  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.
