G.B. acknowledges funding from the Alan Turing Institute and from Royal Society IEC
\NSFC\191147. J.J.T. acknowledges financial support from the Consejería de Transformación
Económica, Industria, Conocimiento y Universidades, Junta de Andalucía and
European Regional Development Funds, Ref. P20_00173. This work is also part of the
Project of I+D+i Ref. PID2020-113681GB-I00, financed by MICIN/AEI/10.13039/
501100011033 and FEDER “A way to make Europe". This research utilized Queen Mary’s Apocrita HPC facility, supported by QMUL Research-IT. https://doi.org/10.5281/zenodo.
438045.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.Alan Turing Institute and from Royal Society IEC \NSFC\191147. J.J.TConsejería de Transformación Económica, Industria, Conocimiento y Universidades, Junta de Andalucía and European Regional Development Funds, Ref. P20_00173Project of I+D+i Ref. PID2020-113681GB-I00, financed by MICIN/AEI/10.13039/ 501100011033 and FEDER “A way to make Europe"QMUL Research-I
