An analytical scheme on complete integrability of 2D biophysical excitable systems

Abstract

We report a method to find the complete integrability and analytical scheme of a well-known two dimensional (2D) excitable biophysical system that includes a coupled non-linear ordinary differential equations (ODEs). We demonstrate the integrability of the autonomous FitzHugh–Nagumo (FHN) dynamical system for the time evolution equations of motion and briefly describe the solution procedure using the extended Prelle–Singer (PS) scheme without perturbation and series solution method. To illustrate the mathematical formulation procedure, both the time independent and time dependent integrals have been derived by using the extended PS method. Finally, the numerical solution of the integrals has been found with the original dynamical model. It also allows us to establish the integrability of another type of an analytical technique for the deterministic system and this explores the quantitative applicability of the method. Interestingly, the method may have significant applications in other physical systems.