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June 2014 Propagation of chaos in neural fields
Jonathan Touboul
Ann. Appl. Probab. 24(3): 1298-1328 (June 2014). DOI: 10.1214/13-AAP950

Abstract

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 the 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 and a singular spatio-temporal Brownian motion. We also show how these intricate equations can be used in practice to uncover mathematically the precise mesoscopic dynamics of the neural field in a particular model where the mean-field equations exactly reduce to deterministic nonlinear delayed integro-differential equations. These results have several theoretical implications in neuroscience we review in the discussion.

Citation

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Jonathan Touboul. "Propagation of chaos in neural fields." Ann. Appl. Probab. 24 (3) 1298 - 1328, June 2014. https://doi.org/10.1214/13-AAP950

Information

Published: June 2014
First available in Project Euclid: 23 April 2014

zbMATH: 1305.60107
MathSciNet: MR3199987
Digital Object Identifier: 10.1214/13-AAP950

Subjects:
Primary: 60B10, 60F99
Secondary: 34C15

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 3 • June 2014
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