Data Selection and Pre-processing
We use ECoG data, recorded with grid electrodes, of six patients with focal epilepsy who underwent long-term ECoG monitoring prior to surgery at the University Medical Centre Utrecht. Data are retrospectively studied and handled coded and anonymously according to the guidelines of the institutional ethical committee. Patient characteristics are provided in Table
1. For each patient, SPES has been performed as part of clinical routine. ECoG data has been recorded using a common reference montage with respect to an extracranial reference electrode. We consider two subsets of ECoG data for each patient: a segment of on-going inter-ictal data, to calculate functional connectivity, and the segment with SPES data.
Table 1
Patient characteristics
1 | 2048 | F(\(2\times 8\); \(4\times 8\)), IH(\(1\times 8\)) | 56 | n | Awake, agile |
2 | 512 | F(\(4\times 8\); \(4\times 8\)) | 56 | n | Awake, quiet |
3 | 2048 | F(\(4\times 8\)), T(\(4\times 8\)), C(\(1\times 8\)), IH(\(1\times 8\)) | 72 | y | Light sleep |
4 | 512 | T(\(6\times 8\); \(1\times 8\); \(1\times 8\)), F(\(2\times 8\)) | 58 | n | Light sleep |
5 | 2048 | T(\(2\times 8\)), C(\(4\times 8\)) | 45 | y | Awake |
6 | 512 | T(\(6\times 8\); \(2\times 8\); \(1\times 8\); \(1\times 8\)), F(\(2\times 8\)) | 89 | y | Awake, language task |
The segments of on-going ECoG data have been recorded just preceding SPES. In this way we are sure that effects of anti-epileptic drugs and situational confounders are similar for the ongoing ECoG and SPES recordings, while any influence of SPES on the connectivity for CC and GC is excluded. We note that by imposing this condition it was not possible to control the cognitive state of the patient as this is a retrospective study. The inter-ictal ECoG segment is sampled at either
\(512~\text {Hz}\) or
\(2048~\text {Hz}\) (see Table
1). An expert clinical neurophysiologist (FSSL) marked artefacts in the raw ECoG recordings, e.g. those arising from the reference electrode. In the next sections we explain how CC and GC networks are obtained from this data. Both methods require specific pre-processing steps. For example, for CC it is usual to band-filter the data, while for GC this is not recommended (Barnett and Seth
2011). Also, it is common to apply differencing before calculating GC, while this is not the case for CC.
The protocol for SPES has been described in (van’t Klooster et al.
2011). Specifically, ten monophasic electrical stimuli are applied to pairs of horizontally adjacent electrodes. The stimuli have a duration of
\(1~\text {ms}\) with an inter-stimulus time of
\(5~\text {s}\) and an intensity of
\(8~\text {mA}\). During SPES ECoG data has been registered at a sampling rate of
\(2048~\text {Hz}\).
For all selected patients, the ECoG grid consisted of one or two large grids, spatially arranged in four or six times eight electrodes, and some additional strips consisting of eight electrodes each. We discarded all data from electrodes not used to stimulate with SPES as well as dysfunctional electrodes. Table
1 shows the selected number of electrodes per patient.
Cross-Correlation
CC networks are non-directional weighted networks constructed from ongoing inter-ictal ECoG data. For consistency, all ongoing ECoG data are resampled to
\(512~\text {Hz}\) if necessary. We band-pass filtered the data to the
\(\theta\)-,
\(\alpha\)- and
\(\beta\)-band, i.e. between 4 and
\(30~\text {Hz}\), following (Sinha et al.
2014). Next, we divided all segments of ECoG data without artefacts into non-overlapping epochs of 20 s (starting from the beginning of each segment and neglecting remaining parts or segments of < 20 s). We selected the last 60 epochs, so 20 min in total, for further analysis. For each of the selected epochs we proceed as follows for every pair of electrodes. First, we estimate the cross-correlation function for all time lags
m with
\(\left| m\right| \le M\) and
M the maximal lag in samples. Next, we set the connection strength as the maximum absolute value of this estimated cross-correlation function. We take a maximal lag of
\(M=26\) samples corresponding to a time of
\(50~\text {ms}\). We average over all 60 epochs to obtain the mean connectivity.
Granger Causality
GC networks are constructed from the same inter-ictal ECoG data as CC networks. In contrast to CC networks, GC networks are directional. The main idea behind GC is that a connection from
x to
y is present if the prediction of the time series of
y improves significantly by incorporating the past of the time series of
x (Bressler and Seth
2011; Ding et al.
2006). In this study we use conditional GC, a multivariate form of GC, which besides the past of the time series
x and
y also uses the past of all other time series to determine the connectivity from
x to
y. This method reduces spurious connectivity, e.g. connections that arise due to common input (Barnett and Seth
2014).
GC relies on fitting multivariate autoregressive models (MVAR models) to the data. The model order
m of this MVAR model determines the length of the history taken into account and must be specified. If we want to capture a history of
\(50~\text {ms}\) at a sampling rate of
\(512~\text {Hz}\), as in "
Cross-correlation" section, we would need
\(m=26\). For such high model orders many unknowns must be estimated in the MVAR model. To avoid overfitting of the model, enough data points and as a consequence long time series must be considered. For such long time series the assumption of (approximate) stationarity is likely to fail. By downsampling the required model order can be reduced, while a longer history can be taken into account (Murin et al.
2016,
2018).
Our complete procedure to calculate GC is as follows. First, we resample the ECoG data to
\(128~\text {Hz}\). Next, first-order differencing is applied to enhance stationarity (Seth
2010). We select 60 epochs of 20 s in the same way as we do for CC (actually the same). Next, we calculate conditional GC in the time domain using the MVGC toolbox (Barnett and Seth
2014). We set the model order to
\(m=7\), which is sufficient to capture
\(50~\text {ms}\) of history. Statistical significance is assessed using the recommended options of the MVGC toolbox, i.e. Granger’s F-test with a significance level of 0.05 and the false discovery rate method to account for multi-hypothesis testing. For each epoch this results in a binary matrix with an entry being one if GC finds a significant connection and zero otherwise. Finally, we obtain the mean connectivity by averaging over all 60 epochs. The resulting network is directional with weights between zero and one.
Localizing Broca’s and Wernicke’s Area
In three patients both the areas of Broca and Wernicke have been covered by the electrode grid. As part of clinical routine the precise locations of those two areas have been determined using electrocortical stimulation mapping (ESM). In ESM pulse trains of 4–7 s, \(50~\text {Hz}\), 0.2–0.3 ms, 4–15 mA (stimulation amplitude was altered to avoid afterdischarges) are applied during a picture naming task and in case of Wernicke’s area also during item presentation in a Token Test. If repeated stimulation interferes with language (either inability to name or understand, or paraphasia) and the cause is not anarthria (sound production is unaffected) the stimulated electrode pair is marked as positive for language. Stimulations are applied to horizontally and, in contrast to SPES, also vertically and diagonally adjacent electrode pairs. An individual electrode is marked positive if it was part of at least two positively marked pairs.
Comparing Networks
To compare CC and GC with SPES connectivity we need to cast the networks in the same form. We obtain binary CC and GC networks by thresholding the edge weights; if the weight of an edge exceeds this threshold, then there is a connection in the dichotomized network. The threshold
\(h^{*}\) is determined using a data-driven approach. This data-driven approach is inspired by both (Rummel et al.
2015) and the definition of outliers in a boxplot. Let
\(Q_{1}\) and
\(Q_{3}\) denote the first and third quartile of the set of all edge weights. Then
\(Q_{3}-Q_{1}\) denotes the inter-quartile range, which is a measure for the spread. We set
\(h^{*}:=\max (Q_{3}+w(Q_{3}-Q_{1}),0.1)\) with
w a parameter. We use
\(w=1.5\), which is the standard choice for defining outliers (Rummel et al.
2015).
The dichotomized GC network and the SPES network are both directional, unweighted networks and hence they can be compared. A non-directional variant of the SPES network is constructed by putting an edge between nodes i and j if either \(i\rightarrow j\) or \(j\rightarrow i\) is present in the directional SPES network. This non-directional SPES network can be compared with the dichotomized CC network.
Next, we test if edges of the CC and GC networks coincide with those in the SPES network using a hypergeometric test for overrepresentation. Under the null hypothesis the connections of the functional network are distributed proportionally over existing and non-existing SPES connections. This hypothesis will be tested against the alternative hypothesis that CC/GC connections are overrepresented in the set of SPES connections. In other words, we test whether it is more likely to find a CC/GC connection between two nodes if there is a SPES connection between these two nodes.
The probability of finding
k CC/GC connections in a set of
\(n_{s}\) SPES connections (and consequently
\(n_{s}-k\) non-existing CC/GC connections) is, under the null hypothesis, given by a hypergeometric distribution:
$$\begin{aligned} p_{n_s,n_{f}}(k)=\left. \genfrac(){0.0pt}0{n_{f}}{k}\genfrac(){0.0pt}0{n-n_{f}}{n_{s}-k}\Bigg /\genfrac(){0.0pt}0{n}{n_{s}}\right. , \end{aligned}$$
with
\(n_f\) the total number of CC/GC connections and
n the total number of possible connections. We have
\(n=N_{el}(N_{el}-1)\) for the comparison between GC and SPES and
\(n=N_{el}(N_{el}-1)/2\) for the comparison between CC and SPES. Let
\(n_{sf}\) denote the number of connections in both the SPES and the CC/GC network. Under the null hypothesis, the probability
P to have
\(n_{sf}\) or more CC/GC connections in the set of SPES connections is given by:
$$\begin{aligned} P=\sum _{k=n_{sf}}^{\min \left\{ n_{s},n_{f}\right\} } p_{n_{s},n_{f}}(k). \end{aligned}$$
We will reject the null hypothesis if
\(P<0.01\).
We also investigate the dependence of our results on the threshold for CC/GC. Let h be the threshold for the CC or GC network. Take \(a_{c}(h)\) as the fraction of positive agreement between the SPES and CC/GC network, i.e. the number of connections that arise in both the SPES and the CC/GC network dichotomized using threshold h divided by the number of SPES connections. If \(a_{c}\) is one all connections in the SPES network are also part of the CC/GC network. If \(a_{c}\) is zero then none of the SPES connections are part of the CC/GC network. Equivalently, define \(a_{nc}(h)\) as the fraction of negative agreement, i.e. the number of non-existing SPES and CC/GC connections as a fraction of the total number of non-existing SPES connections. If \(a_{nc}\) is one then all non-existing SPES connections are also non-existing in the CC/GC network in which case all connections in the CC/GC network are part of the SPES network. Further, we calculate the total agreement, i.e. the number of agreeing connections and non-existing connections as fraction of the total number of possible connections. We define \(h_{ma}\) as the threshold maximizing the total agreement.
Finally, we study connectivity between electrodes in Broca’s and Wernicke’s area in all three networks. We examine the number of connections found between both areas as a fraction of \(n_{bw}\), the maximal number of possible connections between electrodes in Broca’s and Wernicke’s area. For the directional networks, i.e. SPES and GC, \(n_{bw}\) is given by \(2n_bn_w\) and for the CC network by \(n_bn_w\), where \(n_b\) and \(n_w\) denote the number of electrodes in Broca’s and Wernicke’s area respectively.