  We propose Local Dirac Synchronization which uses the Dirac operator to
capture the dynamics of coupled nodes and link signals on an arbitrary network.
In Local Dirac Synchronization, the harmonic modes of the dynamics oscillate
freely while the other modes are interacting non-linearly, leading to a
collectively synchronized state when the coupling constant of the model is
increased. Local Dirac Synchronization is characterized by discontinuous
transitions and the emergence of a rhythmic coherent phase. In this rhythmic
phase, one of the two complex order parameters oscillates in the complex plane
at a slow frequency (called emergent frequency) in the frame in which the
intrinsic frequencies have zero average. Our theoretical results obtained
within the annealed approximation are validated by extensive numerical results
on fully connected networks and sparse Poisson and scale-free networks. Local
Dirac Synchronization on both random and real networks, such as the connectome
of Caenorhabditis Elegans, reveals the interplay between topology (Betti
numbers and harmonic modes) and non-linear dynamics. This unveils how topology
might play a role in the onset of brain rhythms.
