Phase-isostable descriptions of coupled oscillator networks

“相位缩减模型，其中振荡器动态被缩减为它们在极限环上的相位的动态，在描述耦合振荡器网络的动力学行为方面非常成功，在这种情况下，单个振荡器具有强烈吸引的极限环，并且网络中振荡器之间的耦合很弱[1]。然而，对于许多生物（特别是神经）振荡器网络，这些建模假设是不适当的;数值模拟和其他建模技术的结果揭示了在仅考虑振荡器相位的框架中无法捕获的动态[2]。阶段缩减模型描述已知动态的失败引出了一个问题，即是否也跟踪一些概念的替代框架，与极限环的距离沿着极限环上的相位，可以更准确地描述观测到的动态。最近有人提出了这样一个使用相位等稳坐标的框架[3，4]，但它描述一系列观察到的网络行为的能力还没有得到彻底的研究。本项目的目的是首先研究在何种条件下相位等稳框架可以精确地描述单个振子对外部强迫的响应。在对框架进行任何必要的改进后，将开发相位等稳坐标中的网络方程，并用于研究超出弱耦合限制的网络动力学。网络分析将利用网络动力系统的任何对称性，允许预测何时可能存在某些簇和嵌合体状态。可以预见的是，相位等稳范式也将能够揭示丰富多样的更复杂的动态网络行为。"

"Phase-reduced models, where oscillator dynamics are reduced to the dynamics of their phase on limit cycle, have been extremely successful in describing dynamical behaviours of networks of coupled oscillators in the case where individual oscillators possess a strongly attracting limit cycle and coupling between oscillators in the network is weak [1]. However, for many biological (and particularly neural) oscillator networks these modelling assumptions are not appropriate; numerical simulations and results from other modelling techniques have revealed dynamics that cannot be captured in a framework where only phases of oscillators are considered [2].The failure of phase reduced models to describe the known dynamics begs the question of whether an alternative framework that also tracks some notion of distance from the limit cycle, along with the phase on limit cycle, can more accurately describe observed dynamics. Such a framework, using phase-isostable coordinates has recently been proposed [3,4], but its ability to describe a range of observed network behaviours has yet to have been thoroughly investigated. The aims of this project will be to first investigate conditions under which the phase-isostable framework can accurately characterise the response of single oscillators to external forcing. After making any required refinements to the framework, network equations in phase-isostable coordinates will be developed and used to investigate network dynamics beyond the weak coupling limit. The network analysis will utilise any symmetries of the network dynamical system allowing for prediction of when certain cluster and chimera states may exist. It is anticipated that the phase-isostable paradigm will also be able to reveal a rich variety of more complex dynamical network behaviour."