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Quelques équations de champ moyen en neuroscience

Geoffroy Hermann 1 
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : This thesis deals with the study of the dynamical properties of large neu- ronal networks. We study neurons described by their firing rate with a linear intrinsic dynamics, and take into account several types of microscopic noise impacting the behavior of individual neurons. The "mean field" approach consists in studying the limit of the system of stochastic differential equations describing the network, when the number of neurons tends to infinity. The noise is either additive, or multiplicative if it affects the synaptic weights, and these ones are either fixed at the beginning, or dynamic. Therefore we obtain three types of equations that we study in this thesis. One of the main result is that in each case the propagation of chaos property holds. We analyze par- ticularly the influence of the noise on the dynamics (we show for example its role in the creation of cycles) and we discuss the implications in neuroscience.
Keywords : mean field champ moyen
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Geoffroy Hermann. Quelques équations de champ moyen en neuroscience. Mathématiques générales [math.GM]. Ecole Polytechnique X, 2011. Français. ⟨tel-00850271⟩

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