Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework

To each Boolean function f: (0, 1}(n) -> {0, 1}(n) and each x is an element of {0, 1}(n), we associate a signed directed graph G(x). and we show that the existence of a positive circuit in G(x) for some v is a necessary condition for the existence of several fixed points in the dynamics (the sign of a circuit being defined as the product of the signs of its edges), and that the existence of a negative circuit is a necessary condition for the existence of an attractive cycle. These two results are inspired by rules for discrete models of genetic regulatory networks proposed by the biologist R. Thomas. The proof of the first result is modelled after a recent proof of the discrete Jacobian conjecture. (c) 2008 Elsevier Inc. All rights reserved.