Existence and nonexistence of traveling pulses in a lateral inhibition neural network

Yixin  Guo; Aijun  Zhang

doi:10.3934/dcdsb.2016020

Article Contents

2016, Volume 21, Issue 6: 1729-1755. Doi: 10.3934/dcdsb.2016020

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Existence and nonexistence of traveling pulses in a lateral inhibition neural network

Yixin Guo^1, and
Aijun Zhang^1,

1.
Department of Mathematics, Drexel University, Philadelphia, PA 19104

Received: April 2015

Revised: June 2016

Published: August 2016

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Abstract

We study the spatial propagating dynamics in a neural network of excitatory and inhibitory populations. Our study demonstrates the existence and nonexistence of traveling pulse solutions with a nonsaturating piecewise linear gain function. We prove that traveling pulse solutions do not exist for such neural field models with even (symmetric) couplings. The neural field models only support traveling pulse solutions with asymmetric couplings. We also show that such neural field models with asymmetric couplings will lead to a system of delay differential equations. We further compute traveling 1--bump solutions using the system of delay differential equations. Finally, we develop Evans functions to assess the stability of traveling 1--bump solutions.

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Mathematics Subject Classification: Primary: 34B05, 34B15, 34D20, 35Q92, 92B20.

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