Bifurcations in Boolean Networks - Chris Kuhlman; Henning Mortveit; David Murrugarra; Anil Kumar (Discrete Mathematics & Theoretical Computer Science, 2011)
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 $Δ$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $Δ ≥ 2$ they may have long periodic orbits. The limiting case of $Δ = 2$ 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.