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We consider a new class of non Markovian processes with a countable number ofinteracting components, both in discrete and continuous time. Each component isrepresented by a point process indicating if it has a spike or not at a giventime. The system evolves as follows. For each component, the rate (incontinuous time) or the probability (in discrete time) of having a spikedepends on the entire time evolution of the system since the last spike time ofthe component. In discrete time this class of systems extends in a non trivialway both Spitzer's interacting particle systems, which are Markovian, andRissanen's stochastic chains with memory of variable length which have finitestate space. In continuous time they can be seen as a kind of Rissanen'svariable length memory version of the class of self-exciting point processeswhich are also called "Hawkes processes", however with infinitely manycomponents. These features make this class a good candidate to describe thetime evolution of networks of spiking neurons. In this article we present acritical reader's guide to recent papers dealing with this class of models,both in discrete and in continuous time. We briefly sketch results concerningperfect simulation and existence issues, de-correlation between successiveinterspike intervals, the longtime behavior of finite non-excited systems andpropagation of chaos in mean field systems.

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