Stability for Hawkes processes with inhibition

Abstract

We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.

Multivariate Hawkes processes on inhomogeneous random graphs

“…One can also introduce inhibition through a signed multiplying factor (that may depend or not on the neuron), see for instance [29,28,57]. Some works have also parted the whole population into two subclasses of neurons, the excitatory ones and the inhibitory ones [59,30]. In the latter, the inhibition is made thanks to a (small) multiplicative factor onto the intensity of the excitatory population.…”