Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience
In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochasti...
| Autores: | , , , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2011 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/13374 |
| Acceso en línea: | https://hdl.handle.net/2117/13374 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Nonlinear Equacions diferencials parabòliques Equacions diferencials no lineals Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coeffcients depending on the probability ux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study .... |
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