Stationarity properties of neural networks

Abstract

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y (i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y (i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

A Markovian event-based framework for stochastic spiking neural networks

“…This kind of processes has already been studied in the field of queueing theory under the name of the hourglass model. It was first introduced by Marie Cottrell in (Cottrell, 1992), studied by Fricker, Robert and collaborators in (Fricker et al, 1994) and then instantiated and studied in (Asmussen and Turova, 1998;Cottrell and Turova, 2000;Turova, 2000Turova, , 1996. In order to apply this modeling to the kind of networks of interest here, we need to define in each case the reset and the interactions random variables.…”