We study Boolean networks which are simple spatial models of the highly conserved Delta–Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained “patterns” that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms.

我们研究布尔网络，它是高度保守的Delta-Notch系统的简单空间模型。这些模型假设同一细胞中的Notch抑制了每个细胞中的Delta，并且在周围细胞中存在Delta的情况下激活了Notch。我们考虑对表示单元格之间邻居关系的无向图进行完全异步动力学。在这种框架下，可以证明所有吸引子都是系统的固定点，与邻居关系无关，例如，通过使用模型简化版本的已知属性，其中每个单元只定义一个物种。固定点对应于在横向抑制的离散和连续建模中出现的所谓的细粒度“模式”。我们研究定点的可达性，给出了完整模型和简化模型的陷阱空间和吸引盆地的特征。另外，我们使用陷阱空间的特征来研究模式对扰动的鲁棒性。定性分析的结果可以补充和指导基于仿真的方法，并且可以作为研究更复杂机制的基础。