Attractors in continuous and Boolean networks

Random Boolean models of complex regulatory networks are known to exhibit rich dynamical behaviors, including an order/disorder transition. We show that implementation of the nominal Boolean logic of a network using differential equations involving sigmoidal switching functions generically leads to deviations from the Boolean predictions. On simple rings, the ``reliable'' set of Boolean attractors corresponds to the stable attractors of the analogous continuous system. For networks with more complex logic, however, the set of the continuous attractors is determined by non-Boolean characteristics of the switching events. In large random networks, the nature of the order/disorder transition is altered by collective effects associated with compositions of the sigmoidal switching functions.