An interacting neuronal network with inhibition: theoretical analysis and perfect simulation (Q6060118)

The aim in this article is to perform a theoretical investigation followed by a simulation, on purely inhibitory neural networks. These are networks where neurons are represented by their inhibitory state. As stated by the author, different extensions of the Cottrell model, issued in [\textit{M. Cottrell}, Stochastic Processes Appl. 40, No. 1, 103--126 (1992; Zbl 0749.92003)], are used. The model is described in the second section of the article. One defines, \({X_t}^{i,N}\), the state of inhibition of neuron i at time t, \(W_{j->i}\) the inhibition weight of the neuron j on neuron i and \(M_i\) a random Poisson measure. In the article the case where neuron i is inhibitory for neuron j, i.e. \(W_{j->i}\) \(\geq\) 0 is considered. The formula expressing the state of inhibition of neuron i at time t as well as the infinitesimal generator associated to the model are introduced. Further the low of the first jump time of the process is studied. In the third section of the article one investigates conditions of non-evanescence of the process. The Forster-Lyapunov and Doeblin conditions are discussed. One proves the existence of a unique invariant probability measure of the process. The aim in the fourth section is to built a perfect simulation algorithm to show in a different way the recurrence of the considered process. Algorithms for the backward procedure as well as forward procedure together with simulation results are presented.