We consider a new class of non Markovian processes with a countable number of interacting 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 evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.
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We thank D. Brillinger, B. Cessac, S. Ditlevsen, M. Jara, M. Kelbert, Y. Kohayakawa, C. Landim, R.I. Oliveira, S. Ribeiro, L. Triolo, C. Vargas and N. Vasconcelos for many discussions on Hawkes processes, random graphs and neural nets at the beginning of this project.
This work is part of USP project “Mathematics, computation, language and the brain”, FAPESP project “NeuroMat” (grant 2011/51350-6), USP/COFECUB project “Stochastic systems with interactions of variable range” and CNPq project “Stochastic modeling of the brain activity” (grant 480108/2012-9). A.G. is partially supported by a CNPq fellowship (grant 309501/2011-3), A.G. and E.L. have been partially supported by the MathAmSud project “Stochastic structures of large interacting systems” (grant 009/10). E.L. thanks Numec, USP, for hospitality and support.
Dedicated to Errico Presutti, frateddu e mastru.
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Galves, A., Löcherbach, E. Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets. J Stat Phys 151, 896–921 (2013). https://doi.org/10.1007/s10955-013-0733-9
- Biological neural nets
- Interacting particle systems
- Chains of infinite memory
- Chains of variable length memory
- Hawkes process