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Section: New Results

Mean field approaches

A Large Deviation Principle and an Expression of the Rate Function for a Discrete Stationary Gaussian Process

Participants : Olivier Faugeras, James Maclaurin.

We prove a Large Deviation Principle for a stationary Gaussian process over ${ℝ}^b$, indexed by ${\mathbb{Z}}^d$ (for some positive integers $d$ and $b$), with positive definite spectral density and provide an expression of the corresponding rate function in terms of the mean of the process and its spectral density. This result is useful in applications where such an expression is needed.

This work has been accepted for publication in Entropy [20] .

A representation of the relative entropy with respect to a diffusion process in terms of its infinitesimal-generator

Participants : Olivier Faugeras, James Maclaurin.

In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback-Leibler Divergence) $R\left(\mu \right|P)$, where $\mu$ and $P$ are measures on $C([0,T];{ℝ}^d)$. The underlying measure $P$ is a weak solution to a Martingale Problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since $R\left(\mu \right|\left|P\right)$ governs the exponential rate of convergence of the empirical measure (according to Sanov's Theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions.

This work has been accepted for publication in the Journal Entropy [21] .

Asymptotic description of stochastic networks of rate neurons with correlated synaptic weights

Participants : Olivier Faugeras, James Maclaurin.

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. Unlike previous works which made the biologically unplausible assumption that the weights were i.i.d. random variables, we assume that they are correlated. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The result is that the image law through the empirical measure satisfies a large deviation principle with a good rate function. We provide an analytical expression of this rate function. This work has appeared in the Comptes Rendus de l'Academie des Sciences. Serie 1, Mathematique [22] .

We have continued the development, started in [22] , of the asymptotic description of certain stochastic neural networks. We use the Large Deviation Principle (LDP) and the good rate function $H$ announced there to prove that $H$ has a unique minimum, a stationary measure on the set of trajectories ${𝒯}^{\mathbb{Z}}$. We characterize this measure by its two marginals, at time 0, and from time 1 to T. The second marginal is a stationary Gaussian measure. With an eye on applications, we show that its mean and covariance operator can be inductively computed. Finally we use the LDP to establish various convergence results, averaged and quenched. This work has also appeared in the Comptes Rendus de l'Academie des Sciences. Serie 1, Mathematique [23] .

Asymptotic description of stochastic networks of integrate-and-fire neurons

Participants : François Delarue [UNS, LJAD] , James Inglis [EPIs TOSCA and NeuroMathComp] , S Rubenthaler [UNS, LJAD] , Etienne Tanré [EPI TOSCA] .

J. Inglis, together with F. Delarue (Univ. Nice – Sophia Antipolis), E. Tanré (Inria Tosca ) and S. Rubenthaler (Univ. Nice – Sophia Antipolis) completed their study of the mean-field convergence of a highly discontinuous particle system modeling the behavior of a spiking network of neurons, based on the integrate-and-fire model. Due to the highly singular nature of the system, it was convenient to work with a relatively unknown Skorohod topology. The resulting article [46] has been accepted for publication in Stochastic Processes and Related Fields.

Asymptotic description of stochastic networks of spiking neurons with dendrites

Participants : James Inglis [EPIs TOSCA and NeuroMathComp] , Denis Talay [EPI TOSCA] .

J. Inglis and D. Talay introduced in [49] a new model for a network of spiking neurons that attempted to address several criticisms of previously considered models. In particular the new model takes into account the role of the dendrites, and moreover includes non-homogeneous synaptic weights to describe the fact that not all neurons have the same effect on the others in the network. They were able to obtain mean-field convergence results, using new probabilistic arguments.

Asymptotic description of stochastic networks of realistic neurons and synapses

Participants : Mireille Bossy [EPI TOSCA] , Olivier Faugeras, Denis Talay [EPI TOSCA] .

In this note, we clarify the well-posedness of the limit equations to the mean-field N -neuron models proposed in [1] and we prove the associated propagation of chaos property. We also complete the modeling issue in [1] by discussing the well-posedness of the stochastic differential equations which govern the behaviour of the ion channels and the amount of available neurotransmitters.

This work has been submitted for publication to a Journal and is available as [40] .

On the Hamiltonian structure of large deviations in stochastic hybrid systems

Participants : Paul Bressloff [Prof. University of Utah, Inria International Chair] , Olivier Faugeras.

We develop the connection between large deviation theory and more applied approaches to stochastic hybrid systems by highlighting a common underlying Hamiltonian structure. A stochastic hybrid system involves the coupling between a piecewise deterministic dynamical system in ${ℝ}^d$ and a time-homogeneous Markov chain on some discrete space $\Gamma$. We assume that the Markov chain on $\Gamma$ is ergodic, and that the discrete dynamics is much faster than the piecewise deterministic dynamics (separation of time-scales). Using the Perron-Frobenius theorem and the calculus-of-variations, we evaluate the rate function of a large deviation principle in terms of a classical action, whose Hamiltonian is given by the Perron eigenvalue of a $\left|\Gamma \right|$-dimensional linear equation. The corresponding linear operator depends on the transition rates of the Markov chain and the nonlinear functions of the piecewise deterministic system. The resulting Hamiltonian is identical to one derived using path-integrals and WKB methods. We illustrate the theory by considering the example of stochastic ion channels. This work has been submitted for publication to a Journal and is available as [41] .