Mean-field limits for non-linear Hawkes processes with excitation and inhibition

Abstract

We study a multivariate, non-linear Hawkes process Z N on the complete graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1− p). We take the meanfield limit of Z N , leading to a multivariate point process Z. If p = 1 2 , we rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z are independent. In the critical case, p = 1 2 , we rescale by N 1/2 and obtain a limit intensity, which solves a stochastic convolution equation and all components of Z are conditionally independent.

Interacting Hawkes processes with multiplicative inhibition

“…The case 1a. is a straightforward consequence of the definition of U Φ in (29) together with (40). Turn now to case 1b.…”

“…In this context, the main strategy followed in the literature so far has been to model inhibition by introducting kernels h j→i (•) in ( 5) taking negative values [20,41,42,43,4] or positive kernels h (e.g. of Erlang type) multiplied by random and possibly negative coefficients [25,26,40]. This is what we could qualify as an additive inhibition, in the sense that the intensity of a neuron is the (temporal) additive superposition of several memory kernels h i→j , in the same spirit as the scheme suggested by (3) for integrate and fire models.…”

Mean-field limits for non-linear Hawkes processes with inhibition on a Erdős-Rényi-graph

“…The Erdős-Rényi-model [7] is one of the first and simplest models for random graphs, where the existence of edges between any two vertices is indicated by independent Bernoulli random variables with common probability q. In [25], mean-field limits for the multivariate, non-linear Hawkes process with excitation and inhibition on a complete graph are derived. The main goal of this paper is to generalize these limit results to a q-Erdős-Rényi-Graph graph.…”

“…Nonlinear multivariate Hawkes processes have been studied to some extent in the past decades, a summary can be found in the introduction of [25]. In a standard mean-field setting, all components of the Hawkes process share the same firing rate, see e.g.…”

“…In the present paper, we extend the model from [25] by a parameter q, which denotes the fraction of open connections/edges between neurons/vertices. We assume that each vertex/neuron is either excitatory or inhibitory, i.e.…”