Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider a product XX of nn finite intervals of integers, a map FF from XX to itself, the asynchronous state transition graph Γ(F)Γ(F) on XX that Thomas proposed as a model for the dynamics of a network of nn 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 nn components. Then, we establish an upper bound on the number of fixed points for FF, and more generally on the number of attractors in Γ(F)Γ(F), which only depends on XX 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 Γ(F)Γ(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.