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The dynamics of neural fields on bounded domains: an interface approach for Dirichlet boundary conditions

Gökçe, Aytül; Avitabile, Daniele; Coombes, Stephen

Authors

Aytül Gökçe

Daniele Avitabile Daniele.Avitabile@nottingham.ac.uk



Abstract

Continuum neural field equations model the large scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded as well as unbounded domains. Neural field models do not require the specification of boundary conditions. Relatively little attention has been paid to the imposition of neural activity on the boundary, or to its role in inducing patterned states. Here we redress this imbalance by studying neural field models of Amari type (posed on one- and two-dimensional bounded domains) with Dirichlet boundary conditions. The Amari model has a Heaviside nonlinearity that allows for a description of localised solutions of the neural field with an interface dynamics. We show how to generalise this reduced but exact description by deriving a normal velocity rule for an interface that encapsulates boundary effects. The linear stability analysis of localised states in the interface dynamics is used to understand how spatially extended patterns may develop in the absence and presence of boundary conditions. Theoretical results for pattern formation are shown to be in excellent agreement with simulations of the full neural field model. Furthermore, a numerical scheme for the interface dynamics is introduced and used to probe the way in which a Dirichlet boundary condition can limit the growth of labyrinthine structures.

Journal Article Type Article
Journal Journal of Mathematical Neuroscience
Electronic ISSN 2190-8567
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 7
APA6 Citation Gökçe, A., Avitabile, D., & Coombes, S. (in press). The dynamics of neural fields on bounded domains: an interface approach for Dirichlet boundary conditions. Journal of Mathematical Neuroscience, 7, https://doi.org/10.1186/s13408-017-0054-4
DOI https://doi.org/10.1186/s13408-017-0054-4
Keywords neural fields; bounded domain; Dirichlet boundary condition; interface dynamics; piece-wise constant kernel
Publisher URL https://mathematical-neuroscience.springeropen.com/articles/10.1186/s13408-017-0054-4
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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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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