A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics
Cáceres, María J. (Universidad de Granada. Departamento de Matemática Aplicada)
Carrillo de la Plata, José Antonio (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Tao, Louis (Peking University. Plant Genetics Engineering)
Centre de Recerca Matemàtica

Publicación:	Centre de Recerca Matemàtica 2010
Descripción:	25 p.
Resumen:	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.
Derechos:	Aquest 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.
Lengua:	Anglès
Colección:	Centre de Recerca Matemàtica. Prepublicacions
Colección:	Prepublicacions del Centre de Recerca Matemàtica ; 938
Documento:	Article ; Prepublicació ; Versió de l'autor
Materia:	Fokker-Planck, Equació de ; Xarxes neuronals (Informàtica)

25 p, 2.4 MB

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