Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

Shaher Momani,Asad Freihat,Mohammed Al-Smadi

doi:10.1155/2014/276279

Abstract

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

Highlights

Mathematical modeling method of real-life phenomena is widely applied in medicine and biology
We present basic facts, definitions, and notations related to the fractional calculus and multistep generalized differential transform method (MSGDTM)
A multistep generalized differential transform method has been successfully applied to find the numerical solutions of the fractional-order multiple chaotic FitzHughNagumo neurons model

Summary

Introduction

Mathematical modeling method of real-life phenomena is widely applied in medicine and biology. The attention is given to obtain the approximate solution of the fractional-order multiple chaotic FHN neurons model under external electrical stimulation with different gap junctions using the MSGDTM. This method is only a simple modification of the generalized differential transform method (GDTM), in which it is treated as an algorithm in a sequence of small intervals (i.e., time step) for finding accurate approximate solutions to the corresponding systems. Accuracy, and efficiency of the MSGDTM for solving linear and nonlinear fractionalorder equations, we applied this scheme to the fractionalorder model of three coupled chaotic FHN neurons with different gap junctions [20], which is the lowest-order chaotic system among all the chaotic systems.

A Test Problem for the Fractional-Order Chaotic FHN Neurons Model

Conclusions