Multiple Phase Transitions for an Infinite System of Spiking Neurons

Author(s)

Nascimento, A. M. B.

• Document type: Journal article
• Source: Journal of Statistical Physics; v. 188, n. 1, p. 12-pg., 2022-07-01.

Abstract

We consider a stochastic model describing the spiking activity of a countable set of neurons spatially organized into a homogeneous tree of degree d, d >= 2; the degree of a neuron is just the number of connections it has. Roughly, the model is as follows. Each neuron is represented by its membrane potential, which takes non-negative integer values. Neurons spike at Poisson rate 1, provided they have strictly positive membrane potential. When a spike occurs, the potential of the spiking neuron changes to 0, and all neurons connected to it receive a positive amount of potential. Moreover, between successive spikes and without receiving any spiking inputs from other neurons, each neuron's potential behaves independently as a pure death process with death rate gamma >= 0. In this article, we show that if the number d of connections is large enough, then the process exhibits at least two phase transitions depending on the choice of rate gamma: For large values of gamma, the neural spiking activity almost surely goes extinct; For small values of gamma, a fixed neuron spikes infinitely many times with a positive probability, and for "intermediate" values of gamma, the system has a positive probability of always presenting spiking activity, but, individually, each neuron eventually stops spiking and remains at rest forever. (AU)

• FAPESP's process: 13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat
• Grantee: Oswaldo Baffa Filho
• Support Opportunities: Research Grants - Research, Innovation and Dissemination Centers - RIDC