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Modeling the Nerve Conduction in a Myelinated Axon A Brief Review

机译：髓鞘轴突中的神经传导建模简述

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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 R, with known values at ±∞. 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 [3-5] 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 [6-8]. 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.
机译：在本文中，对考虑Fitzhugh-Nagumo方程，前后微分方程（FBDE）的髓鞘轴突中神经传导的数学建模进行了简要回顾。我们寻找在R中定义的FBDE的解决方案，其已知值为±∞。扩展了最初在[1]和[2]中提出的思想，开发了一种数值方法，该方法使用步长和有限差分的方法来求解自治线性FBDE。继续采用这种方法，[3-5]的作者介绍了一些基于步长法，并置法和有限元法的数值方案。这些为线性FBDE开发的方案适用于解决非线性边界值问题，即Fitzhugh-Nagumo方程[6-8]。同种分析方法，是廖（Liao）在1991年提出的算法[9]，在过去的20年中成为解决非线性方程的重要工具，也被用于获得正在研究的方程的数值解[25]。在这里，我们对不同方法进行了简要回顾，以解决模拟神经传导的数值方程。还使用了不同的数据基础，使用径向函数[10,11]来数值求解所研究的方程，类似于[12]中介绍的工作，其中径向函数被认为是从声学中求解非线性方程的方法。计算结果并将其与其他计算方法的结果进行比较。结果令人鼓舞，但仍然有必要继续进行另一组基础功能的实验。

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