Bifurcations in Boolean Networks

Details Bifurcations in Boolean Networks Bifurcations in Boolean Networks

2011-08-15

arxiv.org/abs/1108.2974v2

Mathematics

Dynamical Systems

18 pages, 4 figures, Discrete Mathematics and Theoretical Computer Science 2011

Scientific paper

This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of classical 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 \Delta of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for \Delta >= 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 graph classes.

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