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We consider a new class of non Markovian processes with a countable number ofinteracting components. At each time unit, each component can take two values,indicating if it has a spike or not at this precise moment. The system evolvesas follows. For each component, the probability of having a spike at the nexttime unit depends on the entire time evolution of the system after the lastspike time of the component. This class of systems extends in a non trivial wayboth the interacting particle systems, which are Markovian, and the stochasticchains with memory of variable length which have finite state space. Thesefeatures make it suitable to describe the time evolution of biological neuralsystems. We construct a stationary version of the process by using aprobabilistic tool which is a Kalikow-type decomposition either in randomenvironment or in space-time. This construction implies uniqueness of thestationary process. Finally we consider the case where the interactions betweencomponents are given by a critical directed Erd\"os-R\'enyi-type random graphwith a large but finite number of components. In this framework we obtain anexplicit upper-bound for the correlation between successive inter-spikeintervals which is compatible with previous empirical findings.

Infinite systems of interacting chains with memory of variable length - a stochastic model for biological neural nets