Asymptotic Behavior of Fronts and Pulses of the Bidomain Model

Hiroshi Matano, Yoichiro Mori, Mitsunori Nara, Koya Sakakibara

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)616-649
Number of pages34
JournalSIAM Journal on Applied Dynamical Systems
Issue number1
Publication statusPublished - 2022


  • Allen-Cahn model
  • FitzHugh-Nagumo model
  • Hopf bifurcation
  • bidomain model
  • front and pulse solutions


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