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Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

Avitable, Daniele; Wedgwood, Kyle C. A.

Authors

Daniele Avitable

Kyle C. A. Wedgwood



Abstract

We study coarse pattern formation in a cellular automaton modelling a spatially-extended stochastic neural network. The model, originally proposed by Gong and Robinson (Phys Rev E 85(5):055,101(R), 2012), is known to support stationary and travelling bumps of localised activity. We pose the model on a ring and study the existence and stability of these patterns in various limits using a combination of analytical and numerical techniques. In a purely deterministic version of the model, posed on a continuum, we construct bumps and travelling waves analytically using standard interface methods from neural field theory. In a stochastic version with Heaviside firing rate, we construct approximate analytical probability mass functions associated with bumps and travelling waves. In the full stochastic model posed on a discrete lattice, where a coarse analytic description is unavailable, we compute patterns and their linear stability using equation-free methods. The lifting procedure used in the coarse time-stepper is informed by the analysis in the deterministic and stochastic limits. In all settings, we identify the synaptic profile as a mesoscopic variable, and the width of the corresponding activity set as a macroscopic variable. Stationary and travelling bumps have similar meso- and macroscopic profiles, but different microscopic structure, hence we propose lifting operators which use microscopic motifs to disambiguate them. We provide numerical evidence that waves are supported by a combination of high synaptic gain and long refractory times, while meandering bumps are elicited by short refractory times.

Citation

Avitable, D., & Wedgwood, K. C. A. (in press). Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis. Journal of Mathematical Biology, 75(4), https://doi.org/10.1007/s00285-016-1070-9

Journal Article Type Article
Acceptance Date Oct 16, 2016
Online Publication Date Feb 1, 2017
Deposit Date Mar 3, 2017
Publicly Available Date Mar 28, 2024
Journal Journal of Mathematical Biology
Print ISSN 0303-6812
Electronic ISSN 0303-6812
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 75
Issue 4
DOI https://doi.org/10.1007/s00285-016-1070-9
Keywords Multiple scale analysis ; Mathematical neuroscience ; Refractoriness ; Spatio-temporal patterns ; Equation-free modelling ; Markov chains
Public URL https://nottingham-repository.worktribe.com/output/836881
Publisher URL http://link.springer.com/article/10.1007/s00285-016-1070-9

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