Structure of attractors in randomly connected networks

TL;DR

This paper analyzes the dynamics of randomly connected neural networks, providing a theoretical framework for understanding attractor structures, cycle lengths, and state overlaps, with predictions validated by numerical simulations.

Contribution

It introduces a Markovian stochastic process model for state overlaps in large networks, offering analytical insights into attractor counts and cycle properties.

Findings

Analytical predictions match numerical simulations for large networks.

Derived formulas for total attractors and cycle lengths.

Quantified state concentration probabilities in neural networks.

Abstract

The deterministic dynamics of randomly connected neural networks are studied, where a state of binary neurons evolves according to a discreet-time synchronous update rule. We give a theoretical support that the overlap of systems' states between the current and a previous time develops in time according to a Markovian stochastic process in large networks. This Markovian process predicts how often a network revisits one of previously visited states, depending on the system size. The state concentration probability, i.e., the probability that two distinct states co-evolve to the same state, is utilized to analytically derive various characteristics that quantify attractors' structure. The analytical predictions about the total number of attractors, the typical cycle length, and the number of states belonging to all attractive cycles match well with numerical simulations for relatively large system sizes.