A closed form asymptotic solution for the FitzHugh–Nagumo model

By means of a change of unknown function and independent variable, the Cauchy problem of singular perturbation from electrophysiology, known as the FitzHugh–Nagumo model, is reduced to a regular perturbation problem (Section 1). Then, by applying the regular perturbation technique to the last problem and using an existence, uniqueness and asymptotic behavior theorem of the second and third author, the models of asymptotic approximation of an arbitrary order are deduced (Section 2). The closed-form expressions for the solution of the model of first order asymptotic approximation and for the time along the phase trajectories are derived in Section 3. In Section 4, by applying several times the method of variation of coefficients and prime integrals, the closed-form solution of the model of second order asymptotic approximation is found. The results from this paper served to the author to study (elsewhere) the relaxation oscillations versus the oscillations in two and three times corresponding to concave limit cycles (canards).