Existence and uniqueness of wave fronts in neuronal network with nonlocal post-synaptic axonal and delayed nonlocal feedback connections

Lijun Zhang

doi:10.1186/1687-1847-2013-243

Abstract

An integral-differential model equation, arising from neuronal networks with both axonal and delayed nonlocal feedback connections, is considered in this paper. The kernel functions in the feedback channel we study here include not only pure excitations but also lateral inhibition. For the kernel functions in the synaptic coupling, pure excitations, lateral inhibition, the lateral excitations and more general synaptic couplings (e.g., oscillating kernel functions) are considered. The main goal of this paper is the study of the existence and uniqueness of the traveling wave front solutions. The main method we applied is the speed index functions and principle of linear superposition.

Highlights

IntroductionWe consider the following integral differential model equation |x – y|
In this paper, we consider the following integral differential model equation |x – y|ut + u = α K(x – y)H u y, t – R c – θ dy+ β J(x – y)H u(y, t – τ ) – θ dy, ( . )which was proposed by Hutt [ ] to understand the mechanism of the formation and propagation of activity patterns in neural networks
3.1 Existence and uniqueness of the wave speed At first, we prove the existence of the solution to φ(μ) = θ, and we prove that it is unique when the kernel function J(z) is of type (A) or type (B)

Summary

Introduction

We consider the following integral differential model equation |x – y|. ). which was proposed by Hutt [ ] to understand the mechanism of the formation and propagation of activity patterns in neural networks. The first term and the second one on the right side of equation ) represent the synaptic input by axonal and feedback connections, respectively. The kernel functions K(x) and J(x) are introduced as probability density functions of connection, which may be negative at some points to allow for inhibitory behavior in coupling. The parameters α and β represent the synaptic strength of axonal and nonlocal feedback contributions, respectively. Both the intral-areal nonlocal axonal connections with a transmission delay

Objectives

Conclusion