Institut de Mathématiques de Luminy

Abstract 2005-08

Remy Elisabeth, Ruet Paul, Thieffry Denis.
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 {0, 1}n, we associate a signed directed graph , and we prove that the existence of a positive circuit in for some is a necessary condition for the existence of several fixed points in the dynamics (the sign of a circuit being 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 rules have been proposed by the biologist R. Thomas in the course of his analysis of genetic regulatory networks. The proof of the first one relies on a recent proof of the discrete Jacobian conjecture.

keywords:
Discrete dynamical systems, Regulatory networks, Genetic regulation, Differentiation, Homeostasis, Discrete Jacobian matrix, Jacobian conjecture.

 


Last update : may 13, 2005, EL.