Neural fields with sigmoidal firing rates: Approximate solutions

Coombes, Stephen; Schmidt, Helmut

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.

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

• Journal Article Type: Article
• Publication Date: Dec 1, 2010
• Deposit Date: Feb 23, 2010
• Publicly Available Date: Dec 1, 2010
• Journal: Discrete and Continuous Dynamical Systems
• Print ISSN: 1078-0947
• Electronic ISSN: 1553-5231
• Publisher: American Institute of Mathematical Sciences (AIMS)
• Peer Reviewed: Not Peer Reviewed
• Volume: 28
• Issue: 4
• Pages: 1369-1379
• DOI: https://doi.org/10.3934/dcds.2010.28.1369
• Keywords: integro-differential equations; neural field models; sigmoidal firing rate; approximation theory
• Public URL: https://nottingham-repository.worktribe.com/output/1012490
• Publisher URL: https://www.aimsciences.org/article/doi/10.3934/dcds.2010.28.1369