SIAM Review, Volume 66, Issue 4, Page 617-617, November 2024.
Neural oscillations are periodic activities of neurons in the central nervous system of eumetazoa. In an oscillatory neural network, neurons are modeled by coupled oscillators. Oscillatory networks are employed for describing the behavior of complex systems in biology or ecology with respect to the connectivity of the network components or the nonlinear dynamics of the individual units. Phase-locked periodic states and their instabilities are core features in the analysis of oscillatory networks. In “Oscillatory Networks: Insights from Piecewise-Linear Modeling,” Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, and Yi Ming Lai review techniques for studying coupled oscillatory networks. They first discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. Then they consider nonsmooth piecewise-linear (PWL) systems, for which periodic orbits are easily obtained. Saltation operators are used for modeling the propagation of perturbations through switching manifolds in the analysis of the dynamics and bifurcations at the network level. Applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds illustrate the power of these methods. PWL modeling has been applied for a long time in engineering. Recently, it has been introduced in other fields, such as social sciences, finance, and biology. For many modern applications in science, piecewise models are much more versatile than the classical smooth dynamical systems. In neuroscience, PWL functions enable explicit calculations which are infeasible in the original smooth system. This includes discontinuous dynamical systems, which are used to model impacting mechanical oscillators, integrate-and-fire models of spiking neurons, and cardiac oscillators. On the other hand, the price to pay is the retrieval of new conditions for the existence, uniqueness, and stability of solutions. The paper discusses the application of PWL models to a large variety of applications from engineering and biology. It will be of interest to many readers.

中文翻译：

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 调查和审查


SIAM 评论，第 66 卷，第 4 期，第 617-617 页，2024 年 11 月。

神经振荡是真后生动物中枢神经系统中神经元的周期性活动。在振荡神经网络中，神经元由耦合振荡器建模。振荡网络用于描述生物学或生态学中复杂系统的行为，这些行为与网络组件的连接性或单个单元的非线性动力学有关。锁相周期性状态及其不稳定性是振荡网络分析的核心特征。在“振荡网络：来自分段线性建模的见解”中，Stephen Coombes、Mustafa Şayli、Rüdiger Thul、Rachel Nicks、Mason A. Porter 和 Yi Ming Lai 回顾了研究耦合振荡网络的技术。他们首先讨论了相位减少、相位幅度减少和平滑动力学系统的主稳定函数。然后，他们考虑了非光滑分段线性 （PWL） 系统，对于这种系统，周期性轨道很容易获得。盐化算子用于在网络层面的动力学和分岔分析中，通过切换流形对扰动的传播进行建模。在神经系统、心脏系统、机电振荡器网络以及牛群合作中的应用说明了这些方法的力量。PWL 建模在工程中应用了很长时间。最近，它已被引入其他领域，例如社会科学、金融和生物学。对于科学中的许多现代应用，分段模型比经典的平滑动力学系统要通用得多。在神经科学中，PWL 函数支持显式计算，这在原始平滑系统中是不可行的。 这包括非连续动力学系统，用于模拟撞击机械振荡器、脉冲神经元的积分和发射模型以及心脏振荡器。另一方面，要付出的代价是找回解的存在、唯一性和稳定性的新条件。本白皮书讨论了 PWL 模型在工程和生物学中的各种应用中的应用。许多读者会对此感兴趣。