Fluctuation limits for mean-field interacting nonlinear Hawkes processes

Abstract

We investigate the asymptotic behavior of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctuations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales. We also discuss extensions of our results to multi-class systems.

Introduction

The main purpose of this paper is to give a short proof of a functional central limit theorem for the fluctuations of a network of interacting Hawkes processes in the large population limit. Interacting Hawkes processes have been intensively studied in recent years as mathematical models for the statistical analysis of biological neural networks (cf. [2], [3], [4], [5], [7] and references therein). Results on their asymptotic behavior in the large population limit have been obtained in particular in the latter four references. Various functional laws of large numbers and (functional) central limit theorems (CLT henceforth) have been obtained therein in the respective settings. In particular, a CLT by simultaneously taking both, the number of neurons (or size of the network) and time to infinity has been obtained in [5] and a functional CLT for age-dependent Hawkes processes in [2]. In contrast to the usual proof of functional CLTs via tightness properties, we will represent the fluctuation process as the unique solution of a stochastic Volterra integral equation driven by càdlag̀ martingales that are shown to converge to a rescaled Brownian motion via the martingale CLT. This allows in addition to the corresponding result in [2] for an explicit representation of the limiting fluctuation process (see Remark 2.3 for a precise comparison).

Let us introduce our precise setting. Consider an ${\mathbb{R}}_{+}$-indexed filtered probability space $\left(\Omega ,\mathcal{F},\mathbb{F},\mathbb{P}\right)$ and let $N_i$, $i\in \mathbb{N}$, be a sequence of iid $\mathbb{F}$-Poisson random measures with intensity measure $dzds$ on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$. Let $f$ and $h$, modeling spike rate resp. synaptic weights, be two real-valued functions satisfying the following set of assumptions (A):

• $f\in C^2\left(\mathbb{R},{\mathbb{R}}_{+}\right)$, $f$ and $f^{\prime }$ are Lipschitz continuous,

• $h\in C^1\left({\mathbb{R}}_{+},\mathbb{R}\right)$, $h\left(0\right)=0$

Then, for each $N\in \mathbb{N}$, there exists a pathwise unique solution $Z^N=\left(Z_1^N,\dots ,Z_N^N\right)$ of the integral equation

$$Z_i^N\left(t\right)={\int }_0^t{\int }_0^{\infty }{1}_{\left\{z\le {\lambda }^N\left(s\right)\right\}}{N}_i\left(dz\times ds\right),t\ge 0,1\le i\le N,$$$${\lambda }^N\left(t\right)=f\left(\frac{1}{N}\sum _{i=1}^N{\int }_0^{t-}h(t-s)d{Z}_i^N\left(s\right)\right),t>0,$$

(cf. [4, Theorem 6]). Moreover, $Z^N$ is an interacting non-linear Hawkes process with parameters $\left(f,h,N\right)$ in the sense of [4, Proposition 3(a), Definition 1]). In particular, $Z^N=\left(Z_1^N,\dots ,Z_N^N\right)$ is a family of $\mathbb{F}$-counting processes satisfying

$$\mathbb{P}\left(\Delta Z_i^N\left(t\right)=1,\Delta Z_j^N\left(t\right)=1\right)=0,i\ne j,t\ge 0,$$

with compensator (or cumulative intensity process)

$${\Lambda }^N\left(t\right)={\int }_0^t{\lambda }^N\left(s\right)ds,t\ge 0,$$

for each $1\le i\le N$.

Delattre et al. prove in [4, Theorem 8] the following propagation of chaos result: for any $N\in \mathbb{N}$, there exists a family of iid Poisson processes ${\bar{Z}}_i$, $1\le i\le N$, with compensator $m$ being the unique solution of

$$m\left(t\right)={\int }_0^{t}f\left({\int }_0^{s}h(s-u)dm\left(u\right)\right)ds,t\ge 0,$$

such that for all $T>0$ there exists some constant $C\left(T\right)$

$$\mathbb{E}\left[{\Vert Z_i^N-{\bar{Z}}_i\Vert }_T\right]\le \frac{C\left(T\right)}{\sqrt{N}},1\le i\le N,N\in \mathbb{N}.$$

Here, ${\Vert y\Vert }_T≔{sup}_{t\in [0,T]}\left|f\left(t\right)\right|$ denotes the supremum norm of a function $y:[0,T]\to \mathbb{R}$. This particularly yields a first order approximation of the Hawkes process $Z^N$ with the help of $N$ iid Poisson processes, each with intensity

$$\lambda \left(t\right)=\frac{dm}{dt}\left(t\right)=f\left(x_t\right),t\ge 0.$$

with

$$x_t={\int }_0^{t}h(t-s)dm\left(s\right),t\ge 0.$$

As an immediate consequence (cf. Proposition A.1) the following functional law of large numbers for interacting non-linear Hawkes processes holds:

$${\Vert \frac{1}{N}\sum _{i=1}^N\left(Z_i^N-m\right)\Vert }_T\underset{N\to \infty }{\overset{L^1}{⟶}}0,T>0.$$

In this paper we are now interested in the corresponding asymptotic fluctuations

$$Y^N\left(t\right)=\frac{1}{\sqrt{N}}\sum _{i=1}^N\left(Z_i^N\left(t\right)-m\left(t\right)\right),t\ge 0,N\in \mathbb{N}.$$

The limiting behavior of (10) is linked to the asymptotic behavior of the associated intensity processes ${\lambda }^N$ that can be written as

$${\lambda }^N\left(t\right)=f\left(x_t+\frac{X^N\left(t\right)}{\sqrt{N}}\right),t\ge 0,N\in \mathbb{N},$$

with

$$X^N\left(t\right)={\int }_0^{t}h(t-s)d{Y}^N\left(s\right)={\int }_0^t{h}^{\prime }(t-s)Y^N\left(s\right)ds,$$

where we used integration by parts and $h\left(0\right)=Y^N\left(0\right)=0$. A Taylor expansion of $f$ at $x_t$ yields the linear approximation

$$\sqrt{N}\left({\lambda }^N\left(t\right)-\lambda \left(t\right)\right)\approx f^{\prime }\left(x_t\right)X^N\left(t\right)=f^{\prime }\left(x_t\right){\int }_0^t{h}^{\prime }(t-s)Y^N\left(s\right)ds,$$

which is the key to our analysis. Our main results are contained in Section 2. We will first prove a weak convergence result for $Y^N$ and identify its limit in Theorem 2.1, subsequently prove the weak convergence of $X^N$ in Corollary 2.2 and finally the weak convergence of $\sqrt{N}\left({\lambda }^N\left(t\right)-\lambda \left(t\right)\right)$ in Theorem 2.4. We also discuss in Section 2.1 extensions of these convergence results to multi-class systems. Section 3 contains some preliminary results used in the analysis of Section 2. Finally, in Section 4, we apply our results to a second order approximation of the Hawkes process $Z^N$ by a system of $N$ identically distributed Cox processes.