Existence of Wave Front Solutions of an Integral Differential Equation in Nonlinear Nonlocal Neuronal Network

Lijun Zhang,C M Khalique,Jie Yuan,Linghai Zhang

doi:10.1155/2014/753614

Lijun Zhang, C M Khalique + Show 2 more

Open Access

https://doi.org/10.1155/2014/753614

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Abstract

An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions). The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.

Highlights

We study the existence and uniqueness of the wave front solution to model equation (5); that is to say, we study the solution which tends from 0 to α as z goes from −∞ to ∞
We have investigated the existence and uniqueness of the wave front solution of the integral-differential model equation (5) arising from neuronal networks
We reduced the nonlinear-integral differential equation (5) into a simpler solvable linear differential equation (21) by using the special property of Heaviside gain function and the hypothesis of the wave front solution that U(0) = θ, U(z) < θ for z ∈ (−∞, 0), and U(z) > θ for z ∈ (0, +∞)

Summary

Introduction

Hutt considered both the nonlocal axonal connections and nonlocal feedback connections with a time delay and examined briefly the dependence of the speed of the front on various parameters in this model equation He assumed the same firing threshold for all neurons function and the transfer function was chosen to be the Heaviside step function. Zhang [11] studied the existence, uniqueness, and stability of traveling wave solutions to the model equation (5) for three typical classes of kernel functions. It is reasonable and meaningful to proceed with the study of the model equation (5) with oscillatory kernel functions Motivated by their pioneer works [11,12,13, 18,19,20]; in this paper, we aim to study the existence and uniqueness of the wave front solutions of IDE (5) with more general kernel functions. The main idea in this paper is employing the speed index functions (the main idea in [11, 19] and other pioneering works)

Preliminary Analysis

Existence and Uniqueness of the Wave Front

Conclusion and Some Discussions