Asymptotic description of stochastic neural networks. I - existence of a Large Deviation Principle

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. The dynamics of the neurons is described by a set of stochastic differential equations in discrete time. The neurons interact through the synaptic weights which 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 unrealistic 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 in terms of the spectral representation of certain Gaussian processes

Asymptotic description of stochastic neural networks. II - Characterization of the limit law

We continue the development, started in 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 mu_e, a stationary measure on the set of trajectories. 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

Euler Characteristic in Odd Dimensions

It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes

The meanfield limit of a network of Hopfield neurons with correlated synaptic weights

We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of Hopfield neurons. The randomness in the network is modelled by random couplings which are centered Gaussian correlated random variables. We prove that the annealed law of the empirical measure satisfies a large deviation principle without any condition on time. We prove that the good rate function of this large deviation principle achieves its minimum value at a unique Gaussian measure which is not Markovian. This implies almost sure convergence of the empirical measure under the quenched law. We prove that the limit equations are expressed as an infinite countable set of linear non Markovian SDEs.Comment: 102 page