Date of Award

Spring 1-1-2013

Document Type


Degree Name

Doctor of Philosophy (PhD)


Applied Mathematics

First Advisor

Juan G. Restrepo

Second Advisor

James D. Meiss

Third Advisor

Keith Julien

Fourth Advisor

David M. Bortz

Fifth Advisor

Alain Karma


The study of cardiac alternans, a phenomenon characterized by beat-to-beat alternations of activity in cardiac tissue that has been directly linked with sudden cardiac arrest, has become an important area of research at the intersection of biology, physics, and mathematics. In this Thesis, we derive and study the spatiotemporal dynamics of a reduced model describing the beat-to-beat evolution of calcium-driven alternans in a periodically-paced cable of tissue. This work can be thought of as an extension of the seminal work of Echebarria and Karma [Physical Review Letters, 88:208101, 2002; Physical Review E, 76:051911, 2007], which was the first analytical treatment of alternans in tissue. While Echebarria and Karma considered the case of alternans driven by a voltage-mediated instability and neglected the effect of calcium dynamics, we extend this approach to the important case of a calcium-mediated instability and account for the effect of bi-directional coupling between voltage and calcium dynamics. Our reduced model consists of two bi-directionally coupled integro-difference equations that describe the amplitude of alternans in calcium and voltage along a cable of tissue. In agreement with detailed ionic models, our reduced model yields three solution regimes separated by two bifurcations. The three regimes are described by (i) no alternans, (ii) smooth wave patterns, and (iii) discontinuous patterns. Due to the smoothing effect of electrotonic coupling on voltage, discontinuous patterns are non-physical in voltage-driven alternans, and thus can only be observed when the instability is mediated by the calcium dynamics. We study spatial properties and dynamics of solutions in each regime, as well as several novel properties of solutions in the third regime. We find that solutions in the third regime are subject to unique memory and hysteresis effects, which are not present in the solutions in the second regime. In addition to symmetrizing the shape of profiles about the phase reversals, or node, we find a novel phenomenon we call unidirectional pinning, a mechanism where nodes can be moved towards, but not away from, the pacing site when parameters are changed. Furthermore, we find that while the spatial wavelength of solutions in the smooth regime scales sub-linearly with the conduction velocity (CV) length scale, the spatial wavelength of solutions in the discontinuous regime scales linearly with this length scale. Due to the tendency for nodes to cause conduction blocks in tissue, we hypothesize that intracellular calcium-driven alternans are more arrhythmogenic than previously believed since they cannot be expelled from the cable due to unidirectional pinning. We complement these analytical results with numerical studies of a detailed, biologically robust ionic model of a cable of cells. We show that our reduced model captures the behavior of these detailed ionic models and can in fact predict their dynamics, and that detailed ionic models display the novel properties found in the reduced model, including unidirectional pinning. This work extends our theoretical understanding of alternans to include the important effect of calcium dynamics. Finally, we make concluding remarks about physiological implications and experiment suggestions as well as discuss possible extensions and future work.