The time dependent behavior of the Kuramoto model, describing synchronization, has been studied numerically on small-world graphs. We determined the desynchronziation behavior , by solving this model via the 4th order Runge-Kutta algorithm on a large, weighted human connectome network and compared the results with those of a two-dimensional lattice, with additional random, long-range links. In the latter case a mean-field critical transition is expected and here we provide numerical results for the synchonization/desynchonization duration distributions. We find power-law tails, characterized by a critical exponent \tau_d \simeq 1.6(1). In case of the connectome we assumed a homeostatic state, by the application of normalized incoming weights . 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 desynchronization durations, with \tau_t \simeq 1.2(1), away from experimental values for the brain. Additionally, we changed the signs of outgoing weights of 20%DIFDELCMD < \\%%% %DIF < of randomly selected nodes, to mimic a model with inhibitory interactions. In this case the at the crossover point we found \tau_t \simeq 1.9(2), which is in the range of human brain experiments.abstract \le 
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. %DIFDELCMD < \\%%% %DIF < of randomly selected nodes, to mimic a model with inhibitory interactions. In this case the at the crossover point we found \tau_t \simeq 1.9(2), which is in the range of human brain experiments.abstract 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.
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time dependent behavior of the Kuramoto model, describing synchronization, has been studied numerically on small-world graphs. We determined the desynchronziation behavior , by solving this model via the 4th order Runge-Kutta algorithm
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
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and compared the results with those of a two-dimensional lattice, with additional random, long-range links. In the latter case a mean-field critical transition is expected and here we provide numerical results for the synchonization/desynchonization duration distributions. We find power-law tails, characterized by a critical exponent \tau_d \simeq 1.6(1). In case of the connectome we assumed a homeostatic state, by the application of normalized incoming weights
, containing 804092 nodes, in an assumed homeostatic state
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desynchronization
synchronization
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Additionally, we changed the signs of outgoing weights of 20
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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
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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.