# Synchronization of Phase Oscillators on the Hierarchical Lattice

## 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.

## References

- 1.Acebrón, J.A., Bonilla, L.L., Pérez Vicente, C.J., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys.
**77**, 137–185 (2005)ADSCrossRefGoogle Scholar - 2.Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep.
**469**, 93–153 (2008)ADSMathSciNetCrossRefGoogle Scholar - 3.Bertini, L., Giacomin, G., Poquet, C.: Synchronization and random long time dynamics for mean-field plane rotators. Probab. Theory Relat. Fields
**160**, 593–653 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Dahms, R.: Long-time behavior of a spherical mean field model. PhD thesis at Technical University Berlin (2002) (unpublished)Google Scholar
- 5.Pra, P.Dai, Hollander, F.den: McKean–Vlasov limit for interacting random processes in random media. J. Stat. Phys.
**84**, 735–772 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 6.Ha, S., Slemrod, M.: A fast-slow dynamical systems theory for the Kuramoto type phase model. J. Differ. Equ.
**251**, 2685–2695 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 7.Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
- 8.Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics, pp. 420–422. Lecture Notes in Physics 39, Springer, Berlin (1975)Google Scholar
- 9.Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
- 10.Laforgia, A., Natalini, P.: Some inequalities for modified bessel functions, J. Ineq. App. 2010, article 253035, 1–10 (2010)Google Scholar
- 11.Luçon, E.: Oscillateurs couplés, désordre et renormalization. PhD thesis, Université Pierre et Marie Curie-Paris VI (2012)Google Scholar
- 12.Luçon, E., Poquet, C.: Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model. Ann. Inst. Henri Poincaré Probab. Stat.
**53**, 1196–1240 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Sakaguchi, H.: Cooperative phenomena in coupled oscillators systems under external fields. Prog. Theor. Phys.
**79**, 39–46 (1988)ADSMathSciNetCrossRefGoogle Scholar - 14.Segura, J.: Bounds for ratios of modified Bessel functions and associated Turin-type inequalities. J. Math. Anal. Appl.
**372**, 516–528 (2011)CrossRefzbMATHGoogle Scholar - 15.Sonnenschein, B., Schimansky-Geier, L.: Approximate solution to the stochastic Kuramoto model. Phys. Rev. E
**88**(052111), 1–5 (2013)Google Scholar - 16.Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D
**143**, 1–20 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Strogatz, S.H., Mirollo, R.E.: Stability of incoherence in a population of coupled oscillators. J. Stat. Phys.
**63**, 613–635 (1991)ADSMathSciNetCrossRefGoogle Scholar - 18.Strogatz, S.H., Mirollo, R.E., Matthews, P.C.: Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping. Phys. Rev. Lett.
**68**, 2730–2733 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 19.Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol.
**16**, 15–42 (1967)CrossRefGoogle Scholar - 20.Winfree, A.T.: The Geometry of Biological Time. Springer, New York (1980)CrossRefzbMATHGoogle Scholar