Analytical properties of the perturbed FitzHugh-Nagumo model

Analytical properties of the well-known FitzHugh-Nagumo model are studied. It is shown that the standard FitzHugh-Nagumo model does not pass the Painlevé test in the general case and does not have any meromorphic solutions because there are not any expansions of the general solution in the Laurent series. We demonstrate that the introduction of a nonlinear perturbation into the standard system of equations does not lead to the Painlevé property as well. However, in this case there are expansions of the general solution of the system of equations in the Laurent series for some values of parameters. This allows us to look for some exact solutions of the system of the perturbed FitzHugh-Nagumo model. We find some exact solutions of the perturbed FitzHugh-Nagumo system of equations in the form of kinks. These exact solutions can be used for testing numerical simulations of the system of equations corresponding to the FitzHugh-Nagumo model.