Dynamic System of Neurons on a Complete Graph: Synchronization

TL;DR

This paper models a complete graph of neurons as a dynamical system, analyzing conditions under which their states synchronize over time, with potential applications in understanding neural network synchronization.

Contribution

It introduces a mathematical framework for neuron synchronization on a complete graph, proving the existence of time-invariant state sets under certain conditions.

Findings

Neurons' states form a time-invariant set on the circle

Synchronization occurs under specific conditions

States eventually rotate uniformly on the circle

Abstract

The net of N ``physical'' neurons is considered as a dynamical system. These neurons form a complete graph. The state of any neuron is its electric potential. The potential linearly increases until reaches its maximal value. Then it falls to zero and the neuron sends spikes to all other neurons. Having got a spike any neuron changes its state by some given function. We study the problem of a synchronization of the net. We glue the maximal value of potential to zero and so consider the state of any neuron as a point on the circle. We prove that under some conditions the states of neurons considered as points on the circle finally (when time turns to infinity) form a set which is time-invariant modulo its uniform rotation along the circle.