A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by con...

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Detalles Bibliográficos
Autores: Cáceres, María J., Carrillo de la Plata, José Antonio, Tao, Louis
Tipo de recurso: artículo
Fecha de publicación:2010
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:60938
Acceso en línea:https://ddd.uab.cat/record/60938
Access Level:acceso abierto
Palabra clave:Fokker-Planck, Equació de
Xarxes neuronals (Informàtica)
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spelling A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamicsCáceres, María J.Carrillo de la Plata, José AntonioTao, LouisFokker-Planck, Equació deXarxes neuronals (Informàtica)To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.Centre de Recerca MatemàticaCentre de Recerca Matemàtica 22010-01-0120102010-01-01Articlehttp://purl.org/coar/resource_type/c_6501AOhttp://purl.org/coar/version/c_b1a7d7d4d402bcceinfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/60938reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.https://creativecommons.org/licenses/by-nc-nd/2.5/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:609382026-06-06T12:50:31Z
dc.title.none.fl_str_mv A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
title A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
spellingShingle A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
Cáceres, María J.
Fokker-Planck, Equació de
Xarxes neuronals (Informàtica)
title_short A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
title_full A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
title_fullStr A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
title_full_unstemmed A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
title_sort A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
dc.creator.none.fl_str_mv Cáceres, María J.
Carrillo de la Plata, José Antonio
Tao, Louis
author Cáceres, María J.
author_facet Cáceres, María J.
Carrillo de la Plata, José Antonio
Tao, Louis
author_role author
author2 Carrillo de la Plata, José Antonio
Tao, Louis
author2_role author
author
dc.contributor.none.fl_str_mv Centre de Recerca Matemàtica
dc.subject.none.fl_str_mv Fokker-Planck, Equació de
Xarxes neuronals (Informàtica)
topic Fokker-Planck, Equació de
Xarxes neuronals (Informàtica)
description To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.
publishDate 2010
dc.date.none.fl_str_mv 2
2010-01-01
2010
2010-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AO
http://purl.org/coar/version/c_b1a7d7d4d402bcce
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/60938
url https://ddd.uab.cat/record/60938
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/2.5/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/2.5/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Centre de Recerca Matemàtica
publisher.none.fl_str_mv Centre de Recerca Matemàtica
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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