Canard bifurcation in the FitzHugh-Nagumo model for spikes generation in neurons

Abstract

We provide a new insight of the all-or-none-spike behavior of the solutions of the FitzHugh-Nagumo model for axon-current. Using the various tools introduced by nonstandard analysis for studying canards behavior we show how the firing of oscillations takes place by the appearance of a canard-without-head cycle that splits into two "concentric" cycles, the larger being stable and becoming a canard-with-head and the the smaller being unstable and collapsing down to the equilibrium, all this taking place within an exponentially small parameter interval.