Rhythmic and non-rhythmic attractors in asynchronous random Boolean networks

In multi-component, discrete systems, such as Boolean networks and cellular automata, the scheme of updating of the individual elements plays a crucial role in determining their dynamic properties and their suitability as models of complex phenomena. Many interesting properties of these systems rely heavily on the use of synchronous updating of the individual elements. Considerations of parsimony have motivated the claim that, if the natural systems being modelled lack any clear evidence of synchronously driven elements, then random asynchronous updating should be used by default. The introduction of a random element precludes the possibility of strictly cyclic behaviour. In principle, this poses the question of whether asynchronously driven Boolean networks, cellular automata, etc., are inherently bad choices at the time of modelling rhythmic phenomena. This paper focuses on this subsidiary issue for the case of Asynchronous Random Boolean Networks (ARBNs). It defines measures of pseudo-periodicity between states and sufficiently relaxed statistical constraints. These measures are used to guide a genetic algorithm to find appropriate examples. Success in this search for a number of cases, and the subsequent statistical analysis lead to the conclusion that ARBNs can indeed be used as models of co-ordinated rhythmic phenomena, which may be stronger precisely because of their in-built asynchrony. The same technique is used to find non-stationary attractors that show no rhythm. Evidence suggests that the latter are more abundant than rhythmic attractor. The methodology is flexible, and allows for more demanding statistical conditions for defining pseudo-periodicity, and constraining the evolutionary search.