EEG recordings are frequently contaminated with physiological and nonphysiological artifacts. The physiological artifacts include electrical potentials generated by biological activities, which are largely generated by eye movements, muscle and cardiac activities, and sweat. The nonphysiological artifacts can originate from anywhere near the EEG recording system, such as the electrode-scalp interface, devices, and adjacent environment. Identifying discriminative features by analyzing the EEG signals requires the artifacts to be removed. Below we introduce some techniques for cleaning EEG signals that can remove these artifacts.

Unfiltered EEG signals are usually contaminated with artifacts such as 60-Hz line noise, muscle activities, background noise, and very-low-frequency noise. These contaminants tend to predominate within specific frequency bands, and so they can be effectively reduced by applying appropriate band-pass filtering.

The bandwidths of filters applied to EEG signals need to reflect the artifact-specific frequency ranges. For example, highpass filters (e.g., >0.1 Hz) are applied to remove low-frequency drifts, while lowpass filters (e.g., <50 Hz) are applied to remove high-frequency noise such as that due to muscle activity. A notch filter is commonly used to eliminate power line noise, since is only present within a very narrow frequency band. In addition to the frequency bands of artifacts, the frequency range of the signals of interest should be considered when selecting the filters, such as slow oscillations (0.3-1 Hz) in non-REM (rapid eye movement) sleep and high-frequency oscillations (>80 Hz) in epilepsy.

Bad EEG segments are those that are extremely contaminated with artifacts in the entire channel, and they should be removed from the raw EEG signals. For example, excessive electromyography bursts that are widely spread over 20 Hz and instantaneous popping bursts with relatively large amplitudes around the forehead, ears, and neck are considered as bad segments. The common approach to remove bad segments is the manual inspection of individual segments by well-trained experts. Another common approach is to use automatic rejection criteria, such as when the peak-to-peak amplitude exceeds a certain threshold.

Independent components analysis (ICA) is a source-separation technique that attempts to identify independent sources of variance.8,9 Applying ICA to multichannel EEG data involves breaking down the EEG time-series data into a set of independent components (ICs), which include brain, eye, cardiac, and muscle activities. Artifacts can be rejected with minimal loss of the actual brain activity by subtracting artifact-related ICs from EEG signals.10,11 Artifact-related ICs can be identified based on their topographic characteristics, across-trial temporal distributions, and frequency distributions. Artifact-embedded ICs can be distinguished from brain-related ICs either manually or automatically. After artifact-embedded ICs have been removed, the remaining ICs are reorganized into artifact-removed EEG data.

A frequency-domain analysis is important for characterizing specific rhythmic or oscillating patterns of neural activity. Spectral density estimations can quantify brain oscillations in the form of the signal power or energy distribution across frequency. The Fourier transform is used to identify the frequency components in a temporal signal. Welch’s method is the most-popular method for estimating the spectral density in EEG analysis.12 This method first divides the signal of N samples into K data segments of length M overlapping by D points. The periodogram of each segment is then calculated by computing the discrete Fourier transform. Finally, the spectral density is estimated by averaging the periodograms of all segments.

Some aspects need to be considered when applying Welch’s method to EEG data. The EEG segments should be long enough to provide the required frequency resolution. The appropriate length of the overlapping samples, which subsequently influences K and the variance of the spectral estimate, should also be considered. Choosing the appropriate window function is crucial for optimizing spectral estimation. A rectangular window provides the best ability to resolve adjacent sinusoids (i.e., high resolution), but its wideness might mask important details even at lower levels (i.e., low dynamic range). Nonrectangular windows such as the Hamming and Hann windows can redistribute the spectral leakage to places where it causes the least harm (i.e., high dynamic range), at the expense of increasing this leakage in the vicinity of the original component (i.e., low resolution).

One of the most-informative features of EEG data is the functional connectivity underlying the interregional neural interactions in the brain. EEG connectivity analysis can provide important evidence of functional interactions between the neural systems operating in each frequency band. Below we introduce several promising connectivity measures used in EEG data analysis.

A coherence analysis can assess the linear relationship between two time series within each frequency bin.13 Assume that X(t) and Y(t) represent the EEG signals measured by electrodes X and Y, respectively. Time-domain signals X(t) and Y(t) are first converted into the frequency domain using the fast Fourier transform or wavelet transform. For each frequency bin f, the individual spectral power density S_xx(f) and S_yy(f) and their cross-spectrum power density S_xy(f) are then estimated. Coherency function K_xy(f) is calculated as the ratio between the cross-spectrum power density and the individual spectral power density of the two signals:

Finally, the coherence in frequency bin f is computed as

The coherence value ranges from 0 to 1, where a value of 0 indicates no linear dependence between X(t) and Y(t) at frequency f, and a higher coherence value indicates the presence of greater statistical dependence between the two signals.

Coherence has been widely used in EEG connectivity studies, but it has some limitations: 1) it can only assess the linear dependence between two signals, and hence any nonlinearity in the relationship is ignored, 2) it is significantly influenced by the signal amplitudes, and 3) it cannot dissociate volume conduction from true interactions between the brain regions.

Phase-synchronization-based measures utilize the phase components of EEG signals within a given frequency band. If two oscillatory activities are synchronized with a constant phase difference, it can be assumed that phase synchronization exists. Among the various connectivity methods, the phase-locking value (PLV) is the most commonly used measure of phase synchronization.14,15 To analyze the PLV, bandpass filtering is applied to transform EEG signals into narrowband signals within specific frequency bands. The instantaneous phase at each time point is calculated from the narrowband signal and its Hilbert transform as follows:

To quantify the degree of phase synchronization, the PLV between two electrodes j and k is calculated for each time point t by averaging the phase difference over N segments as follows:

Where N represents the total number of segments, n denotes a specific segment, and ϕj denotes the phase of signals of electrode j at time t of the nth segment. It is then determined whether the PLVs between two specific electrodes indicate the presence of significant phase synchronization. This involves checking whether the PLV change under investigation is meaningful with respect to the PLVs of surrogate data (obtained by randomly shuffling the segments). The level of significance is determined using the distribution of the PLVs calculated from the surrogate data.14,15 The significance cutoff is generally set at 1%, which means that the phase synchronization is significantly increased if it is higher than the first percentile of the PLV values from surrogate data.

Long-range phase synchronizations between EEG signals can be calculated using the weighted phase-lag index (wPLI), which is based on the imaginary component of the cross-spectrum between a pair of EEG signals and is known to be minimally affected by volume conduction.16 A shorttime Fourier transform is used to extract the instantaneous phase perturbation of the oscillations in EEG signals. The cross-spectrum between EEGs from electrodes i and j (designated as X_i,j) is calculated using the extracted complex-valued Fourier spectra vector Z as follows:

The wPLI between two electrodes is calculated as follows:

where $\mathfrak{J}\left\{X_{i,j}\right\}$is the imaginary part of cross-spectrum X between signals i and j, and sgn(∙) denotes the sign function. wPLI ranges from 0 to 1, where 0 indicates no coupling and 1 indicates perfect phase synchrony between the two signals (Fig. 1).

A network is defined as a set of nodes (vertices) and edges (connections), where the edge between two vertices represents some kind of interaction or connection between them (Fig. 2). Graph theory is a mathematics technique that can be used to understand the topological properties of complex brain networks, such as their modularity, hierarchy, and centrality, and the distribution of network hubs. There is a wide variety of graph-theory measures, among which the essential ones are the degree distribution, clustering coefficient, and path length.17 First, the degree refers to the number of connections that link a node to the rest of the network, and its distribution is defined as likelihood p(k) that a randomly chosen vertex will have degree k. Random networks have a symmetrically centered degree distribution, whereas complex networks generally have non-Gaussian distributions with a long tail toward high degree values. Second, clustering coefficient C is a measure of the clustering of local networks. When the nearest neighbors of a vertex are directly connected to each other, they can be considered to form a cluster. Clustering coefficient C_i is therefore defined as the ratio of the number of real edges between neighbors of a vertex i to the maximum possible number of edges. Random networks have low clustering coefficients, whereas regular and complex networks are characterized by high clustering coefficients. Third, the characteristic path length is a graph measure to characterize the global network structure. It is defined as the minimum number of edges that have to be traveled between two vertices, which indicates how easily (or efficiently) information is transferred within the network.

Further information on other graph-theory measures can be found in the literature.18 It is important to choose an appropriate graph-theory measure that reflects the network characteristics of interest. The network measures can be calculated and the differences between groups or conditions can be compared to assess whether and how the network structure has changed, such as in pathological conditions.