Phase Description of Oscillations in Singularly-Perturbed Systems

我们想研究典型的神经元活动的不规则振荡。对于这种不规则的振荡，不稳定的呼吸节奏就是一个经典的例子。利用所建立的方法，振荡的简化描述建立在它们的特征相位的划分上。在应用中，这种相位描述可用于检测振荡之间非常弱的相互作用。然而，标准相位法只允许处理高度规则的振荡。因此，我们建议的中心目标是充分扩展相描述方法，以应用于现实的神经元活动，这显示出大量的不规则性。特别地，我们要处理两个神经元振荡的例子。所谓的噪声诱发振荡(1)只存在于外部波动的情况下，这些波动使系统脱离实际上不振荡的状态。在不考虑扰动的情况下，不能像在标准相位法中那样描述这种振荡。因此，必须使用另一个起点。为此，我们想为呼吸的描述扩展最近开发的方法。所谓的爆破(2)是一种混合模式振荡，在某种程度上，两个振荡合二为一，是由快、慢时间尺度的相互作用产生的。在耦合震波网络中，由于耦合强度较弱，标准相位描述已经失效。因此，我们计划设计一个更充分的相位描述，通过取平均值来实现对快速模式的简化处理。在应用程序中，当网络出现在中枢神经系统中时，我们的目标是描述各种同时发生的同步模式。

We want to study the typically irregular oscillations of neuronal activity. For such irregular oscillations, the unsteady rhythm of breathing is a classical example. Using the established method, a simplified description of oscillations builds on their division into characteristic phases. In applications, such a phase description can be used for the detection of very weak interactions between oscillations. However, the standard phase method allows only for a treatment of highly regular oscillations. Therefore, the central goal of our proposal is a sufficient extension of the methods of phase description for an application to realistic neuronal activity, which shows substantial irregularity. In particular, we want to treat two examples of neuronal oscillations. So called noise-induced oscillations (1) exist only in the presence of external fluctuations that arouse the system out of a state that actually does not oscillate. This type of oscillation cannot be described under disregard of the perturbation as it would be tried in the standard phase method. Therefore, another starting point has to be used. For this, we want to extend a recently developed method for the description of breathing. So called bursting (2) is a type of mixed mode oscillation, in a way two oscillators in one, that arises from an interplay of fast and slow time scales. In networks of coupled bursters, the standard phase description breaks down already for weak coupling strength. We, therefore, plan to design a more adequate phase description that achieves a simplified handling of the fast mode by taking an average. In the application to burster networks as they appear in the central nervous system, our goal is to describe the variety of simultaneously occurring synchronization patterns.