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Regularization of Ill-Posed Point Neuron Models
Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rate function is employed, then standard ODE theory implies that such models are well-posed...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer Berlin Heidelberg
2017
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5509800/ https://www.ncbi.nlm.nih.gov/pubmed/28707194 http://dx.doi.org/10.1186/s13408-017-0049-1 |
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author | Nielsen, Bjørn Fredrik |
author_facet | Nielsen, Bjørn Fredrik |
author_sort | Nielsen, Bjørn Fredrik |
collection | PubMed |
description | Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rate function is employed, then standard ODE theory implies that such models are well-posed and can thus, approximately, be solved with finite precision arithmetic. We investigate whether the solution of this well-posed model converges to a solution of the ill-posed limit problem as the steepness parameter of the firing rate function tends to infinity. Our argument employs the Arzelà–Ascoli theorem and also yields the existence of a solution of the limit problem. However, we only obtain convergence of a subsequence of the regularized solutions. This is consistent with the fact that models with a Heaviside firing rate function can have several solutions, as we show. Our analysis assumes that the vector-valued limit function v, provided by the Arzelà–Ascoli theorem, is threshold simple: That is, the set containing the times when one or more of the component functions of v equal the threshold value for firing, has zero Lebesgue measure. If this assumption does not hold, we argue that the regularized solutions may not converge to a solution of the limit problem with a Heaviside firing function. |
format | Online Article Text |
id | pubmed-5509800 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2017 |
publisher | Springer Berlin Heidelberg |
record_format | MEDLINE/PubMed |
spelling | pubmed-55098002017-07-31 Regularization of Ill-Posed Point Neuron Models Nielsen, Bjørn Fredrik J Math Neurosci Research Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rate function is employed, then standard ODE theory implies that such models are well-posed and can thus, approximately, be solved with finite precision arithmetic. We investigate whether the solution of this well-posed model converges to a solution of the ill-posed limit problem as the steepness parameter of the firing rate function tends to infinity. Our argument employs the Arzelà–Ascoli theorem and also yields the existence of a solution of the limit problem. However, we only obtain convergence of a subsequence of the regularized solutions. This is consistent with the fact that models with a Heaviside firing rate function can have several solutions, as we show. Our analysis assumes that the vector-valued limit function v, provided by the Arzelà–Ascoli theorem, is threshold simple: That is, the set containing the times when one or more of the component functions of v equal the threshold value for firing, has zero Lebesgue measure. If this assumption does not hold, we argue that the regularized solutions may not converge to a solution of the limit problem with a Heaviside firing function. Springer Berlin Heidelberg 2017-07-14 /pmc/articles/PMC5509800/ /pubmed/28707194 http://dx.doi.org/10.1186/s13408-017-0049-1 Text en © The Author(s) 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
spellingShingle | Research Nielsen, Bjørn Fredrik Regularization of Ill-Posed Point Neuron Models |
title | Regularization of Ill-Posed Point Neuron Models |
title_full | Regularization of Ill-Posed Point Neuron Models |
title_fullStr | Regularization of Ill-Posed Point Neuron Models |
title_full_unstemmed | Regularization of Ill-Posed Point Neuron Models |
title_short | Regularization of Ill-Posed Point Neuron Models |
title_sort | regularization of ill-posed point neuron models |
topic | Research |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5509800/ https://www.ncbi.nlm.nih.gov/pubmed/28707194 http://dx.doi.org/10.1186/s13408-017-0049-1 |
work_keys_str_mv | AT nielsenbjørnfredrik regularizationofillposedpointneuronmodels |