Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks

Abstract

Consider the following nonlinear scalar integral differential equation arising from delayed synaptically coupled neuronal networks\begin{eqnarray*}\frac{\partial u}{\partial t}+f(u)&=&\alpha\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y)H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\&+&\beta\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H(u(y,t-\tau)-\Theta){\rm d}y\right]{\rm d}\tau.\end{eqnarray*}This model equation generalizes many important nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks.The kernel functions $K$ and $W$ represent synaptic couplings between neurons in synaptically coupled neuronal networks.The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions),but also lateral inhibitions (modeled with Mexican hat kernel functions) and lateral excitations (modeled with upside down Mexican hat kernel functions).In this nonlinear scalar integral differential equation, $u=u(x,t)$ stands for the membrane potential of a neuron at position $x$ and time $t$.The integrals represent nonlocal spatio-temporal interactions between neurons.   &nbsp We have accomplished the existence and stability of three traveling wave fronts $u(x,t)=U_k(x+\mu_kt)$ of the nonlinear scalar integral differential equation in an earlier work [42],where $\mu_k$ denotes the wave speed and $z=x+\mu_kt$ denotes the moving coordinate, $k=1,2,3$.In this paper, we will investigate how the neurobiological mechanisms represented by the synaptic couplings $(K,W)$,by the probability density functions $(\xi,\eta)$, by the synaptic rate constants $(\alpha,\beta)$ and by the firingthresholds $(\theta,\Theta)$ influence the wave speeds $\mu_k$ of the traveling wave fronts.We will define several speed index functions and use rigorous mathematical analysis to investigate the influence of the neurobiological mechanisms on the wave speeds.In particular, we will compare wave speeds of the traveling wave fronts of the nonlinear scalar integral differential equationwith different synaptic couplings and with different probability density functions;we will accomplish new asymptotic behaviors of the wave speeds; we will compare wave speeds of traveling wave fronts ofmany reduced forms of nonlinear scalar integral differential equations of the above model equation;we will establish new estimates of the wave speeds. All these will greatly improve results obtained in previous work [38], [40] and [41].