Canards existence in Fitzhugh-Nagumo and Hodgkin-Huxley neuronal models (Q1665395)

Summary: 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 \textit{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 \textit{slow} and two \textit{fast} variables. This method enables stating a unique generic condition for the existence of ``canard solutions'' for such four-dimensional singularly perturbed systems which is based on the stability of \textit{folded singularities} ( \textit{pseudo singular points} in this case) of the \textit{normalized slow dynamics} deduced from a well-known property of linear algebra. This unique generic condition is identical to that provided in previous works. Application of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables showing the existence of ``canard solutions'' in such systems.