Spike trains statistics in Integrate and Fire Models: exact results

TL;DR

This paper reviews exact results on spike train statistics in leaky Integrate-and-Fire neuron networks, showing they follow a Gibbs distribution with an explicitly computable potential, and discusses implications for modeling real neural data.

Contribution

It provides a rigorous characterization of spike train statistics as a Gibbs distribution with an explicit potential, highlighting limitations of Ising models and proposing controlled Markovian approximations.

Findings

Spike train statistics are characterized by a Gibbs distribution with an explicit potential.

The Gibbs potential is the log of the conditional intensity with infinite memory.

Ising models are limited as they do not account for time correlations and have finite range.

Abstract

We briefly review and highlight the consequences of rigorous and exact results obtained in \cite{cessac:10}, characterizing 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 distribution, whose potential is explicitly computable. This establishes, on one hand, a rigorous ground for the current investigations attempting to characterize real spike trains data with Gibbs distributions, such as the Ising-like distribution, using the maximal entropy principle. However, it transpires from the present analysis that the Ising model might be a rather weak approximation. Indeed, the Gibbs potential (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). 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 depend on the state of the neurons at the \textit{same time}, ruling out the Ising model as a candidate for an exact characterization of spike trains statistics. However, Markovian approximations can be proposed whose degree of approximation can be rigorously controlled. In this setting, Ising model appears as the "next step" after the Bernoulli model (independent neurons) since it introduces spatial pairwise correlations, 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.