Understanding the dynamics of biological and neural oscillator networks through mean-field reductions: a review

Abstract

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, the function of these networks depends on the collective dynamics: Synchrony of oscillations is probably amongst the most prominent examples of 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 emergent 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 primarily relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches - commonly known as the Ott-Antonsen and Watanabe-Strogatz reductions - that allow to simplify the analysis by bridging small and large scales: To obtain reduced model equations, a subpopulation in an oscillator network is replaced by a single variable that describes its collective state exactly. The resulting equations are next-generation models: Rather than being heuristic, they 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 to understand how network structure and interaction shapes 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 neural disease.