Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China.
South China Institute for Stem Cell Biology and Regenerative Medicine, Guangzhou Institutes of Biomedicine and Health, Chinese Academy of Sciences, Guangzhou, Guangdong 510530, China.
Chaos. 2022 Aug;32(8):083117. doi: 10.1063/5.0097874.
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.
由考夫曼引入的布尔网络最初旨在作为一种原型模型，用于深入了解基因调控动态，现已成为理解各种以二进制状态变量描述的复杂系统的范例。然而，在某些情况下，例如在生物学中，基础动力系统的二进制状态描述是不充分的。我们提出了随机三元网络，并研究了与变量的三元离散化相关的一般动力特性。我们发现，三元动力学可以是有序的，也可以是无序的，具有正的李雅普诺夫指数，并且它们在参数空间中的边界可以通过解析确定。决定边界的一个关键动力事件是出现了一个额外的平衡点，我们为此提供了数值验证。我们还发现，在塑造系统动力学方面起着关键作用的节点在参数空间的不同区域具有特征性的不同行为，值得注意的是，这些区域之间的边界与有序和无序动力学之间的边界相吻合。总的来说，我们的三元网络框架通过能够对更丰富和更复杂的动力行为进行定量描述，极大地扩展了经典的布尔范例。
