Transition to chaos in models of genetic networks

Abstract

Randomly constructed networks of N elements governed by piecewise linear differential equations have been proposed as models for neural and genetic networks. In this model an element is labelled “on” if it is above a threshold, and “off” otherwise. For each element, there is a rule (truth table) specified by the values of K input elements that determines whether it will switch its state (from 1 to 0 or from 0 to 1) at some future time. Previous studies of these networks have demonstrated the existence of steady state, periodic, and chaotic attractors. The probability that the output in a truth table for a given gene is 1 (or 0), corresponding to an increased tendency for a gene’s activity to be repressed or expressed, is designated as p. Recent studies have demonstrated a transition from steady states to chaotic dynamics, with an intervening region of periodic dynamics, when p is decreased from 1.0 to 0.5. A probabilistic model of the dynamics yielded a critical relation between p and K that separates steady state behaviour from deterministic chaos. Here we present numerical data supporting the theoretical prediction of the relation between critical values of p and K. We also present numerical evidence for the existence of extremely long transients.