# Synchronization of phase oscillators on the hierarchical lattice

Garlaschelli, Diego and den Hollander, Frank and Meylahn, Janusz and Zeegers, Benthen Synchronization of phase oscillators on the hierarchical lattice. Working Paper arXiv (Submitted)

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## 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 oscillator has a natural frequency, drawn independently from a common probability distribution. In addition, 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. Also these block communities are given a natural frequency, drawn independently from a common probability distribution that depends on their hierarchical scale. 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 show 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 identify three universality classes for the synchronization levels on successive hierarchical scales, with explicit characterizations in terms of the sequence of interaction parameters and the sequence of natural frequency probability distributions. We show that disorder reduces synchronization when the natural frequency probability distributions are symmetric and unimodal, with the reduction gradually vanishing as the hierarchical scale goes up.

Item Type: Working Paper (Working Paper) arXiv:1703.02535 Networks Q Science > QC Physics Economics and Institutional Change Caterina Tangheroni 09 Mar 2018 13:18 09 Mar 2018 13:18 http://eprints.imtlucca.it/id/eprint/3999