A model of Poissonian interactions and detection of dependence

TL;DR

This paper introduces a statistical model for Poissonian interactions between point processes, specifically applied to neural activity, and develops an adaptive wavelet-based testing procedure to detect dependencies with strong theoretical guarantees.

Contribution

It proposes a novel wavelet thresholding test for dependence in Poisson processes, with proven optimality and practical effectiveness in neuroscience applications.

Findings

Test maintains level and power under assumptions

Adaptive minimax separation rate achieved

Simulation results confirm good practical performance

Abstract

This paper proposes a model of interactions between two point processes, ruled by a reproduction function h, which is considered as the intensity of a Poisson process. In particular, we focus on the context of neurosciences to detect possible interactions in the cerebral activity associated with two neurons. To provide a mathematical answer to this specific problem of neurobiologists, we address so the question of testing the nullity of the intensity h. We construct a multiple testing procedure obtained by the aggregation of single tests based on a wavelet thresholding method. This test has good theoretical properties: it is possible to guarantee the level but also the power under some assumptions and its uniform separation rate over weak Besov bodies is adaptive minimax. Then, some simulations are provided, showing the good practical behavior of our testing procedure.