Survey and Review

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.