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

Permutation tests have been proposed by Albert et al. (2015) to detect dependence between point processes, modeling in particular spike trains, that is the time occurrences of action potentials emitted by neurons. Our present work focuses on exhibiting a criterion on the separation rate to ensure that the Type II errors of these tests are controlled non asymptotically. This criterion is then discussed in two major models in neuroscience: the jittering Poisson model and Hawkes processes having $M$ components interacting in a mean field frame and evolving in stationary regime. For both models, we obtain a lower bound of the size $n$ of the sample necessary to detect the dependency between two neurons.