Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity

Abstract

The Hawkes process is a multivariate past-dependent point process used to model the relationship of event occurrences between different phenomena. Although the Hawkes process was originally introduced to describe excitation interactions, which means that one event increases the chances of another occurring, there has been a growing interest in modeling the opposite effect, known as inhibition. In this paper, we propose a maximum likelihood approach to estimate the interaction functions of a multivariate Hawkes process that can account for both exciting and inhibiting effects. To the best of our knowledge, this is the first exact inference procedure designed for such a general setting in the frequentist framework. Our method includes a thresholding step in order to recover the support of interactions and therefore to infer the connectivity graph. A benefit of our method is to provide an explicit computation of the log-likelihood, which enables in addition to perform a goodness-of-fit test for assessing the quality of estimations. We compare our method to classical approaches, which were developed in the linear framework and are not specifically designed for handling inhibiting effects. We show that the proposed estimator performs better on synthetic data than alternative approaches. We also illustrate the application of our procedure to a neuronal activity dataset, which highlights the presence of both exciting and inhibiting effects between neurons.

A neural network based model for multi-dimensional nonlinear Hawkes processes

“…The parameters of the neural networks utilized to model the kernels in the NNNH method are optimized by maximizing the log-likelihood through the stochastic gradient descent method, as described in Section 3.1. We utilize the WH method [Bacry and Muzy, 2014] and the Bonnet Multivariate method [Bonnet et al, 2022] as our comparative models. While the WH method is non-parametric, the Bonnet Multivariate approach assumes a parametric structure for the kernels in the Hawkes process.…”

“…Bonnet et al [2021] use a negative exponential function to model inhibitive kernels for a univariate nonlinear Hawkes process and use maximum likelihood estimation to determine the optimal parameters. Bonnet et al [2022] extend this approach to a multivariate nonlinear Hawkes process. Lemonnier and Vayatis [2014] develop the Markovian Estimation of Mutually Interacting Process (MEMIP) method, which utilizes weighted exponential functions to determine kernels for the nonlinear Hawkes process.…”