Local Negative Circuits and Cyclic Attractors in Boolean Networks with at most Five Components

We consider the following question on the relationship between the asymptotic behaviors 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 to be 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.