Poisson processes provide a basic model for describing events occurring over time, when these events do not interact with each other, such as phone calls or customer arrivals. This model becomes less suitable when past events influence the probability of appearance of future events, as in the case of earthquake aftershocks or neuronal interactions. Hawkes processes provide a natural framework for modeling such temporal dependencies, in particular through self-excitation.

In this talk, we will first introduce Poisson processes and then Hawkes processes, before focusing on statistical inference, i.e. parameter estimation from observed data. We will also discuss model comparison to decide which model is better suited to the data.

Finally, we will discuss some extensions allowing for more complex temporal interactions, such as inhibition or memory of variable length, and present an application to real neuronal data.