Chapter 2. The Model^ 29with=^gij [Ei(t) — Ei (t —^ (2.15)being the sum of coupling currents over all cells i which influence cell j.This form of the coupling currents for modelling nonadjacent cell interaction is con-sistent with (2.9) for adjacent cells. For two adjacent cells, cell i and cell j, the gapjunction between the cells constitutes the entire propagation path for impulses travel-ling in either direction; therefore, letting g be the conductance of this intercell membrane,the following is true:g =Furthermore, the impulse propagation times between the cells will be negligible; there-fore, the following assumption can be made:ri,j = 0.Then (2.13) which describes the influence of cell i on cell j becomes:^i cj = g [Ei(t) — Ei (t)].^ (2.16)which is (2.9) given earlier.2.5 Range of Coupling ConductancesNumerical computations of cell interaction were run over a range of coupling conduc-tances from gi,i = 0 mS/cm2 (no coupling) to = 0.1 mS/cm2 . Ypey [39], in astudy of interactions between general cardiac cells uses a value of 0.003 mS/cm 2 forthe maximum coupling conductance while Lambert and Chay [5] use values as highas 3.5 mS/cm2 in coupling their simple 2-variable model cells. The upper limit usedthroughout this paper, Ai = 0.1 mS/cm2 , corresponds to a gap junction membraneChapter 2. The Model^ 30resistance of 10 k1t/cm 2 and is the value used by Michaels et al. [30] in their study ofthe interactions of SA node cells modelled using the same Yanagihara et al. equations[38] as are used here.2.6 Propagation Time DelaysPropagation time delays appropriate to the types of cells coupled in a particular com-putation are incorporated into the coupling currents flowing between the cells. Forexample, for studies involving the influence of an SA node cell on an AV node cell, T„,,, isassigned a value representative of the time required for impulse conduction from the SAnode to the AV node. This is the time required for atrial depolarization which, accordingto Section 1.1.3, has an average value of between 80 and 120 msec in a healthy heart.Similarly, in a computation of normal behavior, ra,p might be assigned a value corre-sponding to the time required for the passage of an impulse through the AV node fromwhich it emerges to the Purkinje fibre network. Under pathological conditions, such aswith damaged tissue, the transmission times between different regions of the heart canbe much longer than their average values and, in severe cases, the damaged tissue maycreate a blockage so that impulses are not conducted at all from one region to another.The propagation time delays can be chosen according to the particular situation beingmodelled.Chapter 3Phasic Interactions of Pacemaker Cells3.1 Phase ResettingWhen two pacemaker cells interact, impulses which originate at one cell may inducesubthreshold depolarizations in another causing all subsequent action potentials of thelatter cell to be advanced or delayed in comparison with its undisturbed oscillation[28,39]. These alterations, or phase shifts, in the rhythms of interacting cells are a pre-requisite to entrainment and have been observed both experimentally and theoretically.Phase resetting has been demonstrated by Jalife and Antzelevitch [22] for rabbit SAnode cells and by Guevara, Shrier, and Glass [13] for embryonic chick ventricular heartcell aggregates. Numerous computer simulations have also been done. Winfree [37] hasstudied phase resetting for a wide variety of biological oscillators. Michaels et al. [30]using the Yanagihara et al. [38] SA node cell model and McAllister, Noble, and Tsien[28] using their Purkinje fibre cell model have reproduced experimental results quiteclosely. These studies, however, have been based on the interactions of similar adjacentcells. For the present model, the phenomenon of phase resetting will be analyzed forvarious pairs of the three different types of pacemaker cells. Because these cells are notnecessarily physically adjacent, the propagation time between them will be significantand the strength of the coupling will be direction dependent.The effect of an incoming pulse on the cycle length of an oscillator depends not onlyon the intensity and duration of the stimulus but also on its timing. If the arrival of31Chapter 3. Phasic Interactions of Pacemaker Cells^ 32the impulse occurs while the cell is in a refractory state, its next action potential willbe delayed. If, on the other hand, the impulse arrives at a time when the membraneimpedance is low, e.g., during phase 0 or the latter part of phase 4, then the subsequentaction potential may be advanced. The arrival time of an impulse from one cell atanother cell is determined by two factors: the relative frequencies of the cells and theimpulse propagation time between them. In the remainder of this chapter, impulsepropagation times will be systematically adjusted to study the effects of the arrival timeof one cell's action potentials on the firing times of another cell.When a relatively fast cell (one with a shorter period of oscillation) is coupled to aslower cell, more than one action potential of the fast cell will occur in each cycle ofthe slow cell. Therefore, to study the effects of a single action potential of one cell onthe cycle length of another cell, the two cells were permitted to interact only duringa single action potential phase (Figure 1.1) of the cell providing the stimulus. Theaction potential phase of a cell is defined as the period beginning when the membranepotential crosses its threshold value during phase 0 depolarization and ending when thepotential first reaches its minimum value. For consistency, this type of pulsed couplingwas also used for studies of the effects of slow cells on fast cells. Continuous coupling(the continuous interaction of cells from the time of onset of the stimulus to the endof the computation), however, probably reflects the true physical situation in a realheart more closely. Michaels et al. [30] consider the influence of one cell on another asconsisting of two parts: the "phasic" influence during the action potential phase and thecontinuous or "tonic" influence over the period of phase 4 depolarizaton. Using bothpulsed and continuous coupling to study phase resetting of SA node cells, they foundthat, qualitatively, the results were similar for both types of coupling and concludedthat the entrainment phenonema observed during continuous coupling were "primarilya function of the phasic influence" ...."of one pacemaker on the activity of the other" . TheChapter 3. Phasic Interactions of Pacemaker Cells^ 33primary method of coupling in this chapter will be pulsed coupling although continuouscoupling will also be done to compare results.3.2 MethodsTo study the effects of a single action potential of model cell i on the intrinsic cyclelength of model cell j, the cells were coupled as follows. First, the influence of j on i waseliminated, i.e., the conductance gj,i was set to zero so that no coupling current wouldflow from j to i. Thus i would oscillate unperturbed. On the other hand, the couplingconductance gij was maintained at a sufficiently high value during the action potentialphase of i and set equal to zero otherwise. This achieved the desired pulsed coupling.The propagation time Ti,i was then adjusted in steps so the action potential of i was 'felt'at different times, or phases, Oi within a cycle of j. A cycle of j begins from the peakor maximum potential of one action potential and ends with the peak of the subsequentaction potential. The phases, c6j, were measured relative to the time corresponding tothe peak of the first action potential of the perturbed cycle of j. Denoting this time byti and the time at which the potential of i crosses its threshold value by ti thresho,,, thephase Oi satisfies:03 = (tithreahold + 1 73) t where 0 < 03 < Ti .Figure 3.1 indicates the terminology used to describe phase interactions and shows anaction potential of i which causes an abbreviation of the perturbed cycle of j. Com-putations were caried out for numerous values of the propagation time r i j such thatthe resulting phases q assumed values ranging from 0 to Th the intrinsic cycle lengthof j (q = 0, indicates that the action potential of i reached j at exactly the time, ti ,corresponding to the peak of the first action potential of the perturbed cycle of j). Inthis way the effect of the arrival of an action potential from i at any time within a cycleA T;IAt ! threshold(4— 0;  — •tjChapter 3. Phasic Interactions of Pacemaker Cells^ 34Figure 3.1: Terminology used to describe the influence of an action potential of cell ion cell j. Schematic on left shows unidirectional coupling from i to j. Dashed traceindicates control action potential (no interaction). Solid traces indicate activity whencells are coupled. Cell i oscillates unperturbed and stimulates cell j causing its secondaction potential to be advanced. Terminology used for delays (not shown) is analogous.Chapter 3. Phasic Interactions of Pacemaker Cells^ 35of j could be analyzed. After each computation, it was possible to scan the cycle inwhich the stimulus occurred and measure the phase shift ATi in the cycle length. Thesephase shifts were measured as the perturbed cycle length minus the intrinsic length;therefore, a positive phase shift (positive OTC) corresponds to a delay while a negativephase shift (negative OTC) corresponds to an advance in the occurrence of the subse-quent action potential of j. The shifts OT, were then plotted against Oi (where bothwere expressed as percent of the intrinsic cycle length, Ti ) in what is known as a phaseresponse curve (PRC). Some sample PRCs were obtained for various values of couplingconductances between different types of cells with different intrinsic cycle lengths. Thecoupling conductance values were chosen from the range of conductances of this study(0.0-0.1 mS/cm2 ) and were sufficiently high to cause a measurable phase shift in theperturbed cell's cycle length.3.3 An SA Node Cell and a Purkinje Fibre CellAlthough in a real heart an SA node cell is never physically adjacent to a Purkinjefibre cell, the activity of one may affect the activity of the other through the cell-to-cellpropagation of impulses. With the present model, this type of interaction is studied bythe incorporation of propagation time delays in the coupling currents flowing betweenthe cells. This form of coupling and the method of the preceding section were used tostudy the phasic interactions of a model SA node cell (s) and model Purkinje fibre cell(p). The intrinsic periods of the cells were T8 = 599.53 msec and Tp = 748.13 msec whichcorrespond to frequencies of approximately 100 and 80 cycles/min, respectively. Thesevalues represent a normal healthy oscillation rate for an SA node cell and a significantlyhigh oscillation rate for a Purkinje fibre cell. This is a situation which is common inarrhythmias such as ventricular tachycardia and fibrillation.Chapter 3. Phasic Interactions of Pacemaker Cells^ 36The phase response curves of Figure 3.2 summarize, for two different values of thecoupling conductance gp,., the effects of an action potential of p on the cycle length ofs. Although PRCs were obtained for several values of gp , s , only two have been included.These PRCs use values of gp , 8 which are large enough to induce measurable phase shiftsin the cycle of s and produce PRCs which demonstrate, by the differences in their shapes,the effect of the coupling conductance. For gpo, = 0.005 mS/cm2 , the PRC shows thataction potentials from p which arrive during approximately the first one-third of the cycleof s cause a delay, whereas those which arrive later cause an advance of the subsequentfiring of s. When the coupling conductance was decreased to gi,„„ = 0.002 mS/cm2the shape of the PRC was maintained. Also, during this first one-third of the cycleof s where both curves indicate delays (positive .6, T,), they are less for the case ofcoupling with the smaller conductance. Similarly, where both curves show advances, themagnitudes of the phase shifts are less for weaker coupling. These results are expected,since with decreased conductance, p has less influence on the cycle length of s. Forthis reason also, with the smaller conductance, delays occur over a larger portion ofthe cycle (up to 40%). With the larger value of g po , an action potential of p is able tocause an advancement of the excitation of s, but a weaker stimulus still may cause onlysubthreshold depolarizations. These subthreshold depolarizations do not excite the celland are followed by a period of repolarization to a more negative membrane potentialfrom which the increase to the threshold value must begin again. The time requiredfor this depolarization and subsequent repolarization delays the occurrence of the nextaction potential of s. Conversely, a stronger stimulus will be capable of exciting s while itis relatively more refractory which occurs early in its peak-to-peak cycle, thus advancesoccur for lower 0, with a larger conductance.The lower graph of Figure 3.2 shows the membrane potential of s plotted as a func-tion of percent of its intrinsic cycle length over a complete peak-to-peak cycle prior to1 .-^1 )—60 I^An unperturbed cycle of s forT,=599.53 msec80TP = 748.13 msecIs = 599.53 msec40i—20+ gp ,5 =0.0050 g r‘.=0.002—4020^40^60O s (731-s )0 80 10060Chapter 3. Phasic Interactions of Pacemaker Cells^ 370 20 40 %T. 60 80 100Figure 3.2: Upper graph shows phase response curves summarizing the effects of a singleaction potential from p on the cycle length of s for two values of gp,, (mS/cm2 ). Lowergraph shows the intrinsic electrical activity of s over a complete peak-to-peak cycle.Chapter 3. Phasic Interactions of Pacemaker Cells^ 38coupling. This plot provides a means of associating the cell's intrinsic electrical activitywith any value of 0 8 of the corresponding PRC. For example, it is clear that for bothvalues of gp , 8 the earliest phase advance of the action potential of s does not occur untils is undergoing phase 4 depolarization and membrane impedance is low.For the same two model cells, the PRCs of Figure 3.3 demonstrate the effect of anaction potential of s on the cycle length of p for two values of the coupling conductancegs ,p . As for the previous case, PRCs were obtained for several values of gs ,p , however,only two have been included. The values shown are sufficiently high that, in both cases,s induces a measurable influence on the cycle length of p, yet the effects are differentenough in each case to demonstrate the effect of the coupling conductance on the shapeof the PRC. Again, for smaller conductance, corresponding to a decrease in the influenceof s on p, the general shape of the PRC is maintained and, where either both curves showdelays or both show advances, the corresponding phase shifts are smaller in magnitudefor the smaller value of g 8 ,p . Also, as in the preceding case with stronger coupling, delaysoccur over a smaller portion of p's cycle because the larger the conductance, the moreeasily and the earlier in p's cycle an action potential of s can cause the advancement ofa subsequent action potential of p. Comparing the electrical activity of p (lower graphof Figure 3.3) at various phases to its PRC indicates that the largest phase delays occurduring phase 3 repolarization when the cell is in a highly refractory state following itsexcitation. Furthermore, the transition from maximal delay to maximal advance occursnear the end of phase 3 repolarization and beginning of phase 4 depolarization when therefractory period comes to an end. Action potentials of s arriving early in p's cycle, ata time corresponding to the initial rapid rate of repolarization from the peak (phase 1),cause slight delays. McAllister, Noble, and Tsien [28] suggest that these delays mightbe due to a delayed repolarization. Following this region, there is an interval over whichthe rate of decrease in the membrane potential of p lessens and during which a stimulus40 20-An unperturbed cycle of p forT,=748.13 msec-20-p, -40-- 60-- 80--100 -^ .■11=1=111■IvChapter 3. Phasic Interactions of Pacemaker Cells^ 390^20^40^60^80^100013 (%T p )0 20^40^%T^60^80^100Figure 3.3: Upper graph shows phase response curves summarizing the effects of a singleaction potential from s on the cycle length of p for two values of gs ,p (mS/cm2 ). Lowergraph shows the intrinsic electrical activity of p over a complete peak-to-peak cycle.Chapter 3. Phasic Interactions of Pacemaker Cells^ 40from s causes slight advances. The magnitude of these phase shifts which occur earlyin p's cycle are small, in agreement with the fact that a cardiac cell is most immune tostimulii immediately after excitation when it is in its absolute refractory state.The biphasic response curves obtained here are similar in shape to those obtainedexperimentally by, amongst others, Jalife et al. [22] and numerically by Michaels et al.[30]. When either p stimulates s or vice-versa, the resulting PRCs (Figures 3.2, 3.3)demonstrate that, as the interaction between the cells is strengthened by increasing thecoupling conductance, the transition from maximal delays to maximal advances occursover a narrower range of phase values. This behavior was also found experimentally byGuevara et al. [13] for embryonic chick ventricular heart cells. When aggregates of thesecells were stimulated by increasingly strong current pulses, the transition from maximaldelay to maximal advance occurred much more abruptly.3.4 An SA Node Cell and an AV Node CellPhase response curves were also obtained for the case of interaction between an SA nodecell (s) and an AV node cell (a) where the same model SA node cell with an intrinsicoscillation period given by 718 = 599.53 msec as in the preceding simulations was used andcoupled to a model AV node cell with Ta = 793.00 msec. These cycle lengths correspondto frequencies of approximately 80 and 75 cycles/min, respectively and represent normalrates for healthy SA node and AV node cells. The PRC of Figure 3.4 summarizes theeffects of an action potential of s on the intrinsic cycle length of a for g 8 ,a = 0.05 mS/cm2while the PRC of Figure 3.5 demonstrates, for the same two cells and g„,, = 0.05 mS/cm2 ,the effects of the excitation of a on the cycle length of s (this value of the couplingconductances is one of several values which causes measurable changes in the perturbedcell's cycle length and for which PRCs were obtained). The PRCs are almost identical40200E -20d- 40- 60- 80Chapter 3. Phasic Interactions of Pacemaker Cells^ 4160Ts = 599.53 msecT. = 793.00 msecgs,. = 0.05 mS/cm22040- 40- 60-800 20^40^60^80^100Oa (%Ta )40^60^80%T.^ 100Figure 3.4: Upper graph shows phase response curve summarizing the effects of a singleaction potential from s on the cycle length of a. Lower graph shows the intrinsic electricalactivity of a over a complete peak-to-peak cycle.0 202 0 -4 0 -T. = 793.00 msecTs = 599.53 msecg a,s = 0.05 mS/cm2-40-U—60 -—80iI^'^I-4'1An unperturbed cycle of s forT.=-599.53 msecChapter 3. Phasic Interactions of Pacemaker Cells^ 42^6 0 —^0^20^40 60^80^10095s (%Ts )0^20^40^60%T3Figure 3.5: Upper graph shows phase response curve summarizing the effects of a singleaction potential from a on the cycle length of s. Lower graph shows the intrinsic electricalactivity of s over a complete peak-to-peak cycle.80^100Chapter 3. Phasic Interactions of Pacemaker Cells^ 43in shape. This is not unexpected since, except for the slower rate of depolarization ofthe AV node cell, the action potentials of the two cells are very similar (Figure 1.2). Inboth cases, discharges arriving early prolong, while those arriving later shorten, the cyclein which the stimulus occurs. As for the preceding PRCs, the transition from delay toadvance occurs early in the phase 4 depolarization portion of the cycles in both cases. Afurther observation is that Figure 3.5 which shows the effects of the action potentials ofa on s is very similar to Figure 3.2 which shows the effects of action potentials of p on .s.This indicates, as one would expect, that the source of the stimulus is not as importantin determining the form of the resulting PRC as the type of cell being stimulated.3.5 Fast and Slow CellsPrevious studies [30,39] have obtained phase response curves showing that slower cellswere much more affected by depolarizing currents than faster cells and that the phaseshifts induced in a slow cell by a faster cell were always greater than those induced inthe fast cell by the slower cell. These studies also indicated that the common entrainedperiod for coupled fast and slow pacemakers firing in synchrony was always closer to theintrinsic period of the fast cell than of the slow cell. Thus, the changes in cycle length andsubsequent phase shifts of the PRC were necessarily greater for the slower cell. Theseresults; however, were obtained for similar cells coupled using adjacent-cell-coupling(Section 2.3) where the same current flowing out of one cell flows in to the other. Forthe present model, which allows the coupling of nonadjacent cells, the coupling currentsflowing between the cells are direction dependent and are characterized by values of thecoupling conductances and propagation time delays. The PRCs obtained here indicatethat, with this direction dependent coupling, the phase shifts are not necessarily greaterfor the slower cell.Chapter 3. Phasic Interactions of Pacemaker Cells^ 44Comparing Figure 3.2 with Figure 3.3 reveals that the maximal advance of the slowerPurkinje fibre cell (p) may be larger or smaller than the maximal advance of the fasterSA node cell. Similarly, the maximal delay of p may be greater or less than the maximaldelay of s. The shape of the PRCs, and subsequently the cell which shows the greaterphase shifts, is dependent on the values of the conductances gp,, and g8 ,p . Thus, eithercell can exert the larger influence.Behavior similar to that of previous studies was obtained when a particular cell witha fixed period was coupled to another cell whose period of oscillation was then altered.As an example, Figure 3.6 shows the phase dependent effects of an action potential ofan SA node cell on the cycle lengths of two Purkinje fibre cells, one with a period of748.13 msec and the other with a period of 995.50 msec. The coupling conductancegs ,p remained constant at 0.065 mS/cm 2 , a value that is sufficiently high that actionpotentials at s were able to cause measurable phase-shifts in p's cycle in both cases. Atany given phase, the phase shifts are greater in magnitude for the slower Purkinje fibrecell. Furthermore, both the maximal delay and maximal advance are greater for theslower cell (although the maximal delays of the two cells differ only very slightly).3.6 Phase Response Curves and Zones of EntrainmentThe phase response curve gives some insight into the synchronization properties of anexcitable cell oscillator since it determines the limits of the zones of stable entrainmentfor unidirectional coupling [39]. An action potential of one pacemaker can only advanceor delay an action potential of another cell by a maximum amount indicated by thePRC. If the periods of the two oscillators differ by more than the maximum possiblephase shift, the cells will not be entrained to a common frequency. For example, if arelatively fast cell is providing the stimulus to a slower cell and the corresponding PRC60Is = 599.53 msecg s, ,p = 0.065 mS/cm 240-20-—20-—40- -I- T P =748.13 msec0 Tp=995.50 msecb1^120^40 60cb (%TP P—600 80^100R; 1_; QpChapter 3. Phasic Interactions of Pacemaker Cells^ 45Figure 3.6: Phase response curves summarizing the effects of a single action potentialfrom s on two different Purkinje fibre cells with periods as shown.Chapter 3. Phasic Interactions of Pacemaker Cells^ 46predicts that the maximal advance of the slower cell is less than the difference in thecycle lengths of the cells then the slow cell cannot be entrained to beat synchronouslyto the fast cell. Thus, a phase response curve provides a means of determining boundson the cycle lengths of the cells for which entrainment can occur.3.7 Pulsed versus Continuous CouplingWhen pacemaker cells of the heart interact, the interaction is not restricted to the actionpotential phase. Consequently, the pulsed coupling used in obtaining the preceding PRCsdoes not reflect the true physiological situation. However, it was mentioned earlier thatprevious studies have concluded that the phasic interactions during an action potentialwere most important in determining the ultimate shape of the phase response curveand, therefore, the ultimate rhythm of coupled cells. This section will investigate someof the differences in PRCs obtained under conditions of pulsed and continuous coupling.Continuous coupling refers to a simulation allowing the interaction of the cells from thetime of onset of the stimulus until the occurrence of the subsequent action potential ofthe perturbed cell at which time the computation was usually terminated.Figure 3.7a shows the phase shifts induced in an SA node cell with period T, = 599.53msec due to continuous stimulation by a Purkinje fibre cell with period Tp = 748.13msec. The coupling conductance used was gp,, = 0.005 mS/cm 2 . This value of gp , 8and the same two cells were also used to obtain one of the PRCs shown in Figure 3.2under conditions of pulsed coupling. Qualitatively, the results are similar in both cases.The general shape of the PRCs is the same with delays occurring for action potentialsarriving early and advances occurring for action potentials arriving late in the cycle of s.The striking similarity between the PRCs is expected because the slower Purkinje fibrecell fires only once during any cycle of the faster SA node cell and coupling only during80T = 748.13 msecTa = 599.53 msecgp.s = 0.005 mS/cm 2(a)6040-20—20—40-20^40^60cb s (%1- s)—60 t^0 so 100—20—4020Chapter 3. Phasic Interactions of Pacemaker Cells^ 4780T. = 599.53 msecT = 748.13 msecgs.p = 0.065 mS/cm 24.060 (b)—600 20^40^60gyp (%Tp)80 100Figure 3.7: Phase response curves summarizing the effects of p on .s (a) and s on p (b)under conditions of continuous coupling.Chapter 3. Phasic Interactions of Pacemaker Cells^ 48the action potential phase of p simply eliminates the remaining weaker influence duringphase 4 depolarization. Quantitatively, there are differences in the results obtainedunder conditions of pulsed and continuous coupling. Although the portion of the curvesshowing advances are almost identical, the delays are greater in magnitude for the caseof continuous coupling. This is due to the continuous 'pull' of the slower Purkinje fibrecell on the faster SA node cell during phase 4 depolarization.When a fast cell provides the stimulus for a slower cell, the results obtained forthe two methods of coupling are somewhat different. Figure 3.7b shows, for the sameSA node and Purkinje fibre cells, the effects of the activity of s on the cycle lengthof p under conditions of continuous coupling. The corresponding PRC obtained usingpulsed coupling is shown in Figure 3.3 with 9 ,,,p = 0.05 mS/cm2 . As in the previouscase, the portion of the curves after the transition from delay to advance are almostidentical. However, for small values of Op the PRCs are quite different. When thecoupling is continuous the cycle length of p is significantly shortened, whereas underpulsed coupling the phase shifts were small for small values of O p . These advances whichoccur for continuous coupling are due to the fact that if an action potential of the fastercell (s) arrives early in p's cycle then a second action potential will also arrive within thesame cycle of p. In addition, since T, = 599.53 msec and Tp = 748.13 msec this secondaction potential of s will occur for Op > 80%, at a time when p is in the latter stagesof phase 4 depolarization and membrane impedance is low. Consequently, the firing ofp will be advanced in comparison to its unperturbed oscillation. A further observationis that the phase delays under continuous coupling are smaller due to the 'pull' of thefaster SA node cell during phase 4 of its cycle than under conditions of pulsed coupling.Chapter 4Behavior of a Healthy Heart4.1 Activation of a Nonspontaneous CellUnder normal conditions muscle cells of the ventricles are not capable of spontaneousdepolarization. These cells are excited by action potentials which are initiated by pace-maker cells and arrive through neighbouring Purkinje fibre cells (Figure 4.1) or othermuscle cells. Since Purkinje fibre cells are spontaneous they can act as independentpacemakers and fire at their own intrinsic frequencies or they can fire in response tothe activity of another pacemaker site. Figure 4.2 demonstrates the activation of aventricular myocardial cell (v) by an adjacent Purkinje fibre cell (p) with an intrinsicperiod of 748.13 msec (approximate frequency of 80 cycles/min). This period representsan approximate frequency of 80 cycles/min, a rate which is above the normal range offrequencies of healthy Purkinje fibre cells; however, the results are qualitatively simi-lar to results obtained for several other values of the Purkinje fibre cell period whichwere within the normal range. Results are shown only for the shorter period (748.13msec) in order that a shorter time scale can be used to demonstrate the behavior of thetwo cells. Because the two cells are contiguous, the action potential propagation timebetween them is negligible and the conductance path is simply the intercell membrane(Section 2.3); therefore, the following are assumed:Tp 9v = Tv,p^0,gp,v = gv,p•49Chapter 4. Behavior of a Healthy Heart^ 50Figure 4.1: Schematic representation of the interaction of a Purkinje fibre cell (p) andan adjacent ventricular myocardial cell (v). Arrows indicate flow of coupling currentbetween cells.Letting g = gp,,, = gv ,p , the coupling current added to the model for the Purkinje fibrecell is analogous to (2.9) and is, therefore, given by:icp = g [Ep(t) — EE (t)1•The negative of this current is added to the muscle cell model to describe the influenceof the pacemaker cell.When there is no interaction between the cells, g = 0 (Figure 4.2a), the nonpacemakerventricular myocardial cell is completely silent. Its membrane potential remains at itsrest value of approximately —84 mV while the Purkinje fibre cell fires independentlyat its own rate. When the coupling is weak (Figure 4.2b) subthreshold responses areinduced in the muscle cell. These responses grow in amplitude with increased couplingstrength. With still stronger coupling, each cell influences the other in such a way thata form of arrhythmia develops. Under these circumstances entrainment may occur asin (Figure 4.2d) where, for every 2 cycles of the Purkinje fibre cell there is 1 cycle ofthe muscle cell so that a 2:1 ratio exists between the frequencies of the cells. Whenthe coupling is strong enough, the cells beat synchronously (Figure 4.2e); however, theaction potential of the ventricular myocardial cell is slightly delayed and continues todiffer in shape from the action potential of the Purkinje fibre cell. With further increasesin g, the delay gradually disappears, the nonspontaneous cell begins to exhibit phase 4>E-40--80-- a o-,Ir^-100^40-20-^0^> -20-E- 40-W- SO--80-100^(b) g=0.02st(c) g=0.02520-E - 20--40-- 50--60-1000 1000^20'00^30'00^4000(d) g=0.0259 time (msec)Chapter 4. Behavior of a Healthy Heart^ 51Zo--100^40-(a) g=° ' °I11 111,1ttPti1%tt40--100(e) g=0.03> -20-lalE-40--80 -- 00 -,- 100 ^(f) g=0.03540-20-,0 20001000 400040-20-0> -20-i- 40-W- 80-- 80- 10050 '00^8000time (msec)30 '00(g) g-0.1Chapter 4. Behavior of a Healthy Heart^ 52Figure 4.2: Transmembrane potentials of a Purkinje fibre cell (solid trace) and a ven-tricular myocardial cell (dashed trace) for increasing values of the coupling conductanceg (mS/cm2 ).Chapter 4. Behavior of a Healthy Heart^ 53depolarization of its resting membrane and the action potentials of the two cells becomeincreasingly similar (Figure 4.2f).While the changes in cycle length which accompany the increases in coupling strengthare not as pronounced for the Purkinje fibre cell as for the nonspontaneous musclecell, they are significant. Figure 4.3 shows the period of oscillation of the Purkinjefibre cell as a function of the coupling conductance g. For weak coupling, 0 < g <0.0255 (approximately), the period is less than it was prior to cell interaction due tothe subthreshold depolarizations which the ventricular myocardial cell induces in thepacemaker cell. The period continues to decrease with increases in g until g reaches acritical value of approximately 0.0255 mS/cm 2 . At this point, an action potential is,for the first time, induced in the nonspontaneous cell and there is a sharp increase inthe pacemaker cell's period to a value somewhat higher than the cycle length prior tocoupling. This sharp rise continues over a small range of g where the muscle cell isperiodically activated but does not yet oscillate at the same frequency as p. When greaches approximately 0.026 mS/cm2 , the cells fire at the same rate and for g > 0.026mS/cm2 the period plotted in Figure 4.3 is the common period of both cells. Furtherincreases in coupling strength produce a short interval of decrease followed by a regionof slow increase in the entrained cycle length. Although not shown, the cycle lengthapproaches an upper limit of approximately 850 msec.These results are in close agreement with those of van Capelle et al. [34] who, usinga simple two state variable model of excitable elements, studied the interaction of apacemaker cell and a nonpacemaker cell and also noted that when the cells beat syn-chronously the common oscillation period is somewhat longer than the intrinsic rateof the pacemaker cell. This increase in cycle length is caused by a hyperpolarizing(outward) current flowing from the ventricular myocardial (nonpacemaker) cell to theChapter 4. Behavior of a Healthy Heart800780547607407200.00^0.02^0.04^0.06^0.08^0.10g (mS/cm 2 )Figure 4.3: Period of oscillation of the Purkinje fibre cell as a function of couplingconductance g (mS /cm 2 ).Purkinje fibre (pacemaker) cell. This occurs during the latter part of phase 4 depolar-ization when the membrane potential of the nonpacemaker cell is more negative thanthe membrane potential of the pacemaker cell.4.2 The Production of a HeartbeatImmediately and for a short time after the production of an action potential, a cell is inan absolute refractory state where no stimulus, no matter what its strength, can cause asecond discharge; therefore, when an impulse reaches a cell and causes its excitation, theaction potential generated cannot cause neighbouring cells which immediately precededit in the conduction path to fire. For this reason, in a healthy heart, where actionpotentials are generated at, and spread radially from, the SA node, these impulsestravel through the atria to the AV node, then to the Purkinje fibre network and finallyto the muscles cells of the ventricles where the refractory nature of the surrounding tissueChapter 4. Behavior of a Healthy Heart^ 55Figure 4.4: Schematic representation of the model used to simulate the production of aheartbeat. Arrows indicate flow of coupling currents between cells.causes them to die out. There is no retrograde conduction of impulses in the reversedirection back towards the SA node.In simulating this generation and propagation of action potentials which produceseach heartbeat (where a heartbeat consists of the spread of an action potential fromthe SA node throughout the entire heart causing the atria and ventricles to contract inturn), cells representing the major components of the heart's conduction system werecoupled as follows. An SA node cell (s) with an intrinsic frequency of 79.65 cycles/minwas coupled to an AV node cell (a) with an intrinsic frequency of 75.66 cycles/min. TheAV node cell was coupled to a Purkinje fibre cell (p) which had a frequency of 39.84cycles/min and which, in turn, was coupled to a nonpacemaker ventricular myocardialcell (v) (Figure 4.4). These particular frequencies were chosen as they are representativeof average rates in a healthy heart (Section 2.2.5). The coupling conductance go,,, wasset to zero so that no coupling current would flow from the AV node back to the SAnode. Similarly, gp ,a and gv ,p were set to zero. While preventing reverse conduction, theChapter 4. Behavior of a Healthy Heart^ 56simulation had to, at the same time, ensure that impulses originating at the SA node didin fact reach the cells of the ventricles. Setting gs,,, to a value high enough for an actionpotential conducted from the SA node cell to depolarize the AV node cell membranepotential to at least its threshold value guaranteed AV node excitation. Similarly, g a ,pand gp , v were given sufficiently high values to ensure impulse propagation to the Purkinjefibre cell and the ventricular myocardial cell.To completely characterize the coupling currents, delays corresponding to actionpotential conduction times between pairs of interacting cells must be specified. Becausethe simulation is for a healthy heart, the information in Section 1.1.3 can be used toassign values to T ,a 7 Ta ,p and Tp,v. . The duration of atrial depolarization, which has anaverage value of between 60 and 120 msec, is the time required for an impulse generatedat the SA node to spread over the atria and reach the AV node. In the current simulation,this corresponds to the conduction time TB ,a . Passage through the AV node requiresapproximately 50 msec and is indicative of the conduction time Ta ,p between cells ofthe AV node and the Purkinje fibre system. Finally, the propagation time between aPurkinje fibre cell and a ventricular myocardial cell varies considerably from a valuenear zero, when the two cells are adjacent, to a maximum in the range of 60 — 100 mseccorresponding to the average duration of ventricular depolarization. Table 4.4 reportsthe actual values of the coupling conductances and propagation time delays used in thesimulation.Results of the simulation are demonstrated in Figure 4.5 which shows the electricalactivity of each model cell. The temporal displacement of phase 0 in the action potentialsof Figures 4.5a-d is indicative of the propagation times between the cells. The SA nodecell fires independently at its intrinsic frequency of approximately 80 cycles/minute (toptrace). After the production of each impulse there is a delay due to atrial depolarizationbefore the impulse reaches the AV node cell causing it to fire in response. After a furtherChapter 4. Behavior of a Healthy Heart^ 5740 -20-E -20-„-------)-BO- - 100(a) Sinus node cell40-20--20-a -40-1.„)La-80- 80-- 100^(d) AV node cell40-20-^0 ^-20-a -40-w-BO-- BO- 100 ^40-20-(g) Purkinje fibre cell-20-P -40-W-60--6o--1002000^3000^4000^5000(d) Ventricular myocardial cellFigure 4.5: Electrical activity of SA node, AV node, Purkinje fibre, and ventricularmuscle cells (traces (a) - (d), respectively) during the simulation of a heartbeat. Tracesshow the activity of the cells beginning 2 sec after coupling was introduced when thecells have settled into a regular rhythm.50 100time (msec)Chapter 4. Behavior of a Healthy Heart^ 58Cellsi, jgij(mS/cm2 )ri,j(msec)s, a 0.03 100a, p 0.05 50p,v 0.10 10Table 4.4: Coupling conductance and delay values.delay, representing the propagation time through the AV node to the Purkinje fibrenetwork, an action potential is produced in the Purkinje fibre cell and finally, when theimpulse reaches the nonspontaneous ventricular myocardial cell it too fires. Each cell isexcited in turn as the impulse spreads. This process is repeated for every action potentialproduced at the SA node. Consequently, the AV node and Purkinje fibre cells no longerfire at their own intrinsic frequencies of approximately 75 and 40 cycles/min, respectively,but are entrained to the SA node with its faster rate of spontaneous depolarization.Similarly, the nonspontaneous muscle cell which, prior to coupling produced no actionpotentials, now fires at the SA node frequency.Figure 4.6 plots the time corresponding to the peak or maximum membrane potentialof each cell along the horizontal axis, and the time between the peak of an action potentialat the SA node and the peak of the subsequent action potential of each of the other threemodel cells along the vertical axis. The time corresponding to the peak of an actionpotential of a particular cell, cell i, is denoted ti . For each action potential initiated atthe SA node there is a corresponding action potential induced in each of the other threecells indicating that all cells beat synchronously. The common cycle length (given bythe horizontal distance between the firings of each cell) is approximately 753 msec whichis the intrinsic oscillation period of the SA node cell. For two cells, cell i and cell j,the latency between the activation of cell i and the next activation of cell j is denotedChapter 4. Behavior of a Healthy Heart^ 59200—0000000000000150 =^ p^AAA 0000000000001007 0• is^ i sAO tv0 • ^•^• ,^.^.^•^JO .^. 0. ,^.•0 2000 4000^6000 8000 10000time (msec)Figure 4.6: Peak action potential times of the AV node, Purkinje fibre, and ventricularmyocardial cells relative to the peak action potential times of the SA node during asimulation of the production of a heartbeart. t i is the time corresponding to the peakof an action potential of cell i, where i is one of s,a,p, or v.50L.Chapter 4. Behavior of a Healthy Heart^ 60/i,i and is represented by the vertical distances between cell firing times. The values ofthese peak-to-peak delays are approximately:to — t8 = 102,la P = tp —ta = 53,= tv — tp = 14.These values are slightly higher than the action potential propagation times used in thesimulation (Table 4.4) due to the fact that once an impulse reaches a cell there is a smalldelay before the cell is excited and its membrane potential reaches its maximum.Chapter 5Modulated Ventricular Parasystole5.1 Cardiac Arrhythmias and Ectopic PacemakersArrhythmias or irregular cardiac rhythms, are caused by abnormalities in both impulseinitiation and conduction. Abnormal impulse initiation may refer to an alteration inthe rate of the primary pacemaker or the initiation of impulses at a nonsinus location(ectopic focus). Subsidiary pacemaker cells normally remain latent because the SA(sinus) node produces impulses at a rate faster than other cardiac cells; however, undervarious pathological conditions a cell or group of cells other than those of the SA node canbecome a site of pacemaker activity. For example, when the rate of impulse productionat the SA node is reduced significantly so that it is smaller than the rate of a secondarypacemaker, the ectopic focus can exhibit its spontaneous activity.When an ectopic pacemaker produces an action potential in addition to the primarypacemaker, there will be a collision of the two impulses. The action potentials maycollide while propagating in opposite directions causing both to be extinguished. Theyalso may travel in similar directions in which case the second impulse will be delayedbecause of the refractory nature of the tissue. This interaction and competition betweenpacemaker sites for control of the myocardium results in various arrhythmias.61Chapter 5. Modulated Ventricular Parasystole^ 625.2 Modulated Ventricular ParasystoleMany of the more serious cardiac arrhythmias result from a situation known as parasys-tole which involves the simultaneous activity of two (rarely more) pacemaker sites. Inpure parasystole one pacemaker functions completely independently of the other, whilein modulated parasystole each pacemaker is affected by the activity of the other. Usu-ally, one of the pacemaker sites is the SA node and the other an ectopic focus which,although it can be located anywhere in the heart, is usually found in the ventricles. Thisis known as modulated ventricular parasystole.Figure 5.1 shows a schematic representation of a simple model used to simulate mod-ulated ventricular parasystole. Since most muscle cells are not capable of spontaneousdepolarization, the ventricular ectopic focus is assumed to be located in the Purkinjefibre system. Thus, the simulation involves a model SA node cell (s) coupled to a modelPurkinje fibre cell (p). Atrial depolarization has an average duration of between 80 and120 msec and an impulse requires, on average, a further 50 msec to pass through theAV node (Section 1.1.3). The sum of these two durations gives an indication of the timerequired for an action potential generated at the SA node to reach the Purkinje fibresof the ventricles, thus, in these simulations Ts ,p is given a value of 150 msec. Makingthe assumption that impulses propagating in the reverse direction are slowed somewhat,perhaps due to the refractory nature of the tissue, rp,, was assigned a value of 160 msec.The interaction of the cells for various values of the coupling conductances, g s ,p and gp ,s ,were studied and simulations repeated using model cells with different intrinsic cyclelengths, T, and Tp .When two pacemakers of different intrinsic oscillation periods are coupled, and thestrength of the coupling is increased, the ratio of their periods approaches unity [30,39].For the present simulation, this means that as one (or both) of g s ,p and gp , s is increased,Chapter 5. Modulated Ventricular Parasystole^ 63Figure 5.1: Schematic diagram of the model used to simulate modulated ventricularparasystole. A model SA node cell (s) is coupled to a nonadjacent model Purkinje fibrecell (p). Arrows indicate flow of coupling currents between cells.the ratio Tp/T, approaches 1. Near certain values of the coupling conductances, g,,p andgp,,, the cells become entrained so that their mean cycle lengths are related as simpleintegral values. This type of entrainment in which there are m periods of .s to every nperiods of p is denoted m:n entrainment and is a commonly occurring form of cardiacarrhythmia. If the influence of one cell on the other is strong enough, the cells willbeat synchronously (1:1 entrainment). Furthermore, the cell with the stronger influencewill lead the other and determine the entrained oscillation period. In all computationsof modulated ventricular parasystole, the cells were allowed to interact for at least 6seconds or until a definite pattern of s : p firings could be recognized. The average timebetween the action potentials of each cell was calculated for each cycle of the pattern.The period of the cell was then calculated as the mean of these averages over all observedcycles of the pattern.5.3 The Lead CellIn a simulation involving two cells, i and j, the latency, denoted li,j, between theirimpulses is defined as the time from the peak or maximum voltage of an action potentialT1:1••4^Chapter 5. Modulated Ventricular Parasystole^ 64t i^t,14---+-1,,Figure 5.2: Action potentials of cells i and j at 1:1 entrainment. The solid and dashedcurves represent the activities of cell i and cell j, respectively.of i to the peak of the next action potential of j. When the cells have synchronizedto 1:1 entrainment their action potentials alternate and they oscillate with a commonfixed period denoted That is, 7; = T; = In addition, the time, 1 ; ,; , betweenevery action potential of i and subsequent action potential of j is constant over theentire simulation. Similarly, the latency, / i,i , between every action potential of j andsubsequent action potential of i is constant throughout the simulation (Figure 5.2). In atwo-cell interaction, the lead cell at 1:1 entrainment is that cell whose action potentialscause the excitation of the other most quickly after their arrival at the other cell. If anaction potential induced in j is caused by the arrival of the preceding impulse originatingChapter 5. Modulated Ventricular Parasystole^ 65at i then the excitation of j must occur after the impulse from i reaches j. That is,>^(5.1)If j is activated before the arrival of the impulse from i then its excitation cannot havebeen caused by the arrival of the incoming current from i; therefore, (5.1) constitutes anecessary condition for i to be the lead cell.Over the range of coupling conductances studied (g8 ,9 E [0, 0.1] (mS/cm2 ), gp,. E[0, 0.01] (mS/cm2 )) in the computations of modulated ventricular parasystole, it wasfound that the entrained period was always greater than the round-trip propagationtime from one cell to the other and back again. That is,^T1:1 ^+ 19, 8 -? 7.8 ,P^TP,8*^ (5.2)Thus, at least one of the following must be true:and/or19, 8 > TP, 818 ,9 ^T ^ 8 ,P(5.3)(5.4)The lead cell is determined by comparing the peak-to-peak times between the actionpotentials of the two cells with the impulse propagation times 78 ,p and Tms . If 18 ,p <then s violates (5.1) and is not the lead cell; however, from (5.3), p will satisfy thenecessary condition (5.1). Each activation of s occurs after the arrival of an impulsefrom p. In this case p is said to be the lead cell and the entrainment is referred to as 1:1p entrainment. Similarly, if lP 8 < r9 , 8 then p violates the necessary condition (5.1) andso cannot be the lead cell; however, s will satisfy (5.1) and the entrainment is referredto as 1:1 s entrainment.Finally, if bothChapter 5. Modulated Ventricular Parasystole^ 66and18,P > T,P -1P1 8 > 7-P, 8(5.5)(5.6)then, in this case, the lead cell is the one which causes the other to be activated themost quickly after the arrival of its impulse. If18,p - 7-8,P < 1P,8 ^Tp,8 (5.7)then s is designated as the lead cell (1:1 s entrainment). On the other hand, if^Tp,8 < 18,P - T8 1P^ (5.8)then p is said to be the lead cell (1:1 p entrainment).5.4 Tp/T, = 3For the first set of simulations of modulated ventricular parasystole, a model SA nodecell with an intrinsic cycle length T8 = 502.44 msec (frequency = 119.42 cycles/min)was coupled to a model Purkinje fibre cell with an intrinsic cycle length T p = 1506 msec(frequency = 39.84 cycles/min) so that an approximate 3:1 ratio existed between theperiods Tp and T, prior to coupling.5.4.1 Regions of Synchronous and Asynchronous BehaviorFigure 5.3 shows the regions of synchronous and asynchronous behavior for a range ofthe coupling conductances g.,p and gp,, where 0 < g,,p < 0.1 mS/cm 2 and 0 < gp,, < 0.01mS/cm2 . For reasons to be outlined in Section 5.7, the Purkinje fibre cell exerts a greaterinfluence on the SA node cell for equal values of the coupling conductances gs ,p and gp , 8than vice-versa; therefore, to obtain a comparable level of influence of p on s, the rangeChapter 5. Modulated Ventricular Parasystole^ 67of values of gp , s used in the simulations of modulated parasystole was smaller than therange of values of g s ,p . The four curves represent cubic spline interpolations of discretedata points given by coupling conductance pairs (g,,p , gp ,8). Approximately 200 suchcoupling conductance pairs were used to obtain the curves separating the regions ofentrainment. Each computation was run until a definite pattern of s : p firings could berecognized and there was little or no variation in the periods of the cells between cyclesof the pattern. Much of the curves were obtained by fixing gs,p at a particular value andthen systematically increasing gm.. If two values of gp , s were found, one for which thecells beat synchronously and one for which they did not, a third computation using theaverage gp , 8 value was done. This method of bisection was continued until the boundaryseparating the regions of synchrony and asynchrony was located within approximately0.0001 mS/cm2 . The dashed curves indicate values of the conductances which resulted in1:1 entrainment while the solid curves represent values of the conductances for which thecells beat asynchronously. The actual boundaries of the regions lie somewhere betweenthe pairs of dashed and solid curves. The lower plot of Figure 5.3 shows a particulararea of the larger plot at a scale which emphasizes the existence of the four curves.When there is no interaction between s and p (gs ,p = 0 = gp , ^ ) the cells fire attheir own intrinsic frequencies of 119.42 and 39.06 cycles/minute, respectively. At weakcoupling (gs ,p < 0.05192 mS/cm2 , gp ,s < 0.004 mS/cm2 (approximately)), correspondingto the region labelled 'not 1:1', neither cell exhibits enough influence on the other tocause 1:1 entrainment.When gp ,s = 0 mS/cm 2 , p has no influence on s so that when gs ,p is sufficiently high(> 0.05192 mS/cm 2 ) to cause synchrony, p is entrained to s. The positive slope of thecurves defining the left boundary of the 1:1 s region indicates that as gp , s is increased,the effects which p begins to exert on s require that g8 ,p also increase in order that theinfluence of s on p is sufficient to maintain the 1:1 s entrainment. Conversely, as g s ,p isTs = 502.44 msecT = 1506.00 msec1:1 so.00s ^0.0525^0.0530 0.0535^0.0540 0.0545^0.055068Chapter 5. Modulated Ventricular Parasystole0.0101:1p(a)0.008"E 0.006V)E•• 0 • 004CI)INot 1:10.002 1:1Not 1:10.0000.00 0.01^0.02^0.03^0.04^0.05^0.06^0.07^0.08^0.09^0.10gsp (mS/cm2 )gsp (MS/CM2)Figure 5.3: Regions of entrainment for Tp/T, = 3.Chapter 5. Modulated Ventricular Parasystole^ 69increased over the range [0.05192, 0.055495], the stronger influence s exerts on p resultsin an expansion of the region of 1:1 s entrainment. Finally, when the coupling currentfrom s to p is strong enough (gs ,p > 0.0555 mS/cm2 ) the cells are at 1:1 s entrainmentover the entire range of gp , s from 0 to 0.01 mS/cm2 .The Purkinje fibre cell also can be the lead cell. When gp , s is greater than somecritical minimum value near 0.004 mS/cm 2 and the ratio gp , s /gs ,p is sufficiently high, sfires synchronously with p (region labelled 1:1 p).For high values of gp,,, the regions of synchrony to p and to s appear to be almostcontiguous. Moving horizontally along a fixed value of gp , s from the 1:1 p region to the1:1 s region by increasing gs,p causes dramatic changes in the periods Ti, and T.. Forexample, with gp ,s at its maximum value of 0.01 mS/cm2 and gs ,p = 0.55495 mS/cm 2 ,the common cycle length is 1255 msec. The time between each action potential at sand each action potential at p is given by /.,p = 1063 msec while the latency fromthe activation of p to the activation of s is given by 1p,. = 172 msec; therefore, fromSection 5.3, p is the lead cell. Keeping gp , s fixed at 0.01 and increasing gs ,p by only0.000005 to a value of 0.0555 results in 1:1 s entrainment with a common cycle lengthof only 409.5 msec and latencies given by /., p = 204.0 msec and /p,. = 205.5 msec. It ishypothesized that between these two values, g.,i, = 0.055495 mS/cm2 and g34, = 0.0555mS/cm2 , which correspond to two very different types of 1:1 entrainment with the cyclelength differing by approximately 745.5 msec, there are values of g34, for which the cellsare not synchronized and that the region marked 'not 1:1' extends upwards separatingthe 1:1 p and 1:1 s regions.Chapter 5. Modulated Ventricular Parasystole^ 705.4.2 1:1 s EntrainmentWhen s is the lead cell in 1:1 entrainment (5.3), the entrained oscillation period is closerto the intrinsic cycle length of s than to the intrinsic cycle length of p. Figure 5.4 demon-strates the changes which occur in Tp and T, as the cells approach 1:1 s entrainment forthe specific case where gp , s is fixed at 0.001 mS/cm2 and gs ,p is increased from 0 to 0.1mS/cm2 . This corresponds to moving along a horizontal line at gp,, = 0.001 mS/cm2 inFigure 5.3a. When g s ,p = 0 mS/cm 2 , the cycle lengths of s and p are 530.25 msec and1505.8 msec, respectively, so that TpIT„ = 2.84 (Figure 5.4b). As g s ,p is increased themean cycle lengths decrease although the changes are much more pronounced in T p thanin T8 . When gs ,p reaches 0.051919 mS/cm 2 , there is an abrupt decrease in Tp and finally,for gs ,p = 0.05192 mS/cm2 , p fires with the same period (474 msec) as s and Tp/T, = 1.The latency 18 ,p between an action potential of .s and subsequent activation of p is 219msec while /p,, = 255 msec; therefore, .s is the lead cell. With further increases in g s ,pto 0.1 mS/cm2 , 1:1 s entrainment continues with slight decreases in the entrained cyclelength to a value of 454 msec, a value which is less than the intrinsic oscillation periodsof both s and p.Figure 5.5 shows, for gp ,s fixed at 0.001 mS/cm 2 , the effects of increases in g s ,p on theaction potentials of the two cells. In (a) with gs ,p = 0 there are almost three completeaction potentials of s for each action potential of p while in (b), where gs ,p = 0.03mS/cm2 , there is 2:1 entrainment. Increasing g 8 ,p to 0.051918 mS/cm2 results in 3:2entrainment and for g s ,p = 0.1 mS/cm2 the cells exhibit 1:1 s entrainment.5.4.3 1:1 p EntrainmentWhen the Purkinje fibre cell is the lead cell in 1:1 entrainment (5.3), the entrainedoscillation period is closer to the intrinsic cycle length of p than to the intrinsic cycleChapter 5. Modulated Ventricular Parasystole^ 710.04^0.06g s,2, (mS/cm2 )(a)0.08^0.10gp,s=0. 001 (mS/cm 2)0.00^0.023^CIDED^0.5^0.00 1 '^1^'^"^I^1^1^1^'0.04 0.06 0.08gsp (MS/CM2)(b)0.02 0.10Figure 5.4: Changes in cycle lengths Tp and T, for gp,,, = 0.001 mS/cm 2 and increasinggs ,p (mS/cm2).I40—20—0 5:4 —20——40—1.a.1—60—80—•—100^I^ i SI^$I II^ 12$^A ItY.---1 ,/9I $• 1....^$ ... ... .... ""'^$ 0. O.. WOSi• ••••^ ko da. . ....Iwall•.1 OD40—20—0^I—20-1 I I—40— I—60——80—r—100 I (a) ge2,=0.0E —20—40—60—80—100(d) gcp =0.1 time (msec)(c) g.. p =0.051918II III I •I •I^•ELatt /40-120-0—207—60——80--1130 ^(b) g,, p =0.0 3Chapter 5. Modulated Ventricular Parasystole^ 72Figure 5.5: Action potentials of cells .s and p for gp,, = 0.001 mS/cm2 and increasinggs ,p (mS/cm2 ). Solid curve is the SA node cell. Dashed curve is the Purkinje fibre cell.Chapter 5. Modulated Ventricular Parasystole^ 73length of s. Figure 5.6 shows the changes which occur in Tp and T. as the cells approach1:1 p entrainment. Here, as a specific example, g s ,p is fixed at a value of 0.04 mS/cm 2and gp ,s is increased over the range [0,0.01] mS/cm 2 . This corresponds to moving upwardalong a vertical at g8 ,p = 0.04 mS/cm2 in Figure 5.3.When gp , s = 0 mS/cm 2 the mean cycle lengths of s and p are 502.71 and 1005.3 msec,respectively, and the cells exhibit 2:1 entrainment. As gp , s increases, the cycle lengths Tpand T. increase (in contrast to the previous case of 1:1 s entrainment where increasedcoupling caused decreased cycle lengths). The ratio Tp/718 , however, remains constantat approximately 2 until gp , s reaches a value near 0.004 mS/cm 2 when, due to an abruptincrease in the period of s, there is a corresponding decrease in Tp/Ts . With anotherslight increase to gp , s = 0.00465 mS/cm2 , the cycle lengths become equal and s firessynchronously with p with a common period of 1186 msec. At this point, the latencies/8 ,p and 1p,. are 982 and 204 msec, respectively; therefore, p is the lead cell. Withadditional increases in gp,,, 1:1 p entrainment continues with corresponding increasesin the entrained cycle length which reaches a value of 1261.5 msec when gp , s = 0.01mS/cm2 . Throughout the region of 1:1 p entrainment the common cycle length is closerto the intrinsic cycle length of p than to the intrinsic cycle length of s. Furthermore,in this case, the entrained period takes on values between the intrinsic cycle lengths(T8 = 502.44 msec, Tp = 1506 msec) in contrast to the case of 1:1 s entrainment wherethe common period at synchrony was always less than the intrinsic cycle lengths of sand p.Figure 5.7 shows the changes in the action potentials of s and p for gs ,p fixed at0.04 mS/cm2 and increasing values of gp ,s . In (a) there is 2:1 entrainment. In (b) and(c) the cells undergo 5:3 and 4:3 entrainment patterns, respectively. Finally, (d), withgp , s = 0.0047 mS/cm2 , demonstrates 1:1 p entrainment.Earlier it was noted that these patterns of m:n entrainment are common in clinical10.008^0.0101400-1200-..--,0o woo0E-00t.• 800-n_600-400x. x—x-x—x—x-x—x",^I^,^,^1^,^,^,^1^,0.002 0.004 0.006gp,s (mS/cm2)(a)0.0002.5-g s,22=0.04 (mS/crn 2 ) gyp=0.04 (mS/cm 2 )0 Tsx TPChapter 5. Modulated Ventricular Parasystole^ 740.5^0.000.^1 1^1^.^I^1^.^I^10.002^0.004^0.006 0.008 0.010gps (MS/CM 2)(b)Figure 5.6: Changes in cycle lengths Tp and T, for gs ,p = 0.04 mS/cm 2 and increasinggp,s (mS/cm2).^....--------1-- w^1^1^I^I I, 1^,^,%....— -" % ...— ''^% ...a1 `a11(b) g p ,=0.004340—120— II%1E> —20—''--- —40-,1.4—60——80——100a1a^i1 1I^I.., .... —.. ....a V^a1 ^% _... ... IIIa ..---—(c) g r , s=0.004540—Chapter 5. Modulated Ventricular Parasystole^ 75I^1^ia, ‘ _,40—20—0> —20E—40-W—80——80—100^; ^1^.1II k 1^L lit 1^Pii t h InI,^,1,^h1% , I ,1 t , 11^,^1 % ,1 ,^, 1 , 11 , •1^, , 1 1^,U'^t ^1 %,,,, ./.2, k / JAI ^ ^/,..i? , . 1/ /, ,^.., ,^,^1 , .^•^ ^•..,..._ - -^,,,.. -^... _ ,,,,....-^.. • _ -(a) 9 2,,,=0•0■aIa^a 2?%I /^1^I■ •••• --^% / % .... ..•Ia40-20—0^> —20-I %ELa— 80—40—../....„..)1^1I^%.(19%—80—..—100^20—^IIOAI> —20—^1 %E^„......1iE -40— ILai—60—^I)^l— 80—1001L.Ia^/t ... /% .... ..1II%a■I%I^%1I1II^ia^_—I'II^%I^■%1^^1^N —^,....^si^''''.sala aI^i1% 1aI/II11II1Ia^I^1I... 1 l% .... •••••^%. ---a/ —1I^aatI11% , .... -...1fa(d) gp.,=0.0047 time (msec)Figure 5.7: Action potentials of cells 8 and p for ga ,p = 0.04 mS/cm2 and increasing gp,,(mS/cm2 ). Solid curves indicate SA node cell. Dashed curves indicate Purkinje fibrecell.Chapter 5. Modulated Ventricular Parasystole^ 76!..^:•••••••mannuninur ow ins won unworn srammun1111111111111111211111IDEIMIIIIIIIIMENDRIMOUHIPIEIIIIIIMIPIIIIIMIIIIMIEMPIMMIEN411.1=2,21.1 71=11 "•^ •^•Figure 5.8: ECG recording showing 4:3 entrainment. Arrows indicate atrial contractionsand the large spikes indicate ventricular contractions (taken from [6], p.284).electrocardiography (ECG). Figure 5.8 shows an ECG recording from a patient expe-riencing a 4:3 entrainment arrhythmia. The arrows indicate atrial depolarization whilethe large peaks correspond to ventricular depolarizations. This is analogous to the resultof Figure 5.7c where there are 4 SA node action potentials (atrial contractions) for every3 Purkinje fibre action potentials (ventricular contractions).5.5 Tp/T, = 2For the second set of calculations of modulated ventricular parasystole the same modelPurkinje fibre cell was coupled to an SA node cell with a period of oscillation 1.25 timeslonger than in Section 5.4, i.e., the intrinsic cycle lengths of p and s were Tp = 1506.0and T. = 753.25 msec respectively. Hence, the ratio of the intrinsic periods Tp/T, wasapproximately 2. These cycle lengths correspond to approximate intrinsic frequenciesof 80 and 40 cycles/min which are normal healthy rates for Purkinje fibre and SA nodecells, respectively.Chapter 5. Modulated Ventricular Parasystole^ 775.5.1 Regions of Synchronous and Asynchronous BehaviorFigure 5.9 shows the regions of synchronous and asynchronous behavior of the two cellsfor the coupling conductances g s ,p and gm8 over the same range of values as in Section 5.4.A comparison with Figure 5.3 shows that the behavior is very similar to the casewhere Tp/T, = 3. Again, for small values of g s ,p and gp ,8 the cells fire asynchronously.For sufficiently high values of g,,p/gp , 8 with g8 ,p greater than some critical value near0.036 mS/cm2 , the cells exhibit 1:1 s entrainment. Also when gp , s is greater than someminimum value near 0.0004 mS/cm 2 and the ratio gp,./gs ,p is sufficiently large, p becomesthe lead cell and 1:1 p entrainment occurs. As for the previous set of computations, itappears that the region of asynchronous behavior extends upward to separate the regionsof 1:1 s and p entrainment.The main difference between this and the previous case lies in the relative sizes ofthe regions of synchronous and asynchronous behavior of the cells. The 1:1 s regionhas expanded over lower values of gs ,p to include a region defined only for very smallgp,, and where the slope of the curves defining its boundary are almost zero. The sloperemains near zero until gs ,p reaches approximately 0.055 mS/cm 2 , (a value only slightlyhigher than the corresponding value in the previous set of calculations with TplTs = 3)when the boundary becomes almost vertical. More noticeably, the 1:1 p region has alsoexpanded from the previous case. Here entrainment to p occurs for sufficiently highvalues of the ratio gp , s /gs ,p where gp , s is greater than some critical value near 0.0004mS/cm2 whereas previously, for Tp/Ts = 3, the critical value was almost ten timeslarger (,:-.. 0.004 mS/cm 2 ). This expansion of the regions of 1:1 entrainment suggests thewell-known result that the closer the intrinsic cycle lengths of the cells, the more easily(for weaker coupling) synchronization is obtained [39].1: 1Not 1:10.002 -:.0 .0 0 0 —)0.00 0.02/ Not 1:10.04^0.06Chapter 5. Modulated Ventricular Parasystole^ 780.010omos -:00"..01E 0' 006—tnE1/, 0.0047El,^-0)Ts = 753.25 msecTP = 1506.00 msec1:1p1 :1 sANITVoilmt0.10I^.^,^.0.08gs p (m SA m2 )Figure 5.9: Regions of entrainment for Tp/T, = 2.Chapter 5. Modulated Ventricular Parasystole^ 795.6 Tp/T. = 0.75In each of the previous two sets of computations of modulated ventricular paraystole,a model Purkinje fibre cell with an intrinsic period of oscillation of 1506 msec wascoupled to an SA node cell with a shorter period so that Tp/T. > 1. For the final setof simulations a model Purkinje fibre cell with an intrinsic cycle length of 501.11 msec(frequency = 120 cycles/minute) was allowed to interact with a model SA node cell withan intrinsic cycle length of 753.25 msec (This is the same model SA node cell used inSection 5.5 with Tp/T. = 2.). In this case, the ratio of the intrinsic periods of the cellsis given by TpIT. = 0.75.5.6.1 Regions of Synchronous and Asynchronous BehaviorFigure 5.10 illustrates the values of the coupling conductances gs ,p and gp , s for whichthe cells fire synchronously and asynchronously. For the values of g s ,p and gp , s studied,the cells do not exhibit 1:1 .s entrainment at all and the 1:1 p entrainment region hasexpanded over the full range of gs ,p shown. The positive slope of the lower boundarydefining this region demonstrates that, as the influence of p on s increases, so does therange of values of g s ,p for which s is entrained to p; or, conversely, as the strength of thecoupling from s to p increases so must the coupling from p to .9 in order for p to remainthe lead cell in 1:1 p entrainment.5.7 The Greater Influence of the Purkinje Fibre CellFor the three sets of computations corresponding to Tp/Ts = 3, 2, and 0.75, synchronyto the Purkinje fibre cell occurs for much smaller values of gp,, than the values of gs ,prequired to cause synchrony to s. For example, when Tp/T. = 3 (Figure 5.3) 1:1 p en-trainment occurs for gp,, > 0.0043 mS/cm 2 , whereas, 1:1 .s entrainment does not occurChapter 5. Modulated Ventricular Parasystole 800.0100.008-Ts = 753.25 msecTP = 501. 11 msec.0"E 0 006-V)0.004-g,tr)1:1 p0.0020.0000.00Not 1:1 NOT 1:11:10.02 0.04 0.06 0.08 0.10gsip (mS/cm2 )Figure 5.10: Regions of entrainment for Tp/T, = 0.75.Chapter 5. Modulated Ventricular Parasystole^ 81unless gs ,p > 0.05192 mS/cm2 . Similarly, in the case when Tpas = 2 (Figure 5.9), syn-chrony to p occurs for gp,, just slightly greater than 0.0004 mS/cm 2 ; however, synchronyto s occurs only when gs ,p > 0.036 mS/cm 2 . Lastly, when Tp/T, = 0.75 (Figure 5.10),1:1 s entrainment does not occur for any of the values of g a ,p studied. In all cases, for s tobe the lead cell at 1:1 s entrainment, gs ,p must be over ten times greater than the valueof gp,, required for p to be the lead cell. These findings indicate that for equal values ofthe conductances, p has a much stronger influence on s than .9 has on p. Two factorswhich affect the activation of a cell by an external current are the strength and durationof the stimulus. Because the most negative potential of the Purkinje fibre cell (P.,' —85mV) is much less than the minimum potential of the SA node cell (=:-.,' —64 mV) andbecause the threshold for p is —60 mV, the potential of p will be less than the potentialof s for almost the entire duration of phase 4 depolarization of p. This will induce in.s a depolarizing current which will contribute to its excitation. Furthermore, the rateof depolarization of the Purkinje fibre cell's membrane potential during phase 0 is veryrapid and will have a more pronounced effect on s than the slower rate of depolarizationof s will have on p.5.8 The Lead Cell and the Entrained Oscillation PeriodIn previous studies, Ypey et al. [39] used a simple Hodgkin-Huxley-type model of apacemaker cell, and Michaels et al. [30] used the same SA node model due to Yanagiharaet al. [38] to study interactions of identical adjacent cells with different intrinsic periods.They found that at 1:1 entrainment the cell with the shorter intrinsic cycle length wasalways the lead cell. Furthermore, the entrained cycle length was, not only closer to theintrinsic period of oscillation of the lead cell, but had a value between the intrinsic cyclelengths of the coupled cells. For the present model, the results also suggest that theChapter 5. Modulated Ventricular Parasystole^ 82entrained period of oscillation is closer to that of the lead cell; however, here either cellcan be the lead cell. This result is also supported by the phase response curves obtainedin Chapter 3. There it was found that the shape of the PRCs, and subsequently thecell which undergoes the smaller phase shifts (from the results of the current chapter,this cell is the lead cell), was dependent on the values of the conductances gp,,,, and gs ,p .The computations of modulated ventricular parasystole also demonstrated that the cyclelength at 1:1 entrainment was not restricted to values between the intrinsic periods of sand p. For example, for the case where Tp/TS = 3 and gs ,p = 0.04 mS/cm2 , the entrainedperiod in the 1:1 .s region was smaller than the intrinsic cycle lengths of both s and p(Figure 5.4).5.9 The Effect of Changes in Tp and T.A comparison of Figure 5.3 where Tp/Ts = 3 and Figure 5.9 where Tp/TS = 2 indicatesthat for Tp > T. the closer the intrinsic periods of s and p, the more easily the cells aresynchronized. Both regions of 1:1 s entrainment and 1:1 p entrainment became largerwith the decrease in Tp/TS suggesting that the closer the intrinsic periods of oscillationthe weaker the coupling strength required for synchrony. Decreasing the ratio Tp/TS to0.75 resulted in a further increase in the size of the 1:1 p region; however, the 1:1 sregion ceased to exist for the range of gs ,p studied. The difference here is that p fires ata faster rate than s and combining this with the fact that p exerts a greater influenceon s than s on p, means that 1:1 s entrainment is more difficult to achieve.Chapter 6Discussion6.1 Summary of the Heart ModelVarious aspects of the electrophysiology of the heart, such as phase resetting and mu-tual entrainment, due to the initiation and conduction of cardiac action potentials havebeen investigated through the dynamic interactions of model cells representing the ma-jor components of the heart's conduction system. These cells of the SA node, AVnode, Purkinje fibre system, and ventricular myocardium were modelled using Hodgkin-Huxley-type oscillators. The electrical activities of the SA node, Purkinje fibre, andventricular myocardial cells were simulated using the models devised by Yanagihara etal. [38], McAllister et al. [28], and Beeler and Reuter [1], respectively. Because the actionpotentials of an AV node cell are very similar to those of an SA node cell, the electricalactivity of the AV node cell was simulated using a model derived from the Yanagiharaet al. model for the SA node cell. In order that the model heart be capable of simulatingthe wide range of both normal and pathological functionings typical of a real heart,certain physiologically based parameters in the model equations were altered to producemodel cells which exhibit a wide range of frequencies. The model allowed the couplingof any number of the four types of cells where the frequencies of the pacemaker cellswere chosen from predetermined sets of values. Cell interaction was simulated by theaddition of purely resistive coupling currents flowing between interacting cells where theform of the coupling current was dependent on whether or not the cells were assumed83Chapter 6. Discussion^ 84to be physically adjacent. An action potential travelling from one cell to another non-adjacent cell will likely take a different path over the tissue separating the cells than animpulse travelling in the reverse direction. Furthermore, the time required for this im-pulse propagation between such nonadjacent cells is significant relative to the durationof the action potentials. These facts were incorporated into the model by means of direc-tion dependent coupling conductances and conduction time delays. For adjacent cells,where the intercell membrane constitutes the entire propagation path, the conductiontimes are negligible and the coupling conductances simply represent the conductanceof the gap junction membrane. Consequently, in this case, the coupling currents werereduced to the form used in previous studies of adjacent cell interaction [5,30,39] wherethe coupling currents flowing between two model cells were of opposite sign but equalmagnitude. Then with the four types of model cells of varying frequencies and a methodof coupling them, various aspects of cell interaction were investigated throughout thechapters of this thesis.6.2 Discussion of ResultsThe changes in cycle length caused by the subthreshold depolarizations induced in onepacemaker by another are a prerequisite to the entrainment of the cells. This phase-resetting phenomenon was investigated in Chapter 3 through the use of phase-responsecurves for various pairs of interacting cells using pulsed-coupling. Biphasic responsecurves similar to those obtained experimentally were produced. As in previous studies,the shape of a PRC was maintained with a decrease in coupling conductance. Also, asexpected, over most of the cycle, the phase shifts were less with weaker cell interaction.The exception was a small range of phase values where delays occur for lower conduc-tances and the transition to advances had already been made with the stronger stimulus.Chapter 6. Discussion^ 85It also was found that with an increase in conductance the transition from maximal cy-cle prolongation to maximal cycle abbreviation occurred over a much narrower rangeof phases. This is analogous to previous experimental results obtained for chick heartcells [13]. The PRCs for fast and slow cells were also studied. In contrast to previousworks studying interactions of similar cells, the phase shifts induced in a slow cell by afaster one were not necessarily larger than the corresponding phase shifts induced in thesame fast cell by the slower one. Again this is due to the direction dependent couplingcurrents and the unequal influences exerted by different types of cells. Behavior demon-strating the greater sensitivity of a slower cell was obtained, however, when a particularcell was coupled to another cell whose intrinsic frequency was subsequently altered. Inthis case, the phase shifts were indeed greater at a given phase for the slower cell. Fi-nally, comparisons were made between phase response curves obtained under conditionsof pulsed and continuous coupling with expected results. PRCs summarizing the effectsof a slower cell on a faster cell were similar for both methods of coupling. The differ-ences lie in the size of the phase delays which were greater under continuous couplingdue to the continuous 'pull' of the slower cell. For the case of a fast cell stimulating aslower one, the PRC obtained under continuous coupling was somewhat different fromthe corresponding pulsed-coupling PRC. This is due to the fact that for certain phasesmore than one discharge of the faster cell will occur during the perturbed cycle of theslower cell. Also, the continuous influence of the fast cell caused delays to be less for thePRC obtained under conditions of continuous coupling.Most muscle cells are nonspontaneous and are only activated by the external stimu-lation of neighbouring cells. For ventricular muscle cells the external stimulation may beprovided by Purkinje fibre cells. This activation of a nonpacemaker cell was computedfor the present model in Chapter 4 and it was found that the level of activation of themuscle cell was dependent on the strength of the interaction. Also, as in an earlier studyChapter 6. Discussion^ 86by van Capelle [34] which uses a simple two-state variable model, the common cyclelength at synchrony was somewhat lower than the intrinsic period of oscillation of thepacemaker cell.A second property of a healthy heart is that the SA node acts as the primary pace-maker and other cells are entrained to its regular rhythm. Impulses generated at the SAnode spread throughout the atria and then the ventricles causing first atrial and thenventricular contraction. This production of a heartbeat was mimicked for the presentmodel in Chapter 4 by the coupling of one of each type of model cell with intrinsicfrequencies and propagation time delays representative of those in a healthy heart. Re-sults demonstrated that each impulse initiated at the SA node was transmitted from onemodel cell to another causing the excitation of each shortly after its stimulation. Eachcell fired at the SA node rate. Also, the time between the activation of one cell and asucceeding cell in the conduction path was within milliseconds of the specified actionpotential propagation time between the same cells.Both of the functions of a healthy heart discussed in Chapter 4 are possible becauseof the ability of cardiac cells to become entrained to a common frequency. Entrainment,however, can also be the cause of many irregular rhythms of the heart. A particularpathological situation known as modulated ventricular parasystole which is character-ized by the formation of a ventricular ectopic focus (site of pacemaker activity other thanthe SA node) was studied in Chapter 5. A simple two-cell model in which an SA nodecell (s) was coupled to a Purkinje fibre cell (p) was used throughout the computations.The results confirmed the well-known result that when two pacemakers of different in-trinsic oscillation periods are coupled and the strength of the coupling increased, theratio of their periods approaches unity [30,39]. Furthermore, around certain values of thecoupling conductances, stable patterns of m:n entrainment (m cycles of one pacemakerChapter 6. Discussion^ 87for every n cycles of the other) which are common in the clinical analysis of electrocar-diograms were observed. The results differed from previous studies of similar adjacentcell interaction, in that the faster cell did not always determine the cycle length at 1:1entrainment. This is due to the fact that the coupling currents are characterized bydirection dependent conductances, g.,p and gp,., and also that different cells do not exertequal influences on each other. For example, it was found that for equal values of gs,pand gp ,s the Purkinje fibre cell exerts a much greater influence on the SA node cell thanvice-versa. Cells of different intrinsic oscillation periods were made to interact for differ-ent values of the coupling conductances and regions of synchronous and asynchronousbehavior plotted. It was found that when gp ,s was greater than some minimum value andthe ratio of the coupling conductances gp ,s /g8 ,p was sufficiently high (indicating that theinfluence of p on s was greater than the influence of s on p), then p became the lead celland the common period at synchrony was closer to the intrinsic cycle length of p thanof s. Similarly, when gs ,p was greater than some minimum value and the ratio g,,p/gp , swas sufficiently high then 1:1 s entrainment occurred and the common cycle length wascloser to the intrinsic cycle length of s than of p. For the case of interactions betweensimilar adjacent cells, previous studies [30,39] have found that the common cycle lengthwas, both closer to the intrinsic period of the faster cell, and always between the intrinsicperiods of the two cells. For the present model, the common cycle length was alwayscloser to the intrinsic oscillation period of the lead cell but was not restricted to valuesbetween the periods of the two cells. When the intrinsic period of s was less than that ofp (s firing at a faster rate than p) and the ratio Tp/Ts decreased, the regions of 1:1 s and1:1 p entrainment increased in size, supporting the general rule: the closer the intrinsicperiods of interacting cells the less coupling is required for synchrony [39]. However, theincrease in size of the region of entrainment to p was greater than the increase in sizeof the 1:1 s entrainment region. This exemplified the fact that the Purkinje fibre cellChapter 6. Discussion^ 88exerts a stronger influence on the SA node cell than vice-versa. Further evidence of thiswas discovered when the ratio Tp/T8 was decreased to a value less than 1. In this case,p fires at a faster rate than s and it was found that, over the range of conductancesstudied, p was never 1:1 s entrained.6.3 Suggestions for Future ResearchDue to the vast number of cardiac arrhythmias and entrainment phenomena readilyobservable in a real heart, the present study has only begun to explore the propertiesof impulse initiation and conduction among interacting cells. As a result, the follow-ing also are just a few of the ways the model can be used to investigate further theelectrophysiology of the heart.The AV node provides the only passage for impulses travelling between the atria andthe ventricles; therefore, its malfunction may completely dissociate atrial and ventricularcontraction. Studies of arrhythmias caused by the malfunction of the AV node have beendone using simplified models, for example by Keener [25]. The present model which usesthe more physiologically realistic Hodgkin-Huxley-type oscillators and provides a methodof coupling nonadjacent cells could be used to study these arrhythmias which lead tothe independent functioning of cells of the atria and ventricles.Ventricular fibrillation, an arrhythmia which occurs under conditions of ventricularparasystole (at least two ventricular ectopic foci discharging independently of the SAnode), has been of much interest in recent years. This arrhythmia, which leads torapid death, has often been associated with aperiodic dynamics and chaos. There hasbeen much controversy over the existence of chaotic modes separating the stable modesof m:n entrainment [24,15]. Glass et al. [9], using circle maps to simulate ventricularparasystole, have theoretically predicted regions of aperiodic dynamics which also haveChapter 6. Discussion^ 89been observed experimentally. They claim that, even if experimental noise were notpresent, these regions would still exist. Ypey et al. [39]; however, state that their resultson the mutual entrainment of pacemaker cells suggest that chaotic modes do not existbetween the regions of stable m:n entrainment. For the present model, the existence ofaperiodic dynamics, also could be investigated for ventricular parasystole. For the simplemodel of Chapter 6, this would require subdividing the regions of asynchronous behavior(labelled 'not 1:1') of the various synchrony diagrams into zones of m:n entrainment.The difficulty lies in the fact that some stable m:n entrainment patterns, such as, say,1021 : 1019 entrainment, require long simulation runs to detect and may be mistakenlyclassified as aperiodic.Simulations involving larger numbers of model cells would allow the study of thespread of the cardiac action potential over a tissue as well as provide some insight intothe mechanisms which cause the mutual synchronization of small clusters of adjacentcells. However, the computational cost and time of such an endeavor could becomeexorbitant.The use of the highly nonlinear Hodgkin-Huxley-type oscillators to represent the cellsof the model meant that extensive numerical computations were required to solve thesystems of equations. One of the long range objectives of this research is to reduce thesystem of cell model equations to a form which is more tractable analytically and yetretains physiological relevancy. A possible approach is to adapt the averaging techniqueemployed by Ermentrout and Kopell [27] in their studies of coupled biological oscillators.Finally, it was briefly mentioned that the phase response curve may be useful inpredicting zones of entrainment for unidirectional interaction [39]. This, as well as itspotential use in studies of mutual entrainment, is an area worthy of further investigation.Bibliography[1] Beeler, G.W. and Reuter, H. Reconstruction of the action potential of ventricularmyocardial fibres. J. Physiol. Lond., 268, 177-210, 1977.[2] Brown, H.F., Kimura, J., Noble, D., Noble, S.J., and Taupignon, A. The ioniccurrents underlying pacemaker activity in rabbit sino-atrial node: experimentalresults and computer simulations. Proc. R. Soc. Lond., B222, 329-347, 1984.[3] Brown, H.F., Kimura, J., Noble, D., Noble, S.J., and Taupignon, A. The slowinward current, i si, in the rabbit sino-atrial node investigated by voltage clamp andcomputer simulation. Proc. R. Soc. Lond., B222, 305-328, 1984.[4] Burne, R.M. and Levy, M.N. Cardiovascular Physiology. 5th ed., C.V. Mosby Com-pany, 1986.[5] Lambert, M.H. and Chay, T.R. Cardiac arrhythmias modelled by Ca;-inactivatedCa 2+ channels. Biol. Cybern., 61, 21-28, 1989.[6] Chung, P.F. Principles of Cardiac Arrhythmias. 4th ed., Williams & Wilkins, Bal-timore, 1989.[7] Cranfield, P.F. The Conduction of the Cardiac Impulse. Futura Publishing Com-pany, Inc., New York, 1975.[8] DiFrancesco, D. and Noble, D. A model of cardiac electrical activity incorporatingionic pumps and concentration changes. Phil. Trans. R. Soc. Lond. [Biol.], 307,353-398, 1984.90Bibliography^ 91[9] Glass, L., Goldberger, A.L., Courtemanche, M., and Shrier, A. Nonlinear dynamics,chaos and complex cardiac arrhythmias. Proc. R. Soc. Lond., A413, 9-26, 1987.[10] Glass, L., Hunter, P., and McCulloch, A., editors. Theory of Heart, Biomechanics,Biophysics, and Nonlinear Dynamics of Cardiac Function. Springer-Verlag, NewYork, 1991.[11] Glass, L., Guevara, M.R., Belair, J., and Shrier, A. Global bifurcations of a peri-odically forced biological oscillator. Phys. Rev. A, 29, 1348-1357, 1984.[12] Glass, L., and Mackey, M.C. From Clocks to Chaos: The Rhythms of Life. PrincetonUniversity Press, Princeton, 1988.[13] Guevara, M.R., Shrier, A., and Glass, L. Phase resetting of spontaneously beatingembryonic ventricular heart cell aggregates. Am. J. Physiol., 251, H1298-H1305,1986.[14] Guevara, M.R., Shrier, A., and Glass, L. Phase-locked rhythms of periodicallystimulated heart cell aggregates. Am. J. Physiol, 254, H1-H10, 1988.[15] Goldberger, A.L., Bhargava, V., West B.J., and Mandell, A.J. Some observationson the question : Is ventricular fibrillation "chaos"?. Physica D, 19, 282-289, 1986.[16] Heethaar, R.M., Pao, Y.C., and Ritman, E.L. Computer aspects of three-dimensional finite element analysis of stresses and strains in the intact heart. Comp.Biomed. Res., 10, 271-285, 1977.[17] Heller, L.J. and Mohrman, D.E. Cardiovascular Physiology. McGraw-Hill, Inc., NewYork, 1981.Bibliography^ 92[18] Honerkamp, J. The heart as a system of coupled nonlinear oscillators. J. Math. Biol.,18, 69-88, 1983.[19] Horowitz, A., Lanir, Y., Yin, F.C.P., Perl, M., Sheinman, I., and Strumpf, R.K.Structural three-dimensional constitutive law for the passive myocardium. ASMEJ. Biomech. Eng., 200-207, 1988.[20] Hunter, P.J., McCulloch, A.D., Nielsen, P.M.F., and Smaill, B.H. A finite elementmodel of passive ventricular mechanics. Computational Methods in BioengineeringASME, New York, 387-397, 1988.[21] Ikeda, N., Yoshizawa, N.S., and Sato, T. Difference equation model of ventricularparasystole as an interaction of pacemakers based on the phase response curve. J.Theor. Biol., 103, 439-465, 1983.[22] Jalife, J. and Antzelevitch, C. Phase resetting and annihilation of pacemaker activ-ity in cardiac tissue. Science, 206, 695-697, 1979.[23] Janz, R.F. and Grimm, A.F. Finite element model for the mechanical behavior ofthe left ventricle. Circ. Res., 30, 244-252, 1972.[24] Kaplan, D.T. and Cohen, R.J. Is fibrillation chaos?. Circ. Res., 67, 886-892, 1990.[25] Keener, J.P. On cardiac arrythmias: AV conduction block. J. Math. Biology, 12,215-225, 1981.[26] Keener, J.P. and Glass, L. Global bifurcations of a periodically forced oscillator. J.Math. Biol., 21, 175-190, 1984.[27] Ermentrout, G.B. and Kopell, N. Multiple pulse interactions and averaging in sys-tems of coupled oscillators. J. Math. Biol., 29, 195-217, 1991.Bibliography^ 93[28] McAllister, R.E., Noble, D., and Tsien, R.W. Reconstruction of the electrical ac-tivity of cardiac Purkinje fibres. J. Physiol. Lond., 251, 1-59, 1975.[29] McCulloch, A.D. Deformation and stress in the passive heart. Ph.D. thesis, Univer-sity of Auckland, New Zealand, 1986.[30] Michaels, D.C., Matyas, E.P., and Jalife, J. Dynamic interactions and mutual syn-chronization of sinoatrial node pacemaker cells. Circ. Res., 58, 706-720, 1986.[31] Noble, D. A modification of the Hodgkin-Huxley equations applicable to Purkinjefibre action and pacemaker potentials. J. Physiol. Lond., 160, 317-352, 1962.[32] Noble, D. The surprising heart: A review of recent progress in cardiac electrophys-iology. J. Physiol. Lond., 353, 1 -50, 1984.[33] Pao, Y.C., Ritman, E.L., and Wood, E.H. Finite-element analysis of left ventricularmyocardial stresses. J. Biomech., 7, 469-477, 1974.[34] van Capelle, Frans J.L. and Durrer, D. Computer simulation of arrhythmias in anetwork of coupled excitable elements. Circ. Res., 47, 454-466, 1980.[35] van der Pol, B. and van der Mark, J. The heartbeat considered as a relaxationoscillation, and an electrical model of the heart. Phil. Mag. (Series 7), 6, 763-775,1928.[36] Vinsen, C.A., Gibson, D.G., and Yettram, A.L. Analysis of left ventricular behaviorin diastole by means of finite element method. Br. Heart. J., 41, 60-67, 1979.[37] Winfree, A.T. The Geometry of Biological Time. Springer, New York, 1980.Bibliography^ 94[38] Yanagihara, K., Noma, A., and Irisawa, H. Reconstruction of sino-atrial node pace-maker potential based on the voltage clamp experiments. Jpn. J. Physiol., 30,841-857, 1980.[39] Ypey, D.L., Van Meerwijk, W.P.M., Ince, C., and Groos, G. Mutual entrainmentof two pacemaker cells. A study with an electronic parallel conductance model.J. Theor. Biol., 86, 731-755, 1980.Appendix AThe Cell ModelsThe SA and AV nodes, Purkinje fibre, and ventricular myocardial cells have been mod-elled using the Hodgkin-Huxley formulation in which the rate of change of the trans-membrane potential of a cell is described by a governing equation of the form:dEi(t)^1dt^=^Zionic(A.1)where t is the time in msec, j denotes the cell type and is one of: s,a,p, or v indicatingSA node, AV node, Purkinje fibre, and ventricular myocardial cells, respectively, E; (t)is the membrane potential in mV (expressed as the inside potential minus the outsidepotential) of cell j at time t, C is the membrane capacitance in FF/cm2 , and i ionic is thetotal ionic current in pA/cm2 flowing out of cell j.In the sections to follow, the equations describing the ionic current, iion ic , of eachof the four types of cells are given in detail. The equations for the SA node, Purkinjefibre, and ventricular myocardial cells are, for the most part, exactly as they are in theoriginal papers [1,28,38] with the exception of a few minor changes in notation. Theequations for the AV node cell are a modification of the equations for the SA nodecell. In each of the models, ionic current components which are time dependent includegating variables for activation and/or inactivation. Such a gating variable x follows thefirst-order kinetics:dxTit ax (1- x)-,3x x (A.2)95Appendix A. The Cell Models^ 96A.1 The Sinus Node CellYanagihara et al. [38] describe the total ionic current of the SA node cell by:Zionic = Zsi + iNa + iK + ih + il.^ (A.3)The individual components of the ionic current are given by the following equationswhere E8 is the transmembrane potential of the SA node cell at a particular time.Slow inward current:i 8i = 12.5 {exp[(E8 — 30)/15] — 1} (0.95d + 0.05) (0.95f + 0.05)^(A.4)where the gating variables d and f satisfy (A.2) and the rate constants are given by:ad#d==1.045 x 10-2 (E8 + 35)^3.125 x 10 -2 E8 (A.5)(A.6)1 — exp[—(E8 + 35)/2.5] + 1 — exp(—E8 /4.8)'4.21 x 10 -3 (E8 — 5)exp[(E8 — 5)/2.5] — 1'of = 3.55 x 10-4 (E8 + 20) (A.7)exp[(E8 + 20)/5.633] — 1'l3f = 9.44 x 10-4 1E8 + 60 1 (A.8)1 + exp[—(E8 + 29.5)/4.16] .Sodium current:iNa = 0.5 m3 h (E8 — 30) (A.9)where the gating variables 171 and h satisfy (A.2) and the rate constants are given by:am = Es 4- 37 (A.10)1 — exp[—(E8 + 37)/10] 'Om = 40 exp[-5.6 x 10-2 (E8 + 62)],^(A.11)ah = 1.209 x 10-3exp[—(E8 + 20)/6.534],1Oh =exp[—(E. + 30)/10) + 1] .(A.12)(A.13)Appendix A. The Cell Models^ 97Hyperpolarization current:ih = 0.4 q (E8 + 25)^ (A.14)where the gating variable q satisfies (A.2) and the rate constants are given by:aq/3q3.4 x 10-4 (E, + 100) = ^ + 4.95 x 10-5 ,exp[(E. + 100)/4.4] — 1—^5 x 10 -4 (E. + 40) + 8.45 x 10-5 .1— exp[—(E, + 40)/6](A.15)(A.16)Potassium current:ix -0.7p lexp[0.0277(E8 + 90)] — 1}exp[0.0277(E, + 40)](A.17)where the gating variable p satisfies (A.2) and the rate constants are given by:9 x 10-3aP = ^ + 6 x 10-41 + exp[—(E8 — Eap)/9.71]^,2.25 x 10-4flp = (E8 — Epp ){exp[(E, — Epp )/13.3] — 1}(A.18)(A.19)Since slight changes in Eap and Eiji, produce significant changes in the oscillation fre-quency of the SA node cell, these parameters were used to produce model cells withvarying cycle lengths. The actual values used along with the resulting frequencies aregiven in Section 2.2.5.Appendix A. The Cell Models^ 98Leak current:it = 0.8 {1 — exp[—(E8 60)/20]}.^ (A.20)A.2 The AV Node CellA model for AV node cells was created from the equations for the SA node cells [38].The total ionic current for the AV node cell is given by (A.3) where, with the exceptionof the individual current components are described by the same equations as for theSA node cell. The slow inward current, i e i , however, was scaled by the constant, i 8 i , toslow the rate of depolarization which is not as fast for AV node cells as it is for SA nodecells (Figure 1.2). Thus, for the AV node cell, i 8i is given by:isi = 1 81 12.5 {exp[(Ea — 30)/15] — 1} (0.95d + 0.05) (0.95f -I- 0.05)^(A.21)where Ea denotes the cell membrane potential and the subscript a indicates that thecell is an AV node cell. The gating variables d and f and their rate constants are asfor the SA node cell. The intrinsic oscillation frequency of the model AV node cell isdetermined by the value of the scaling constant i si and, as for the SA node cell, the valuesof the constants Ea„, and Epp of iK. The actual values used for these three constants arereported in Section 2.2.5.A.3 The Purkinje Fibre CellThe McAllister, Noble, and Tsien model for the Purkinje fibre cell [28] describes themembrane current by the following:ionic = iNa^isi^Z qr^ZK2^ix1^ix2^ZNab^iClb•^(A.22)Appendix A. The Cell Models^ 99The ionic current components are described below where E„, denotes the membrane po-tential of the Purkinje fibre cell at a specific time.Excitatory Sodium current:where the gating variables m and=iNa = m3h (Ep — 40)h satisfy (A.2) and the rate constants1.13 x 10-7exp[—(Ep + 10)/5.43],2.5(A.23)are given by:(A.24)(A.25)exp[-0.082(Ep + 10)] + 1'arn^=(Ep -I- 47)(A.26)1 — exp[-0.1(Ep + 47)] '/3„,^= 9.86 exp[—(Ep + 47)/17.86]. (A.27)Secondary inward current:i si = 0.8 (Ep — 70) d f + 0.04 (Ep — 70) d' (A.28)where the gating variables d and f satisfy (A.2) and the rate constants are given by:ad =0.002 (Ep + 40) (A.29)1 — exp[-0.1(Ep^40)]'13dof=-=0.02 exp[-0.0888(Ep^40)],0.000987 exp[-0.04(Ep^60)],(A.30)(A.31)0.02Qf = (A.32)exp[-0.087(Ep + 26)] + 1'd' = 1 + exp[-0.15(Ep^40)] -1 . (A.33)2.8 {exp[0.04(Ep + 110)] — 1}exp[0.08(Ep + 60)] + exp[0.04(Ep + 60)]'0.001(Ep — Ek)1 — exp[-0.2(Ep — Ek)]'0.00005 exp[-0.067(Ep — Ek)],(A.35)(A.36)(A.37)iK2 =ak =Ok =Appendix A. The Cell Models^ 100Pacemaker potassium current:iK2 = iK2 k^(A.34)where the gating variable k 1 satisfies (A.2) and the rate constants are given by:where the variable Ek is used to determine the oscillation frequency of the Purkinjefibre cell. The values used and the resulting oscillation frequencies are reported in Sec-tion 2.2.5.Plateau potassium currents:The first plateau potassium current is:. 1.2 x i lexp[0.04(Ep + 95)] — 1}exp[0.04(Ep + 45)]where the gating variable x 1 satisfies (A.2) and the rate constants are given by:(A.38)as,133i =0.0005 exp[(Ep + 50)/12.1] 1 + exp[(Ep + 50)/17.5] '0.0013 exp[ — (Ep + 20)/16.67]1 + exp[ — (Ep + 20)/25] •(A.39)(A.40)'In the original paper [28], the gating variable is denoted s rather than k, however, in this study, sis reserved to indicate an SA node cell.ix' =- (zh-2 12.8) + {1 — expr —0.04(4 + 30)]}0.2 (Ep + 30) (A.49)Appendix A. The Cell Models^ 101The second plateau potassium current is:ix2 = x 2 (25 + 0.385Ep )^(A.41)where the gating variable x 2 satisfies (A.2) and the rate constants are given by:axe 0.000127 = 1 + exp[— (Ep + 19)/5] ' (A.42). 0.0003 exp[— (Ep + 20)/16.67] 1 + exp[ — (Ep + 20)/25] • (A.43)Transient chloride current:iqr = 2.5 q r (Ep + 70)^ (A.44)where the gating variables q and r satisfy (A.2) and the rate constants are given by:0.008 Ep1 - exp(-0.1Ep)'13q = 0.08 exp[-0.08884],^ (A.46)ar = 0.00018exp[-0.04(Ep + 80)],^(A.47)0 fir =^ .02 exp[-0.087(Ep + 26)] + 1.(A.48)Outward background current:aq = (A.45)Inward background current carried by sodium ions:iNa,b = 0.105 (Ep — 40).^ (A.50)Appendix A. The Cell Models^ 102Background current carried by chloride ions:iC1,6 = 0.01 (4 + 70).^ (A.51)A.4 The Ventricular Myocardial CellThe model, due to Beeler and Reuter [1], for the ventricular muscle cell describes theionic current by:ionic = ZNa 2 K1^isi•^ (A.52)The individual current components are described by the following equations where Et,denotes the membrane potential of the ventricular muscle cell at a specific instance intime.Sodium current:iNa = (4M3hi 0.003)(E, — 50)^ (A.53)where the gating variables m, h, and j satisfy (A.2) and the rate constants are given by:—[E„ --F 47]an, (A.54)exp[-0.1(E, + 47)] — 1 '13,7, = 40 exp[-0.056(E,^72)], (A.55)ah = 0.126 exp[-0.25(E„^77)], (A.56)1.7= (A.57)exp[- 0.082(E„ + 22.5)] + 1'a0.055 exp[-0.25(E, + 78 )] (A.58)exp[-0.2(Et, + 78)] + 10.3= (A.59)exp[-0.1(Et, + 32)] + 1 •Appendix A. The Cell Models^ 103Time-independent potassium current:ixi = 0 .35 1^4 { exp[0.04(E, + 85)] — 1} 0.2(E, + 23) +exp[0.08(E, + 53)] + exp[0.04(E„ + 53)] 1 — exp[-0.04(Ei, + 23)] 1 •(A.60)Outward current:0.8x1 lexp[0.04(Et, + 77)] — 1}Zvi^exp[0.04(Et, + 35)]where the gating variable x 1 satisfies (A.2) and the rate constants are given by:0.0005 exp[0.083 (Et, + 50)] exp[0.057(Et, + 50)] + 1ax i =(A.61)(A.62)= 0.0013 exp[-0.06(Z, 20 )] exp[-0.04(Et, + 20)] + 1 • (A.63)Slow inward current:= 0.09d f (Ey —^ (A.64)where the gating variables d and f satisfy (A.2) and the rate constants are given by:0.095 exp[-0.01(E„ — 5)] =^ (A.65)exp[-0.072(Et, — 5)] + 113c1^0.07 exp[-0.017(E, + 44)] (A.66)exp[0.05(E, + 44)] + 1of =/3/ ==d[Ca]i/dt =0.012 exp[-0.008(E, + 28)] (A.67)exp[0.15(Et, -I- 28)] + 10.0065 exp[- 0.02(E„ + 30 )] (A.68)exp[-0.2(Ee, + 30)] + 1—82.3 — 13.02871nrab^(A.69)+ 0.07{10-7 — [Ca]i},^(A.70)Appendix A. The Cell Models^ 104where this last equation models the movement of calcium which flows into the musclecell and is subsequently removed by an uptake mechanism that reduces the intracellularcalcium concentration to 10 -7 M.

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