Qualitative Analysis of certain Reaction-Diffusion Systems of the FitzHugh-Nagumo type

Abstract: This article aims to provide insights into the qualitative analysis of some nonlinear Reaction-Diffusion (RD) systems arising in Neuroscience. We first introduce a non-homogeneous FitzHugh-Nagumo (nhFHN) featuring excitability and oscillatory properties. Then, we discuss the qualitative analysis of a toy model related to nhFHN. In particular, we focus on the convergence of solutions of the toy model toward different solutions (fixed point, periodic) and show the existence of a cascade of Hopf bifurcations. Finally, we connect this analysis to the nhFHN system.