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Influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks. (English) Zbl 1197.35077

Summary: The main purpose of this paper is to apply mathematical analysis to investigate the influence of sodium currents on the speeds of traveling wave fronts. The authors use speed index functions to investigate the speeds of traveling wave fronts of some scalar integral differential equations arising from synaptically coupled neuronal networks. The mathematical model equation is
\[ u_t + f(u) = \alpha \int_{\mathbb R} K(x-y)H\left(u\left(y, t - \tfrac{1}{c}|x-y|\right)-\theta \right)\,dy, \]
where \(0<c\leq \infty\), \(\alpha> 0\) are constants, satisfying the condition \(0< 2f(\theta)< \alpha \). The function \(f(u)\) represents sodium currents, the function \(K\) denotes synaptic coupling in a neuronal network, and the Heaviside step function \(H\) is defined by \(H(u)=0\) for all \(u<0\), \(H(0)= \frac{1}{2}\) and \(H(u)= 1\) for all \(u> 0\).

MSC:

35C07 Traveling wave solutions
35R09 Integro-partial differential equations
45K05 Integro-partial differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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