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The hypothesis, that cortical dynamics operates near criticality alsosuggests, that it exhibits universal critical exponents which marks theKuramoto equation, a fundamental model for synchronization, as a primecandidate for an underlying universal model. Here, we determined thesynchronization behavior of this model by solving it numerically on a large,weighted human connectome network, containing 804092 nodes, in an assumedhomeostatic state. Since this graph has a topological dimension $d < 4$, a realsynchronization phase transition is not possible in the thermodynamic limit,still we could locate a transition between partially synchronized anddesynchronized states. At this crossover point we observe power-law--tailedsynchronization durations, with $\tau_t \simeq 1.2(1)$, away from experimentalvalues 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 wefound global coupling control-parameter dependent exponents $1 < \tau_t \le 2$,overlapping with the range of human brain experiments. We also studied theeffects of random flipping of a small portion of link weights, mimicking anetwork with inhibitory interactions, and found similar results. Thecontrol-parameter dependent exponent suggests extended dynamical criticalitybelow the transition point.

Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs