Power-law distribution of phase-locking intervals does not imply critical interaction

Neural synchronization plays a critical role in information processing, storage, and transmission. Characterizing the pattern of synchronization is therefore of great interest. It has recently been suggested that the brain displays broadband criticality based on two measures of synchronization, phase-locking intervals and global lability of synchronization, showing power-law statistics at the critical threshold in a classical model of synchronization. In this paper, we provide evidence that, within the limits of the model selection approach used to ascertain the presence of power-law statistics, the pooling of pairwise phase-locking intervals from a noncritically interacting system can produce a distribution that is similarly assessed as being power law. In contrast, the global lability of synchronization measure is shown to better discriminate critical from noncritical interaction.

Figure 1

Plot (a) shows the evolution of order parameter $r$ for the Kuramoto model with cyan solid circles (error bars show standard deviations). The coupling parameter $K$ increases along the $x$ axis. The hollow purple diamonds show $\Delta Kr$, the change in order parameter multiplied by coupling (error bars not shown for readability). The time-averaged number of synchronized pairs $N_{SP}$ is shown with hollow green squares (error bars show standard deviations), and the difference in $N_{SP}$, $\Delta N_{SP}$, is indicated by solid blue triangles (error bars not shown for readability). The peaks in $\Delta Kr$ and $\Delta N_{SP}$ can be used to indicate the location of the critical point for a specific system, which for this selection of natural frequencies occurs at around $K=2$. This effective coupling value of $K=2$ will be used throughout the paper. Note that, for the Kuramoto model, the order increases with rising coupling. Plot (b) displays the corresponding measures $r$, $\Delta Kr$, $N_{SP}$, and $\Delta N_{SP}$ for the independent pairs model. There is no change in order parameter with coupling, indicating that the oscillators are not critically coupled to a mean field.

Figure 2

The evolution of phase difference between the oscillators in a two-oscillator Kuramoto system, plotted using our analytic expression (blue), and a simulation of the Kuramoto model by Euler's method (red). The two phase calculations are perfectly superimposed. The root mean square error (RMSE) is shown for different coupling values, for a single simulation. Panels (a), (b), (c) have $C<1$ (where $C$ is defined in Sec. 2b), but coupling is increased progressively. The phase evolves periodically. Panel (d) is the same pair of oscillators, but for $C>1$. There is a brief transient before the oscillators fully synchronize with a constant level of phase difference. The initial phase separation has been set to $△=0$ without loss of generality.

Figure 3

Power spectra for a system of 44 Kuramoto oscillators, with natural frequencies drawn from a $\mathcal{N}\left(60\pi ,20\pi \right)$ distribution and three distinct levels of coupling: (a) $K=0$, (b) $K=2$, the effective critical coupling for this specific finite Kuramoto system, as seen from Fig. 1a, and (c) $K=4$. The vertical numbered lines represent wavelet scales $3\text{--}11$.

Figure 4

Distribution of PLIs in a system of 44 Kuramoto oscillators, with natural frequencies drawn from a $\mathcal{N}\left(0,1\right)$ distribution and four levels of coupling: $K=0$, $K=K_c\simeq 1.6$, $K=2$, and $K=4$ (from top left, clockwise). A power law of exponent $-2$ is shown by a dotted black line. The colored lines represent wavelet scales $3\text{--}11$ (see key).

Figure 5

Distribution of GLS in a system of 44 Kuramoto oscillators, with natural frequencies drawn from a $\mathcal{N}\left(0,1\right)$ distribution and four levels of coupling: $K=0$, $K=K_c\simeq 1.6$, $K=2$, and $K=4$ (from top left, clockwise). A power law of exponent $-1$ is shown by a dotted black line. The colored lines represent wavelet scales $3\text{--}11$ (see key).

Figure 6

Distribution of PLIs in the independent pairs model, with natural frequencies drawn from a $\mathcal{N}\left(0,1\right)$ distribution and four levels of coupling: $K=0$, $K=K_c\simeq 1.6$, $K=2$, and $K=4$ (from top left, clockwise). A power law of exponent $-2$ is shown by a dotted black line. The colored lines represent wavelet scales $3\text{--}11$ (see key).

Figure 7

Distribution of GLS in the independent pairs model, with natural frequencies drawn from a $\mathcal{N}\left(0,1\right)$ distribution and four levels of coupling: $K=0$, $K=K_c\simeq 1.6$, $K=2$, and $K=4$ (from top left, clockwise). A power law of exponent $-1$ is shown by a dotted black line. The colored lines represent wavelet scales $3\text{--}11$ (see key).