Hawkes Processes with Variable Length Memory: Existence, Inference and Application to Neuronal Activity

TL;DR

This paper introduces a new class of nonlinear Hawkes processes with variable length memory, proves their existence, and develops a likelihood-based inference method, demonstrated on synthetic and neuronal data.

Contribution

It generalizes classical Hawkes processes to include variable memory length, with proven existence and a new inference approach for neuroscience applications.

Findings

Successfully modeled neuronal activity with variable memory Hawkes processes.

Developed a likelihood maximization method for identifying memory dynamics.

Validated approach on synthetic and real neuronal datasets.

Abstract

Multivariate Hawkes processes are past-dependant point processes originally introduced to model excitation effects, later extended to a nonlinear framework to account for the opposite effect, known as inhibition. Motivated by applications in neuroscience, where the memory of a neuron may reset upon firing, we introduce a new class of nonlinear Hawkes processes with variable length memory. Our model generalises classical Hawkes processes, with or without inhibition, describing the situation where the probability of an event occurring within a given subprocess may depend differently on the history before and after its last event. In particular, if the subprocess does not depend on the history before its last event, it is said to have a variable length memory. Our main contributions are to prove existence of such processes, and to derive a workable likelihood maximisation method, capable of identifying both classical and variable memory dynamics. We demonstrate the effectiveness of our approach both on synthetic data, and on a neuronal activity dataset.