Prof. Eva LÖCHERBACH
Département de Mathématiques Faculté des Sciences et Techniques Université de Cergy-Pontoise
TITLE: Modeling spike trains by systems of interacting Hawkes processes with (or without) memory of variable length
Tuesday, December 13 at 15:30 Part I Wednesday, December 14 at 15:00 Part II Friday, December 16 at 11:00 Part III
at Dipartimento Matematica e Applicazioni, Università di Napoli FEDERICO II, Complesso di Monte Sant'Angelo, Via Cintia, Napoli.
ABSTRACT: Let us consider a class of non Markovian processes with a countable number of interacting components, both in discrete and continuous time. Each component is represented by a point process indicating if this component has a spike or not at a given time. The system evolves as follows. For each component, the rate (in continuous time) or the probability (in discrete time) of having a spike depends on the entire time evolution of the system since the last spike time of the component. In discrete time this class of systems extends in a non trivial way both Spitzer's interacting particle systems, which are Markovian, and Rissanen's stochastic chains with memory of variable length which have finite state space. In continuous time they can be seen as a kind of Rissanen's variable length memory version of the class of self-exciting point processes which are also called ``Hawkes processes'', however with infinitely many components. These features make this class a good candidate to describe the time evolution of networks of spiking neurons. In the lectures, I will present the model, both in discrete and in continuous time. I will then sketch results concerning perfect simulation and existence issues, de-correlation between successive interspike intervals, the longtime behavior of finite systems and propagation of chaos in mean field systems. ------