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

Abstract

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 836733 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 τt ≃ 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: τt ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 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.

Pattern Recognition in Epileptic EEG Signals via Dynamic Mode Decomposition

“…The phase angle in the third column in Figure 2 for frequencies whose magnitude is zero is ignored because it is worthless. The absolute value of the phase angle over 7 Hz of the first signal is calculated to be approximately 2.8723 and that of the second signal is The Kuramoto model, which explains the synchronization phenomena of the oscillation of coupled signals, analyzes brain connectivity and dynamics [11]. This synchronization phenomenon can be confirmed by comparing the phase difference of a specific frequency band between interacting signals.…”

“…However, the magnitude of the DMD mode allows you to visually identify specific frequency bands with abnormal phase transitions whether or not it is in the high frequency. The Kuramoto model, which explains the synchronization phenomena of the oscillation of coupled signals, analyzes brain connectivity and dynamics [11]. This synchronization phenomenon can be confirmed by comparing the phase difference of a specific frequency band between interacting signals.…”

“…where c i is the ith entry of c given in Equation (11). Meanwhile, the DMD mode power is defined by φ i 2 2 [14,20].…”

“…On the other hand, in FFT, the phase difference between these two channels cannot be confirmed. The first column in Figure 2 shows the phase angle of the DMD mode obtained from the computation of ℑ(log ), where is the DMD mode in Equation (11) and ℑ(⋅) is the imaginary part of a complex number (Note that…”

“…On the other hand, in FFT, the phase difference between these two channels cannot be confirmed. The first column in Figure 2 shows the phase angle of the DMD mode obtained from the computation of (log Φ), where Φ is the DMD mode in Equation (11) and (•) is the imaginary part of a complex number (Note that x(t) = φ • e iθt = e i(θt+ (log φ))+ (log φ) , where (•) is the real part of a complex number.). The second column (b) in Figure 2 displays the magnitude of the mode computed by |Φ|.…”