Abstract. Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suffer reversals of phase at various points along the fiber, a phenomenon called (spatially) discordant alternans. Either stationary or moving patterns are possible. Using a weak approximation, Echebarria and Karma proposed an equation to describe the spatiotemporal dynamics of small-amplitude alternans in a class of simple cardiac models, and they showed that an instability in this equation predicts the spontaneous formation of discordant alternans. To study the bifurcation, they computed the spectrum of the relevant linearized operator numerically, supplemented with partial analytical results. In the present paper we calculate this spectrum with purely analytical methods in two cases where a small parameter may be exploited: (i) small dispersion or (ii) a long fiber. From this analysis we estimate the parameter ranges in which the phase reversals of discordant alternans are stationary or moving.

Variation in cardiac pacing cycles, as seen, for example, in heart rate variability, has been observed for decades. Contemporarily, various mathematical models have been constructed to investigate the electrical activity of paced cardiac cells. Yet there has not been a study of these cardiac models when there is variation in the pacing cycles such as noise. We present a method that uses the stochasticity of pacing cycles to determine approximate models of the dynamics of cardiac cells, and use these models to detect bifurcations to alternans.

Abstract. While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose stability, as the pacing rate is increased, through either a Hopf or steady-state bifurcation. Which bifurcation occurs first depends on parameters in the equation, and for one critical case both modes bifurcate together at a degenerate (codimension 2) bifurcation. For parameters close to the degenerate case, we investigate the competition between modes, both numerically and analytically. We find that at sufficiently rapid pacing (but assuming a 1:1 response is maintained), steady patterns always emerge as the only stable solution. However, in the parameter range where Hopf bifurcation occurs first, the evolution from periodic solution (just after the bifurcation) to the eventual standing wave solution occurs through an interesting series of secondary bifurcations.Mathematics Subject Classification. 35B32, 92C30.

Instabilities in cardiac dynamics have been widely investigated in recent years. One facet of this work has studied chaotic behavior, especially possible correlations with fatal arrhythmias. Previously chaotic behavior was observed in various models, specifically in the breakup of spiral and scroll waves. In this paper we study cardiac dynamics and find spatiotemporal chaotic behavior through the Echebarria-Karma modulation equation for alternans in one dimension. Although extreme parameter values are required to produce chaos in this model, it seems significant mathematically that chaos may occur by a different mechanism from previous observations.