Analytical Properties and Solutions of the FitzHugh – Rinzel Model

The FitzHugh – Rinzel model is considered, which differs from the famous FitzHugh – Nagumo model by the presence of an additional superslow dependent variable. Analytical properties of this model are studied. The original system of equations is transformed into a third-order nonlinear ordinary differential equation. It is shown that, in the general case, the equation does not pass the Painlevé test, and the general solution cannot be represented by Laurent series. Using the singular manifold method in terms of the Schwarzian derivative, an exact particular solution in the form of a kink is constructed, and restrictions on the coefficients of the equation necessary for the existence of such a solution are revealed. An asymptotic solution is obtained that shows good agreement with the numerical one. This solution can be used to verify the results in a numerical study of the FitzHugh – Rinzel model.