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Neural fields with sigmoidal firing rates: Approximate solutions

Coombes, Stephen; Schmidt, Helmut

Authors

Helmut Schmidt pmxhs@nottingham.ac.uk



Abstract

Many tissue level models of neural networks are written in the language of nonlinear integro-differential equations. Analytical solutions have only been obtained for the special case that the nonlinearity is a Heaviside function. Thus the pursuit of even approximate solutions to such models is of interest to the broad mathematical neuroscience community. Here we develop one such scheme, for stationary and travelling wave solutions, that can deal with a certain class of smoothed Heaviside functions. The distribution that smoothes the Heaviside is viewed as a fundamental object, and all expressions describing the scheme are constructed in terms of integrals over this distribution. The comparison of our scheme and results from direct numerical simulations is used to highlight the very good levels of approximation that can be achieved by iterating the process only a small number of times.

Journal Article Type Article
Publication Date Dec 1, 2010
Journal Discrete and Continuous Dynamical Systems
Print ISSN 1078-0947
Electronic ISSN 1937-1632
Publisher American Institute of Mathematical Sciences
Peer Reviewed Not Peer Reviewed
Volume 28
Issue 4
Pages 1369-1379
APA6 Citation Coombes, S., & Schmidt, H. (2010). Neural fields with sigmoidal firing rates: Approximate solutions. Discrete and Continuous Dynamical Systems - Series A, 28(4), 1369-1379. https://doi.org/10.3934/dcds.2010.28.1369
DOI https://doi.org/10.3934/dcds.2010.28.1369
Keywords integro-differential equations
neural field models
sigmoidal firing rate
approximation theory
Publisher URL https://www.aimsciences.org/article/doi/10.3934/dcds.2010.28.1369
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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