Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis

Abstract

When dealing with classical spike train analysis, the practitioner often performs goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model (Yana et al. in Biophys. J. 46(3):323–330, 1984; Brown et al. in Neural Comput. 14(2):325–346, 2002; Pouzat and Chaffiol in Technical report, http://arxiv.org/abs/arXiv:0909.2785, 2009). In doing so, there is a fundamental plug-in step, where the parameters of the supposed underlying model are estimated. The aim of this article is to show that plug-in has sometimes very undesirable effects. We propose a new method based on subsampling to deal with those plug-in issues in the case of the Kolmogorov–Smirnov test of uniformity. The method relies on the plug-in of good estimates of the underlying model that have to be consistent with a controlled rate of convergence. Some nonparametric estimates satisfying those constraints in the Poisson or in the Hawkes framework are highlighted. Moreover, they share adaptive properties that are useful from a practical point of view. We show the performance of those methods on simulated data. We also provide a complete analysis with these tools on single unit activity recorded on a monkey during a sensory-motor task.Electronic Supplementary MaterialThe online version of this article (doi:10.1186/2190-8567-4-3) contains supplementary material.

Inference of functional connectivity in Neurosciences via Hawkes processes

“…The proof is an easy adaptation of Theorem 2 of [1], which is originally stated for n = 1, to the case of n multivariate Hawkes processes (see also [13]). …”

“…A more extensive study can be found in [1] and an application on real neuronal data has been done in [13].…”

“…With respect to T 2 −T 1 and n, their orders of magnitude are typically log(T 2 − T 1 ) or log(n) (see [1], [13]). …”

“…It has been proved in [1] where the case n = 1 is investigated, that if the process is stationary and if the interaction functions are non negative, under some technical assumptions on both the process and the dictionary, then c is of the order of T 2 −T 1 with high probability. In [13], the n trials set-up is considered and it has been proved that even if the process is not stationary and under technical assumptions, then the order of magnitude of c is n with high probability. In practice, since G n is a symmetric non negative observable matrix, it is easy to compute its smallest eigenvalue to assess the quality of the estimate.…”

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

“…Hawkes processes provide good models of this synaptic integration phenomenon by the structure of their intensity processes, see (1) below. We refer to Chevallier et al (2015) [7], Chornoboy et al (1988) [8], Hansen et al (2015) [24] and to Reynaud-Bouret et al (2014) [35] for the use of Hawkes processes in neuronal modeling. For an overview of point processes used as stochastic models for interacting neurons both in discrete and in continuous time and related issues, see also Galves and Löcherbach (2016) [22].…”