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Neural fields with sigmoidal firing rates: Approximate solutions

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


Cite as: http://hdl.handle.net/21.11116/0000-0003-4F01-4
Abstract
Many tissue level models of neural networks are written in the language of nonlinear integro-differentialequations. Analytical solutions have only been obtained for the special case that the nonlinearity is a Heavisidefunction. Thus the pursuit of even approximate solutions to such models is of interest to the broad mathemati-cal 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 Heavi-side is viewed as a fundamental object, and all expressions describing the scheme are constructed in terms ofintegrals over this distribution. The comparison of our scheme and results from direct numerical simulationsis used to highlight the very good levels of approximation that can be achieved by iterating the process only asmall number of times.