Local negative circuits and cyclic attractors in Boolean networks with at most five components

Abstract: We consider the following question on the relationship between the asymptotic behaviours of asynchronous dynamics of Boolean networks and their regulatory structures: does the presence of a cyclic attractor imply the existence of a local negative circuit in the regulatory graph? When the number of model components $n$ verifies $n \geq 6$, the answer is known to be negative. We show that the question can be translated into a Boolean satisfiability problem on $n \cdot 2^n$ variables. A Boolean formula expressing the absence of local negative circuits and a necessary condition for the existence of cyclic attractors is found unsatisfiable for $n \leq 5$. In other words, for Boolean networks with up to $5$ components, the presence of a cyclic attractor requires the existence of a local negative circuit.