We studied the effects of spatial inhomogeneities on inwardly rotating spiral waves in a typical type of oscillatory medium using the complex-Ginzburg-Landau equation. With a small degree of the inhomogeneity in the medium, the slower inward spiral always suppressed a faster spiral; when the inhomogeneity exceeded a critical value, however, a transition occurred to the coexistence of multiple inward spirals, insulated by regions of highly disordered wave break. The occurrence of this transition is examined theoretically and shown to be due to the Eckhaus instability.

This dissertation is dedicated to understanding the dynamical mechanisms underlying early afterdepolarization (EAD)-mediated arrhythmias. EADs are a richly mined area of research that has resulted in little therapeutic benefit at the bedside since their first description in the early 1980's. In an effort to provide a fresh perspective on EADs, I take an experimental and computational non-linear dynamics approach to studying these arrhythmias that plague millions worldwide and are associated with significant cost burden and high mortality. Here, I show that the dual Hopf-homoclinic bifurcation mechanism for generating EADs may be responsible for EAD-mediated arrhythmias in cardiac disease, providing new therapeutic targets and expanding the current concept of reduced repolarization reserve. Also, I provide evidence that certain pathological cardiac tissue substrates may exhibit the unique dynamical property of bistability (ie. switch-like behavior) in the form of biexcitability, the ability to switch between INa- and ICa-mediated propagation. This unique property of biexcitability may provide significant mechanistic insight into the seemingly paradoxical properties of certain EAD-mediated arrhythmias such as Torsades de Pointes (TdP), that is uniquely characterized by a slow rate and undulating QRS complexes on ECG. Lastly, I show that activation of KATP channels (inward rectifying K channels that open when cellular ATP levels fall) "flips the switch" from ICa- to INa-mediated propagation. Consistent with clinical observations of TdP and polymorphic ventricular tachycardia (PVT), the majority of ICa-dependent reentries spontaneously terminate following KATP channel activation, with a minority of ICa-dependent reentries converting to INa-dependent reentries. These results suggest that in the clinical setting, KATP channels may serve as an "emergency repolarization reserve" to spontaneously terminate most episodes of ICa-dependent arrhythmias such as PVT and TdP, but may also facilitate the occasional degeneration of these arrhythmias to INa-dependent VT/VF. These findings provide a fresh perspective on EAD-mediated arrhtyhmias that may lead to novel therapeutics at the bedside.

Abnormal calcium (Ca) cycling conditions in cardiac myocytes have been identified as a major contributor to lethal cardiac arrhythmias that lead to sudden cardiac death. Under these conditions, Ca waves can emerge spontaneously from within the subcellular Ca release unit (CRU) network in Ca overloaded myocytes and facilitate the formation of delayed afterdepolarization (DAD) and early afterdepolarization (EAD) arrhythmia triggers at the cellular scale. Whether DADs and EADs can overcome the electronic sink effect of resting neighboring myocytes to propagate and disrupt the normal conduction of action potentials in cardiac tissue highly depends upon the timing and spatial organization of myocytes exhibiting spontaneous Ca waves. The exact mechanisms that govern these events are still under much debate and not fully understood. The goal of our studies in this body of work is two-fold. The first is to understand how a broad spectrum of spontaneous Ca release events, including Ca waves, are able to emerge collectively from within the CRU network. The second is to understand how the effects of subcellular Ca release events traverse scales to influence the morphology of DADs and EADs at the cellular scale and promote arrhythmias at the tissue scale. In order to achieve these ends, we have utilized a multidisciplinary approach that draws upon and integrates ideas, concepts, and techniques largely from experimental electrophysiology, nonlinear dynamics, and mathematical modeling. This approach has allowed tremendous versatility in the way arrhythmias can be conceptualized, quantitatively assessed, and logically predicted. As such, we have discovered a universal mechanism of criticality that governs the transition from stochastic Ca sparks to deterministic Ca waves, a general theory for Ca wave entrainment that can explain subcellular pacemaker site emergence, novel complex EAD morphologies that emerge due to the interaction between the stochastic effects of subcellular Ca release and the deterministic relationships among voltage-gated ion channels, that the distribution of latencies to the onset of Ca waves among myocytes is the most important factor promoting DADs in cardiac tissue, and that DADs and EADs can promote arrhythmias not only by generating triggers but by enhancing tissue substrate vulnerability.

Cardiac myocytes are excitable cells in which an external current stimulus depolarizes the membrane potential to elicit an action potential. This action potential then triggers calcium release from intracellular stores, which mediates contraction. Conversely, intracellular calcium also modulates membrane currents, affecting action potential morphology and action potential duration (APD). The interactions between action potential and calcium, termed excitation-contraction coupling, give rise to a rich spectrum of nonlinear dynamics, especially at rapid heart rates, which are important for cardiac contraction and the development of lethal arrhythmias. In this study, we developed a nonlinear iterated map model to investigate the dynamics of cardiac excitation-contraction coupling in a periodically stimulated cell. We first studied the nonlinear dynamics due to APD restitution, a functional relation between APD and its preceding diastolic interval. We then studied the nonlinear dynamics due to intracellular calcium cycling when total cell calcium is constant or varies at a beat-to-beat basis. Finally, we studied the nonlinear dynamics due to the bidirectional coupling of the two dynamical systems. Saddle-node bifurcations leading to bistability, period-doubling bifurcations leading to alternans, and period-doubling routes to chaos can independently occur in both action potential or intracellular calcium cycling subsystems as heart rate increases. A Hopf bifurcation leading to quasiperiodicity occurs when the two dynamical systems are coupled. Although these dynamics are predicted from low-dimensional iterated maps, the approach here provides valuable information which can be used as a basis to explore dynamical features of physiologically detailed ionic models, to illuminate experimental findings, and to design experimentally testable predictions for new biological experiments.