Multiple Phase Transitions for an Infinite System of Spiking Neurons

TL;DR

This paper models a neural network on a homogeneous tree, revealing multiple phase transitions in spiking activity depending on the death rate parameter, with different long-term behaviors such as extinction or persistent activity.

Contribution

It introduces a stochastic model of infinite neurons on a tree, demonstrating multiple phase transitions in neural activity based on the death rate parameter.

Findings

High death rate leads to almost sure extinction of activity.

Low death rate results in neurons spiking infinitely often with positive probability.

Intermediate rates cause neurons to eventually stop spiking, despite ongoing system activity.

Abstract

We consider a stochastic model describing the spiking activity of a countable set of neurons spatially organized into a homogeneous tree of degree $d$, $d \geq 2$; 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 $\gamma \geq 0$. 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.