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The McKean’s caricature of the FitzHugh-Nagumo model. I: The space-clamped system. (English) Zbl 1036.34047

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. A generalization of the model leads us to formulate two conjectures concerning the number of limit cycles for the piecewise linear Liénard equations.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34A36 Discontinuous ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92C20 Neural biology
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