We analyze spatial patterns on networks of cells where adjacent cells inhibit each other through contact signaling. We represent the network as a graph where each vertex represents the dynamics of identical individual cells and where graph edges represent cell-to-cell signaling. To predict steady-state patterns we find equitable partitions of the graph vertices and assign them into disjoint classes. We then use results from monotone systems theory to prove the existence of patterns that are structured in such a way that all the cells in the same class have the same final fate. To study the stability properties of these patterns, we rely on the graph partition to perform a block decomposition of the system. Then, to guarantee stability, we provide a small-gain type criterion that depends on the input-output properties of each cell in the reduced system. Finally, we discuss pattern formation in stochastic models. With the help of a modal decomposition we show that noise can enhance the parameter region where patterning occurs.

中文翻译：

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在横向抑制系统中预测模式的图形划分方法。

我们分析了细胞网络中相邻细胞通过接触信号互相抑制的空间格局。我们将网络表示为图，其中每个顶点代表相同单个细胞的动力学，图边缘代表细胞间信号。为了预测稳态模式，我们找到图顶点的等价分区，并将它们分配为不相交的类。然后，我们使用单调系统理论的结果来证明模式的存在，这种模式的结构使得同一类中的所有单元格都具有相同的最终命运。为了研究这些模式的稳定性，我们依靠图形分区对系统进行块分解。然后，为了保证稳定性，我们提供了一个小增益类型准则，该准则取决于简化系统中每个像元的输入输出特性。最后，我们讨论了随机模型中的模式形成。借助模态分解，我们表明噪声可以增强发生图案化的参数区域。