Topological Phases in Nonlinear Oscillatory Systems

拓扑学是指物质的一种性质，它是实现稳健系统的指导原则，对连续变形和各种噪声源不敏感。一个标准的例子是在凝聚态物理中讨论的量子化霍尔电导。在量子系统和经典系统中都有拓扑相的分析，但在生物应用的背景下分析的较少，尽管生物系统对固有随机涨落的稳健性还没有完全了解。我们想要探索拓扑保护机制是否也在生物系统中发挥一定的作用。我们考虑了三个可以进行自主振荡的非线性系统：(I)一个由前馈和反馈环组成的单元，它相当于遗传和神经网络中的标准基序；(Ii)一个具有分层异宿动力学的单元，适合描述瞬时过程，特别是在大脑中；以及(Iii)一个改进的阻遏器，设计用于模拟瞬时细胞动力学。当属于这一类的动力单元在空间网格上耦合时，我们选择不同的耦合，这保证了拓扑相的观察。我们分析了期望的边模，用拓扑不变量来刻画它们，并求出了拓扑同步作为系统参数、实现的手性、非线性相互作用强度以及所涉及的有效哈密顿量的厄米或非厄米性质的函数的依赖关系。在生物网络的背景下，异宿网络中稳定的时钟、稳定的生物量传输和稳定的过渡路径将是理想的特征，因为它们的稳定性是基于对各种噪声源的非常有效的拓扑保护。

Topology refers to a property of matter that is much desired as a guiding principle to realize robust systems, insensitive to continuous deformations and various sources of noise. A standard example is the quantized Hall conductance discussed in condensed matter physics. Topological phases have been analyzed both in quantum and classical systems, but less in the context of biological applications, although the robustness of biological systems to the inherent stochastic fluctuations is not fully understood. We want to explore whether topological protection mechanisms play also some role in biological systems. We consider three nonlinear systems which can perform autonomous oscillations: (i) a unit composed of a feed-forward and a feed-back loop that amounts to a standard motif in genetic and neural networks; (ii) a unit with hierarchical heteroclinic dynamics, suited to describe transient processes in particular in the brain; and (iii) a modified repressilator, designed to model transient cell dynamics. When dynamical units, belonging to one of these types, are coupled on spatial grids, we choose different couplings, which promise the observation of topological phases. We analyze the expected edge modes, characterize them by topological invariants, and pursue the dependence of topological synchronization as a function of the system parameters, the implemented chirality, the nonlinear interaction strength, and the Hermitian or non-Hermitian properties of the involved effective Hamiltonians. In the context of biological networks, stable clocks, stable biomass transport and stable transition paths in heteroclinic networks will be desirable features when their stability is based on the very efficient topological protection against the various sources of noise.