What do neurons, fireflies and dancing the Nutbush have in common?

Synchronised phenomena are all around us, whether it is human clapping and dancing, or the way fireflies flash, or how our neurons and heart cells interact. However, it is something not fully understood in engineering and science.

Associate Professor Joseph Lizier, expert in complex systems at the University of Sydney, said: "We know the feeling of dancing in step to the 'Nutbush' in a crowd -- or the awkward feeling when people lose time clapping to music. Similar processes occur in nature, and it is vital that we better understand how falling in and out of sync actually works.

"Being in sync in a system can be very good; you want your heart cells to all beat together rather than fibrillate. But being in sync can also be very bad; you don't want your brain cells to all fire together in an epileptic seizure."

Associate Professor Lizier and colleagues at the Max Planck Institute in Leipzig, Germany have published new research on synchronisation in the Proceedings of the National Academy of Sciences (PNAS).

The paper sets out the mathematics of how the network structure connecting a set of individual elements controls how well they can synchronise their activity. It is a critical insight into how these systems operate, because in most real-world systems, no one individual element controls all the others. And nor can any individual directly see and react to all the others: they are only connected through a network.

Associate Professor Lizier, from the Centre of Complex Systems and the School of Computer Science in the Faculty of Engineering, said: "Our results open new opportunities for designing network structures or interventions in networks. This could be super useful in stabilising electricity in power grids, vital for the transition to renewables, or to avoid neural synchronisation in the brain, which can trigger epilepsy."

To understand how these systems work, the researchers studied what are known as "walks" through a network in a complex system. Walks are sequences of connected hops between individual elements or nodes in the network.

Associate Professor Lizier said: "Our maths examines paired walks: where you start at one node and set off on two walks with randomly chosen hops between nodes for a specified number of steps. Those two walks might end up at the same node (convergent walks) or at different nodes (divergent walks).

"Our main finding is that the more commonly paired walks on a network are convergent, the worse the quality of synchronisation on that network structure would be."

This is good news for the brain, where synchronisation is not desirable as it can cause epilepsy . The brain's highly modular structure means it has a high proportion of convergent walks, which naturally push it away from epilepsy.

"We can even draw an analogy to social media with the echochamber phenomenon," said co-author Jürgen Jost, whose group also works on social network dynamics. "Here we see sub-groups reinforcing their own messages, via convergent walks within their own group, but not necessarily synchronising to the wider population."

The findings represent a major step forward in the theory of how the structure of complex networks affects their dynamics or how they compute, such as how brain structure underpins cognition.

The research was supported by the Australian Research Council Discovery Early Career Researcher Award (DECRA) grant DE160100630, the The University of Sydney, Sydney Research Accelerator (SOAR) award, the Alexander von Humboldt Foundation and the NSF Grant Division of Mathematical Sciences (DMS)-0804454 Differential Equations in Geometry. The research used the University of Sydney's high-performance computing cluster Artemis in generating the paper's results for this paper.

The researchers declare no competing interests.