Modeling the Nerve Conduction in a Myelinated Axon

Abstract

In this paper it is done a brief review about the mathematical modeling of the nerve conduction in a myelinated axon considering the Fitzhugh-Nagumo equation, a forward-backward differential equation (FBDE). We look for a solution of this FBDE defined in $\mathbb {R}$, with known values at $\pm \infty$. Extending the idea initially presented in [1] and [2], was developed a numerical method to solve an autonomous linear FBDE using the method of steps and finite differences. Continuing this approach, the authors of [3,4,5] introduced some numerical schemes based on method of steps, collocation and finite element method. These schemes developed for linear FBDEs were adapted to solve the nonlinear boundary value problem, the Fitzhugh-Nagumo equation [6,7,8]. The homotopy analysis method, algorithm proposed Liao in 1991 [9], became in an important tool to solve non linear equations during the last two decades, was also applied to get the numerical solution of the equation under study [25]. Here, it is done a brief review of different aproaches to solve nunerically equations that models nerve conduction. Also is used a different data basis using radial functions [10, 11] to solve numerically the equation under study, similarly to the work presented in [12], where radial functions are considered to solve a nonlinear equation from acoustics. The results are computed and compared with the ones from other computational methods. The results are promising but it still necessary to continue with the experiments with another sets of basis funtions.