Cardiac Alternans Arising from an Unfolded Border-Collision Bifurcation

TL;DR

This paper investigates the complex transition to cardiac alternans, revealing hybrid smooth and nonsmooth bifurcation behaviors through analytical solutions of an unfolded border-collision model, enhancing understanding of cardiac dynamics.

Contribution

It introduces an analytical approach to unfolded border-collision bifurcations, explaining hybrid behaviors observed in cardiac tissue experiments.

Findings

Hybrid smooth/nonsmooth bifurcation behavior explained

Analytical solutions characterize the crossover dynamics

Guidance for future cardiac model development

Abstract

Following an electrical stimulus, the transmembrane voltage of cardiac tissue rises rapidly and remains at a constant value before returning to the resting value, a phenomenon known as an action potential. When the pacing rate of a periodic train of stimuli is increased above a critical value, the action potential undergoes a period-doubling bifurcation, where the resulting alternation of the action potential duration is known as alternans in the medical literature. Existing cardiac models treat alternans either as a smooth or as a border-collision bifurcation. However, recent experiments in paced cardiac tissue reveal that the bifurcation to alternans exhibits hybrid smooth/nonsmooth behaviors, which can be qualitatively described by a model of so-called unfolded border-collision bifurcation. In this paper, we obtain analytical solutions of the unfolded border-collision model and use it to explore the crossover between smooth and nonsmooth behaviors. Our analysis shows that the hybrid smooth/nonsmooth behavior is due to large variations in the system's properties over a small interval of the bifurcation parameter, providing guidance for the development of future models.