Matthieu Gilson, Center for Brain and Cognition, Universitat Pompeu Fabra, Barcelona | Propagation of moments in Hawkes networks

Abstract

The present study provides a mathematical description of high-order moments of spiking activity in a recurrently-connected network of Hawkes processes. It extends previous studies that have explored the case of a (linear) Hawkes network driven by deterministic rate functions to the case of a stimulation by external inputs (rate functions or spike trains) with arbitrary correlation structure. Our approach describes the spatio-temporal filtering induced by the afferent and recurrent connectivities using operators of the input moments. This algebraic viewpoint provides intuition about how the network ingredients shape the input-output mapping for moments, as well as cumulants. This mathematical framework provides a basis for the study of learning in recurrently connected networks when high-order spiking statistics are involved (e.g., triplet STDP).