Separation rates for the detection of synchronization of interacting point processes in a mean field frame. Application to neuroscience

We develop and study a statistical test to detect synchrony in spike trains. Our test is based on the number of coincidences between two trains of spikes. The data are supplied in the form of n pairs (assumed to be independent) of spike trains. The aim is to assess whether the two trains in a pair are also independent. Our approach is based on previous results of Albert et al. (2015, 2019) and Kim et al. (2022) that we extend to our setting, focusing on the construction of a non-asymptotic criterion ensuring the detection of synchronization in the framework of permutation tests. Our criterion is constructed such that it ensures the control of the Type II error, while the Type I error is controlled by construction. We illustrate our results within two classical models of interacting neurons, the jittering Poisson model and Hawkes processes having M components interacting in a mean field frame and evolving in stationary regime. For this latter model, we obtain a lower bound of the size n of the sample necessary to detect the dependency between two neurons.