Bifurcations in Boolean Networks

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 → 1 (up-threshold) and 1 → 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 ∆ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for ∆ ≥ 2 they may have long periodic orbits. The limiting case of ∆ = 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.