Dirac synchronization is rhythmic and explosive

TL;DR

This paper introduces Dirac synchronization, a topological signal synchronization on networks that exhibits explosive transitions and rhythmic phases, providing insights into brain rhythms and topological dynamics.

Contribution

It presents a novel model of topological signals on networks using the Dirac operator, revealing explosive synchronization and rhythmic phases.

Findings

Dirac synchronization is explosive with hysteresis.

The model shows a rhythmic phase with non-stationary order parameters.

Analytical phase diagram explains topological explosive synchronization.

Abstract

Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate Dirac synchronization, describing locally coupled topological signals defined on the nodes and on the links of a network, and treated using the topological Dirac operator. The dynamics of signals defined on the nodes is affected by a phase lag depending on the dynamical state of nearby links and vice versa. We show that Dirac synchronization on a fully connected network is explosive with a hysteresis loop characterized by a discontinuous forward transition and a continuous backward transition. The analytical investigation of the phase diagram provides a theoretical understanding of this topological explosive synchronization. The model also displays an exotic coherent synchronized phase, also called rhythmic phase, characterized by non-stationary order parameters which can shed light on topological mechanisms for the emergence of brain rhythms.