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Phase Transition for Infinite Systems of Spiking Neurons

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Abstract

We prove the existence of a phase transition for a stochastic model of interacting neurons. The spiking activity of each neuron is represented by a point process having rate 1 whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all presynaptic neurons since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to 0 and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effect leading to an abrupt loss of potential occurring at random times driven by an independent Poisson point process of rate \( \gamma > 0 .\) For this process we prove the existence of a value \(\gamma _c\) such that the system has one or two extremal invariant measures according to whether \(\gamma > \gamma _c \) or not.

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Acknowledgements

Many thanks to Marzio Cassandro and Christophe Pouzat for illuminating discussions, constructive criticism and useful information on neurobiology. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01) and it is part of USP project Mathematics, computation, language and the brain and of FAPESP project Research, Innovation and Dissemination Center for Neuromathematics (Grant 2013/07699-0). AG is partially supported by CNPq fellowship (Grant 311 719/2016-3).

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Authors and Affiliations

  1. Universidad de Buenos Aires, Buenos Aires, Argentina

    P. A. Ferrari

  2. Universidade de São Paulo, São Paulo, Brazil

    A. Galves

  3. University of Miami, Coral Gables, USA

    I. Grigorescu

  4. Université Paris Seine, Cergy, France

    E. Löcherbach

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  1. P. A. Ferrari
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  2. A. Galves
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  3. I. Grigorescu
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  4. E. Löcherbach
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Correspondence to A. Galves.

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Ferrari, P.A., Galves, A., Grigorescu, I. et al. Phase Transition for Infinite Systems of Spiking Neurons. J Stat Phys 172, 1564–1575 (2018). https://doi.org/10.1007/s10955-018-2118-6

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  • DOI: https://doi.org/10.1007/s10955-018-2118-6

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