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The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions

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 bounda...

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Autores principales: Gökçe, Aytül, Avitabile, Daniele, Coombes, Stephen
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Berlin Heidelberg 2017
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5658324/
https://www.ncbi.nlm.nih.gov/pubmed/29075933
http://dx.doi.org/10.1186/s13408-017-0054-4
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author Gökçe, Aytül
Avitabile, Daniele
Coombes, Stephen
author_facet Gökçe, Aytül
Avitabile, Daniele
Coombes, Stephen
author_sort Gökçe, Aytül
collection PubMed
description 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.
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spelling pubmed-56583242017-11-20 The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions Gökçe, Aytül Avitabile, Daniele Coombes, Stephen J Math Neurosci Research 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. Springer Berlin Heidelberg 2017-10-26 /pmc/articles/PMC5658324/ /pubmed/29075933 http://dx.doi.org/10.1186/s13408-017-0054-4 Text en © The Author(s) 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
spellingShingle Research
Gökçe, Aytül
Avitabile, Daniele
Coombes, Stephen
The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions
title The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions
title_full The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions
title_fullStr The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions
title_full_unstemmed The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions
title_short The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions
title_sort dynamics of neural fields on bounded domains: an interface approach for dirichlet boundary conditions
topic Research
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5658324/
https://www.ncbi.nlm.nih.gov/pubmed/29075933
http://dx.doi.org/10.1186/s13408-017-0054-4
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