Synchrony in Gene Regulatory Networks

Synchrony in a Boolean network model of the L-arabinose operon in Escherichia coli

The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean network model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a …

Networks with asymmetric inputs: lattice of synchrony subspaces

We consider coupled cell networks with asymmetric inputs and study their lattice of synchrony subspaces. For the particular case of 1-input regular coupled cell networks we describe the join-irreducible synchrony subspaces for their lattice of synchrony subspaces, first in terms of the eigenvectors and generalized eigenvectors that generate them, and then by giving a characterization of the possible patterns of the associated balanced colourings. The set of the join-irreducible synchrony subspaces is join-dense for the lattice, that is, the lattice can be obtained by sums of those join-irreduc…

Synchrony and Anti-synchrony in Weighted Networks

We consider weighted coupled cell networks, that is networks where the interactions between any two cells have an associated weight that is a real valued number. Weighted networks are ubiquitous in real-world applications. We consider a dynamical systems perspective by associating to each network a set of continuous dynamical systems, the ones that respect the graph structure of the network. For weighted networks it is natural for the admissible coupled cell systems to have an additive input structure. We present a characterization of the synchrony subspaces and the anti-synchrony subspaces fo…

Synchrony Branching Lemma for Regular Networks

Coupled cell systems are dynamical systems associated to a network and synchrony subspaces, given by balanced colorings of the network, are invariant subspaces for every coupled cell systems associated to that network. Golubitsky and Lauterbach (SIAM J. Applied Dynamical Systems, 8 (1) 2009, 40-75) prove an analogue of the Equivariant Branching Lemma in the context of regular networks. We generalize this result proving the generic existence of steady-state bifurcation branches for regular networks with maximal synchrony. We also give necessary and sufficient conditions for the existence of ste…

网络动力学中的通用平衡同步模式

Generic balanced synchrony patterns in network dynamics

本文研究了网络动力学中同步模式的平衡特性，证明了在通用向量场下，同步模式总是由网络结构对称性继承，从而验证了多个猜想。

基于对称性的树网络同步分类

A Symmetry-Based Classification of Synchrony in Tree Networks

本文研究了树网络中同步性的对称性分类，证明了树型网络中不存在异域同步性，并分析了叶子对稳定性的影响。