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Interface dynamics in planar neural field models

Coombes, Stephen; Schmidt, Helmut; Bojak, Ingo

Authors

Helmut Schmidt pmxhs@nottingham.ac.uk

Ingo Bojak i.bojak@bham.ac.uk



Abstract

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves. © 2012 Coombes et al.

Journal Article Type Article
Publication Date May 11, 2012
Journal Journal of Mathematical Neuroscience
Electronic ISSN 2190-8567
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 2
Issue 1
Pages 1-46
APA6 Citation Coombes, S., Schmidt, H., & Bojak, I. (2012). Interface dynamics in planar neural field models. Journal of Mathematical Neuroscience, 2(1), 1-46. https://doi.org/10.1186/2190-8567-2-9
DOI https://doi.org/10.1186/2190-8567-2-9
Keywords neural field
interface dynamics
Publisher URL https://mathematical-neuroscience.springeropen.com/articles/10.1186/2190-8567-2-9
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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