A Graph Partitioning Approach to Predicting Patterns in Lateral Inhibition Systems

作者：Ferreira Ana S Rufino^*; Arcak Murat

来源：SIAM Journal on Applied Dynamical Systems, 2013, 12(4): 2012-2031.

DOI：10.1137/130910142

• large-scale systems
• nonlinear systems
• developmental biology
• pattern formation

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 an identical individual cell 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.

• 出版日期2013