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

G�k�e, Ayt�l; Avitabile, Daniele; Coombes, Stephen

Abstract

© 2017, The Author(s). 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.

Citation

Gökçe, A., Avitabile, D., & Coombes, S. (2017). The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions. Journal of Mathematical Neuroscience, 7(1), Article 12. https://doi.org/10.1186/s13408-017-0054-4

• Journal Article Type: Article
• Acceptance Date: Oct 10, 2017
• Online Publication Date: Oct 26, 2017
• Publication Date: Dec 1, 2017
• Deposit Date: Aug 21, 2017
• Publicly Available Date: Oct 26, 2017
• Journal: Journal of Mathematical Neuroscience
• Electronic ISSN: 2190-8567
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 7
• Issue: 1
• Article Number: 12
• DOI: https://doi.org/10.1186/s13408-017-0054-4
• Keywords: neural fields; bounded domain; Dirichlet boundary condition; interface dynamics; piece-wise constant kernel
• Public URL: https://nottingham-repository.worktribe.com/output/889961
• Publisher URL: https://mathematical-neuroscience.springeropen.com/articles/10.1186/s13408-017-0054-4