Combinatorics

Physics

Graph

Neuroscience

Topology (electrical circuits)

Functional connectivity

Mathematics

Connectome

Random graph

Biology

Statistical physics

Synchronization networks

Phase transition

Criticality

Mathematical analysis

Philosophy

Hausdorff dimension

Lebesgue covering dimension

Acoustics

Nuclear physics

Critical point (mathematics)

Condensed matter physics

Kuramoto model

Exponent

Lattice (music)

Synchronization (alternating current)

Linguistics

Complex network

Scientific Reports

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.

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