  The hypothesis, that cortical dynamics operates near criticality also
suggests, that it exhibits universal critical exponents which marks the
Kuramoto equation, a fundamental model for synchronization, as a prime
candidate for an underlying universal model. Here, we determined the
synchronization behavior of this model by solving it numerically on a large,
weighted human connectome network, containing 804092 nodes, in an assumed
homeostatic state. Since this graph has a topological dimension $d < 4$, a real
synchronization phase transition is not possible in the thermodynamic limit,
still we could locate a transition between partially synchronized and
desynchronized states. At this crossover point we observe power-law--tailed
synchronization durations, with $\tau_t \simeq 1.2(1)$, away from experimental
values for the brain. For comparison, on a large two-dimensional lattice,
having additional random, long-range links, we obtain a mean-field value:
$\tau_t \simeq 1.6(1)$. However, below the transition of the connectome we
found global coupling control-parameter dependent exponents $1 < \tau_t \le 2$,
overlapping with the range of human brain experiments. We also studied the
effects of random flipping of a small portion of link weights, mimicking a
network with inhibitory interactions, and found similar results. The
control-parameter dependent exponent suggests extended dynamical criticality
below the transition point.

Disordered Systems and Neural Networks;Statistical Mechanics;Biological Physics;Neurons and Cognition