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Journal of Statistical Physics

, Volume 174, Issue 1, pp 188–218 | Cite as

Synchronization of Phase Oscillators on the Hierarchical Lattice

  • D. Garlaschelli
  • F. den Hollander
  • J. M. MeylahnEmail author
  • B. Zeegers
Article

Abstract

Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction between the oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.

Keywords

Hierarchical lattice Phase oscillators Noisy Kuramoto model Block communities Renormalization Universality classes 

Mathematics Subject Classification

60K35 60K37 82B20 82C27 82C28 

Notes

Acknowledgements

DG is supported by EU-Project 317532-MULTIPLEX. FdH and JM are supported by NWO Gravitation Grant 024.002.003–NETWORKS. The authors are grateful to G. Giacomin for critical remarks.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • D. Garlaschelli
    • 1
    • 2
  • F. den Hollander
    • 3
  • J. M. Meylahn
    • 3
    Email author
  • B. Zeegers
    • 3
  1. 1.IMT School of Advanced StudiesLuccaItaly
  2. 2.Lorentz Institute for Theoretical PhysicsLeiden UniversityLeidenThe Netherlands
  3. 3.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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