{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/24n975980","DOIUrl":null,"url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 617-617, November 2024. <br/> 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.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"9 1","pages":""},"PeriodicalIF":10.8000,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Review","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/24n975980","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍:
Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter.
Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.