Phase Transition for Infinite Systems of Spiking Neurons |
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Authors: | P. A. Ferrari A. Galves I. Grigorescu E. Löcherbach |
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Affiliation: | 1.Universidad de Buenos Aires,Buenos Aires,Argentina;2.Universidade de S?o Paulo,S?o Paulo,Brazil;3.University of Miami,Coral Gables,USA;4.Université Paris Seine,Cergy,France |
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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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