How do reference montage and electrodes setup affect the measured scalp EEG potentials?

With head model and electrode coordinates, the lead field matrix ${\bf K}$ is calculated with the size of ${{N}_{{\rm e}}}$ (number of channels) by ${{N}_{{\rm s}}}$ (number of dipole sources). Given the activity strengths x of all dipole sources at one moment, the potentials over all electrodes are generated by the equation,

$$\begin{align} \displaystyle{\bf v_\infty}={\bf K}\cdot{\bf x}.\nonumber \end{align} \tag{ 1 }$$

It should be noted that the lead field matrix K is with infinity reference (IR) [32, 33]. Namely, the association between source activities and multichannel potentials is based on a neutral reference at infinity. Here, the EEG generative model is a linear combination of the activities of all dipole sources at each instant. Thus, the EEG potentials are fully determined by the sources at the same instant under the given head model and electrodes setup. Since reference problem is due to the activity of a dipole source arriving at both active electrode and reference site, it is therefore unnecessary to consider the temporal process. We just consider the potentials at one instant in this study. In the following, all the dipole sources are evaluated in turn to minimize the biased effect due to the subjective selection of source positions. Thus, totally ${{N}_{{\rm e}}}$ by ${{N}_{{\rm s}}}$ standard potentials are generated for the error comparison. Certainly, due to the linearity of the equation (1), any combination of different sources can be evaluated similarly.

Realistically, it is inevitable to mix the sensor noise into the recorded EEG signal. We adopt the zero-mean multivariate Gaussian distribution to generate the sensor noise e that may come from various unknown sources. Thus, the EEG generative model (1) changes to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\{\begin{array}{@{}llllllllllllllllllllll@{}} {{{\tilde{{\bf v}}}}_{\infty }}={{{\bf v}}_{\infty }}+{\bf e},{\bf e}\sim N({\bf 0},{{{\bf I}}_{{{N}_{{\rm e}}}}}) \nonumber \\ {\rm SNR}=10{{\log }_{10}}({\left\Vert {{{\bf v}}_{\infty }} \right\Vert_{2}^{2}}/{\left\Vert {\bf e} \right\Vert_{2}^{2}}\;) \nonumber \\ \end{array} \right.\nonumber \end{align} \tag{ 2 }$$

where 0 is a ${{N}_{{\rm e}}}$ by 1 zeros vector and ${{{\bf I}}_{{{N}_{{\rm e}}}}}$ are a ${{N}_{{\rm e}}}$ by ${{N}_{{\rm e}}}$ identity matrix; ${\rm SNR}$ is the power ratio of the pure EEG signal ${{{\bf v}}_{\infty }}$ to the sensor noise e in dB unit; $\left\Vert \cdot \right\Vert_{2}^{2}$ means the square of the ${{l}_{2}}$ norm.

We investigated 11 channel numbers and two types of electrodes layouts which may affect the accuracy of the EEG potentials. They are 10, 16, 21, 32, 64, 85, 96, 128, 335 channels with '10-x' layout, and 129, 257 channels with 'GSN' layout displayed in figure 2. In detail, about the 3D electrode Cartesian coordinates, 10, 16 and 21 channels are picked from 10-20 system [20]; 32, 64, 96 and 128 channels with '10-10 c' layout are selected from ActiCHamp 128Ch Standard (Brain Products Co. Ltd, Germany); 129 and 257 channels are from the HydroCel^™ Geodesic Sensor Net E001 (Electrical Geodesics, Inc., USA); 64 channels with '10-10 n' layout, 85 channels with '10-10 r' layout and 335 channels are from the 10-05 system [18].

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In general, all the reference transforming processes can be formulated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{{\bf v}}_{r}}={\bf T}\cdot {{{\bf v}}_{\infty }}\nonumber \end{align} \tag{ 3 }$$

where ${{{\bf v}}_{\infty }}$ is the EEG potentials with infinity reference, T denotes reference transforming matrix that is specific reference dependent, ${{{\bf v}}_{{\rm r}}}$ is the potentials transformed with a reference notated by $r$.

2.4.1. Recording references.

The principle of mono-polar recording reference is to subtract the potential at the mono-polar physical reference electrode from each of the active electrodes. Thus,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf T}={{{\bf T}}_{r}}, {{{\bf T}}_{r}}={{{\bf I}}_{{{N}_{{\rm e}}}}}-{\bf 1}\cdot {{{\bf f}}^{{\bf T}}}, \ {\bf f}={{[ 0 ~ \cdots ~ 1 ~ \cdots ~ 0]}^{{\bf T}}}\nonumber \end{align} \tag{ 4 }$$

where 1 is a ${{N}_{{\rm e}}}$ by $1$ ones vector, and f is a ${{N}_{{\rm e}}}$ by $1$ vector with only one non-zero entry (i.e. 1) at the corresponding row of a mono-polar reference, such as LE, Fz, Pz, Oz, and Cz.

2.4.2. Re-references.

Re-reference is a further step based on the measured EEG potentials where the online recording reference process was involved inherently.

Linked mastoids (LM)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf T}=({\bf I}-{\bf 1}\cdot {{{\bf f}}^{{\bf T}}})\cdot {{{\bf T}}_{r}}, \ {\bf f}={{\left[ 0 ~ \cdots ~ 0.5 ~ \cdots ~ 0.5 ~ \cdots ~ 0 \right]}^{{\bf T}}}\nonumber \end{align} \tag{ 5 }$$

where f is the vector with two non-zero entries (i.e. 0.5) at the corresponding rows of left and right mastoids.

Average reference (AR)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf T}=({\bf I}-{\bf 1}\cdot {{{\bf f}}^{{\bf T}}})\cdot {{{\bf T}}_{r}}, \ {\bf f} = {{\left[{1}/{{{N}_{{\rm e}}}} ~ \cdots ~ {1}/{{{N}_{{\rm e}}}}\right]}^{{\bf T}}}\nonumber \end{align} \tag{ 6 }$$

where f is with all the entries being ${1}/{{{N}_{{\rm e}}}}\;$.

Reference electrode standardization technique (REST)

Plugging the equations (1) and (4) into (3), we have ${{{\bf v}}_{r}}={{{\bf T}}_{r}}\cdot {\bf K}\cdot {\bf x}$. According to the equivalent source principle [11, 25], the scalp potential ${{{\bf v}}_{\infty }}$ generated by x through lead field K can be generated equivalently by the equivalent sources x₁ through lead field ${{{\bf K}}_{1}}$, i.e.,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{{\bf v}}_{\infty }}={\bf K}\cdot {\bf x}={{{\bf K}}_{1}}\cdot {{{\bf x}}_{1}}.\nonumber \end{align} \tag{ 7 }$$

Here, ${{{\bf x}}_{1}}$ could be the equivalent distributed sources over a 2D cortical sheet that covers the actual sources inside, the 3D distributed sources overlapped with the actual sources, or the multipolar sources at the origin of a coordinate system [25]. For REST, although the true sources x and true lead field K are unknown, we can take the supposed equivalent sources x₁ and the related lead field K₁ [11, 12]. In this work, 3D grid dipole sources with orthogonal orientation and cortical dipole sources with fixed radial orientation are adopted as x₁ for the spherical head model and the realistic head model, respectively.

As the EEG inverse solution is reference independent [34, 35], the approximate estimate of x₁ is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\hat{{\bf x}}}_{1}}={{({{{\bf T}}_{r}}\cdot {{{\bf K}}_{1}})}^{+}}\cdot {{{\bf v}}_{r}}.\nonumber \end{align} \tag{ 8 }$$

By substituting the equations (8) and (3) into (7), the EEG potentials ${{{\bf v}}_{\infty }}$ with infinity reference could be reconstructed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{{\bf v}}_{{\rm rest}}}={{{\bf K}}_{1}}\cdot {{\hat{{\bf x}}}_{1}}={{{\bf K}}_{1}}\cdot {{({{{\bf T}}_{r}}\cdot {{{\bf K}}_{1}})}^{+}}\cdot {{{\bf T}}_{r}}\cdot {{{\bf v}}_{\infty }}.\nonumber \end{align} \tag{ 9 }$$

Therefore, the reference transforming matrix for REST is ${\bf T}={{{\bf K}}_{1}}\cdot {{({{{\bf T}}_{r}}\cdot {{{\bf K}}_{1}})}^{+}}\cdot {{{\bf T}}_{r}}$.

For clarity, all the reference montages are summarized in table 1.

Table 1. Reference transforming based on the infinity reference.

Unified reference process	${{{\bf v}}_{r}}={\bf T}\cdot {{{\bf v}}_{\infty }}$
Reference classification	Online recording reference	Offline re-references
Reference montages	LE, Fz, Cz, Pz, Oz	REST	LM	AR
Reference transformation matrix	${{{\bf T}}_{r}}={\bf I}-{\bf 1}\cdot {{{\bf f}}^{{\bf T}}}$	${{{\bf K}}_{1}}\cdot {{({{{\bf T}}_{r}}\cdot {{{\bf K}}_{1}})}^{+}}\cdot {{{\bf T}}_{r}}$	$({\bf I}-{\bf 1}\cdot {{{\bf f}}^{{\bf T}}})\cdot {{{\bf T}}_{r}}$
Linear combination weights $\boldsymbol{f}$	(4)		(5)	(6)

Essentially, the performance of REST depends on the exactitude of head model, since ${{{\bf K}}_{1}}$ is changed from K accordingly with the equivalence between ${{{\bf x}}_{1}}$ and x. Assuming that ${{{\bf K}}_{1}}$ is approximate to K, we are curious about whether REST is sensitive to a perturbation of head model. In previous studies, the perturbation is subjectively added to electrode locations [15] or by means of changing the conductivities [11, 12]. This subjective perturbation is retested over 64 channels ('10-10 c' layout) by recalculating ${{\tilde{{\bf K}}}_{1}}$ as the perturbated ${{{\bf K}}_{1}}$ with various combinations of two head shapes, 11 skull conductivities and five source configurations detailed in section 3.7.

To be more general, ${{{\bf K}}_{1}}$ is given an objective perturbation supposed to be together from imprecise estimates of head shape, conductivity, electrode location and other unknown factors that are difficult to concretely quantify in practice. By putting the equations (1) and (9) together, it becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\{\begin{array}{@{}lllllllllllllllllllll@{}} {{{\bf v}}_{\infty }}={\bf K}\cdot {\bf x} \nonumber \\ {{{\bf v}}_{{\rm rest}}}={{{\bf K}}_{1}}\cdot {{({{{\bf T}}_{r}}\cdot {{{\bf K}}_{1}})}^{+}}\cdot {{{\bf T}}_{r}}\cdot {{{\bf v}}_{\infty }} \nonumber \\ \end{array} \right.\nonumber \end{align}$$

Ideally, both ${\bf K}$ and ${{{\bf K}}_{1}}$ should be computed by accurate head model and electrode layout. Here, ${\bf K}$ is calculated with the accurate head model and true sources, and ${{{\bf K}}_{1}}$ should be calculated with a disturbed head model and assumed equivalent sources. To add perturbation, zero mean multivariate Gaussian noise ${\bf e}_{1}^{\,j}$ is injected to the lead field vector ${\bf k}_{1}^{\,j}$ corresponded to the $j{\rm th}(\,j=1,...,{{N}_{{\rm s}}})$ dipole source of ${{{\bf K}}_{1}}$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\{\begin{array}{@{}lllllllllllllllllllllllll@{}} \tilde{{\bf k}}_{1}^{\,j}={\bf k}_{1}^{\,j}+{\bf e}_{1}^{\,j},{\bf e}_{1}^{\,j}\sim N({\bf 0},{{{\bf I}}_{{{N}_{{\rm e}}}}}) \nonumber \\ {\rm SN}{{{\rm R}}_{1}}=10{{\log }_{10}}({\left\Vert {\bf k}_{1}^{\,j} \right\Vert_{2}^{2}}/{\left\Vert {\bf e}_{1}^{\,j} \right\Vert_{2}^{2}}\;) \nonumber \\ \end{array} \right.\nonumber \end{align}$$

where ${\rm SN}{{{\rm R}}_{1}}$ denotes the power of ${\bf k}_{1}^{\,j}$ to the power of noise ratio in dB unit.

Accordingly, the REST reconstruction model is transformed to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\{\begin{array}{@{}llllllllllllllllllllllllllll@{}} {{{\bf v}}_{\infty }}={\bf K}\cdot {\bf x} \nonumber \\ {{{\tilde{{\bf v}}}}_{{\rm rest}}}={{{\tilde{{\bf K}}}}_{1}}\cdot {{({{{\bf T}}_{r}}\cdot {{{\tilde{{\bf K}}}}_{1}})}^{+}}\cdot {{{\bf T}}_{r}}\cdot {{{\bf v}}_{\infty }} \nonumber \\ \end{array} \right..\nonumber \end{align} \tag{ 10 }$$

Here, we take the change from ${{{\bf K}}_{1}}$ to ${{\tilde{{\bf K}}}_{1}}$ as the model perturbation.

Given by a head model, the values of lead field are varied with channel number and electrodes layout. By setting the activity of single dipole source with unit strength and the others being zero, the multichannel EEG potentials numerically equal to the lead field vector corresponding to that dipole source. Therefore, the EEG potentials are affected by the electrodes setup. To evaluate the effects, the relative error is taken as a measure for all channels or the partial sensors over one local scalp region. The relative error of the potentials generated by the $j{\rm th}$ dipole is as follows,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm re}(\,j)={\sqrt{\sum\nolimits_{i=1}^{n}{{{({{{\bf v}}_{r}}(i)-{{{\bf v}}_{\infty }}(i))}^{2}}}}}/{\sqrt{\sum\nolimits_{i=1}^{n}{{{({{{\bf v}}_{\infty }}(i))}^{2}}}}}\;\nonumber \end{align}$$

where $n$ is the channel number (i.e. array size) or the number of partial sensors over one local scalp region, ${{{\bf v}}_{\infty }}$ is the forward EEG potentials with one electrodes setup, ${{{\bf v}}_{r}}$ is the EEG potentials with the specific reference. To avoid the bias of source position, we calculate the mean relative error as to numerous dipole sources at the population level rather than a single dipole at the individual level. In the source space or a source region of interest, the relative error (RE) is redefined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm RE}={\sum\nolimits_{j=1}^{{{N}_{{\rm d}}}}{{\rm re}(\,j)}}/{{{N}_{{\rm d}}}}\;\nonumber \end{align}$$

where ${{N}_{{\rm d}}}$ is the total number of the dipoles in the source space or one source region. Furthermore, the standard deviation (SD) is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm SD}=\sqrt{{\sum\nolimits_{j=1}^{{{N}_{{\rm d}}}}{({\rm re}(\,j)}-{\rm RE}{{)}^{2}}}/{({{N}_{{\rm d}}}-1)}\;}\nonumber \end{align}$$

to evaluate the robustness as to the dipole source position.

The image with scaled colors in figure 3 shows the REs of the potentials over all sensors as to eight reference montages. Without re-reference, the RE is above 61.5% by the mono-polar references (LE, Fz, Pz, Oz, Cz); it reduces to around 41% after re-reference by LM. Especially, the utilization of AR and REST leads to less biased potentials with RE being around 17% and 2.7%, respectively. This means (1) re-reference is indispensable to correct the large biased impacts of mono-polar recording reference; (2) both REST and AR have superiorities to LM; (3) REST is the best among all the reference montages to correct the potentials over all sensors. Eight topographies around the central one with infinity reference (IR) illustrate the effects of reference montages over 257 channels. The central one shows the distribution and strength of standard potentials generated with five cortical dipoles. After reference transforming, the potentials with eight references are plotted in the same colorbar as that with IR for comparison. Apparently, relatively large color change in the topographies referenced by LE, Fz, Pz, Oz, and Cz indicates the distortion effects of online recording reference, while three topographies in the bottom show the correction by offline re-references.

From this section, the comparison will be focused on REST and AR due to their superiorities. The effect of channel number is evaluated in terms of sparse arrays and dense arrays. In addition to all sensors (AS), we calculated the REs over partial sensors as well. Based on the topological conventions, AS is portioned into partial sensors, that is, frontal sensors (FS), temporal sensors (TS), parietal sensors (PS) and occipital sensors (OS). 2D electrodes layout in figure 4 shows how we divided 64 sensors via the red lines. Due to the symmetry between the left TS and the right one, only the RE over the left TS is considered.

Since Y-axes in figure 4 are in 10-base logarithm scale, it is therefore evident that AR has much higher REs than REST over AS and over partial sensors with dense arrays. Over partial sensors with sparse arrays, the REs of REST are still much lower than the REs of AR with 21 and 32 channels; with 10 and 16 channels, the REs of REST are large but still less than the REs of AR. This means that REST enables to outperform AR even with quite sparse arrays. The less REs of REST than AR are found not only over AS but also over partial sensors. These indicate that REST may always correct the EEG potentials with a better performance than AR without the effect of channel number.

The REs of AR do not present the trends of linearly decreasing with increased channel numbers. It implies that dense channels may fail to improve the performance of AR. Namely, channel number or sensor density is not critical to AR. This finding is not in agreement with the general consensus that the advantage of AR only comes into a large number of electrodes and extensive coverage of the head [36]. Lastly, the SDs of REST are always smaller than the SDs of AR. Together to say, REST is more robust than AR in repeated simulations regardless of channel number and scalp region.

Much attention should be paid to the dense arrays from 128 to 335 channels, considering the entirely dissimilar electrodes layouts. Figure 5 shows the comparisons of REs between'10-x' layout with 'BP128', 'BP335' setups and 'GSN' layout with 'EGI129', 'EGI257' setups. Each setup is displayed on the spherical and realistic head surfaces to offer an intuitive view of its placement. 'BP128' and 'BP335' are following '10-10 c' and '10-05' systems, with only upper surficial area of the head evenly covered by plenty of electrodes. By contrast, in addition to the different electrodes positioning rule, extensive areas such as the cheek and neck are covered with 'EGI129' and 'EGI257'. REs in figure 5 as to four setups are 0.50%, 0.14%, 0.05%, 0.27% for 's REST' and 19.56%, 11.50%, 5.67%, 15.65% for 's AR', respectively. It is apparent that both REST and AR produce lower RE as to 'GSN' layout compared with '10-x' layout. This shows that the extensive coverage benefits to REST but mainly brings about the great improvements to AR. With almost same channel number but electrodes positions rearranged from 'BP128' to 'EGI129' layout, the RE of AR declines from 19.56% to 11.50%. Whereas, adding 200 more electrodes to 'BP128' and forming into 'BP335' with a quite high spatial sampling and dense coverage, we found that the RE of AR only showed a slight drop from 19.56% to 15.65%. However, the RE of AR with 'EGI257' reduces to be nearly 1/3 of the RE of 'BP335' but with only around 80 (i.e. 335-257) channels removed. Similar results are obtained if the realistic head model (r) is used in place of the three-layer concentric spherical head model (s). These comparisons reveal that on one hand, the key factor to the effect of AR is extensive coverage rather than the large channel number; on the other hand, REST has the appreciably superiority to AR when playing with large number of electrodes.

Given electrodes setup and head model, the EEG potential is still affected by the dipole source position. We classify 2600 cortical dipoles and 1269 * 3 orthogonal grid dipoles into four regions based on the conventional division of brain volume. The semi-sphere in figure 6 shows how all dipole sources are divided into fontal dipoles (FD), temporal dipoles (TD), parietal dipoles (PD) and occipital dipoles (OD), individually. The bar plots in figure 6 are with respect to the REs of potentials over all sensors generated by the dipoles located in one source region of interest. The REs generated by FD, TD, PD and OD are shown in the bar plots from top left to bottom right in turn. The REs of REST and AR are shown in blue and yellow bars, respectively. Overall, REST always produces much less REs than AR as to whichever dipole source region.

In addition to source position, its activated orientation may have serious effects on the accuracy of acquired potentials. Figure 7 shows the REs of the potentials generated by a radial cortical dipole or a grid dipole activated along one of XYZ orthogonal directions, separately. The REs between REST and AR do not manifest such considerable difference in X and Y orientations as that in Z orientation, whereas it shows more prominent similarity between REST and AR in Y orientation than that in X orientation. For the radial cortical dipoles, the extent of RE difference between REST and AR seems to be the compromise of the differences along X, Y and Z orientations. These results may be related to the goodness of symmetry of electrodes position. Close attention is paid back to figure 2 where we see the perfect symmetry between the left and right regions, quite good but imperfect symmetry for the front and back regions, extremely poor symmetry for the top and bottom regions if the 2D electrodes layouts converse to 3D layouts over a spherical or realistic head model shown in figure 5. Additionally, the similar REs as to dipole source orientation reveals that REST is robust to correct the potentials. Much larger REs in radial and Z orientation than X and Y orientations of AR may warn us that AR is subject to dipole source orientation and it is only suitable for dipole sources oriented in the transverse plane.

Figure 8 shows the comparison of sensitivity to sensor noise between REST and AR. The case without sensor noise is corresponded to the SNR  =  Inf dB. From brick red triangles to blue ones, SNRs are 40, 30, 20, 10, 8, 4 and 2 dB, respectively. Markers in the same color are connected to each other to display the waving trends of REs with the change of channel number. It is obvious that the gaps between the dash curves in figure 8(a) are larger than the gaps in figure 8(b). The dash curves in figure 8(a) are lower than the curves in figure 8(b), except for the extensive coverage by 257 channels with SNR  <  20 dB and the cases of SNR  ⩽  4 dB where AR reaches a little better effect than REST with some array sizes, such as 21, 64, 96 and 257. These suggest that (1) REST may be of larger sensitivity to sensor noise than AR; (2) REST could be preferred if the EEG is mixed with little noise (3) AR may be acceptable for high noisy situation, due to the possible denoising effects by simply averaging.

Figure 9(a) displays three-layer concentric spherical head model (s) and ICBM152 realistic head model (r). We conducted the same simulation and analysis by using 'r' head model as that by using 's' head model. The REs comparison between REST and AR based on the potentials generated by the 'r' and 's' head models are shown in figure 9(b). Big gap between the dash curves in red and purple proves that 'r REST' still enables to obtain quite lower REs than 'r AR' for all the channel numbers. It suggests that the validity of REST may not rely on the complexity of head shape. To check if REST depends on the exactitude of head shape, skull conductivity and source configuration when the truths of these three factors are unknown for someone who underwent the EEG recording, the standard potentials with infinity reference were generated with 'r' head shape, skull to brain conductivity being 0.0125, source configuration as (1) in the black bar of figure 9(c), but the EEG potentials were reconstructed by REST with various combinations of head shape ('s', 'r'), skull to brain conductivity ratio (0.0001, 0.0005, 0.0016, 0.0032, 0.0063, 0.0125, 0.025, 0.05, 0.1, 0.5, 1), and source configuration (2)–(6). Apparently, the superiority of REST over AR can be kept with the simple 's' head shape, skull to brain conductivity being 0.0001, source configuration as the quite different source configurations (2)–(6). It is reasonable to see that the REs of REST decreased with better approximation of head shape, skull to brain conductivity ratio, and source configuration adopted in simulation. To mimic more general situations of the imprecise head model, we tested head model perturbation for 'r REST' by injecting noise. The test results are shown in figure 9(d). Obviously, 'r REST' still enables to have lower REs than 'r AR' even with twice noise (−3dB) injected into the lead field ${{{\bf K}}_{1}}$ used for REST. Overall, these results may reveal that REST is insensitive to the perturbation of head model and its advantage can be kept even with the imprecise head model.