Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur

Méthode stochastique en neurosciences

  • 24 Mai 2017
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This thesis explores the chaotic behavior of systems of interacting particles,with an emphasis on applications in neuroscience. It finds conditions on the neural network such that, in the large network size limit, the particles be-have as if they are independent. This property is known as propagation of chaos. In the first part of this thesis, this property is proved to hold uniformly for any positive time, in a general model that has a Lipschitz coupling term. In the second part, the propagation of chaos is proved to hold over a fixed time interval. The model is of Wilson-Cowan type with electrical synaptic connections. This model lies outside the scope of classical propagation of chaos results. The propagation of chaos result is obtained by taking advantage of the fact that the mean-field equations are Gaussian, with mean and covariance governed by a system of ODEs. This allows one to use Borell's inequality to prove that its tail decays exponentially. Finally, exploratory numerical tests are made on the mean-field ODEs in order to understand how the system's bifurcations depend on the parameters.