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

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. Multivariate Hawkes processes are past-dependant point processes originally introduced tomodel excitation effects, later extended to a nonlinear framework to account for the opposite effect, known as inhibition. Our model generalises classical Hawkes processes, with or without inhibition, focusing on the situation where the probability of an event occurring within a given subprocess depends solely on the history since its last event. 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.