Gyorgy Andras, Arcak Murat
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720 USA.
IEEE Trans Netw Sci Eng. 2018 Jan-Mar;5(1):55-64. doi: 10.1109/TNSE.2017.2730261. Epub 2017 Jul 21.
Two of the most common pattern formation mechanisms are Turing-patterning in reaction-diffusion systems and lateral inhibition of neighboring cells. In this paper, we introduce a broad dynamical model of interconnected modules to study the emergence of patterns, with the above mentioned two mechanisms as special cases. Our results do not restrict the number of modules or their complexity, allow multiple layers of communication channels with possibly different interconnection structure, and do not assume symmetric connections between two connected modules. Leveraging only the static input/output properties of the subsystems and the spectral properties of the interconnection matrices, we characterize the stability of the homogeneous fixed points as well as sufficient conditions for the emergence of spatially non-homogeneous patterns. To obtain these results, we rely on properties of the graphs together with tools from monotone systems theory. As application examples, we consider patterning in neural networks, in reaction-diffusion systems, and contagion processes over random graphs.
两种最常见的模式形成机制是反应扩散系统中的图灵模式形成以及相邻细胞的侧向抑制。在本文中，我们引入了一个相互连接模块的广义动力学模型来研究模式的出现，上述两种机制是该模型的特殊情况。我们的结果不限制模块的数量或其复杂性，允许具有可能不同互连结构的多层通信通道，并且不假设两个相连模块之间的连接是对称的。仅利用子系统的静态输入/输出特性和互连矩阵的谱特性，我们刻画了均匀不动点的稳定性以及空间非均匀模式出现的充分条件。为了获得这些结果，我们依赖于图的性质以及单调系统理论中的工具。作为应用示例，我们考虑神经网络中的模式形成、反应扩散系统中的模式形成以及随机图上的传染过程。
