Large attractors in cooperative bi-quadratic Boolean networks. Part I

TL;DR

This paper constructs large cooperative Boolean networks with bounded indegree and outdegree that exhibit exponentially long periodic orbits, challenging assumptions about stability in such systems.

Contribution

It demonstrates the existence of large cooperative Boolean networks with bounded degrees that can have exponentially long periodic orbits, revealing new insights into their dynamic complexity.

Findings

Existence of cooperative Boolean networks with long periodic orbits

Bounded indegree and outdegree do not prevent exponential cycle lengths

Large stable dynamics can occur in biologically plausible network topologies

Abstract

Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960's as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper, we consider systems with positive feedback interconnections among all variables (known as cooperative systems), which in a continuous setting guarantees a very stable dynamics. We show that for an arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional cooperative Boolean networks in which both the indegree and outdegree of each variable is bounded by two, and which nevertheless contain periodic orbits of length at least c^n. In Part II of this paper we will prove an inverse result showing that any system with such a dynamic behavior must in a sense be similar to the example described.