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How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks? (English) Zbl 1210.35018

Summary: We are concerned with existence, uniqueness, and stability of the traveling wave of a nonlocal model equation which incorporates spatial temporal delay due to the finite propagation velocity of action potentials along axons. In particular, we investigate how wave shape, speed, and stability vary as the synaptic coupling and the model parameters change. The synaptic coupling may be of pure excitation, lateral inhibition, or lateral excitation. We introduce two concepts: the speed index function and the stability index function. One interesting point is that we can define the stability index function through the speed index function. By using this relationship, the stability of the traveling wave can be analyzed easily. These concepts (the speed index function and the stability index function) may play very important roles in rigorous mathematical analysis of traveling waves of nonlinear singularly perturbed systems of integral differential equations. The analysis and results on the speed, the speed index function, and the stability index function can be applied to dynamical systems and computational neuroscience.

MSC:

35B25 Singular perturbations in context of PDEs
35R10 Partial functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
92C20 Neural biology
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