# 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 and Continuous Dynamical Systems - B, 2011, 16 (3) : 1003-1037. doi: 10.3934/dcdsb.2011.16.1003
##### References:
 [1] Fatihcan M. Atay and Axel Hutt, Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM Journal on Applied Mathematics, 65 (2004), 644-666. doi: 10.1137/S0036139903430884. [2] Fatihcan M. Atay and Axel Hutt, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM Journal on Applied Dynamical Systems, 5 (2006), 670-698. doi: 10.1137/050629367. [3] Stephen Coombes, Gabriel J. Lord and Markus R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178 (2003), 219-241. doi: 10.1016/S0167-2789(03)00002-2. [4] Stephen Coombes and Markus R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 3 (2004), 574-600. doi: 10.1137/040605953. [5] Axel Hutt and Fatihcan M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203 (2005), 30-54. doi: 10.1016/j.physd.2005.03.002. [6] Axel Hutt and Fatihcan M. Atay, Effects of distributed transmission speeds on propagating activity in neural populations, Physical Review E, Statistical, nonlinear, and soft matter physics, 73 (2006), 021906. doi: 10.1103/PhysRevE.73.021906. [7] Felicia Maria G. Magpantay and Xingfu Zou, Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections, Mathematical Biosciences and Engineering, 7 (2010), 421-442. doi: 10.3934/mbe.2010.7.421. [8] David J. Pinto and G. Bard Ermentrout, Spatially structured activity in synaptically coupled neuronal networks. I. traveling fronts and pulses, II. Lateral inhibition and standing pulses, SIAM Journal on Applied Mathematics, 62 (2001), I. 206-225, II. 226-243. [9] Hugh R. Wilson and Jack D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5. [10] Hugh R. Wilson and Jack D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13 (1973), 55-80. doi: 10.1007/BF00288786. [11] Eiji Yanagida and Linghai Zhang, Speeds of traveling waves of some integral differential equations, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 347-373. doi: 10.1007/s13160-010-0021-x. [12] Linghai Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, Journal of Differential Equations, 197 (2004), 162-196. doi: 10.1016/S0022-0396(03)00170-0. [13] Linghai Zhang, Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks, Journal of Dynamics and Differential Equations, 17 (2005), 489-522. doi: 10.1007/s10884-005-5404-3. [14] Linghai Zhang, How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?, SIAM Journal on Applied Dynamical Systems, 6 (2007), 597-644. [15] Linghai Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks, Advances in Mathematics Research. Editor-in-Chief: Albert R. Baswell. Nova Science Publishers Inc. New York. ISBN: 978-1-60876-265-1. 10 (2010), 53-204. [16] Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner, Influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks, Physica D, 239 (2010), 9-32. doi: 10.1016/j.physd.2009.09.022.

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##### References:
 [1] Fatihcan M. Atay and Axel Hutt, Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM Journal on Applied Mathematics, 65 (2004), 644-666. doi: 10.1137/S0036139903430884. [2] Fatihcan M. Atay and Axel Hutt, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM Journal on Applied Dynamical Systems, 5 (2006), 670-698. doi: 10.1137/050629367. [3] Stephen Coombes, Gabriel J. Lord and Markus R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178 (2003), 219-241. doi: 10.1016/S0167-2789(03)00002-2. [4] Stephen Coombes and Markus R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 3 (2004), 574-600. doi: 10.1137/040605953. [5] Axel Hutt and Fatihcan M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203 (2005), 30-54. doi: 10.1016/j.physd.2005.03.002. [6] Axel Hutt and Fatihcan M. Atay, Effects of distributed transmission speeds on propagating activity in neural populations, Physical Review E, Statistical, nonlinear, and soft matter physics, 73 (2006), 021906. doi: 10.1103/PhysRevE.73.021906. [7] Felicia Maria G. Magpantay and Xingfu Zou, Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections, Mathematical Biosciences and Engineering, 7 (2010), 421-442. doi: 10.3934/mbe.2010.7.421. [8] David J. Pinto and G. Bard Ermentrout, Spatially structured activity in synaptically coupled neuronal networks. I. traveling fronts and pulses, II. Lateral inhibition and standing pulses, SIAM Journal on Applied Mathematics, 62 (2001), I. 206-225, II. 226-243. [9] Hugh R. Wilson and Jack D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5. [10] Hugh R. Wilson and Jack D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13 (1973), 55-80. doi: 10.1007/BF00288786. [11] Eiji Yanagida and Linghai Zhang, Speeds of traveling waves of some integral differential equations, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 347-373. doi: 10.1007/s13160-010-0021-x. [12] Linghai Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, Journal of Differential Equations, 197 (2004), 162-196. doi: 10.1016/S0022-0396(03)00170-0. [13] Linghai Zhang, Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks, Journal of Dynamics and Differential Equations, 17 (2005), 489-522. doi: 10.1007/s10884-005-5404-3. [14] Linghai Zhang, How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?, SIAM Journal on Applied Dynamical Systems, 6 (2007), 597-644. [15] Linghai Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks, Advances in Mathematics Research. Editor-in-Chief: Albert R. Baswell. Nova Science Publishers Inc. New York. ISBN: 978-1-60876-265-1. 10 (2010), 53-204. [16] Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner, Influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks, Physica D, 239 (2010), 9-32. doi: 10.1016/j.physd.2009.09.022.
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