Bifurcation analysis of periodic action potentials of nerve cells in the FitzHugh-Nagumo model

We study the two-variable FitzHugh-Nagumo reaction-diffusion system for neuron excitation. The periodic action potentials of the nerve cells can be treated as the periodic traveling waves in one dimension. That motivates us to study the existence and the stability of periodic traveling waves in a one-parameter family of solutions. It is observed that periodic traveling waves change their stability by a stability change of Eckhaus type in a two-dimensional parameter plane. We determine the stability boundary between stable and unstable periodic traveling waves. We also calculate essential spectra of the periodic traveling waves.