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Local Dirac Synchronization on networks.

Lucille Calmon1, Sanjukta Krishnagopal2, Ginestra Bianconi1

  • 1School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.

Chaos (Woodbury, N.Y.)
|April 1, 2023
PubMed
Summary
This summary is machine-generated.

We introduce Local Dirac Synchronization, a method using the Dirac operator to model coupled network dynamics. This approach reveals how network topology influences the emergence of synchronized brain rhythms.

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Area of Science:

  • Complex systems
  • Network science
  • Dynamical systems

Background:

  • Coupled dynamical systems exhibit complex behaviors, including synchronization.
  • Understanding the role of network topology in emergent dynamics is crucial.
  • The Dirac operator offers a novel framework for analyzing network dynamics.

Purpose of the Study:

  • To propose and analyze a new synchronization mechanism, Local Dirac Synchronization (LDS).
  • To investigate the influence of network topology on synchronization phenomena.
  • To explore the potential connection between network structure and the emergence of brain rhythms.

Main Methods:

  • Utilizing the Dirac operator to model coupled node and link dynamics.
  • Analyzing harmonic and non-linear modes of the system.
  • Employing theoretical analysis within the annealed approximation.
  • Validating results through numerical simulations on various network types (fully connected, sparse, scale-free, real-world).

Main Results:

  • LDS exhibits discontinuous transitions and a rhythmic coherent phase with increasing coupling.
  • A slow emergent frequency arises in the synchronized state.
  • Theoretical predictions are confirmed by numerical simulations across diverse networks.
  • The interplay between network topology (Betti numbers, harmonic modes) and non-linear dynamics is demonstrated.

Conclusions:

  • Local Dirac Synchronization provides a framework for understanding collective synchronization in networks.
  • Network topology significantly influences the onset and characteristics of synchronization.
  • This work suggests a potential mechanism for how brain structure contributes to brain rhythms.