A global bifurcation organizing rhythmic activity in a coupled network

What is it about?

Many important systems in science and technology are modeled by large networks of interacting dynamical systems. Our ability to predict epileptic seizures, to effectively control a power grid, or to coordinate a group of robots rely on our understanding of the principles underlying collective behavior in coupled networks. The Kuramoto model of coupled phase oscillators provides a framework for studying synchronization, an important mode of collective dynamics in large networks. Motivated by applications in neuroscience, we modified the Kuramoto model by placing the individual oscillators near excitability threshold and studied the onset of synchronization in the modified model. We show that at the heart of the transition to synchronization lies a novel heteroclinic bifurcation, which separates two distinct types of collective oscillations.

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Photo by Hal Gatewood on Unsplash

Photo by Hal Gatewood on Unsplash

Why is it important?

A bifurcation represents a qualitative change of the state of a dynamical system. In modeling, it may indicate a critical event such as the onset of a pathological state in the brain, a blackout in a power grid, or a climate change depending on the model at hand. The boa constrictor bifurcation analyzed in this work is a new addition to the list of heteroclinic bifurcations, which play an important role in many applications.