Stable and unstable attractors in Boolean networks

Boolean networks at the critical point have been a matter of debate for many years as, e.g., the scaling of numbers of attractors with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We point out here that these results are obtained using deterministic parallel update, where a large fraction of attractors are an artifact of the updating scheme. This limits the significance of these results for biological systems where noise is omnipresent. Here we take a fresh look at attractors in Boolean networks with the original motivation of simplified models for biological systems in mind. We test the stability of attractors with respect to infinitesimal deviations from synchronous update and find that most attractors are artifacts arising from synchronous clocking. The remaining fraction of attractors are stable against fluctuating delays. The average number of these stable attractors grows sublinearly with system size in the numerically tractable range.