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

Avitable, Daniele and Wedgwood, Kyle C. A. (2017) Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis. Journal of Mathematical Biology, 75 (4). pp. 885-928. ISSN 0303-6812

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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.

Item Type: Article
Keywords: Multiple scale analysis ; Mathematical neuroscience ; Refractoriness ; Spatio-temporal patterns ; Equation-free modelling ; Markov chains
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1007/s00285-016-1070-9
Depositing User: Eprints, Support
Date Deposited: 03 Mar 2017 11:55
Last Modified: 14 Oct 2017 22:27
URI: http://eprints.nottingham.ac.uk/id/eprint/41032

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