Hopf Bifurcation in an Oscillatory-Excitable Reaction-Diffusion Model with Spatial Heterogeneity

We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity near a bifurcation. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of an Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.