Daniele Avitable
Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis
Avitable, Daniele; Wedgwood, Kyle C. A.
Authors
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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Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0
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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by/4.0
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