Spectral analysis of traveling waves for nonlocal evolution equations

1AbstractTraveling waves of electrical activity are ubiquitous in biological neuronal networks. Traveling waves in the brain are associated with sensory processing, phase coding, and sleep. The neuron and network parameters that determine traveling waves’ evolution are synaptic space constant, synaptic conductance, membrane time constant, and synaptic decay time constant. We used an abstract neuron model to investigate the propagation characteristics of traveling wave activity. We formulated a set of evolution equations based on the network connectivity parameters. We numerically investigated the stability of the traveling wave propagation with a series of perturbations with biological relevance.

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On a class of nonlocal evolution equations with the p[u(x,t)]-Laplace operator

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2020.103165 ◽

2020 ◽

Vol 56 ◽

pp. 103165

Author(s):

Stanislav Antontsev ◽

Sergey Shmarev

Keyword(s):

Laplace Operator ◽

Evolution Equations ◽

Nonlocal Evolution Equations

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Stokes flow singularities in a two-dimensional channel: a novel transform approach with application to microswimming

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0198 ◽

2013 ◽

Vol 469 (2157) ◽

pp. 20130198 ◽

Cited By ~ 6

Author(s):

Darren G. Crowdy ◽

Anthony M. J. Davis

Keyword(s):

Spectral Analysis ◽

Fourier Transform ◽

Boundary Value Problems ◽

Stokes Flow ◽

Evolution Equations ◽

Two Dimensional ◽

Third Order ◽

Transform Method ◽

Independent Variables ◽

Simply Connected

A transform method for determining the flow generated by the singularities of Stokes flow in a two-dimensional channel is presented. The analysis is based on a general approach to biharmonic boundary value problems in a simply connected polygon formulated by Crowdy & Fokas in this journal. The method differs from a traditional Fourier transform approach in entailing a simultaneous spectral analysis in the independent variables both along and across the channel. As an example application, we find the evolution equations for a circular treadmilling microswimmer in the channel correct to third order in the swimmer radius. Significantly, the new transform method is extendible to the analysis of Stokes flows in more complicated polygonal microchannel geometries.

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Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

International Journal of Differential Equations ◽

10.1155/2014/625271 ◽

2014 ◽

Vol 2014 ◽

pp. 1-13

Author(s):

Severino Horácio da Silva ◽

Jocirei Dias Ferreira ◽

Flank David Morais Bezerra

Keyword(s):

Evolution Equation ◽

Lower Semicontinuity ◽

Evolution Equations ◽

Upper Semicontinuity ◽

Global Attractors ◽

Functional Parameter ◽

Normal Hyperbolicity ◽

Nonlocal Evolution Equations

We show the normal hyperbolicity property for the equilibria of the evolution equation∂m(r,t)/∂t=-m(r,t)+g(βJ*m(r,t)+βh), h,β≥0,and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameterJ.

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