Topological Phases in Nonlinear Oscillatory Systems

拓扑是指物质的一种性质，它是实现鲁棒系统的指导原则，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学中讨论的量子化霍尔电导。拓扑相位已经在量子和经典系统中进行了分析，但在生物应用的背景下较少，尽管生物系统对固有随机波动的鲁棒性尚未完全理解。我们想探讨拓扑保护机制是否也在生物系统中发挥一定的作用。我们考虑三个非线性系统，可以执行自主振荡：（i）一个单元组成的前馈和反馈回路，相当于一个标准的基序在遗传和神经网络;（ii）一个单位与分层异宿动力学，适合于描述瞬态过程，特别是在大脑中;和（iii）一个修改represilator，设计用于模拟瞬态细胞动力学。当动力学单元，属于这些类型之一，耦合在空间网格上，我们选择不同的耦合，这保证了拓扑相位的观察。我们分析了预期的边缘模式，其特征在于它们的拓扑不变量，并追求的依赖性的拓扑同步作为系统参数的函数，实施手征，非线性相互作用强度，和厄米特或非厄米特属性所涉及的有效哈密顿。在生物网络的背景下，稳定的时钟，稳定的生物量运输和稳定的过渡路径在异宿网络将是理想的功能时，他们的稳定性是基于非常有效的拓扑保护，对各种来源的噪音。

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.