Phase transition for infinite systems of spiking neurons

FERRARI, PABLO A.; A. GALVES; ILIE GRIGORESCU; EVA LOCHERBACH

JOURNAL OF STATISTICAL PHYSICS

SPRINGER

Lugar: Berlin; Año: 2018 vol. 172 p. 1564 - 1575

0022-4715

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 wheneverits membrane potential is larger than a threshold value. This membrane potential evolves intime and integrates the spikes of all presynaptic neurons since the last spiking time of theneuron. When a neuron spikes, its membrane potential is reset to 0 and simultaneously, aconstant 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 atrandom times driven by an independent Poisson point process of rate γ > 0. For this processwe prove the existence of a value γc such that the system has one or two extremal invariantmeasures according to whether γ >γc or not.