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

## TL;DR

This paper conducts a qualitative analysis of nonlinear Reaction-Diffusion systems, specifically the FitzHugh-Nagumo type, exploring solution behaviors, bifurcations, and their implications in Neuroscience.

## Contribution

It introduces a non-homogeneous FitzHugh-Nagumo model and analyzes a related toy model to understand solution convergence and bifurcation phenomena.

## Key findings

- Solutions converge to fixed points or periodic orbits
- Existence of a cascade of Hopf bifurcations
- Insights into neural excitability and oscillations

## 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.