## venerdì, giugno 26, 2009

### PP09 (VIII)

This week we have started the analysis of interactions between point processes. The oldest model in this context is probably due to Hawkes and it is based on the following idea.

We want to model a family point processes, each with its own events

t_j^n, \qquad j,n \in \mathbb N

where j is the index for the event and n is the index for the process. The simplest idea is to assume that each of the processes is Cox one, i.e. an inhomogeneous Poisson process whose rate function changes randomly in time. We then let depend the rate function on the realization itself of the point process, by convolving the realization with a causal kernel.

This procedure is known as a Hawkes process, and it is very useful. It only has a drawback for neuronal modeling: you cannot model inhibition, since your kernels have to be positive, but if you are interested in this problem, you probably should take a look at the lecture.