Unlike the static structural networks based on anatomical connections between brain areas (Hilgetag and Kaiser, 2004
; Sporns and Kötter, 2004
; Bullmore and Sporns, 2009
), we examine here dynamic, functional networks deduced from ECoG data. We determine these networks using a simple association measure: the cross correlation. We choose this measure for two main reasons: simple linear and sophisticated nonlinear measures appear to perform equally well when applied to ECoG data (Mormann et al., 2005
; Ansari-Asl et al., 2006
; Kreuz et al., 2007
; Osterhage et al., 2007
), and for the cross correlation we can derive an analytic and computationally efficient significance test (Kramer et al., 2009
). Applying a frequency domain measure — the coherence — to the data produces similar results (Supplementary Material
To construct functional networks from the ECoG data, we implement the following procedure. First we divide the ECoG data into ~1 s windows (with ~0.5 s overlap) beginning 120 s before seizure onset and ending after seizure termination. We choose this window size to preserve weak stationary in the data, but did find similar results with different window sizes and overlaps (Supplementary Material
). We then compute the cross correlation — and test the significance — between all electrode pairs for the ECoG data within each window. Finally, we threshold the results of the significance tests to construct a functional network with an associated measure of uncertainty. A detailed discussion of the correlation measure and appropriate statistical tests may be found in (Kramer et al., 2009
). We analyze these networks to show that while some properties (such as the overall level of network synchronization) are preserved during the seizure, others change dramatically.
Network synchronization decreases — while signal energy increases — during seizure progression
In we show example networks constructed from the ECoG data. While the data can be displayed on a 3-dimensional reconstruction of the cortical surface () it is more useful to show these data as a circular network; each subject had electrodes which entered the cortical surface and therefore remain hidden in the cortical surface reconstructions. Accordingly, to observe the connectivity of the entire network, we arrange the electrodes in a ring and connect electrodes exhibiting significant coupling with an edge (i.e., a directionless link between nodes). suggest an obvious way in which the network topologies change in time; there is a dramatic variation in the number of edges (a movie illustrating this variation in another subject is provided in the Supplementary Material
). In this example, at seizure onset and termination the networks possess many more edges (i.e., become more synchronized) than during the middle portion of the seizure. To quantify the changing number of edges observed, we compute the density of each network. This measure ranges from 0 (a desynchronized network containing no edges) to 1 (a hypersynchronized network with edges connecting all possible pairs of nodes). In this example, the density increases briefly just after ictal onset, decreases to preictal values during the ictus itself, and again increases before seizure termination (). For comparison, we also show the simultaneous ECoG activity recorded at a single electrode, and for three, two -second intervals from multiple electrodes for this seizure (). Surprisingly, the large amplitude voltage oscillations characteristic of a seizure occur after the density returns to preictal levels.
To quantify the changing voltage dynamics and network synchronization during the seizure, we analyze an ensemble of 48 seizures collected from 11 patients. We first normalize time for each seizure by dividing the ictal period into ten intervals of equal length. In doing so, we assume that seizures undergo characteristic progressions that can be stretched or compressed in time; repeating the analysis with time intervals of fixed duration and different seizure lengths produces similar results (Supplementary Figure 2
). We then compute the density for all subjects and seizures within preictal, ictal, and postictal intervals and plot the average density per interval adjusted for differences in subjects with a blocked one-way ANOVA (). Only after ictal onset (interval label I1) and before termination (I9, I10) do we observe a significant increase in density above preictal levels during the seizure (see Methods). During the middle portion of the seizure (intervals I2-I8) the density returns to preictal levels. We find similar changes in density for a subset of 10 patients with focal seizure onsets (i.e., all patients except Patient F
in ), and for subsets of patients and seizures grouped by seizure type (Supplementary Figure 1
). We also show in the signal energy (see Methods) within each interval adjusted for patient differences. In contrast to the density, we find a significant increase in overall signal energy for all ictal and postictal intervals compared to the preictal level. The increase in signal energy is not surprising; seizure activity typically manifests as large amplitude voltage oscillations at the macroscopic spatial scale recorded in the ECoG or EEG (electroencephalogram). To support these large amplitude oscillations, we expect increased synchrony at the microscopic spatial scale of individual neurons. Although increased synchrony at the microscopic spatial scale supports the large amplitude ECoG rhythms, we find that synchrony decreases (i.e., density decreases) between macroscopic brain areas during seizure.
The network topology and signal energy also change dramatically at seizure termination. The density increases substantially in the postictal interval compared to preictal or ictal values, while the voltage fluctuations decrease. A low amplitude, slow wave rhythm (visible in ) that appears broadly throughout the brain dominates the postictal activity and contributes to the increased network density. Low frequency rhythms often appear in the postical period (Kaibara and Blume, 1988), although the exact mechanisms that support this activity are unknown (Fisher and Schachter, 2000
). We note that the density increase begins before seizure termination (see I9 and I10 of ), consistent with previous observations of increased synchrony in the late seizure stage (Topolnik et al., 2003
; Schindler et al., 2007
Dominant networks fracture, then reform, during seizure propagation
Although the voltage dynamics change dramatically during the seizure, our first measure of the network topology — the density — does not. Perhaps the ictal networks change in more subtle ways, reorganizing, rather than trimming or growing, their edges? To address this possibility, we examine the network components within each interval. Briefly, a component is a connected subnetwork of mutually reachable nodes (i.e., any node in the component can reach any other node in the same component by following a sequence of edges). The average number of nontrivial components — components that consist of two or more nodes — adjusted for differences in subjects increases significantly (see Methods) during the middle portion of the seizures ().
Figure 2 The largest network components fracture during the seizure. (A) The average number of connected components (adjusted for differences in subjects) increases during the course of a seizure; points plotted in red indicate a statistically significant increase (more ...)
Thus, as the density returns to preictal levels during the seizure (intervals I2-I8 in ), more nontrivial components emerge. Does one connected subnetwork dominate and contain most of the nodes? Or, are the nodes more evenly distributed between smaller subnetworks? To investigate this, we determine the percentage of nodes within each of the components for the ensemble of seizures (). At ictal onset, nearly half of all nodes reside in the largest component, which therefore dominates the network. As the seizure progresses, the largest component fractures — nodes “leave” this component and become isolated or form other, smaller subnetworks. Finally, just before seizure termination, a majority of nodes rejoin to establish a single dominant component.
We illustrate the fracturing and reforming of the largest subnetwork for a single subject and seizure in . At ictal onset (I1), a majority of nodes join the largest component which here covers portions of the frontal, parietal, and temporal lobes — almost the entire extent of the lateral neocortex as well as subcortical brain regions. During the seizure (I6, middle row), the largest component shrinks and collections of nontrivial components emerge. Finally, at ictal termination (I10, bottom row) nodes rejoin the giant component which again dominates the network. In this example, and for the population of subjects, the ictal subnetworks merge at onset, fracture during seizure, and rejoin just before termination.
Small-world topologies of the largest subnetwork emerge during preictal and ictal intervals
The largest subnetworks, which incorporate between 30% and 60% of the nodes during the seizure, play a prominent role in the network topology. To examine the properties of these dominant subnetworks, we determine the characteristic path length and clustering coefficient for the largest component of each network. Because the size of the largest subnetwork changes in time (as nodes break-off and re-join the dominant component) we scale the observed values by those expected for a one-dimensional lattice with the same number of nodes and average degree (Watts and Strogatz, 1998
). We find that the (scaled) characteristic path lengths in the observed networks remain less than one for all intervals, while the (scaled) clustering coefficients tend to exceed one during seizure (). Combined, these results suggest that, for most of the seizure, the largest preictal and ictal subnetworks exhibit small-world topologies (greater clustering coefficients yet smaller path lengths than the associated one-dimensional lattices) for all examined intervals (Netoff et al., 2004
; Ponten et al., 2007
). Although the ictal subnetworks exhibit small-world topologies, the properties of these topologies evolve in time. Just after seizure onset (I3), both the path length and clustering coefficient tend to increase (the former increases significantly), and the networks move in the direction of becoming a more regular lattice with mesh-like connections between nodes, in agreement with previously reported observations near seizure onset (Ponten et al., 2007
; Schindler et al., 2008
). Just before seizure termination (I10), the (scaled) path lengths and clustering coefficients decrease dramatically, suggesting the networks evolve towards a more random configuration, as observed in (Schindler et al., 2008
Figure 3 The topological properties of the dominant subnetwork evolve during the seizure. We plot the scaled characteristic path length [L/L(0), dashed curve] and scaled clustering coefficient [C/C(0), solid curve] for each interval (adjusted for differences in (more ...)
Network topologies become more similar during — and between — seizures
Most patients with epilepsy experience seizures with clinical manifestations (e.g., stereotyped motions) that are similar across seizures within an individual. The voltage activity observed during seizures also appears stereotyped. To determine whether the networks that appear during the seizure do so in stereotyped ways, we apply two measures focused on different aspects of the topological similarities (). The top two rows show example networks extracted from the preictal interval (label −1) and a middle ictal interval (label I6) from a single seizure and subject. The bottom two rows show another set of networks extracted from the same subject and intervals but from a different seizure. Visual inspection suggests more variability in the preictal networks over time. In addition, the ictal networks appear roughly consistent in the two seizures; notice, for example, the concentration of edges in the lower left regions of the networks.
Figure 4 Networks become more similar during — and between — seizures. (A) Two examples of networks from a preictal interval (unshaded) and ictal interval (shaded) from two different seizures of a single subject. Visual inspection suggests that (more ...)
We quantify these observations with two measures. The first — the intra-seizure similarity — examines the variability of networks within a fixed interval of a chosen seizure. To compute this measure we compare each network within an interval (i.e., within interval “-1” of the first seizure of a chosen subject) to all other networks within the same interval and seizure. The intra-seizure similarity is large when the variability in the networks within the interval is small. We apply the intra-seizure similarity measure to each interval for all subjects and seizures, and plot the average results for each interval (adjusting for differences in subjects) in . During seizure (intervals I1 to I9) the intra-seizure similarity increases significantly (see Methods) compared to preictal values; ictal topologies within each interval become more similar (or exhibit less variability) than the preictal networks. We note the dramatic decrease in similarity during the postictal interval. Although large subnetworks dominate this period (), the topologies of these networks exhibit high variability.
To determine the consistency of networks between seizures, we apply a second measure: the inter-seizure similarity (see Methods). In this measure we choose an interval (e.g., “−1”) and compare networks from the same interval and subject across different seizures. For example, we compare each network in the first and third rows (or second and forth rows) of . We repeat this procedure for each patient and note that all patients studied here had at least two seizures. Compared to the intra-seizure similarity, the inter-seizure measure is smaller (); we expect more variability (and less topological consistency) between different seizures of a subject. Yet, we observe that ictal networks in early and middle intervals become significantly more similar than preictal networks from seizure-to-seizure. We conclude that similar ictal networks appear from seizure-to-seizure for a patient, and in that sense the network topologies that emerge, like the voltage rhythms, are consistent.