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

Jean-Marc Ginoux (1) , Jaume Llibre (2)

1 PROTEE - Processus de Transfert et d'Echanges dans l'Environnement - EA 3819
2 UAB - Departament de Matemàtiques [Barcelona]

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.

Mots clés: canard solutions; singularly perturbed dy- namical systems; Geometric singular perturbation theory

Domaines: Géométrie différentielle [math.DG]; Systèmes dynamiques [math.DS]; Dynamique Chaotique [nlin.CD]; Traitement du signal et de l'image [eess.SP]

Jean-Marc Ginoux, Jaume Llibre. Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models. Mathematical Problems in Engineering, 2015, 2015, pp.1 - 17. ⟨10.1155/2015/342010⟩. ⟨hal-01856956⟩