Dirac synchronization is rhythmic and explosive

Lucille Calmon

doi:10.1038/s42005-022-01024-9

Dirac synchronization is rhythmic and explosive

2022-10-17

20

OA

AI

DOI：10.1038/s42005-022-01024-9

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En 中文

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. Topological signals are dynamical variables that can be associated to nodes, links, triangles, etc. Here, the authors formulate Dirac synchronization that uses the Dirac operator to couple locally topological signals defined on nodes and links and show, numerically and analytically, that this gives rise to an explosive synchronization transition and to a coherent rhythmic phase.

Keyword:

PHASE-TRANSITIONS

KURAMOTO

COMMUNICATION

OSCILLATIONS

DYNAMICS

SYSTEMS

DELAY

MODEL

Dirac synchronization is rhythmic and explosive

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