Fig. 1
Example of a functional brain network. If a connectivity measure between a pair of channel recordings (the nodes) exceeds the pre-specified threshold then a connection (i.e., an edge) is placed between the channel recordings. We refer to this set of connections as functional network. When applying graph theoretical measures to scalp EEG, we are faced with two considerable challenges: the fact that all channel recordings depend on a reference recording (the reference problem) and the fact that the electrodes will pick up a mixture of signals arising locally and signal volume conducted from other brain locations (the volume conduction problem)
2.2 Preprocessing
A 50 Hz Notch filter was applied to remove line noise, and subsequently the signals were bandpass filtered between 1 and 45 Hz. Eye artifacts were removed by applying independent component analysis (ICA) from the EEGLAB toolbox of Matlab. In order to assess the possible effects of muscle artifacts, we bandpass filtered the data between 1 and 20 Hz and the results were found to be similar overall; therefore, we present results when the data were preprocessed as mentioned above.
It is known that the montage (i.e., the choice of reference) affects connectivity measures [9] and as a consequence it may affect the corresponding graph-theoretic measures. For this reason, we mathematically converted the input data, which were originally recorded relative to the common cephalic reference, to three different montages: the common reference, the average reference, and the bipolar montage (see Section 2.2.1). We obtained results employing all three montages and compare the results below.
Recording Montages
Scalp EEG recording devices use differential amplifiers to compute the voltage of each EEG channel. A differential amplifier takes as input the measurements of two electrodes and produces the corresponding EEG channel as the difference between the two inputs, after it has been amplified. The choice of input electrodes to each amplifier is known as montage.
In the original recordings obtained with our system, each amplifier takes as input one of the 10–20 system electrodes (Fp1, Fp2, F7, F3, Fz, F4, F8, T3, C3, Cz, C4, T4, T5, P3, Pz, P4, T6, O1, O2, A1, A2) and one reference electrode (REF). This is an example of a common reference montage, since the reference electrode is common to all amplifiers. Additionally, we have mathematically re-referenced the data to Cz, which is often the reference electrode of choice. The average reference montage subtracts the average signal over all channels or a carefully chosen subset of them from the signal at each channel. In this work we used all 19 scalp channels to compute the average.
In the bipolar montage, contrary to the previous two montages, there is no input common to all the time series. Instead, pairs of electrodes placed in nearby locations of the scalp are used to obtain the time series by subtracting the corresponding measurements. In one example of such a montage, electrodes are taken in straight lines from the front to the back of the head, forming the pairs Fp1–F7, F7–T3, T3–T5, T5–O1, Fp2–F8, F8–T4, T4–T6, T6–O2, Fp1–F3, F3–C3, C3–P3, P3–O1, Fp2–F4, F4–C4, C4–P4, P4–O2, Fz–Cz, Cz–Pz.
2.3 Functional Network Construction
We calculated pairwise correlations between all pairs of time series (EEG data in common reference, average reference, or bipolar montage), using the connectivity measures described in Section 2.4. Edges were added between node pairs if the corresponding connectivity measure between each pair exceeded a pre-specified threshold, the value of which was dependent on the employed measure. For each measure we experimented with a range of thresholds, obtaining similar results overall in terms of the overall observed characteristics and the effect of volume conduction and reference. We will henceforth refer to the edges and connections obtained in this way as functional networks. Note that, following the above process, the obtained networks are binary and undirected, similar to the one shown in Fig. 1.
Challenges in Constructing Functional Networks from Scalp EEG Recordings
When applying graph theoretical measures to scalp EEG, we are faced with two considerable challenges in terms of obtaining and interpreting the correlation metrics that are used to construct the network edges (Fig. 1): the fact that all channel recordings depend on a reference recording (the reference problem) and the fact that the electrodes will pick up a mixture of signals arising locally as well as signal volume conducted from other brain regions (the volume conduction problem). With respect to the reference problem, note that any channel recording always reflects a potential difference between a recording electrode and a reference electrode. The latter could take various forms such as a common reference electrode (for instance Cz), a mathematically constructed average reference of all electrode signals, or a nearby recording electrode, as in the case of bipolar recordings. We should note that there is no distinction between recording and reference electrode, in the sense that the reference will never be silent and will contain some electrical brain activity [23]; thus, channel recordings will always be sensitive to signals picked up at both the recording and reference sites. Further, each channel will contain a mixture of signals arising in the vicinity of the recording and reference electrode(s) and signals from far-away sources that conducted through the head volume to the site of the recording and reference electrodes. To sum up, signals that are used to obtain the network edges will reflect a mixture of activity in the vicinity of recording and reference electrode locations and signal volume conducted from elsewhere.
2.4 Connectivity Measures
Common measures for estimating the correlation between pairs of time series include cross-correlation, coherence, synchronization likelihood, Granger causality, directed coherence, mutual information, PLI, and many more; see, e.g., [24] for a review. In this work we compare the performance of six measures of signal correlation. These include cross-correlation and coherence, which quantify linear correlations in the time and frequency domain respectively, and are the most commonly used measures for estimating EEG signal correlations. The remaining four measures were the corrected cross-correlation (i.e., the odd part of the cross-correlation function), imaginary coherence, PLI, and WPLI. These also reflect linear correlations but are less sensitive to the effects of volume conduction than standard correlation and coherence.
Cross-correlation
Given two time series x(t) and y(t), with t ∈ 1 … n, the normalized cross-correlation function between x and y as a function of the lag, τ is given by
where σ x and σ y are the standard deviations of x and y, respectively. C xy is computed for a range of values of the lag τ, which depends on the sampling frequency; e.g., at 200 Hz as in our case, for a required range of [−100 100] msec, τ lies within [−20 20]. Note that when the mean value of both signals is not subtracted beforehand (as done here), the normalized cross-covariance function should be used instead:
C xy takes values between −1 and 1, with 1 indicating the largest positive correlation, −1 the largest negative correlation, and 0 no correlation. For the purposes of constructing the graphs and obtaining the graph theoretical measures, the absolute value of the cross-correlation is computed and the correlation between the two signals is computed as 
, over the desired range of τ.
, over the desired range of τ.Corrected cross-correlation
Cross-correlation often takes its maximum at zero lag in the case of scalp EEG measurements (e.g., see Figs. 3 and 4 below). Consistent zero-lag correlations could be due to volume conduction effects: this is because currents from underlying sources are conducted instantaneously through the head volume to the EEG sensors (i.e., assuming that scalp potentials have no delays compared to their underlying sources (quasi-static approximation)). Thus signals arising from a common source will be simultaneously picked up by different electrodes effecting a spurious zero-lag correlation between the two electrode signals. It should be noted though that zero-lag correlations could also be due to a third common source or even true direct physiological interactions. In principle, true direct interactions between any two physiological sources will typically incur a nonzero delay due to transmission speed, provided that the sampling frequency is high enough to capture such delays. However, consistent nonzero lag correlations are unlikely to be due to the effects of common sources (subsuming volume conduction and reference effects [18]). In order to measure true interactions not occurring at zero lag, we calculate the odd part of cross-correlation, which is a measure of its asymmetry, as defined in [17], by subtracting the negative-lag part of C xy (τ) from its positive-lag counterpart:
provides a lower bound estimate of the nonzero-lag cross-correlations and is notably smaller than C xy .
provides a lower bound estimate of the nonzero-lag cross-correlations and is notably smaller than C xy .Coherence
Coherency may be viewed as the equivalent measure of cross-correlation in the frequency domain; it measures the linear correlation between two signals x and y as a function of the frequency f. It is defined as the cross-spectral density between x and y normalized by the auto-spectral densities of x and y:
where S xy is the cross-spectral density and S xx and S yy are the auto-spectral densities of x and y, respectively.
Coherency is a complex number, as the cross-spectral density is complex, whereas the auto-spectral density function is real. Therefore, in many cases coherence (or the squared coherence), which is defined as the magnitude of coherency (or its square), is employed as a measure of correlation in the frequency domain, i.e.:
For finite data sets, coherency is estimated by using standard power spectral density estimation methods, such as the Welch method, i.e., by dividing the EEG signals into segments of equal length (eight segments with 50 % overlap in our case) and averaging the individual spectral estimates. In the above equation, <ċ> denotes the average over all segments [24].
The value of κ xy (f) ranges between 0 and 1, with 1 indicating perfect linear correlation and 0 no correlation between x and y at frequency f. We calculated the maximum coherence value both for the broadband EEG signals (1–45 Hz) as well as for each EEG frequency band (delta (1–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz), and gamma (30–45 Hz)) separately, in order to assess the degree of correlation between the two signals in these bands.
Imaginary coherence
As stated above, coherency is a complex number. Nolte et al. [18] observed that the imaginary part of coherency is insensitive to volume conduction, while the real part is strongly affected. This is due to the fact that signals arising from the same electrical source in the brain, under the quasi-static approximation, will be volume conducted to any two recording locations with no time delay, thus influencing only the real part of the cross-spectrum. Hence, imaginary coherence is defined as the imaginary part of coherency:
As in the case of coherence, the maximum absolute value in each frequency band quantifies the correlation between the two signals in that band.
Phase lag index
Stam, Nolte, and Daffertshofer [16] proposed the PLI, which is defined as a measure of asymmetry of the phase difference distribution between two signals (in this case, EEG recordings from different electrodes). Following [16], the instantaneous phases were obtained by first bandpass filtering the signals in the frequency bands of interest and then using the Hilbert transform to obtain the phase of the analytic signal. Phase differences (Δϕ) between a given pair of channels were wrapped in the interval –π ≤ Δϕ ≤ π:
where Δϕ is the instantaneous phase difference between x and y.
PLI ranges between 0 and 1, with 0 indicating no correlation and 1 maximal correlation.
WPLI
Vinck et al. [13] argued that PLI’s sensitivity to noise and volume conduction is hindered by the discontinuity of the measure, which is caused by small perturbations turning phase lags into leads and vice versa. To overcome this problem, they defined the WPLI, which modifies PLI by weighting the contribution of observed phase leads and lags by the magnitude of the imaginary component of the cross-spectrum:
Similar to PLI, WPLI ranges between 0 and 1, with 0 indicating no correlation and 1 maximal correlation.
Network Properties
Once a functional network is constructed, properties of either the individual nodes or the individual edges or the network as a whole are examined, aiming to identify consistent changes in the pre-ictal, ictal, and postictal periods. In this section we define a number of network properties that characterize a network as a whole rather than examining individual nodes or edges. In the following, let n denote the number of nodes of the network and N the set of all nodes.
Average degree
The degree, k i , of a node i is defined as the number of neighboring nodes in the network, i.e., the nodes to which node i is connected (equivalently, the number of edges incident to i). A summary of the degrees of a network is given by the average degree, which quantifies how well connected the graph is:
Characteristic path length and global efficiency
Although the average degree reflects the average number of connections any network node may have, it does not provide any information regarding the actual distribution of the edges and, hence, how easy it is for the information to flow in the network. Such information can be captured by the shortest (or geodesic) path length, d i,j , between a pair of nodes, i and j. It is defined as the minimum number of edges that have to be traversed to get from node i to j. Then, the characteristic path length is defined as the average shortest path length over all pairs of nodes in the network:
Unfortunately, the characteristic path length is only reliable when the network is fully connected; note that for any pair of nodes not connected through a path, the shortest path length d i,j = ∞. A workaround for this is to consider only connected pairs of nodes, but this does not reflect the connectivity of the whole network.
To overcome the above problem, Latora and Marchiori [25] defined the efficiency between a pair of nodes as the inverse of the shortest distance between the nodes. When such a path does not exist, the efficiency is zero. Global efficiency is the average efficiency over all pairs of nodes:
Clustering coefficient
A cluster in a network is a group of nodes that are highly interconnected. The measure of clustering coefficient, C i , [26] of a node i is defined as the fraction of existing edges between neighbors of i over the maximal number of such possible connections:
where k i is the degree (i.e., the number of neighbors) of node i, and t i denotes the number of edges between neighbors of i. Then the global clustering coefficient, C, of the network is defined as the mean clustering coefficient among all nodes:
3 Results
In this section, we examine the degree to which each connectivity measure is affected by the choice of reference and volume conduction effects in scalp EEG recordings. We first investigate the extent to which each montage is affected by volume conduction and subsequently use the montage that is less affected for further analyses.
3.1 The Effect of Montage on Connectivity Measures
Ideally one wants to estimate the activity of local cortical generators from EEG recordings and then study their interactions. The choice of reference channel will affect both the local cortical estimates and their interactions. It is known that using a common reference can substantially inflate coherence estimates particularly at smaller distances [23] as a common signal is subtracted from all channels. On the other hand, the average reference is known to produce estimates of coherence that are close to coherence estimates obtained from reference-independent potentials [23], possibly because the average reference montage is thought to approximate the local reference free potentials [23]. However, this advantage of the average reference only comes into play with a large number of electrodes and extensive coverage of the head; in the case of a standard 10–20 system as used here it may provide a poor approximation of the reference-free potentials. In the setting of a very limited number of electrodes, as in the case of standard 10–20 recordings routinely used in the clinical setting, the most pragmatic solution for obtaining estimates of local superficial cortical generators is to use a bipolar montage [23], as—in such cases the use of more sophisticated methods such as source reconstruction or spline Laplacians is not possible. What is clear from these considerations is that the choice of the reference electrode will influence the estimate of the local activity (electrode signal) and the estimate of interactions between pairs of such electrodes. Thus, it is also expected to influence the graph theoretic measures of interest. Below we study the effects of choice of reference electrode (montage) on these measures in our data and compare between such montages.
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