The phase diagram of random threshold networks

TL;DR

This paper explores the phase diagram of random threshold networks, revealing how their dynamical behavior transitions between frozen and chaotic phases, with analysis supported by theoretical derivations and simulations.

Contribution

It introduces a detailed analysis of the phase diagram for threshold networks with real-valued thresholds, including derivations and comparison with simulations.

Findings

Deviations occur at integer thresholds even in large networks

Annealed approximation effectively predicts dynamics for non-integer thresholds

Updating rules influence the accuracy of theoretical models

Abstract

Threshold networks are used as models for neural or gene regulatory networks. They show a rich dynamical behaviour with a transition between a frozen and a chaotic phase. We investigate the phase diagram of randomly connected threshold networks with real-valued thresholds h and a fixed number of inputs per node. The nodes are updated according to the same rules as in a model of the cell-cycle network of Saccharomyces cereviseae [PNAS 101, 4781 (2004)]. Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the "on" and "off" state, and for the sensitivity $\lambda$. The results are compared with simulations of quenched networks. We find that for integer values of h the simulations show marked deviations from the annealed approximation even for large networks. This can be attributed to the particular choice of the updating rule.