Cyclic attractors of nonexpanding q-ary networks

Discrete dynamical systems in which model components take on categorical values have been successfully applied to biological networks to study their global dynamic behavior. Boolean models in particular have been used extensively. However, multi-state models have also emerged as effective computational tools for the analysis of complex mechanisms underlying biological networks. Models in which variables assume more than two discrete states provide greater resolution, but this scheme introduces discontinuities. In particular, variables can increase or decrease by more than one unit in one time step. This can be corrected, without changing fixed points of the system, by applying an additional rule to each local activation function. On the other hand, if one is interested in cyclic attractors of their system, then this rule can potentially introduce new cyclic attractors that were not observed previously. This article makes some advancements in understanding the state space dynamics of multi-state network models with synchronous, sequential, or block-sequential update schedules and establishes conditions under which no new cyclic attractors are added to networks when the additional rule is applied. Our analytical results have the potential to be incorporated into modeling software and aid researchers in their analyses of biological multi-state networks.