Life-threatening cardiac arrhythmias continue to pose a major health problem. Ventricular fibrillation, which is a complex form of electrical wave turbulence in the lower chambers of the heart, stops the heart from pumping and is the largest cause of natural death in the United States. Atrial fibrillation, a related form of wave turbulence in the upper heart chambers, is in turn the most common arrhythmia diagnosed in clinical practice. Despite extensive research to date, mechanisms of cardiac arrhythmias remain poorly understood. It is well established that both spatial disorder of the refractory period of heart cells and triggered activity (TA) jointly contribute to the initiation and maintenance of arrhythmias. TA broadly refers to the abnormal generation of a single or a sequence of abnormal excitation waves from a small submillimeter region of the heart in the interval of time between two normal waves generated by the heart's natural pacemaker (the sinoatrial node). TA has been widely investigated experimentally and occurs in several pathological conditions where the intracellular concentration of free Ca2+ ions in heart cells becomes elevated. Under such conditions, Ca2+ can be spontaneously released from intracellular stores, thereby driving an electrogenic current that exchanges 3Na+ ions for one Ca2+ ion across the cell membrane. This current in turn depolarizes the membrane of heart cells after a normal excitation. If this calcium-mediated "delayed after depolarization'' (DAD) is sufficiently large, it can generate an action potential. While the arrhythmogenic importance of spontaneous Ca2+ release and DADs is well appreciated, the conditions under which they occur in heart pathologies remain poorly understood. Calcium overload is only one factor among several other factors that can promote DADs, including sympathetic nerve stimulation, different expression levels of membrane ion channels and calcium handling proteins, and different mutations of those proteins. How those various factors interact synergistically to promote DADs is not well understood. Furthermore, at an even more basic level, it remains unclear to what degree spontaneous Ca2+ release and the appearance of DADs are deterministic, meaning reproducible under identical conditions, or inherently stochastic like nucleation in the physical context of phase transitions.;In this thesis, we use and further develop a biologically detailed computational model to investigate basic aspects of TA in isolated heart cells (cardiac myocytes). Isolated cells can be obtained by enzymatic dissociation of heart tissue and studied experimentally using standard electrophysiological recording methods and confocal imaging of Ca2+ dynamics. Hence they provide a well controlled setting to investigate the generation of DADs under well controlled conditions. Our computational model captures essential aspects of the hierarchical architecture of ventricular myocytes, which consists of a large number of approximately 20,000 to 50,000 regularly spaced submicron regions containing clusters of 50-100 Ryanodine receptor (RyR) Ca2+ release channels. Each of those regions acts as a discrete "calcium release unit'' (CRU). Therefore our model allows us to address for the first time quantitatively the fundamental question of whether Ca2+ release, which is highly stochastic at the level of a single calcium release unit, is stochastic or deterministic at the whole cell level where the Ca 2+ signal is the summation of releases from a large number of units. Addressing this question is the focus of the first part of this thesis. Our results demonstrate that both the initiation and termination of TA are highly stochastic at the whole cell level due to the spatiotemporal organization of discrete release events into multiple Ca2+ waves. Our results allow us to characterize the probability distributions that govern the number of DADs preceding a triggered action potential and the number of triggered action potentials after termination of periodic stimulation. We show that a limit cycle underlies the bi-directionally coupled dynamics of membrane of voltage and Ca2+ when TA is sustained for long intervals. Furthermore, we construct a simple theoretical model that allows us to relate the shape of those distributions to the statistics and properties of Ca 2+ waves. The second part of this thesis focuses on investigating TA in the context of a specific mutation of a calcium buffering protein calsequestrin (CSQN). This mutation underlies catecholaminergic polymorphic ventricular tachycardia (CPVT), which is a pathophysiological condition that affects a subset of the human population. Our results shed light on the mechanisms by which altered Ca2+ buffering and altered kinetics of RyR Ca 2+ release channels as a direct and indirect effect of this mutation, respectively, promote TA.
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