Neural fields with sigmoidal firing rates: approximate solutions
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...
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| Format: | Article |
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American Institute of Mathematical Sciences
2010
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| Online Access: | https://eprints.nottingham.ac.uk/1233/ |
| _version_ | 1848790565984927744 |
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| author | Coombes, Stephen Schmidt, Helmut |
| author_facet | Coombes, Stephen Schmidt, Helmut |
| author_sort | Coombes, Stephen |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T18:14:39Z |
| format | Article |
| id | nottingham-1233 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:14:39Z |
| publishDate | 2010 |
| publisher | American Institute of Mathematical Sciences |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-12332020-05-04T20:25:23Z https://eprints.nottingham.ac.uk/1233/ Neural fields with sigmoidal firing rates: approximate solutions Coombes, Stephen Schmidt, Helmut 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. American Institute of Mathematical Sciences 2010 Article NonPeerReviewed Coombes, Stephen and Schmidt, Helmut (2010) Neural fields with sigmoidal firing rates: approximate solutions. Discrete and Continuous Dynamical Systems. Series S . ISSN 1937-1632 (Submitted) integro-differential equations neural field models sigmoidal firing rate approximation theory http://www.aimsciences.org/journals/dcdsS/index.htm |
| spellingShingle | integro-differential equations neural field models sigmoidal firing rate approximation theory Coombes, Stephen Schmidt, Helmut Neural fields with sigmoidal firing rates: approximate solutions |
| title | Neural fields with sigmoidal firing rates: approximate solutions |
| title_full | Neural fields with sigmoidal firing rates: approximate solutions |
| title_fullStr | Neural fields with sigmoidal firing rates: approximate solutions |
| title_full_unstemmed | Neural fields with sigmoidal firing rates: approximate solutions |
| title_short | Neural fields with sigmoidal firing rates: approximate solutions |
| title_sort | neural fields with sigmoidal firing rates: approximate solutions |
| topic | integro-differential equations neural field models sigmoidal firing rate approximation theory |
| url | https://eprints.nottingham.ac.uk/1233/ https://eprints.nottingham.ac.uk/1233/ |