The McKean's Caricature of the Fitzhugh-Nagumo Model I. The Space-Clamped System

Within the context of Liénard equations, we present the FitzHugh-Nagumo model with an idealized nonlinearity. We give an analytical expression (i) for the transient regime corresponding to the emission of a finite number of action potentials (or spikes), and (ii) for the asymptotic regime correspond...

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Bibliographic Details
Published inSIAM journal on applied mathematics Vol. 63; no. 2; pp. 459 - 484
Main Author Tonnelier, Arnaud
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2003
Subjects
Action potentials
Applied mathematics
Approximation
Dynamical Systems
Limit cycles
Mathematical discontinuity
Mathematical functions
Mathematics
Oscillation
Phase plane
Polynomials
System theory
Trajectories
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Summary:Within the context of Liénard equations, we present the FitzHugh-Nagumo model with an idealized nonlinearity. We give an analytical expression (i) for the transient regime corresponding to the emission of a finite number of action potentials (or spikes), and (ii) for the asymptotic regime corresponding to the existence of a limit cycle. We carry out a global analysis to study periodic solutions, the existence of which is linked to the solutions of a system of transcendental equations. The periodic solutions are obtained with the help of the harmonic balance method or as limit behavior of the transient regime. We show how the appearance of periodic solutions corresponds either to a fold limit cycle bifurcation or to a Hopf bifurcation at infinity. The results obtained are in agreement with local analysis methods, i.e., the Melnikov method and the averaging method. The generalization of the model leads us to formulate two conjectures concerning the number of limit cycles for the piecewise linear Liénard equations.
ISSN:0036-1399
1095-712X
DOI:10.1137/S0036139901393500