Bifurcations in Boolean Networks

Abstract

This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow $1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.

Keywords

• bifurcation
• bi-threshold
• threshold
• boolean networks
• graph dynamical systems
• synchronous
• asynchronous
• sequential dynamical systems
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-ds] mathematics [math]/dynamical systems [math.ds]
• [nlin.nlin-cg] nonlinear sciences [physics]/cellular automata and lattice gases [nlin.cg]
• [math.math-co] mathematics [math]/combinatorics [math.co]