Lengths of attractors and transients in neuronal networks with random connectivities

TL;DR

This paper analytically derives how the average lengths of attractors and transients in neuronal networks depend on network connectivity, especially near the phase transition where complex dynamics emerge.

Contribution

It provides new scaling laws for attractor and transient lengths in Erdős-Rényi neuronal networks near the phase transition, extending previous work.

Findings

Scaling laws for attractor lengths derived

Scaling laws for transient lengths derived

Focus on phase transition regime

Abstract

We study how the dynamics of a class of discrete dynamical system models for neuronal networks depends on the connectivity of the network. Specifically, we assume that the network is an Erd\H{o}s-R\'{enyi} random graph and analytically derive scaling laws for the average lengths of the attractors and transients under certain restrictions on the intrinsic parameters of the neurons, that is, their refractory periods and firing thresholds. In contrast to earlier results that were reported in \cite{TAWJ}, here we focus on the connection probabilities near the phase transition where the most complex dynamics is expected to occur.