Asymptotic behavior of fronts and pulses of the bidomain model

TL;DR

This paper investigates the instability and asymptotic behavior of planar fronts and pulses in the bidomain model of cardiac electrophysiology, revealing new bifurcation phenomena and developing a numerical scheme for accurate simulations.

Contribution

It introduces a detailed analysis of front and pulse instabilities in the bidomain model, including bifurcation types and geometric explanations, along with a novel numerical scheme for 2D simulations.

Findings

Planar fronts can become unstable and form zigzag patterns.

Hopf bifurcation can be supercritical or subcritical, with coexistence of stable fronts.

Destabilized pulses may disintegrate or form complex patterns.

Abstract

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we investigate the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen-Cahn equation and the bidomain FitzHugh-Nagumo equation in two spatial dimension. In previous work, it was shown that planar fronts of the bidomain Allen-Cahn equation can become unstable in contrast to the classical Allen-Cahn equation. We find that, after the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical, by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. In our computational studies of the bidomain FitzHugh-Nagumo pulse solution, we show that the pulses can also become unstable much like the bidomain Allen-Cahn fronts. However, unlike the bidomain Allen-Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choice of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by the development of a numerical scheme that allows for the accurate computation of the bidomain equation in a two dimensional strip domain of infinite extent.