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

We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive circuits of the interaction graph associated with F. The proof uses and strongly generalizes a theorem of Richard and Comet which corresponds to a discrete version of the Thomas’ conjecture: if the interaction graph associated with F has no positive circuit, then F has at most one fixed point. The obtained upper bound on the number of fixed points also strongly generalizes the one established by Aracena et al for a particular class of Boolean networks.

💡 Research Summary

The paper investigates discrete dynamical systems defined on the Cartesian product X = ∏₁ⁿ Xᵢ of finite integer intervals, with a map F : X → X describing the evolution of a gene regulatory network. The authors focus on the relationship between the structure of the interaction graph associated with F and the number of fixed points (or more generally, attractors) of the system.

First, they introduce the asynchronous state transition graph Γ(F). Each vertex of Γ(F) corresponds to a global state x ∈ X, and there is a directed edge from x to y whenever a single component i can move one step toward its target value fᵢ(x). Fixed points are vertices with no outgoing edges; attractors are minimal trap domains—sets of states that cannot be left once entered. This framework captures the biologically relevant notion of stable or multistable cellular states.

To connect dynamics with graph topology, the authors define a discrete Jacobian matrix F′(x,v) for each pair (x,v) where v ∈ {−1,1}ⁿ indicates a possible direction of change and x+v ∈ X. The entry f_{ij}(x,v) =