Spike trains statistics in Integrate and Fire Models: exact results

ABSTRACT We briefly review and highlight the conse-quences of rigorous and exact results obtained in [1], char-acterizing the statistics of spike trains in a network of leaky Integrate-and-Fire neurons, where time is discrete and where neurons are subject to noise, without restriction on the synaptic weights connectivity. The main result is that spike trains statistics are characterized by a Gibbs dis-tribution, whose potential is explicitly computable. This es-tablishes, on one hand, a rigorous ground for the current in-vestigations attempting to characterize real spike trains data with Gibbs distributions, such as the Ising-like distribution [2], using the maximal entropy principle. However, it tran-spires from the present analysis that the Ising model might be a rather weak approximation. Indeed, the Gibbs poten-tial (the formal “Hamiltonian”) is the log of the so-called “conditional intensity ” (the probability that a neuron fires given the past of the whole network [3, 4, 5, 6, 7, 8, 9, 10]). But, in the present example, this probability has an infinite memory, and the corresponding process is non-Markovian (resp. the Gibbs potential has infinite range). Moreover, causality implies that the conditional intensity does not de-pend on the state of the neurons at the same time, ruling out the Ising model as a candidate for an exact characterization of spike trains statistics. However, Markovian approxima-tions can be proposed whose degree of approximation can be rigorously controlled. In this setting, Ising model ap-pears as the “next step ” after the Bernoulli model (inde-pendent neurons) since it introduces spatial pairwise cor-relations, but not time correlations. The range of validity of this approximation is discussed together with possible approaches allowing to introduce time correlations, with algorithmic extensions