Asymptotic Behavior of Fronts and Pulses of the Bidomain Model

The bidomain model is the standard model for cardiac electrophysiology. This paper investigates 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 dimensions. Previous work showed that planar fronts of the bidomain Allen-Cahn equation could become unstable in contrast to the classical Allen-Cahn equation. 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. Our computational studies of the bidomain FitzHugh-Nagumo pulse solution show that the pulses can also become unstable, 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 choices of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by developing a numerical scheme that allows for the accurate computation of the bidomain equation in a two-dimensional strip domain of an infinite extent.