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

Journal of Mathematical Neuroscience
Aytül GökçeS Coombes

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. Further...Continue Reading

References

Jan 20, 2018·Physical Review. E·Zachary P Kilpatrick, Daniel B Poll
Mar 17, 2018·Journal of Computational Neuroscience·Nikhil KrishnanZachary P Kilpatrick

Citations

Oct 1, 1979·Biological cybernetics·G Bard Ermentrout, J D Cowan
Jan 1, 1975·Biological cybernetics·S Amari
Jan 1, 1972·Biophysical Journal·H R Wilson, J D Cowan
Mar 1, 1995·Neuron·P S Goldman-Rakic
Apr 1, 1997·Brain : a Journal of Neurology·V B Mountcastle
Apr 1, 1996·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·R E GoldsteinD M Petrich
Apr 24, 2001·Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences·Paul C BressloffM C Wiener
Jul 31, 2001·Trends in Neurosciences·X J Wang
Aug 2, 2005·Biological cybernetics·S Coombes
Dec 31, 2005·Physical Review Letters·S E Folias, Paul C Bressloff
Feb 5, 2008·IEEE Transactions on Neural Networks·E M Izhikevich
Apr 29, 2009·NeuroImage·Jean DaunizeauKarl J Friston
Jan 26, 2010·NeuroImage·S Coombes
Apr 7, 2010·Brain Topography·Ingo BojakRolf Kötter
Dec 15, 2010·Neuron·Xiaoying HuangJian-Young Wu
Nov 23, 2011·Biological cybernetics·Julie Goulet, G Bard Ermentrout
Jun 5, 2012·Journal of Mathematical Neuroscience·Paul C Bressloff
Jun 5, 2012·Journal of Mathematical Neuroscience·S CoombesIngo Bojak
Feb 18, 2014·Cognitive Computation·Paul C Bressloff, S Coombes
Aug 19, 2015·Journal of Computational Neuroscience·X Zhao, P A Robinson
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Jun 20, 2017·Physica D. Nonlinear Phenomena·S VisserS Coombes

Related Concepts

Patterns
Bounded by
2-Dimensional
Imbalance
Boundary Elements
Neurons
Neural Stem Cells
Structure of Cortex of Kidney
Spatio-Temporal Analysis
Description

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