Modeling networks of spiking neurons as interacting processes with memory of variable length

Galves, Antonio; Löcherbach, Eva

Journal de la société française de statistique, Tome 157 (2016) no. 1, pp. 17-32.

Nous considérons une nouvelle classe de processus non-markoviens à temps discret ou continu, comportant un nombre dénombrable de composantes en interaction. À chaque instant, chaque composante (neurone) peut prendre deux valeurs, indiquant la présence ou l’absence d’un potentiel d’action (spike). La dynamique du processus est définie de la manière suivante  : pour chaque composante, la probabilité d’avoir un potentiel d’action à l’instant suivant dépend de la trajectoire du système entier depuis l’instant du dernier spike de cette composante. Cette classe de processus stochastiques étend de manière non triviale à la fois les systèmes de particules en interaction, qui sont markoviens, et les chaînes de mémoire variable, qui ont un espace d’états fini. En temps continu, cette classe de processus peut être considérée comme une version des chaînes à mémoire variable pour les processus ponctuels auto-excitants de Hawkes mais avec un nombre infini de composantes. Cette classe de processus constitue ainsi une bonne classe de modèles pour décrire l’évolution temporelle de systèmes neuronaux biologiques. Nous présentons une revue critique des articles récents discutant cette classe de modèles, aussi bien en temps continu qu’en temps discret. Nous exposons brièvement des résultats sur la simulation parfaite, la décorrélation entre intervalles inter spike successifs, le comportement en temps long de systèmes finis ainsi que sur la propagation du chaos dans des systèmes en interaction du type champ moyen.

We consider a new 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 it 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 this article we present a critical reader’s guide to recent papers dealing with this class of models, both in discrete and in continuous time. We briefly 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.

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