Propagation of chaos in neural fields

Details Propagation of chaos in neural fields Propagation of chaos in neural fields

2011-08-11

arxiv.org/abs/1108.2414v3

Mathematics

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Scientific paper

We consider the problem of the limit of bio-inspired spatially extended neuronal networks including an infinite number of neuronal types (space locations), with space-dependent propagation delays modeling neural fields. The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which we can resolve quite fine structure of the neuron's activity in space and where averaging effects occur. The mean-field equations obtained are of a new type: they take the form of well-posed infinite-dimensional delayed integro-differential equations with a nonlocal mean-field term. These results have several theoretical implications in neurosciences we review in the discussion. We also review some results of a companion paper analyzing a particular neuronal model in which the dynamics of these very complex equations can be mathematically analyzed through an exact reduction to deterministic nonlinear delayed integro-differential equations.

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