Beyond Boolean: Ternary networks and dynamics

Boolean networks introduced by Kauffman, originally intended as a prototypical model for gaining insights into gene regulatory dynamics, have become a paradigm for understanding a variety of complex systems described by binary state variables. However, there are situations, e.g., in biology, where a binary state description of the underlying dynamical system is inadequate. We propose random ternary networks and investigate the general dynamical properties associated with the ternary discretization of the variables. We find that the ternary dynamics can be either ordered or disordered with a positive Lyapunov exponent, and the boundary between them in the parameter space can be determined analytically. A dynamical event that is key to determining the boundary is the emergence of an additional fixed point for which we provide numerical verification. We also find that the nodes playing a pivotal role in shaping the system dynamics have characteristically distinct behaviors in different regions of the parameter space, and, remarkably, the boundary between these regions coincides with that separating the ordered and disordered dynamics. Overall, our framework of ternary networks significantly broadens the classical Boolean paradigm by enabling a quantitative description of richer and more complex dynamical behaviors.