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 meanfield 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 meanfield 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 nonlinearity 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.
 Publication:

Journal of Statistical Physics
 Pub Date:
 January 2019
 DOI:
 10.1007/s1095501822085
 arXiv:
 arXiv:1703.02535
 Bibcode:
 2019JSP...174..188G
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 32 pages, 13 figures Updated paper analyzing an approximation rather than the full system in view of the error discovered previously