Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework. We propose novel update modes, the history-based and trapping modes, and provide a comprehensive comparison between them. Furthermore, we show that trapping dynamics, which further generalize the most permissive mode, correspond to a rich class of networks related to transitive dynamics and encompassing commutative networks. Finally, we provide a thorough characterization of the structure of minimal and principal trapspaces, bringing a combinatorial and algebraic understanding of these objects.

翻译：布尔网络作为复杂动态系统的模型被广泛应用，旨在捕捉组件状态随时间变化的因果关系与同步性的关键特征。布尔网络的动力学源自其布尔映射在所谓的更新模式下的应用，该模式规定了网络配置之间可能的转换。本文探索了具有过去配置记忆的更新模式，并提供了一个定义它们的通用框架。我们展示了近期引入的模式（如最容许模式和区间模式）可以自然地在此框架中表达。我们提出了新的更新模式——基于历史的模式与捕获模式，并对它们进行了全面比较。此外，我们证明了捕获动力学（进一步推广了最容许模式）对应于与传递动力学相关并包含交换网络的丰富网络类别。最后，我们对最小与主要陷阱空间的结构进行了深入刻画，为这些对象带来了组合与代数层面的理解。