Boolean Networks (BNs) describe the time evolution of binary states using logic functions on the nodes of a network. They are fundamental models for complex discrete dynamical systems, with applications in various areas of science and engineering, and especially in systems biology. A key aspect of the dynamical behavior of BNs is the number of attractors, which determines the diversity of long-term system trajectories. Due to the noisy nature and incomplete characterization of biological systems, a stochastic asynchronous update scheme is often more appropriate than the deterministic synchronous one. AND-NOT BNs, whose logic functions are the conjunction of literals, are an important subclass of BNs because of their structural simplicity and their usefulness in analyzing biological systems for which the only information available is a collection of interactions among components. In this paper, we establish new theoretical results regarding asynchronous attractors in AND-NOT BNs. We derive two new upper bounds for the number of asynchronous attractors in an AND-NOT BN based on structural properties (strong even cycles and dominating sets, respectively) of the AND-NOT BN. These findings contribute to a more comprehensive understanding of asynchronous dynamics in AND-NOT BNs, with implications for attractor enumeration and counting, as well as for network design and control.

翻译：布尔网络（BNs）利用网络节点上的逻辑函数描述二元状态的时间演化。它们是复杂离散动力学系统的基本模型，在科学与工程的多个领域，尤其是系统生物学中具有广泛应用。布尔网络动力学行为的一个关键方面是吸引子的数量，它决定了系统长期轨迹的多样性。由于生物系统的噪声特性和不完全表征，随机异步更新方案通常比确定性同步更新更为合适。AND-NOT布尔网络（其逻辑函数为文字合取）是布尔网络的一个重要子类，因其结构简单性以及在分析仅掌握组件间相互作用信息的生物系统时的实用性而备受关注。本文针对AND-NOT布尔网络中的异步吸引子建立了新的理论结果。我们基于AND-NOT网络的结构特性（分别为强偶环和支配集），推导出AND-NOT布尔网络中异步吸引子数量的两个新上界。这些发现有助于更全面地理解AND-NOT布尔网络中的异步动力学，对吸引子枚举与计数以及网络设计与控制具有重要意义。