Cyclic attractors of nonexpanding n-ary networks

TL;DR

This paper investigates the dynamics of multi-state biological networks, establishing conditions where additional rules do not create new cyclic attractors, thus aiding accurate modeling of complex biological systems.

Contribution

It provides theoretical conditions ensuring that the application of an additional rule does not introduce new cyclic attractors in multi-state network models.

Findings

Identifies conditions preventing new cyclic attractors from forming.

Analyzes the impact of additional rules on state space dynamics.

Offers insights applicable to biological network modeling software.

Abstract

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 a synchronous update schedule 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.