Traveling wave solutions to a neural field model with oscillatory synaptic coupling types

Alan Dyson; Alan Dyson

doi:10.3934/mbe.2019035

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 2: 727-758. doi: 10.3934/mbe.2019035

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Research article

Traveling wave solutions to a neural field model with oscillatory synaptic coupling types

Alan Dyson ^,

Department of Mathematics, Lehigh University, 14 East Packer Ave., Bethlehem, PA 18015, USA

Received: 26 June 2018 Accepted: 18 November 2018 Published: 15 January 2019

In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits.
- integral differential equations,
- traveling wave solutions,
- existence,
- stability,
- Evans function
Citation: Alan Dyson. Traveling wave solutions to a neural field model with oscillatory synaptic coupling types[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 727-758. doi: 10.3934/mbe.2019035

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Abstract

In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits.

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