Complex phase synchronization in epileptic seizures: Evidence for a devil’s staircase

We describe multifrequency phase synchronization in epileptic seizures. Using magnetoencephalographic recordings from three patients suffering generalized seizures, the evidence is presented that, in addition to the commonly studied 1:1 frequency locking, there exists complex multifrequency coordination that, in some cases, follows a classical “devil’s staircase.” Within the limitations of observing this phenomenon in a clinical experimental setting, these observations reveal that in pathological brain activity, complex frequency locking can be found similar to that identified in certain pathological cardiac re-entrant arrhythmias. This may suggest the existence of similar re-entrant mechanisms active in cerebral neocortex during epileptic seizures.

Figure 1

(Color online) (a) Original MEG trace corresponding to an absence seizure (red in color appears as light gray in grayscale) and one surrogate obtained as described in the text (blue in color appears as dark gray in grayscale). (b) Detail of the absence seizure recorded at one MEG sensor. Note the $3–4\mathrm{Hz}$ spike-and-wave and the higher frequency components embedded in the main oscillation.

Figure 2

(Color online) Phase difference and synchrony indices $\left(R\right)$ between two cortical areas, corresponding to MEG sensors RC32 (located over the right central neocortex) and RF11 (right frontal) for the upper plots, and LC14 (left central) and LC15 (left central, adjacent to the previous LC14) for the lower plots. The time of the seizure (which lasted approximately $12\text{seconds}$, detail shown in Fig. 1) is marked by arrows. Upper plots show the phase synchrony for 1:1 frequency locking, and the lower plots correspond to a 3:4 locking. Central frequency was $5\mathrm{Hz}$. In the phase difference plots, phase locked modes are visualized as horizontal lines.

Figure 3

(Color online) Similar representations as in Fig. 2, showing 1:1 and 3:4 locking for another recording in a different patient. The seizure, in this case, occurs between the 94th and 108th second (marked by arrows). No 1:2 phase locking was detected in this case. Also shown are the $R$ values for the synchronization between two surrogates (“surrogates 1:1”) corresponding to the original signals in the other plots. Note how low the $R$ values are, compared with the original 1:1 graph. The inset on the first panel shows the color-coded 1:1 synchronization for this $2\text{−}\min$ recording (red in color appears as light gray in grayscale is maximum, blue in color appears as dark gray in grayscale is minimum), to show that, about $1.5\min$ before the seizure, there is a period of time with high synchrony. This inset represents the average of all MEG channels (on the $y$ axis); the high synchrony in this time period before the seizure can also be seen in the 1:1 plot for the two specific channels shown in the figure, by the red line segments at $\sim 20–40\sec$, both on the phase difference time series (flat red segments) and the $R$ values (red peaks).

Figure 4

Higher-order synchronization plateaus found during a seizure in patient 3 (left-hand side plot), and patient 2 (right-hand side), the MEG sensors and central frequencies are indicated at the top of the graphs. Sensor LP13 records activity in left central neocortex, and LF51 and LF52 in left frontal lobe. The $x$ axis is the duration (in seconds) of the phase locking region derived from plots like those shown in Figs. 2, 3. Notice that low-order ratios tend to be of longer duration than more complex ones. If, instead of time, a control parameter could be identified and used for the $x$ axis, these plateaus would form a devil’s staircase structure. Lower graph shows the time duration of several phase locking ratios, for two MEG sensors during the absence seizure in patient 3. Sensors RC32 and RF11 record activity in right central and right frontal cortex, respectively. Note here the Arnold tonguelike structure, and again, how the low-order ratios last longer than higher-order ones.