  Many biological and neural systems can be seen as networks of interacting
periodic processes. Importantly, their functionality depends on the emerging
collective dynamics of the network. Synchrony of oscillations is one of the
most prominent examples of such collective behavior and has been associated
both with function and dysfunction. Understanding how network structure and
interactions, as well as the microscopic properties of individual units, shape
the emerging collective dynamics is critical to find factors that lead to
malfunction. However, many biological systems such as the brain consist of a
large number of dynamical units. Hence, their analysis has either relied on
simplified heuristic models on a coarse scale, or the analysis comes at a huge
computational cost. Here we review recently introduced approaches, known as the
Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the
analysis by bridging small and large scales. Thus, reduced model equations are
obtained that exactly describe the collective dynamics for each subpopulation
in the oscillator network via few collective variables only. The resulting
equations are next-generation models: Rather than being heuristic, they exactly
link microscopic and macroscopic descriptions and therefore accurately capture
microscopic properties of the underlying system. At the same time, they are
sufficiently simple to analyze without great computational effort. In the last
decade, these reduction methods have become instrumental in understanding how
network structure and interactions shape the collective dynamics and the
emergence of synchrony. We review this progress based on concrete examples and
outline possible limitations. Finally, we discuss how linking the reduced
models with experimental data can guide the way towards the development of new
treatment approaches, for example, for neurological disease.
