December  2010, 28(4): 1369-1379. doi: 10.3934/dcds.2010.28.1369

Neural fields with sigmoidal firing rates: Approximate solutions

1. 

School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom, United Kingdom

Received  October 2009 Revised  February 2010 Published  June 2010

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: Stephen Coombes, Helmut Schmidt. Neural fields with sigmoidal firing rates: Approximate solutions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1369-1379. doi: 10.3934/dcds.2010.28.1369
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