# American Institute of Mathematical Sciences

October  2011, 16(3): 1003-1037. doi: 10.3934/dcdsb.2011.16.1003

## Influence of neurobiological mechanisms on speeds of traveling wave fronts in mathematical neuroscience

 1 Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015, United States, United States, United States

Received  August 2010 Revised  October 2010 Published  June 2011

We study speeds of traveling wave fronts of the following integral differential equation

$\frac{\partial u}{\partial t}+f(u)\hspace{6cm}$

$=(\alpha-au)\int^{\infty}_0\xi(c)[\int_R K(x-y) H(u(y,t-\frac{1}{c}|x-y|)-\theta)dy]dc$

$+(\beta-bu)\int^{\infty}_0\eta(\tau)[\int_RW(x-y) H(u(y,t-\tau)-\Theta)dy]d\tau.$

This model equation is motivated by previous models which arise from synaptically coupled neuronal networks. In this equation, $f(u)$ is a smooth function of $u$, usually representing sodium current in the neuronal networks. Typical examples include $f(u)=u$ and $f(u)=u(u-1)(Du-1)$, where $D>1$ is a constant. The transmission speed distribution $\xi$ and the feedback delay distribution $\eta$ are probability density functions. The kernel functions $K$ and $W$ represent synaptic couplings between neurons in the neuronal networks. The function $H$ stands for the Heaviside step function: $H(u-\theta)=0$ for all $u<\theta$, $H(0)=\frac{1}{2}$ and $H(u-\theta)=1$ for all $u>\theta$. Here $H$ represents the gain function. The parameters $a \geq 0$, $b \geq 0$, $\alpha \geq 0$, $\beta \geq 0$, $\theta > 0$ and $\Theta > 0$ represent biological mechanisms in the neuronal networks.
We will use mathematical analysis to investigate the influence of neurobiological mechanisms on the speeds of the traveling wave fronts. We will derive new estimates for the wave speeds. These results are quite different from the results obtained before, complementing the estimates obtained in many previous papers [11], [14], [15], and [16].
We will also use MATLAB to perform numerical simulations to investigate how the neurobiological mechanisms $a$, $b$, $\alpha$, $\beta$, $\theta$ and $\Theta$ influence the wave speeds.

Citation: Linghai Zhang, Ping-Shi Wu, Melissa Anne Stoner. Influence of neurobiological mechanisms on speeds of traveling wave fronts in mathematical neuroscience. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 1003-1037. doi: 10.3934/dcdsb.2011.16.1003
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##### References:
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