Cycle attractors for different deterministic updating schemes in Boolean regulation networks

The problem of preserving a limit cycle in a Boolean regulation network when its updating scheme is changed from parallel to block-sequential is studied. A theorem is proved which states that a Boolean regulation network, under certain hypotheses, cannot preserve a limit cycle when its updating scheme is changed from parallel to block-sequential when the network's indegree is less or equal to two. The swarm intelligence optimization technique called the bees algorithm is formulated to learn Boolean regulation networks with predefined limit cycles to generate examples that complement the proposed theorem. The results show that a necessary, but not sufficient, condition to preserve a limit cycle when changing the updating scheme, without violating the hypotheses, is that the network must have nodes with indegree larger than two.