  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.
