Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, 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.
Highlights
Many systems in neuroscience and biology are governed on different levels by interacting periodic processes [1]
To illustrate the mean-field reductions and their applicability, we focus here on networks that are organized into distinctpopulations because of their practical importance.b The mean-field reductions allow one to replace each subnetwork by a set of collective variables to obtain a set of dynamical equations for these variables
The section sets the stage by introducing the notion of a sinusoidally coupled network and we summarize the main oscillator models we relate to throughout the paper; these include the Kuramoto model and networks of Theta neurons (which are equivalent to Quadratic Integrate and Fire (QIF) neurons)
Summary
Many systems in neuroscience and biology are governed on different levels by interacting periodic processes [1]. The exact mean-field reductions we discuss here, the Ott–Antonsen reduction and the Watanabe–Strogatz reduction, can be employed for infinite networks for networks of finitely many oscillators While these reductions only apply to specific classes of systems—and from a mathematical perspective reflect the special structure of these systems—they include models that have been widely used in neuroscience and beyond, such as the Kuramoto model. We apply the reductions and emphasize how they are useful to understand how synchrony and patterns of synchrony emerge in such oscillator networks This includes a number of concrete examples. We conclude with some remarks and highlighting a number of open problems
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