Spike trains as (in)homogeneous Poisson processes or Hawkes processes: non-parametric adaptive estimation and goodness-of-fit tests

This article aims to propose new non-parametric adaptive estimation methods and to adapt other recent similar results to the setting of spike trains analysis. After briefly recalling main features of the homogeneous Poisson model, we focus on two main generalizations of this process : the inhomogeneous Poisson model, which is non-stationary, and the Hawkes model, which can take into account interactions. Goodness-of-fit tests are also proposed and are proved to be of prescribed asymptotical level. They enable us to test these non-parametric models. Various simulations show good performance of the estimation and test procedures. A complete analysis is also performed with these tools on single unit activity recorded on a monkey during a sensory-motor task. We can show that the homogeneous Poisson process hypothesis is always rejected and that the inhomogeneous Poisson process hypothesis is rarely accepted. The Hawkes model seems to fit most of the data.