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 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.