Synchronization measures of bursting data: Application to the electrocorticogram of an auditory event-related experiment

Synchronization measures have become an important tool for exploring the relationships between time series. We review three recently proposed nonlinear synchronization measures and expand their definitions in a straightforward way to apply to an ensemble of measurements. We also develop a synchronization measure in which nearest neighbors are determined across the ensemble. We compare these four nonlinear synchronization measures and show that our measure succeeds in physically motivated examples where the other methods fail. We apply the synchronization measure to human electrocorticogram data collected during an auditory event-related potential experiment. The results suggest a crude model of cortical connectivity.

Figure 1

(a) The electric potential of one ensemble member recorded by one electrode in the ECoG ERP experiment. The stimulus occurs at $t=0\mathrm{ms}$. Note the oscillatory burst between $30\mathrm{ms}$ and $160\mathrm{ms}$. The asterisk marks a point on the oscillatory burst. (b) The embedding of the times series in (a). We chose $d=3$ and $\tau =10$ for illustrative purposes. The asterisk in this figure corresponds to the asterisk in (a). (c) The local neighborhood of the point marked with an asterisk in (b). The five nearest neighbors to this point are marked by triangles. The nearest neighbors are temporally proximal to the fiducial point because the data set is short.

Figure 2

(a) Ten ensemble members embedded using $d=3$ and $\tau =10$. The thickest curve is the ensemble member shown in Figs. 1a, 1b. The point ${\mathbf{x}}^k\left[n^{*}\right]$ is marked with an asterisk. (b) The local neighborhood of the point ${\mathbf{x}}^k\left[n^{*}\right]$. The thickest curve is the trajectory of ${\mathbf{x}}^k\left[n\right]$. The point ${\mathbf{x}}^k\left[n^{*}\right]$ is marked with an asterisk. The thin curves are trajectories of nine other ensemble members. The nearest ensemble neighbors are marked with triangles.

Figure 3

Synchronization measures applied to the unidirectionally coupled nonidentical Henon map. (a) Three synchronization measures: $S\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (dashed line), $H\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (dotted line), and $N\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (solid line). All of the measures are smoothed over a window of size $11$ at each time point. All three measures increase during the interval of nonlinear coupling $(100<n<150)$ between the chaotic time series. (b) The time-shifted synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ smoothed over a two-dimensional window of size $11$ at each time point. Note that the planar axes show time along ensembles $\mathbf{y}$ and $\mathbf{x}$, respectively. In the contour plot, there are five evenly spaced contour levels, ranging from 0.0 (white) to 0.08 (black). Unless defined otherwise, all $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ figures follow this gray-scale scheme. The diagonal line in the figure corresponds to the location of zero time lag. The contour plot shows synchronization occurs with time shift $\eta =0$ during the time interval $100<n<150$.

Figure 4

Nonlinear synchronization measures applied to oscillatory bursting data versus noisy data. (a) A typical ensemble member from the oscillatory bursting data $s^k\left[n\right]$ (solid line) and the noisy data $r^k\left[n\right]$ (dashed line). The weak oscillatory response in $s^k\left[n\right]$ between $100\mathrm{ms}$ and $150\mathrm{ms}$ is hidden in the noise. (b) The ensemble-averaged ERPs of ensemble $s$ (solid) and ensemble $r$ (dashed). The oscillatory response in ensemble $s$ for $100\mathrm{ms}<n<150\mathrm{ms}$ is apparent. (c) Three synchronization measures: $S\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (dashed line), $H\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (dotted line), and $N\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (solid line). All three measures are smoothed over a window of size $11\mathrm{ms}$ at each time point. Both $H\left(\mathbf{x}\right[n]\mid \mathbf{y})$ and $N\left(\mathbf{x}\right[n]\mid \mathbf{y})$ fluctuate between $0.2$ and $0.55$ over the entire time interval and suggest no obvious synchronization between the ensembles, as expected. $S\left(\mathbf{x}\right[n]\mid \mathbf{y})$ increases during the interval $100\mathrm{ms}<n<150\mathrm{ms}$ and therefore suggests an increased synchronization between the ensembles during this interval. This incorrect interpretation is a consequence of the increase in $R\left({\mathbf{x}}^k\right[n\left]\right)$ during the oscillatory burst, as explained in the text.

Figure 5

(a) A typical ensemble member pair from the oscillatory bursting data: $s^k\left[n\right]$ (solid line) and $r^k\left[n\right]$ (dashed line). The weak oscillatory responses of both time series are mostly hidden in the noise. (b) The ensemble-averaged ERPs of $s^k\left[n\right]$ (solid line) and $r^k\left[n\right]$ (dashed line). The oscillatory bursts, hidden in the pair of (a), are revealed here in the ensemble averaged ERPs. (c) Two synchronization measures: $H\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (dotted line) and $N\left(\mathbf{x}\right[n]\mid \mathbf{y})$ (solid line), smoothed over a window of size $11\mathrm{ms}$ at each time point. Neither $H\left(\mathbf{x}\right[n]\mid \mathbf{y})$ nor $N\left(\mathbf{x}\right[n]\mid \mathbf{y})$ accurately captures the synchronization between the two ensemble for $100\mathrm{ms}<n<150\mathrm{ms}$.

Figure 6

The time-shifted synchronization $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to the ensemble of oscillatory bursting data of Fig. 5. $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ was smoothed over a two-dimensional window of size $11$ at each time point. The solid diagonal line corresponds to the location of zero time lag. This measure detects the synchronization between the two ensembles for $100\mathrm{ms}<n<150\mathrm{ms}$.

Figure 7

The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to electrodes $A$ and $B$ from the ECoG ERP experimental data. $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ was smoothed over a two-dimensional window of size $11$ at each time point. The solid diagonal line corresponds to the location of zero time lag. The horizontal and vertical dashed lines correspond to the time of stimulus onset. For this figure there are ten evenly spaced contour levels from 0.01 (white) to 0.19 (black). Note the region of strong synchronization from $40\mathrm{ms}<n<130\mathrm{ms}$ in $A$ and $60\mathrm{ms}<n<110\mathrm{ms}$ in $B$.

Figure 8

The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to electrodes from the ECoG ERP experimental data. In both figures, $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ was smoothed over a two-dimensional window of size $11$ at each time point. The solid diagonal line in both figures corresponds to the location of zero time lag. The horizontal and vertical dashed lines in both figures correspond to the time of stimulus onset. For both figures there are ten evenly spaced contour levels from 0.01 (white) to 0.19 (black). (a) The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to electrodes $C$ and $A$ from the ECoG ERP experimental data. Note that the synchronization is elongated in $C$ and is weaker than the synchronization between electrodes $A$ and $B$. (b) The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to electrodes $C$ and $B$ from the ECoG ERP experimental data. Note that the synchronization is weaker than the synchronization between the other electrode pairs.

Figure 9

The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to shifted simulated data. In both figures, $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ was smoothed over a two-dimensional window of size $11$ at each time point. The solid diagonal line in both figures corresponds to the location of zero time lag. For both figures there are ten evenly spaced contour levels from 0.01 (white) to 0.19 (black). (a) The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to uniformly shifted simulated data. The ensembles $s$ and $r$ consist of oscillatory bursts with the same center time. The oscillatory bursts in each ensemble member pair $s^k\left[n\right]$ and $r^k\left[n\right]$ are shifted in time by the same amount, up to $\pm 20\mathrm{ms}$, and in the same direction. (b) The synchronization measure $T\left(\mathbf{x}\right[n,\eta ]\mid \mathbf{y})$ applied to randomly shifted simulated data. The ensembles $s$ and $r$ consist of oscillatory bursts with the same center time. The oscillatory bursts in each ensemble member pair $s^k\left[n\right]$ and $r^k\left[n\right]$ are shifted in time by different amounts, up to $\pm 10\mathrm{ms}$, and in different directions.