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Dynamic System of Neurons on a Complete Graph: Synchronization

S. A. Pirogov, A. N. Rybko, D. D. Pervouchine, E. N. Petrova

TL;DR

This paper studies synchronization in a complete-graph neural network of $N$ integrate-and-fire-like neurons with potentials on $[0,1]$, recast as $N$ particles on a circle with a monotone jump map $f$ and a uniform drift. It proves existence and uniqueness of stationary configurations for a broad class of monotone updates and characterizes when the network forms a single cluster versus multiple equal-mass clusters, including exponential convergence to the stationary states. In the trapezoidal case (piecewise-linear $f_h$) and in the general monotone case, the authors derive precise conditions on parameters (e.g., $N h$) and provide contraction-based arguments to guarantee convergence. These results provide a rigorous foundation for synchronization phenomena in densely connected neural networks and quantify long-term clustering behavior.

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.

Dynamic System of Neurons on a Complete Graph: Synchronization

TL;DR

This paper studies synchronization in a complete-graph neural network of integrate-and-fire-like neurons with potentials on , recast as particles on a circle with a monotone jump map and a uniform drift. It proves existence and uniqueness of stationary configurations for a broad class of monotone updates and characterizes when the network forms a single cluster versus multiple equal-mass clusters, including exponential convergence to the stationary states. In the trapezoidal case (piecewise-linear ) and in the general monotone case, the authors derive precise conditions on parameters (e.g., ) and provide contraction-based arguments to guarantee convergence. These results provide a rigorous foundation for synchronization phenomena in densely connected neural networks and quantify long-term clustering behavior.

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.
Paper Structure (4 sections, 8 theorems, 53 equations)

This paper contains 4 sections, 8 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Discrete time dynamical system and fixed points
  3. Trapezoidal $\boldsymbol{f(x)}$ dynamics
  4. General case of monotone function $\boldsymbol{f(x)}$ on the circle

Key Result

Theorem 2.1

If the function $V(y)$ is continuous and strictly increasing on the interval $(0,1)$ and $0< V(y) <1$ for $0<y<1$, then the map $Y \longmapsto \widetilde{Y}$ has a unique fixed point in the open symplex $\mathcal{Y}$.

Theorems & Definitions (19)

  • Theorem 2.1
  • proof
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • proof
  • ...and 9 more