Inference of multivariate exponential Hawkesprocesses with inhibition and application toneuronal activity

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.