多随机、时滞因素联合作用下的耦合振子网络时空动力学及其调控机制的研究和应用

本课题希望针对各类由具体生物学背景出发的耦合振子模型（一阶、二阶耦合振子模型）的同步、破同步、新型空间波形的生成机制，进行深层次的理论分析研究，基于新正交基发展低维流形的连续化（振子数量N趋向于无穷）的新约化方法，揭示生成特定时空动力学现象的关键随机因素（随机分布的期望、方差、各高阶矩）和多种形式时滞因素，建立起研究各阶符合生物学现实的耦合振子模型的平均场与连续化动力学研究的全面框架。本课题还将对于各类耦合振子模型，设计实用的（自适应）反馈控制器、噪音调控手段，实现上述各类耦合振子模型的同步、破同步，实现驱使动力学发生本质改变的外加调控，讨论控制手段的能量、时间等因素结合的最优问题。本课题将基于上述基础研究成果，利用相对符合生物学现实的模型，并充分利用和发展相关数据同化、高维参数识别的算法，通过采集和分析具体多尺度数据，解析生物学领域有关振子节律调控与各类生物学特征联系的相关基本问题。

This proposal mainly investigates the positive or negative influences of random switches and time delays in the dynamical behaviors, including synchronization, desynchronization, and various types of spatial waves, emergent the first- or second-order coupled oscillator models. The key factors including statistical quantities (expectation, variance, and moments) and the exact forms of time delays to the formation of the corresponding dynamics will be discovered. Also new reduction techniques along low-dimensional manifold, the conservation law, and the resulting continuous equations will be used to figure out the physical phenomena that are observed in the coupled oscillators of finite number. Furthermore, control and modulations by the external control protocols in terms of some optimized indexes will be developed to realize synchronization, desynchronization, and spatial dynamics transition. More significantly, we will collect datasets at different levels from the animal models with circadian rhythms and datasets of fMRI from subjects of particular mental disorder and control. Based on the oscillator models with the networks inferred from the selected data, we need to connect particular phenotype of rhythms or mental disorders with visible biomarkers, which could confirm some hypothesis of biological significance.