Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model

The standard FitzHugh–Nagumo model for description of impulse from one neuron to another is considered. The system of equations is transformed to a nonlinear second-order ordinary differential equation. It is shown that the differential equation does not pass the Painlevé test in the general case and the general solution of this equation does not exist. The simplest solutions of the system of equations are found. The second-order differential equation is transformed to another asymptotic equation with the general solution expressed via the Jacobi elliptic function. This transformation allows us to obtain the asymptotic solutions of the FitzHugh–Nagumo model. The perturbed FitzHugh–Nagumo model is studied as well. Taking into account the simplest equation method, the exact solutions of the perturbed system of equations are found. The asymptotic solutions of the perturbed model are presented too. The application of the exact solutions for construction of the neural networks is discussed.