EXISTENCE AND UNIQUENESS OF A TRAVELING WAVE FRONT OF A MODEL EQUATION IN SYNAPTICALLY COUPLED NEURONAL NETWORKS

Lianzhong Li; Na Li; Yuanyuan Liu; Linghai Zhang

doi:10.11948/2013012

2013 Volume 3 Issue 2

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Lianzhong Li, Na Li, Yuanyuan Liu, Linghai Zhang. EXISTENCE AND UNIQUENESS OF A TRAVELING WAVE FRONT OF A MODEL EQUATION IN SYNAPTICALLY COUPLED NEURONAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2013, 3(2): 145-167. doi: 10.11948/2013012

Citation:

Lianzhong Li, Na Li, Yuanyuan Liu, Linghai Zhang. EXISTENCE AND UNIQUENESS OF A TRAVELING WAVE FRONT OF A MODEL EQUATION IN SYNAPTICALLY COUPLED NEURONAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2013, 3(2): 145-167. doi: 10.11948/2013012

EXISTENCE AND UNIQUENESS OF A TRAVELING WAVE FRONT OF A MODEL EQUATION IN SYNAPTICALLY COUPLED NEURONAL NETWORKS

1 School of Mathematics and Statistics, Taishan University, Taian, Shandong, 271000, China;
2 College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, China;
3 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China;
4 Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015, USA

Abstract

Abstract

Consider the model equation in synaptically coupled neuronal networks =∂u/∂t + m(u -n) (α -au) ∫₀^∞ ξ(c)[∫_R K(x -y)H (u (y, t -1/c|x -y|) -θ) dy] dc +(β -bu) ∫₀^∞ η(τ)[∫_R K(x -y)H (u (y, t -τ) -θ)dy] dτ.In this model equation, u=u(x, t) stands for the membrane potential of a neuron at position x and time t. The kernel functions K ≥ 0 and W ≥ 0 represent synaptic couplings between neurons in synaptically coupled neuronal networks. The Heaviside step function H=H(u -θ) represents the gain function and it is defined by H(u -θ)=0 for all u<θ, H(0)=1/2 and H(u -θ)=1 for all u>θ. The functions ξ and η represent probability density functions. The function f(u) ≡ m(u-n) represents the sodium current, where m>0 is a positive constant and n is a real constant. The constants a ≥ 0, b ≥ 0, α ≥ 0, β ≥ 0 and θ>0 represent biological mechanisms. This model equation is motivated by previous models in synaptically coupled neuronal networks.
We will couple together intermediate value theorem, mean value theorem and many techniques in dynamical systems to prove the existence and uniqueness of a traveling wave front of this model equation. One of the most interesting and difficult parts is the proof of the existence and uniqueness of the wave speed. We will introduce several auxiliary functions and speed index functions to prove the existence and uniqueness of the front and the wave speed.
- Synaptically coupled neuronal networks /
- traveling wave front /
- existence and uniqueness /
- speed index function /
- wave speed
MSC: 92C20

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