Network dynamics of coupled oscillators and phase reduction techniques

Investigating the dynamics of a network of oscillatory systems is a timely and urgent topic. Phase synchronization has proven paradigmatic to study emergent collective behavior within a network. Defining the phase dynamics, however, is not a trivial task. The literature provides an arsenal of solutions, but results are scattered and their formulation is far from standardized. Here, we present, in a unified language, a catalogue of popular techniques for deriving the phase dynamics of coupled oscillators. Traditionally, approaches to phase reduction address the (weakly) perturbed dynamics of an oscillator. They fall into three classes. (i) Many phase reduction techniques start off with a Hopf normal form description, thereby providing mathematical rigor. There, the caveat is to first derive the proper normal form. We explicate several ways to do that, both analytically and (semi-)numerically. (ii) Other analytic techniques capitalize on time scale separation and/or averaging over cyclic variables. While appealing for their more intuitive implementation, they often lack accuracy. (iii) Direct numerical approaches help to identify oscillatory behavior but may limit an overarching view how the reduced phase dynamics depends on model parameters. After illustrating and reviewing the necessary mathematical details for single oscillators, we turn to networks of coupled oscillators as the central issue of this report. We show in detail how the concepts of phase reduction for single oscillators can be extended and applied to oscillator networks. Again, we distinguish between numerical and analytic phase reduction techniques. As the latter dwell on a network normal form, we also discuss associated reduction methods. To illustrate benefits and pitfalls of the different phase reduction techniques, we apply them point-by-point to two classic examples: networks of Brusselators and a more elaborate model of coupled Wilson–Cowan oscillators. The reduction of complex oscillatory systems is crucial for numerical analyses but more so for analytical estimates and model prediction. The most common reduction is towards phase oscillator networks that have proven successful in describing not only the transition between incoherence and global synchronization, but also in predicting the existence of less trivial network states. Many of these predictions have been confirmed in experiments. As we show, however, the phase dynamics depends to large extent on the employed phase reduction technique. In view of current and future trends, we also provide an overview of various methods for augmented phase reduction as well as for phase–amplitude reduction. Weindicate how these techniques can be extended to oscillator networks and, hence, may allow for an improved derivation of the phase dynamics of coupled oscillators.