Bifurcations and canards in the FitzHugh-Nagumo system: a tutorial in fast-slow dynamics

TL;DR

This paper provides a comprehensive tutorial on the bifurcations and canards in the FitzHugh-Nagumo fast-slow system, combining theoretical analysis with numerical illustrations to understand nerve impulse dynamics.

Contribution

It offers a detailed bifurcation analysis of the FitzHugh-Nagumo system derived from the Hodgkin-Huxley model, highlighting the role of parameters and canard phenomena.

Findings

Identification of key bifurcations affecting system dynamics

Numerical validation of theoretical bifurcation results

Insights into the impact of parameters on nerve impulse modeling

Abstract

In this article, we study the FitzHugh-Nagumo $(1,1)$--fast-slow system where the vector fields associated to the slow/fast equations come from the reduction of the Hodgin-Huxley model for the nerve impulse. After deriving dynamical properties of the singular and regular cases, we perform a bifurcation analysis and we investigate how the parameters (of the affine slow equation) impact the dynamics of the system. The study of codimension one bifurcations and the numerical locus of canards concludes this case-study. All theoretical results are numerically illustrated.