Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models

TL;DR

This paper extends a method for proving canard solutions to four-dimensional neuronal models with two fast and two slow variables, applying it to FitzHugh-Nagumo and Hodgkin-Huxley systems.

Contribution

It introduces a unified generic condition for canard existence in four-dimensional systems with two fast and two slow variables, based on stability of folded singularities.

Findings

Canard solutions exist in FitzHugh-Nagumo model

Canard solutions exist in Hodgkin-Huxley model

Method provides a unified approach for such systems

Abstract

In a previous paper we have proposed a new method for proving the existence of "canard solutions" for three and four-dimensional singularly perturbed systems with only one fast variable which improves the methods used until now. The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with two slow and two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such four-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is identical to that provided in previous works. Applications of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables to show the existence of "canard solutions" in such systems.