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This thesis is devoted to the study of the dynamics of spiral and scroll waves in a mathematical model for cardiac tissue. We study the effects of the presence of heterogeneities on electrical-wave dynamics. The heterogeneities in the medium occur because of the variation in the electrophysiological properties of the constituent myocytes in the tissue, or because of the presence of cells like fibroblasts and pathological myocytes that can trigger early afterdepolarizations (EADs). We study how these heterogeneities can lead to the formation of spiral and scroll waves and how they can affect the stability of the spiral and scroll waves in cardiac tissue. We also investigate the role of abnormal cells, which can trigger pathological excitations like EADs, on the formation of spiral and scroll waves, and how such cells can trigger premature electrical pulses like premature-ventricular-complexes (PVCs) in cardiac tissue. Earlier studies have examined the role of ionic heterogeneities on spiral-wave initiation and their effects on spiral-wave stability. However, none of these studies has calculated, in a controlled way, the effects of individual ion-channel conductances on spiral- and scroll-wave properties, such as the frequency of these waves, and the effects of the spatial gradients, in each ion-channel conductance, on their stability; we present these results in Chapter 2. Although many studies in the past have studied the effects of fibroblast coupling on wave-dynamics in cardiac tissue, a detailed study of spiral-wave dynamics in a medium with a well-defined, heterogeneous distribution of fibroblasts (e.g., with a gradient in the fibroblast density (GFD)) has not been performed; therefore, in Chapter 3 we present the effects of such GFD on spiral- and scroll-wave dynamics. Then, in Chapter 4, we present a systematic study of how a clump of fibroblasts can lead to spiral waves via high-frequency pacing. Some studies in the past have studied the role of early afterdepolarizations (EADs) in the formation of arrhythmias in cardiac tissue; we build on such studies and present a detailed study of the effects of EADs on the formation of spiral waves and their dynamics, in Chapter 5. Finally, in Chapter 6 we provide the results of our detailed investigation of the factors that assist the triggering of abnormal electrical pulses like premature ventricular complexes by a cluster of EAD-capable cells. A brief summary of the chapters is provided below: Chapter 2: In this chapter we investigate the effects of spatial gradients in the ion-channel conductances of various ionic currents on spiral-and scroll-wave dynamics. Ionic heterogeneities in cardiac tissue arise from spatial variations in the electrophysiological properties of cells in the tissue. Such variations, which are known to be arrhythmogenic, can be induced by diseases like ischemia. It is important, therefore, to understand the effects of such ionic heterogeneities on electrical-wave dynamics in cardiac tissue. To investigate such effects systematically, of changing the ion-channel properties by modifying the conductances of each ionic currents, on the action-potential duration (APD) of a myocyte cell. We then study how these changes in the APD affect the spiral-wave frequency ω in two-dimensional tissue. We also show that changing the ion-channel conductance not only changes ω but also the meandering pattern of the spiral wave. We then study how spatial gradients in the ion-channel conductances affect the spiral-wave stability. We find that the presence of this ionic gradient induces a spatial variation of the local ω, which leads to an anisotropic reduction of the spiral wavelength in the low-ω region and, thereby, leads to a breakup of the spiral wave. We find that the degree of the spiral-wave stability depends on the magnitude of the spatial variation in ω, induced by the gradient in the ion-channel conductances. We observe that ω varies most drastically with the ion-channel conductance of rapid delayed rectifier K+ current GKr, and, hence, a spiral wave is most unstable in the presence of a gradient in GKr (as compared to other ion-channel conductances). By contrast, we find that ω varies least prominently with the conductances of the transient outward K+ current Gto and the fast inward Na+ current (GNa); hence, gradients in these conduc-tances are least likely to lead to spiral-wave breaks. We also investigate scroll-wave instability in an anatomically-realistic human-ventricular heart model with an ionic gradient along the apico-basal direction. Finally, we show that gradients in the ion-channel densities can also lead to spontaneous initiation of spiral waves when we pace the medium at high frequency. Chapter 3: In this chapter we study the effects of gradients in the density of fibroblasts on wave-dynamics in cardiac tissue. The existence of fibroblast-myocyte coupling can modulate electrical-wave dynamics in cardiac tissue. In diseased hearts, the distribution of fibroblasts is heterogeneous, so there can be gradients in the fibroblast density (henceforth we call this GFD) especially from highly injured regions, like infarcted or ischemic zones, to less-wounded regions of the tissue. Fibrotic hearts are known to be prone to arrhythmias, so it is important to understand the effects of GFD in the formation and sustenance of arrhythmic re-entrant waves, like spiral or scroll waves. Therefore, we investigate the effects of GFD on the stability of spiral and scroll waves of electrical activation in a state-of-the-art mathematical model for cardiac tissue in which we also include fibroblasts. By introducing GFD in controlled ways, we show that spiral and scroll waves can be unstable in the presence of GFDs because of regions with varying spiral or scroll-wave frequency ω, induced by the GFD. We examine the effects of the resting membrane potential of the fibroblast and the number of fibroblasts attached to the myocytes on the stability of these waves. Finally, we show that the presence of GFDs can lead to the formation of spiral waves at high-frequency pacing. Chapter 4: In this chapter we study the arrhythmogenic effects of lo-calized fibrobblast clumps. Localized heterogeneities, caused by the regional proliferation of fibroblasts, occur in mammalian hearts because of diseases like myocardial infarction. Such fibroblast clumps can become sources of pathological reentrant activities, e.g., spiral or scroll waves of electrical activation in cardiac tissue. The occurrence of reentry in cardiac tissue with heterogeneities, such as fibroblast clumps, can depend on the frequency at which the medium is paced. Therefore, it is important to study the reentry-initiating potential of such fibroblast clumps at different frequencies of pacing. We investigate the arrhythmogenic effects of fibroblast clumps at high- and low-frequency pacing. We find that reentrant waves are induced in the medium more prominently at high-frequency pacing than with low-frequency pacing. We also study the other factors that affect the potential of fibroblast clumps to induce reentry in cardiac tissue. In particular, we show that the ability of a fibroblast clump to induce reentry depends on the size of the clump, the distribution and percentage of fibroblasts in the clump, and the excitability of the medium. We study the process of reentry in two-dimensional and a three-dimensional mathematical models for cardiac tissue. Chapter 5: In this chapter we investigate the role of early afterdepolarizations (EADs) on the formation of spiral and scroll waves. Early after depolarizations, which are abnormal oscillations of the membrane poten-tial at the plateau phase of an action potential, are implicated in the de-velopment of cardiac arrhythmias like Torsade de Pointes. We carry out extensive numerical simulations of the TP06 and ORd mathematical models for human ventricular cells with EADs. We investigate the different regimes in both these models, namely, the parameter regimes where they exhibit (1) a normal action potential (AP) with no EADs, (2) an AP with EADs, and (3) an AP with EADs that does not go back to the resting potential. We also study the dependence of EADs on the rate of at which we pace a cell, with the specific goal of elucidating EADs that are induced by slow or fast rate pacing. In our simulations in two- and three-dimensional domains, in the presence of EADs, we find the following wave types: (A) waves driven by the fast sodium current and the L-type calcium current (Na-Ca-mediated waves); (B) waves driven only by the L-type calcium current (Ca-mediated waves); (C) phase waves, which are pseudo-travelling waves. Furthermore, we compare the wave patterns of the various wave-types (Na-Ca-mediated, Ca-mediated, and phase waves) in both these models. We find that the two models produce qualitatively similar results in terms of exhibiting Na-Ca- mediated wave patterns that are more chaotic than those for the Ca-mediated and phase waves. However, there are quantitative differences in the wave patterns of each wave type. The Na-Ca-mediated waves in the ORd model show short-lived spirals but the TP06 model does not. The TP06 model supports more Ca-mediated spirals than those in the ORd model, and the TP06 model exhibits more phase-wave patterns than does the ORd model. Chapter 6: In this chapter we study the role of EAD-capable cells, and fibroblasts on the trigerring of abnormal electrical pulses called premature ventricular complexes (PVCs). Premature ventricular complexes, which are abnormal impulse propagations in cardiac tissue, can develop because of various reasons including early afterdepolarizations (EADs). We show how a cluster of EAD-generating cells (EAD clump) can lead to PVCs in a model of cardiac tissue, and also investigate the factors that assist such clumps in triggering PVCs. In particular, we study, through computer simulations, the effects of the following factors on the PVC-triggering ability of an EAD clump: (1) the repolarization reserve (RR) of the EAD cells; (2) the size of the EAD clump; (3) the coupling strength between the EAD cells in the clump; and (4) the presence of fibroblasts in the EAD clump. We find that, although a low value of RR is necessary to generate EADs and hence PVCs, a very low value of RR leads to low-amplitude EAD oscillations that decay with time and do not lead to PVCs. We demonstrate that a certain threshold size of the EAD clump, or a reduction in the coupling strength between the EAD cells, in the clump, is required to trigger PVCs. We illustrate how randomly distributed inexcitable obstacles, which we use to model collagen deposits, affect PVC-triggering by an EAD clump. We show that the gap-junctional coupling of fibroblasts with myocytes can either assist or impede the PVC-triggering ability of an EAD clump, depending on the resting membrane potential of the fibroblasts and the coupling strength between the myocyte and fibroblasts. We also find that the triggering of PVCs by an EAD clump depends sensitively on factors like the pacing cycle length and the distribution pattern of the fibroblasts.

Cardiac arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), are among the leading causes of death in the industrialized world. There is growing consensus that these arrhythmias are associated with the formation of spiral and scroll waves of electrical activation in mammalian cardiac tissue; whereas single spiral and scroll waves are believed to be associated with VT, their turbulent analogs are associated with VF. Thus, the study of these waves is an important biophysical problem in-so-far-as to develop an understanding of the electrophysiological basis of VT and VF. In this thesis, we provide a brief overview of recent numerical studies of spiral- and scroll-wave dynamics in mathematical models of mammalian cardiac tissue. In addition to giving a description of how spiral and scroll waves can be initiated in such models, how they evolve, how they interact with conduction and ionic inhomogeneities, how their dynamics is influenced by the size and geometry of the heart, we also discuss how active Purkinje networks and passive fibroblast clusters modify the electrical activity of cardiomyocytes, and the relevance of such studies to defibrillation. In Chapter 2 we present a systematic study of the combined effects of muscle-fiber rotation and inhomogeneities on scroll-wave dynamics in the TNNP (ten Tusscher Noble Noble Panfilov) model for human cardiac tissue. In particular, we use the three-dimensional (3D) TNNP model with fiber rotation and consider both conduction and ionic inhomogeneities. We find that, in addition to displaying a sensitive dependence on the positions, sizes, and types of inhomogeneities, scroll-wave dynamics also depends delicately upon the degree of fiber rotation. We find that the tendency of scroll waves to anchor to cylindrical conduction inhomogeneities increases with the radius of the inho-mogeneity. Furthermore, the filament of the scroll wave can exhibit drift or meandering, transmural bending, twisting, and break-up. If the scroll-wave filament exhibits weak meandering, then there is a fine balance between the anchoring of this wave at the inho-mogeneity and a disruption of wave-pinning by fiber rotation. If this filament displays strong meandering, then again the anchoring is suppressed by fiber rotation; also, the scroll wave can be eliminated from most of the layers only to be regenerated by a seed wave. Ionic inhomogeneities can also lead to an anchoring of the scroll wave; scroll waves can now enter the region inside an ionic inhomogeneity and can display a coexistence of spatiotemporal chaos and quasi-periodic behavior in different parts of the simulation domain. We discuss the experimental implications of our study. In Chapter 3 we present a comprehensive numerical study of plane and scroll waves of electrical activation in two state-of-the-art ionic models for rabbit and pig cardiac tissue. We use anatomically realistic, 3D simulation domains, account for muscle-fiber rotation, and show how to include conduction and ionic inhomogeneities in these models; we consider both localized and randomly distributed inhomogeneities. Our study allows us to compare scroll-wave dynamics, with and without inhomogeneities, in these rabbit-and pig-heart models at a level that has not been attempted hitherto. We begin with a comparison of single-cell action potentials (APs) and ionic currents in the Bers-Puglisi (BP) and modified-Luo-Rudy I (mLRI) models for rabbit- and pig-myocytes, respec-tively. We then show how, for plane-wave propagation in rabbit- and pig-heart models, the conduction velocity CV and wavelength λ depend on the distance of the plane of measurement from the plane containing the heart apex. Without inhomogeneities, and in the parameter r´egime in which these models display scroll waves, the rabbit-heart model supports a single scroll wave, which rotates periodically, whereas the pig-heart model supports two scroll waves, which rotate periodically, but with a slight difference in phase; this is partly because the rabbit-heart model is smaller in size, than the pig-heart one. With randomly-distributed inhomogeneities, we find that the rabbit-heart model loses its ability to support electrical activity, even at inhomogeneity concentra-tions as low as 5%. In the pig-heart model, we obtain rich, scroll-wave dynamics in the presence of localized or distributed inhomogeneities, both of conduction and ionic types; often, but not always, scroll waves get anchored to localized inhomogeneities; and distributed inhomogeneities can lead to scroll-wave break up. In Chapter 4, we present a comprehensive numerical study of spiral-and scroll-wave dynamics in a state-of-the-art mathematical model for human ventricular tissue with fiber rotation, transmural heterogeneity, myocytes, and fibroblasts. Our mathematical model introduces fibroblasts randomly, to mimic diffuse fibrosis, in the ten Tusscher-Noble-Noble-Panfilov (TNNP) model for human ventricular tissue; the passive fibrob-lasts in our model do not exhibit an action potential in the absence of coupling with myocytes; and we allow for a coupling between nearby myocytes and fibroblasts. Our study of a single myocyte-fibroblast (MF) composite, with a single myocyte coupled to Nf fibroblasts via a gap-junctional conductance Ggap, reveals five qualitatively different responses for this composite. Our investigations of two-dimensional domains with a ran-dom distribution of fibroblasts in a myocyte background reveal that, as the percentage Pf of fibroblasts increases, the conduction velocity of a plane wave decreases until there is conduction failure. If we consider spiral-wave dynamics in such a medium we find, in two dimensions, a variety of nonequilibrium states, temporally periodic, quasiperi-odic, chaotic, and quiescent, and an intricate sequence of transitions between them; we also study the analogous sequence of transitions for three-dimensional scroll waves in a three-dimensional version of our mathematical model that includes both fiber rotation and transmural heterogeneity. We thus elucidate random-fibrosis-induced nonequilib-rium transitions, which lead to conduction block for spiral waves in two dimensions and scroll waves in three dimensions. We explore possible experimental implications of our mathematical and numerical studies for plane-, spiral-, and scroll-wave dynamics in cardiac tissue with fibrosis. In Chapter 5 we present a detailed numerical study of the electrophysiological in-teractions between a random Purkinje network and simulated human endocardial tissue, (a) in the presence of, and (b) in the absence of existing electrical excitation in the system. We study the dependence of the activation-onset-time (ta) on the strength of coupling (Dmp) between the myocyte layer and the Purkinje network, in the absence of any external stimulus. Since the connection between the endocardial layer and the Purkinje network occurs only at discrete points, we also study the dependence of ta on the number of Purkinje-myocyte junctions (PMJs) at fixed values of Dmp, in the ab-sence of any applied excitation. We study signal propagation in the system; our results demonstrate the situations of (a) conduction block from the Purkinje layer to the endo-cardial layer, (b) anterograde propagation of the excitation from the Purkinje layer to the endocardium, (c) retrograde propagation of the excitation from the endocardium to the Purkinje layer and (d) development of reentrant circuits in the Purkinje layer that lead to formation of ectopic foci at select PMJs. We extend our study to explore the effects of Purkinje-myocyte coupling on spiral wave dynamics, whereby, we find that such coupling can lead to the distortion and breakage of the parent rotor into multiple rotors within the system; with or without internal coherence. We note that retrograde propa-gation leads to the development of reentrant circuits in the Purkinje network that help to initiate and stabilize ectopic foci. However, in some cases, Purkinje-myocyte coupling can also lead to the suppression of spiral waves. Finally we conduct four representative simulations to study the effects of transmural heterogeneity, fiber rotation and coupling with a non-penetrating Purkinje network on a three dimensional slab of cardiac tissue. Lastly, In Chapter 6, we study reentry associated with inexcitable obstacles in the ionically-realistic TNNP model for human ventricular tissue, under the influence of high-frequency stimulation. When a train of plane waves successively impinge upon an obstacle, the obstacle splits these waves as they tend to propagate past it; the emergent broken waves can either travel towards each other, bridging the gap introduced by the obstacle at the time of splitting, or, they can travel away from each other, resulting in the growth of the gap. The second possibility eventually results in the formation of spiral waves. This phenomenon depends on frequency of the waves. At high frequency, the excitability of the tissue decreases and the broken waves have a tendency to move apart. Hence high-frequency stimulation increases the chances of reentry in cardiac tissue. We correlate the critical period of pacing that leads to reentry in the presence of an inexcitable obstacle, with the period of spiral waves, formed in the homogeneous domain, and study how the critical period of pacing depends on the parameters of the model.