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

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.