Issues and considerations for using the scalp surface Laplacian in EEG/ERP research: A tutorial review.

Schematic sagittal view of the brain with two scalp electrodes (E1, E2). (A) Given a cortical dipole pointing with its negative pole towards E1, a negative potential is measured by the voltage meter for site E1 if site E2 serves as the recording reference (Ref). (B) For the same dipole, a positive potential is measured for E2 if E1is used as the reference. In either case, however, the measured potential difference indicates that E2 is more positive than E1.

Frequency spectra (3–15 Hz) at selected anterior (AFz, F3) and posterior (POz, P3) sites (A) and corresponding peak theta (5–6 Hz) and peak alpha (10–11 Hz) scalp distributions (B, C; mean of shaded frequency ranges in A) of an individual participant during an auditory working memory task. Shown are mean fast Fourier transform (FFT) amplitudes derived from 220 8-s epochs (72-channel EEG montage, 256 samples/s, 20% Hanning taper window, resolution 0.125 Hz) converted to common surface potential [µV] references (NR: nose reference; LM: link-mastoids reference; AR: average reference) and also transformed to current source densities (CSD [µV/cm2]; spherical spline parameters: m = 4, λ = 10−5, 50 iterations; Perrin et al., 1989). Anterior and posterior sites were selected to reflect both the topographical maximum of theta and alpha as well as nearby (off-maximum) locations (marked circles in B and C).

Grand mean waveforms (-60 to 800 ms) at selected midline sites (Cz, Pz) for target (A) and novel (B) stimuli and corresponding topographies of mean P3b (time interval 320–420 ms) and P3a (220–320 ms) amplitudes (C-F; shaded ranges in A and B) recorded during an auditory novelty oddball task (data of 49 healthy adults from Tenke et al., 2010). Compared are common surface potential [µV] references (NR: nose reference; LM: link-mastoids reference; AR: average reference) and the surface Laplacian transformations (CSD [µV/cm2]; spherical spline parameters: m = 4, λ = 10−5, 50 iterations; Perrin et al., 1989). Selected intervals and sites correspond to latency peaks and topographical maxima of P3b and P3a (marked circles in C-F).

Grand mean difference waveforms (-40 to 440 ms) at selected sites (TP9, FC3, Cz, FC6) derived from deviant-minus-standard tones (A) and corresponding mismatch negativity (MMN) topographies (B; mean amplitudes for time interval 100–200 ms as indicated by shaded ranges in A) recorded during a frequency (pitch) MMN paradigm (unpublished data of 8 healthy adults, 72-channel EEG montage, 256 samples/s; see text for further method details). Selected sites correspond to topographical MMN minima and maxima (marked circles in B), which differ between ERP and CSD data (ERP references and CSD transformations as in Fig. 11).

Response- (A) and stimulus-locked (C) grand mean waveforms (N = 17) recorded with a 129-channel geodesic sensor net EEG system (Tucker, 1993) during tonal and phonetic oddball tasks requiring a left or right response button press to target stimuli (data from Kayser and Tenke, 2006b). Baseline corrections for response- and stimulus-locked ERPs are −300 to −100 ms and −100 to 0 ms, respectively. Topographies correspond to a mid-frontal response-related negativity (FRN) peaking at about 50 ms post response (B: mean amplitude during time interval 25–75 ms; shaded ranges in A) and approximately 500 ms post stimulus onset (D: 420–580 ms; shaded ranges in C). Selected sites (see insets for electrode names in A and C) were left mid-central (43), left frontocentral (13), right frontocentral (107) and right mid-central (94) for response-locked waveforms, and left and right central (38, 88), mid-frontocentral (6) and mid-centroparietal (62) for stimulus-locked waveforms, which approximated CSD sink and source maxima (marked circles in B and D; ERP references and CSD transformations as in Fig. 11).

Schematic of interaction between the strength and orientation of a neuronal generator (modeled as a dipole) and choice of EEG reference. Assumed are two hypothetical EEG reference schemes in which reference (-) and recording (+) sites are either placed in line (A, C) or perpendicular (B, D) to the orientation of one dipole (green). A. Two dipoles of equal strength but different orientation will yield different surface potentials in favor of the green dipole which orientation best matches the alignment of reference and recording site. B. The same constellation of dipoles will favor the red dipole if reference and recording site are vertically aligned to the green dipole (i.e., rendering no potential difference between reference and recording site). C. Two dipoles of different strength but equal orientation will favor the green dipole if reference and recording site are aligned in parallel to the orientation of these dipoles. D. However, even in this scenario, a vertical alignment of reference and recording site to the orientation of these dipoles will render the same (i.e., zero) surface potential for either dipole.

Ratio of scalp measure to cortical potential as a function of dipole layer size. A, B: Vertex scalp potentials for a 81-channel 10–10 system EEG montage (average reference) due to simulated dipole layers of varying angular extent, forming superficial spherical caps in a four-shell forward solution (Berg, 2006) for two different ratios of brain-to-skull conductivity with the following conductivities [S/m]: brain = 0.33, skull = 0.0042 (A: ratio = 78.6) or 0.0084 (B: ratio = 39.3), CSF = 1.0, scalp = 0.33. C, D: Corresponding surface Laplacian estimates (CSD: current source density) using spherical splines of different flexibility (m = 2–7; λ = 10−5; Perrin et al., 1989). Note that surface potentials are primarily sensitive to broad dipole layers, whereas the sensitivity of spherical spline surface Laplacian estimates varies with spline flexibility.

Impact of the EEG reference (NR: nose reference; LM: linked-mastoids reference; AR: average reference) on grand mean (N = 44) ERP [µV] waveforms (-100 to 1100 ms, 100 ms pre-stimulus baseline) to foveal presentations of common words (A) or unknown faces (B) recorded during a continuous recognition memory task (data of healthy adults from Kayser et al., 2010). Enlargements of selected sites are shown on the right to highlight the substantial differences in ERP waveforms between references, although these can be observed at all sites. Rectangle enlargements directly compare ERPs referenced to linked mastoids at sites TP9 and TP10 (i.e., left and right mastoid), revealing symmetric activity with opposite polarity.

Selected ERP waveforms (colored lines, PO4: thick, T8: thin) for words (column 1) and faces (column 2) as shown in Fig. 2, and ERP difference waveforms (i.e., faces-minus-words; column 3), comparing three common EEG references (NR: nose reference; LM: linked-mastoids reference; AR: average reference; colors as in Fig. 2). The shaded areas and the black dashed lines depict the differences between the two ERPs shown in each subgraph (i.e., thick-minus-thin colored lines). Relative ERP differences between any two sites are unaffected by the EEG reference, as revealed by identical difference waveforms in each column (dashed lines). It is obvious that any attempt to quantify deflections (i.e., peak and troughs) of the colored lines will yield different results for each EEG reference, notwithstanding their invariance in topography across references.

First and (negative) second (spatial) derivatives of a data series measured at one time point (sample) across nine (scalp) locations (labeled A-I), with negative values plotted upwards. The resulting differences between consecutive data points (1st derivative) are plotted half-way between locations (gray squares). Analogously, the resulting (negative) differences between consecutive difference values (2nd derivative) are plotted half-way between these intermediate locations (i.e., the middle of a 3-point computation; red squares).

Local Hjorth differentiation grid consisting of 3–5 nearest neighbors for 67-channel EEG montage (cf. Fig. 2). Mutual and single (one-directional) neighbors are marked by blue and red arrows for each scalp site.

(A) Topographies of grand mean ERPs [µV] to word stimuli (Fig. 2A) at the peak latency for N1 (144 ms) and P3 (560 ms) referenced to nose (NR), linked mastoids (LM) or the average of all 67 sites (AR). (B) Local Hjorth estimates of these N1 and P3 topographies using differentiation grids consisting of 3–5 (cf. Fig. 5), 8–9, 24–25 or 66 nearest neighbors. All topographies were created from the ERP or local Hjorth values at each sites using a spherical spline surface Laplacian interpolation (m = 2; λ = 0; Perrin et al., 1989).

Data interpolation of discrete data values (green circles, top panel; cf. Fig. 4) and corresponding surface Laplacian estimates (negative second spatial derivative, bottom panel) using spherical splines of different flexibility (m = 2–5; λ = 10−5; Perrin et al., 1989). Because the equivalent spacing of the 9 locations labeled A-I is identical to the scalp surface distances of 9 positions in the coronal plane of the 10–10 system, ranging from T7 to T8 (Jurcak et al., 2007), the hypothetical data series may be conceived as an ERP topography having a negative maximum at vertex (e.g., N1 peak at Cz), with corresponding units of µV (surface potential) and µV/cm2 (surface Laplacian).

A, B. Grand mean ERP [µV] topography of auditory N1 (AR: average reference; unpublished data, N = 164, 112 ms peak amplitude, 80 dB tones presented during a loudness intensity paradigm) and corresponding surface Laplacian estimates (CSD: current source density [µV/cm2]) using spherical splines of different flexibility (m = 2–5; λ = 10−5; Perrin et al., 1989). Topographies were created for the original 72-channel EEG montage (A) and a subset consisting of 31 scalp locations (B). Gray circles mark scalp locations in the coronal plane. Due to the large differences in amplitude range associated with variations of spline flexibility, asymmetric scales were used to optimally represent the scalp distributions of ERP and CSD values; however, the ratios between positive and negative extremes are the same for each scale to preclude distortion of their relative value (i.e., green represents zero in all scales). C, D. Visual N1 and P3 (AR as shown in Fig. 2A) and corresponding CSD topographies of different spline flexibility (m = 2–5; λ = 10−5) using symmetric scales adjusted to the data range for each map.

Surface Laplacian estimates [µV/cm2] employing optimized smoothing for different spherical spline orders (m = 2–5). The optimal value for λ was separately determined for each spline flexibility by computing the cross-validation (CV) criterion from the individual auditory N1 topographies (N = 164) at 112 ms constituting the overall grand mean 72-channel ERP topography (Fig. 8A). Accordingly, the sum of the squared differences between the observed potential and the estimated potential (i.e., using a 71-channel spherical spline interpolation with m flexibility and λ smoothing) was repeatedly computed for each of the 164 topographies at any site for different λ values to determine the minimum of the underlying function corresponding to a particular value of m (cf. Pascual-Marqui et al., 1988; Stone, 1974).