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Mathematical Problems in Engineering
Volume 2015, Article ID 342010, 17 pages
Research Article

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

1Laboratoire LSIS, CNRS, UMR 7296, Université de Toulon, BP 20132, 83957 La Garde Cedex, France
2Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Received 5 August 2015; Accepted 3 December 2015

Academic Editor: Matjaz Perc

Copyright © 2015 Jean-Marc Ginoux and Jaume Llibre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 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 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. 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.