Cooperative Boolean systems with generically long attractors II

TL;DR

This paper investigates how cooperative Boolean networks, especially those with AND/OR functions, limit the complexity of attractors, demonstrating bounds on their lengths and the effects on system sensitivity.

Contribution

It establishes bounds on attractor lengths in cooperative Boolean networks with AND/OR functions and clarifies the impact of cooperativity on system sensitivity.

Findings

Cooperativity prevents strong sensitive dependence on initial conditions.

Attractor lengths are bounded by approximately sqrt(3)^N in certain networks.

The derived upper bound on attractor length is shown to be sharp.

Abstract

We prove that cooperativity in Boolean networks precludes a strong notion of sensitive dependence on initial conditions. Weaker notions of sensitive dependence are shown to be consistent with cooperativity, but if each regulatory functions is binary AND or binary OR, in N-dimensional networks they impose an upper bound of approximately sqrt(3)^N on the lengths of attractors that can be reached from a fraction p approaching 1 of initial conditions. The upper bound is shown to be sharp.