  The present paper provides exact mathematical expressions for the high-order
moments of spiking activity in a recurrently-connected network of linear Hawkes
processes. It extends previous studies that have explored the case of a
(linear) Hawkes network driven by deterministic intensity 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 (with arbitrary
synaptic response kernels) using operators acting on the input moments. This
algebraic viewpoint provides intuition about how the network ingredients shape
the input-output mapping for moments, as well as cumulants. We also show using
numerical simulation that our results hold for neurons with refractoriness
implemented by self-inhibition, provided the corresponding negative feedback
for each neuron only mildly alters its mean firing probability.
