The Annals of Applied Probability

Propagation of chaos in neural fields

Jonathan Touboul

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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.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 3 (2014), 1298-1328.

Dates
First available in Project Euclid: 23 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1398258102

Digital Object Identifier
doi:10.1214/13-AAP950

Mathematical Reviews number (MathSciNet)
MR3199987

Zentralblatt MATH identifier
1305.60107

Subjects
Primary: 60F99: None of the above, but in this section 60B10: Convergence of probability measures
Secondary: 34C15: Nonlinear oscillations, coupled oscillators

Keywords
Mean-field limits propagation of chaos delayed stochastic differential equations infinite-dimensional stochastic processes neural fields

Citation

Touboul, Jonathan. Propagation of chaos in neural fields. Ann. Appl. Probab. 24 (2014), no. 3, 1298--1328. doi:10.1214/13-AAP950. https://projecteuclid.org/euclid.aoap/1398258102


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