Hierarchical abstraction of weakly coupled synchronized oscillator networks

We provide a constructive and numerically implementable proof that synchronized groups of coupled, self-sustaining oscillators can be represented as a single effective Perturbation Projection Vector (PPV) (or Phase Response Curve) phase macromodel - in other words, that a group of synchronized oscillators behaves as a single effective oscillator with respect to external influences. This result constitutes a foundation for understanding and predicting synchronization/timing hierarchically in large, complex systems that arise in nature and engineering. We apply this result hierarchically to networks of synchronized oscillators, thereby enabling their efficient and scalable analysis. We illustrate our theory and numerical methods with examples from electronics (networks of three-stage ring oscillators), biology (Fitzhugh-Nagumo neurons) and mechanics (pendulum clocks). Our experiments demonstrate that effective PPVs extracted hierarchically can capture emergent phenomena, such as pattern formation, in coupled oscillator networks. Copyright (c) 2015 John Wiley & Sons, Ltd.