Estimating the interaction graph of stochastic neural dynamics

In this paper we address the question of statistical model selection for a class of stochastic models of biological neural nets. Models in this class are systems of interacting chains with memory of variable length. Each chain describes the activity of a single neuron, indicating whether it has a spike or not at a given time. For each neuron, the probability of having a spike depends on the entire time evolution of its {\it presynaptic neurons} since the last spike time of the neuron. When the neuron spikes, its potential is reset to a resting level, and all of its postsynaptic neurons receive an additional amount of potential. The relationship between a neuron and its pre- and postsynaptic neurons defines an oriented graph, the {\it interaction graph} of the model. The goal of this paper is to estimate this graph of interactions, based on an observation of the process up to time $n, $ within a growing sequence of observation windows. We prove the consistency of this estimator and obtain explicit error bounds for the probability of wrong estimation of the graph of interactions.