Axe Interdisciplinaire de Recherche de l’Université de Nice – Sophia Antipolis

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Stochastic neural field equations.

Wilhelm Stannat (11 Décembre 2014)

mardi 24 février 2015

Titre : Stochastic neural field equations.
Jeudi 11 décembre 2014 - 9h30 à 12h15
INRIA, Salle Euler Violet
Intervenant : Wilhelm Stannat

Mots Clefs : multiscale analysis, approximation and finite size effects on traveling waves

Neural field equations are used to describe the spatiotemporal evo-
lution of the average activity in a network of synaptically coupled populations
of neurons in the continuum limit. This deterministic description is only accu-
rate in the infinite population limit and the actual finite size of the populations
causes deviations from the mean field behavior.

We will first rigorously derive stochastic neural field equations with noise terms
accounting for these finite size effects. These equations are identified by describing
the evolution of the activity in the finite-size populations by Markov chains and then
determining the limit of their fluctuations. We then introduce a complete mathematical
framework for the analysis of the resulting stochastic neural field equations.

As first steps of a multiscale analysis, a geometrically motivated decomposition of
the stochastic evolution into a randomly moving wave front and fluctuations is derived.
A random ordinary differential equation describing the velocity of the moving wave front
can be deduced and the fluctuations around the wave front turn out to be non-Gaussian,
even if the driving noise term is a Gaussian process.

The presented geometric approach is in principle applicable to describe
the statistics of any macroscopic profile driven by spatially extended noise,
like, e.g., wave fronts and pulses in general stochastic reaction diffusion systems.

The talk is partially based on joint work with Jennifer Krueger and Eva Lang.

References :

J. Krueger, W. Stannat, Front Propagation in Stochastic Neural Fields : A
rigorous mathematical framework, SIAM J. Appl. Dyn. Syst., Vol. 13, 1293-
1310, 2014.

E. Lang, W. Stannat, Finite-Size effects on traveling wave solutions to
neural field equations, 2014, submitted

W. Stannat, Stability of travelling waves in stochastic bistable
reaction-diffusion equations, arXiv:1404.3853