Positive circuits and maximal number of fixed points in discrete dynamical systems
We consider a product X X of n n finite intervals of integers, a map F F from X X to itself, the asynchronous state transition graph ?(F) ? ( F ) on X X that Thomas proposed as a model for the dynamics of a network of n n genes, and the interaction graph G(F) G ( F ) that describes the topology of the system in terms of positive and negative interactions between its n n components. Then, we establish an upper bound on the number of fixed points for F F , and more generally on the number of attractors in ?(F) ? ( F ) , which only depends on X X and on the topology of the positive circuits of G(F) G ( F ) . This result generalizes the following discrete version of Thomas conjecture recently proved by Richard and Comet: If G(F) G ( F ) has no positive circuit, then ?(F) ? ( F ) has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.
positive circuit fixed point discrete dynamical system interaction graph gene network discrete jacobian matrix