Seed-based dual regression: An illustration of the impact of dual regression’s inherent filtering of global signal

Abstract

Background:

Functional connectivity (FC) maps from brain fMRI data are often derived with seed-based methods that estimate temporal correlations between the time course in a predefined region (seed) and other brain regions (SCA, seed-based correlation analysis). Standard dual regression, which uses a set of spatial regressor maps, can detect FC with entire brain “networks,” such as the default mode network, but may not be feasible when detecting FC associated with a single small brain region alone (for example, the amygdala).

New method:

We explored seed-based dual regression (SDR) from theoretical and practical points of view. SDR is a modified implementation of dual regression where the set of spatial regressors is replaced by a single binary spatial map of the seed region.

Results:

SDR allowed detection of FC with small brain regions.

Comparison with existing method:

For both synthetic and natural fMRI data, detection of FC with SDR was identical to that obtained with SCA after removal of global signal from fMRI data with global signal regression (GSR). In the absence of GSR, detection of FC was significantly improved when using SDR compared with SCA.

Conclusion:

The improved FC detection achieved with SDR was related to a partial filtering of the global signal that occurred during spatial regression, an integral part of dual regression. This filtering can sometimes lead to spurious negative correlations that result in a widespread negative bias in FC derived with any application of dual regression. We provide guidelines for how to identify and correct this potential problem.

1. Introduction

Seed-based connectivity analysis (SCA) (Biswal et al., 1995; Fox et al., 2005; Friston, 2002; Huettel et al., 2014; Van Dijk et al., 2010) is commonly employed for deriving brain functional connectivity (FC). FC, often loosely referred to as brain “networks” (Huettel et al., 2014), describes the mapping of coactivated brain regions. Coactivated regions are identified with SCA by mapping the degree of correlation between time courses of brain voxels and the average time course in selected “seed” voxels (Biswal et al., 2010; Fox et al., 2005; Smith et al., 2014). A limitation of SCA is that target networks of interest, such as the “default mode” network, frequently span large, multifocal areas, yet their FC is typically derived from one or more small seeds (Fox et al., 2005); so choice of seed locations within this spatially disparate network can meaningfully impact FC results (Cole et al., 2010).

Dual regression is an alternative method for deriving FC that allows representation of target networks by complete brain maps as regressors for the spatial regression step of dual regression (Biswal et al., 2010; Huettel et al., 2014; Kelly Jr. et al., 2019; Kelly et al., 2010; Smith et al., 2014). These spatial regressors can also represent the spatial distributions of noise sources, whose time courses in the temporal regression step perform a denoising function via their use as nuisance regressors (Kelly Jr. et al., 2019). As originally proposed, spatial regressors for “standard” dual regression are derived from group independent component analysis (ICA) on the temporally concatenated fMRI sessions whose FC is to be computed (Beckmann et al., 2009). However, complete and detailed brain maps become unwieldy in cases where FC is derived with respect to a specific brain structure, such as the amygdala (Hoptman et al., 2010; Morey et al., 2009; Stein et al., 2007). For such cases, a simplified form of dual regression can be applied, seed-based dual regression (SDR), using only a binary spatial map of SCA seed voxels instead of the group-ICA derived spatial regressors.

Although SDR is endorsed on FSL’s website as an equivalent and potentially convenient alternative to SCA (FMRIB Software Library, 2017), some researchers have found that FC results with SDR differ from those with SCA and seem empirically preferable (personal communication from Alessandro Gozzi). Nonetheless, these researchers have been reluctant to publish studies using SDR due to the lack of peer-reviewed studies concerning the method. In our literature review, we found only two applications of SDR, one for clinical research (Cerit, 2015) and one for testing a different method (Chen et al., 2020).

The current paper provides a peer-reviewed study in support of the usefulness of SDR, with a systematic analysis of potential benefits and limitations of SDR, so that SDR can be properly applied in scientific research. We demonstrate that differences between FC results derived from SCA and SDR are caused by partial filtering of the global signal that occurs during spatial regression. We show that in the absence of fMRI preprocessing with global signal regression (GSR) or other denoising strategy that removes global signal, SDR can provide more accurate FC in some cases via intrinsic denoising. In addition, we demonstrate that dual regression implicitly filters global signal, using SDR as an example.

Filtering global signal, for example with GSR, can reveal meaningful anticorrelations, such as those between task-positive and default mode networks (Fox et al., 2005; Fox et al., 2009). However, like other techniques for removing global noise, SDR can negatively bias FC estimates in some cases by generating spurious negative correlations between time courses for seed voxels and other brain voxels (Jezzard et al., 2001; Murphy and Fox, 2017). This potential for negative FC bias is a byproduct of spatial regression and must therefore be considered when using any variant of dual regression.

To demonstrate these properties of SDR, we derive them from theory and confirm them empirically by comparing FC derived with SDR and SCA, with and without global signal regression, using synthetic data and natural task-based fMRI data. We provide illustrative examples that show how spurious negative correlations can in some cases be generated by SDR, as well as by other variants of dual regression, including standard dual regression and single-map dual regression, which uses only a spatial map of interest as its spatial regressor rather than all maps derived from group ICA (Kelly Jr. et al., 2019). Finally, we show how to identify and rectify problematic cases influenced by spurious negative correlations.

2. Theory

In this section we elucidate three main points.

Compared with SCA, SDR reduces the positive bias in functional connectivity due to global signal, because dual regression filters global signal.

This filtering of global signal can be mimicked by a modification of SCA, SCAx, where the mean global signal for brain voxels across volumes is subtracted from the seed time course prior to calculating correlations with voxel time courses. Functional connectivity derived from SCAx is identical to that derived from SDR.

FC derived from SCA becomes identical to that derived from SDR when the global signal is held constant at each time point.

For the discussions that follow, let Y be the N by n matrix presenting the fMRI data from all voxels in the brain, where N and n are the number of voxels and time points, respectively. Consider that each voxel’s functional connectivity derived from SCA can be represented by the correlation (r_si) between the time course for voxel i (y_i; i = 1,2,…,N) and the seed time course $(\mathit{s}=\frac{1}{N_S}{\sum }_{i=1}^{N_S}{\mathit{s}}_i)$, mean of the time courses of N_s voxels $\left\{{\mathit{s}}_1,{\mathit{s}}_2,\dots ,{\mathit{s}}_{N_S}\right\}$ in the seed region:

$$r_{si}=Corr\left(\mathit{s},{\mathit{y}}_i\right)=E\left[\frac{(\mathit{s}-E[\mathit{s}])}{{\sigma }_{\mathit{s}}}\cdot \frac{\left({\mathit{y}}_i-E\left[{\mathit{y}}_i\right]\right)}{{\sigma }_{{\mathit{y}}_i}}\right].$$ (1)

This FC can also be represented by the parameter estimate (β_si) derived from the regression

$${\mathit{y}}_i={\beta }_{si}\mathit{s}+c_i+{\mathit{e}}_{si}$$ (2)

where β_si and c_i are derived from minimizing the sum of the squares of the error terms in e_si. These forms of FC are entirely equivalent: A one-to-one correspondence exists between r_si and β_si, given by

$$r_{si}=\left({\sigma }_{\mathit{s}}/{\sigma }_{{\mathit{y}}_i}\right)\cdot {\beta }_{si}$$ (3)

where σ_s and ${\sigma }_{{\mathit{y}}_i}$ are the sample standard deviations of s and y_i (Cohen, 1988). In addition, FC maps in terms of r_si or β_si convert to identical z-score maps that reflect the probabilities of obtaining larger values by chance for r_si or β_si, respectively (Kelly Jr. et al., 2019; Kelly et al., 2010).

Equation 2 facilitates comparison of SCA and SDR because of its similarity with SDR’s temporal regression,

$${\mathit{y}}_i={\beta }_{mi}\mathit{m}+c_i+{\mathit{e}}_{mi}$$ (4)

where $\mathit{m}={\left[m_1,m_2,\dots ,m_n\right]}^{\top }\in {ℝ}^{n\times 1}$ represents the time course derived from spatial regression, and β_mi and c_i are derived from minimizing the sum of the squares of the error terms in e_mi.

2.1. Dual regression filters global signal

Consider spatial regression using a single spatial map $\mathit{p}={\left[p_1,p_2,\dots ,p_N\right]}^{\top }\in {ℝ}^{N\times 1}$, regressed onto fMRI data whose volume at time t is represented by ${\mathit{x}}_t={\left[x_{t1},x_{t2},\dots ,x_{tN}\right]}^{\top }\in {ℝ}^{N\times 1}$:

$${\mathit{x}}_t=m_{pt}\mathit{p}+c_t+{\mathit{e}}_{pt}$$ (5)

where m_pt and c_t are derived from minimizing the sum of the squares of the error terms in e_pt. m_p = [m_p1, m_p2, …., m_pn] would then represent the time course for dual regression (Equation 5, regressing p onto x₁, x₂, …, x_n). After adding a global signal $\mathit{g}=\left[g_1,g_2,\dots ,g_n\right]\in {ℝ}^{n\times 1}$ uniformly to all brain voxels, resulting in ${\mathit{x}}_t^{\prime }={\mathit{x}}_t+g_t\mathbf{1}$, the standard deviations (${\sigma }_{{\mathit{y}}_i}$ in Equation 3) of ${\mathit{x}}_t^{\prime }$ and x_t would be equal; and from Equation 3 the corresponding time course derived from spatial regression, ${\mathit{m}}_p^{\prime }$, would not differ from m_p because the correlations between ${\mathit{x}}_t^{\prime }$ and p would remain the same as for x_t and p:

$$r_{{\mathit{x}}_t^{\prime }p}=Corr\left({\mathit{x}}_t^{\prime },\mathit{p}\right)=Corr\left({\mathit{x}}_t+g_t\mathbf{1},\mathit{p}\right)=Corr\left({\mathit{x}}_t,\mathit{p}\right)=r_{x_{t}p}$$ (6)

This means that time courses derived from spatial regression remain the same after preprocessing with global signal subtraction. This principle also applies for standard dual regression, which would not affect m_p if spatial regressors added to Equation 5 are mutually orthogonal (Kelly Jr. et al., 2019).

Because y_i retains the global signal, while it is reduced or possibly negative (anticorrelated) in m_p, the positive bias in functional connectivity due to global signal is reduced with dual regression compared with SCA. However, as for other methods that filter global signal (Murphy and Fox, 2017), the possibility of spuriously inducing negative correlations must be considered when deriving FC with dual regression.

2.2. SCAx yields FC results identical to those derived from SDR

If g represents the mean brain global signal, then FC for SCAx at each voxel i can be represented by

$$r_{s^{*}i}=Corr\left(\mathit{s}-\mathit{g},{\mathit{y}}_i\right)$$ (7)

Appendix A shows that $r_{s^{*}i}$ is the same as the r-value for FC derived from SDR for each voxel. Equation 7 bears a similarity to FC for SCA after global signal subtraction (GSS), which becomes

$$r_{s^{*}i}=Corr\left(\mathit{s}-\mathit{g},{\mathit{y}}_i-\mathit{g}\right)$$ (8)

In the general case, $r_{s^{*}i}$ differs from r_si and $r_{s^{*}{i}^{*}}$ (see 4.1. Illustrative Example).

2.3. SDR and SCA yield identical FC results for constant global signal

If the mean voxel value across all brain voxels does not vary with time, then g can be replaced with a constant term $g_{c}1\in {ℝ}^{n\times \mathbf{1}}$ in equation 7, yielding

$$r_{s^{*}i}=Corr\left(\mathit{s}-g_c\mathbf{1},{\mathit{y}}_i\right)=Corr\left(\mathit{s},{\mathit{y}}_i\right)=r_{si}$$ (9)

In summary, FC derived from SDR is always identical to that derived from SCAx and is also identical to that derived from SCA in the special case where fMRI preprocessing causes the mean brain voxel intensity for each volume to be held constant. Examples of such preprocessing include global signal regression, global signal subtraction, and global signal normalization (Liu et al., 2017)(Appendix B). For dual regression in general, the fact that the time course derived from spatial regression is independent of the global signal, i.e., does not change after adding global signal, implies that spatial regression filters global signal. For SDR, this filtering results in a derived time course that is directly and positively proportional to the time course derived from SCA minus the global signal (SCAx), i.e., = k(s – g) for constant k > 0 (Appendix A). Consequently, the corresponding FC maps for SDR and SCAx are identical.

3. Materials and methods

3.1. Overview

To demonstrate potential advantages of using SDR in place of SCA, we evaluated performance of SDR compared with SCA in three experiments, based on how well peak values in FC maps (z-score) agreed with peak values in a known standard-of-comparison (SOC) map (Fig. 1). The measure of agreement was proportion detected (PD), the proportion of “true” voxels in the SOC that are detected in the FC map after maximum-height thresholding the FC map to make the number of false positive and false negative voxels equal (Kelly Jr. et al., 2019; Kelly et al., 2010). We chose this measure of FC detection accuracy, because it is independent of factors that can elevate the number of voxels selected at any given threshold, such as autocorrelation (Davey et al., 2013; Friston et al., 1995; Woolrich et al., 2001), global signal (Satterthwaite et al., 2013), and structured noise (Satterthwaite et al., 2012; van Dijk et al., 2012).

Fig. 1.

Comparison of seed-based correlation analysis (SCA) with seed-based dual regression (SDR), before and after global signal regression (GSR), using synthetic (Experiments 1–2) and natural (Experiment 3) fMRI data. Flowchart key: Inputs–blue rectangular boxes. Actions–yellow boxes with rounded corners.

In Experiments 1 and 2, we tested the capacity to detect an artificial signal of interest in synthetic data that also contained a single, globally distributed artificial noise signal as well as locally-distributed artificial “structured” noise signals. In Experiment 1, the globally distributed noise signal was uniform throughout the brain, while in Experiment 2 it was concentrated in regions of the brain that contained the signal of interest, thereby better simulating natural fMRI data, where physiological noise is generally more intense in gray matter (Krüger and Glover, 2001). Experiment 3 was the natural condition, using task-fMRI data and establishing an SOC derived with the general linear model (GLM) from the timing of the experimental paradigm. We performed each experiment before and after global signal regression.

All procedures were approved by local Institutional Review/Ethics Boards. We used publicly available data that we had used in previous studies. Additional details are described in Kelly Jr. et al., 2019 and Kelly et al., 2010.

3.2. Regression

Regression procedures were performed using the GLM (Huettel et al., 2014; McKeown, 2000), with FSL programs FEAT, fsl_glm, fsl_regfilt, or dual_regression, based on the equation

$$\mathit{y}={\beta }_1{\mathit{x}}_1+{\beta }_2{\mathit{x}}_2+{\beta }_3{\mathit{x}}_3+\dots +{\beta }_n{\mathit{x}}_n+c+\mathit{\epsilon },$$

where the best approximation for the values of vector y (the dependent variable, representing a BOLD signal) for a given set of vectors {x₁, x₂, …, x_n} (n independent variables, representing either spatial maps or time courses) was found by choosing the set of regression coefficients {β₁, β₂, …, β_n} and constant (c) that minimized the sum of squares of the residual values in vector ε.

3.3. Experiment 1

3.3.1. Participants and selection of downloaded data

Publicly available resting-state fMRI data were downloaded through the 1000 Functional Connectomes Project (Biswal et al., 2010), Milwaukee-B dataset (Li, 2009), from 46 adult participants. An artificial signal time course used in previous studies (Erhardt et al., 2011; Kelly Jr. et al., 2019) was derived from publicly available MATLAB code downloaded from the Machine Learning for Signal Processing Laboratory at http://mlsp.umbc.edu/simulated_fmri_data.html.

3.3.2. Experimental designs for data collection

Six minutes of resting-state fMRI data were collected from subjects instructed to close their eyes and relax inside the scanner.

3.3.3. Data acquisition

Resting-state fMRI data were acquired on a 3T GE Signa scanner with a standard transmit-receive head coil, using a single-shot gradient echo-planar imaging (EPI) pulse sequence with TR = 2 s, TE = 25 ms, flip angle 90 degrees, 180 volumes, 36 contiguous sagittal 4 mm-thick slices per volume, field-of-view (FOV) = 24 × 24 cm, acquisition matrix 64 × 64, and voxel size = 3.75 × 3.75 × 4 mm. High-resolution SPGR 3D axial T1-weighted images were acquired with TR = 10 ms, TE = 4 ms, TI = 450 ms, flip angle = 12 degrees, 144 slices, slice thickness = 1.0 mm, and acquisition matrix 256 × 192.

3.3.4. Image preprocessing

FMRI image preprocessing was performed using FSL (version 5.0, on Ubuntu 12.04), FMRIB’s Software Library (www.fmrib.ox.ac.uk/fsl) (Jenkinson et al., 2012; Smith et al., 2004; Woolrich et al., 2009), involving non-brain removal (Smith, 2002); motion correction (Jenkinson et al., 2002); high-pass temporal filtering using Gaussian-weighted least-squares straight line fitting with σ = 100 seconds; spatial smoothing using a Gaussian kernel with full-width half-maximum 6 mm; co-registration to high-resolution T1-weighted images; and normalization of all images to standard space (MNI, Montreal Neurological Institute atlas, resolution 3 × 3 × 3 mm) with affine registration (Jenkinson et al., 2002; Jenkinson and Smith, 2001). To allow for T1 equilibrium, the first five volumes of fMRI data were discarded. Data from only 45 subjects were used because of excessive motion in one of the resting-state data runs (mean framewise displacement > 0.2 mm).

3.3.5. Method comparison

To elucidate mechanisms that may contribute to superior performance for SDR, we compared SCA and SDR in their ability to detect a simple, artificial “network” of brain areas sharing common signal. Synthetic fMRI datasets for 45 “subjects” were constructed to contain 1) an artificial network of voxels with highly correlated time courses; 2) artificial global noise; 3) artificial structured noise; and 4) random background noise (approximately Gaussian) derived from resting-state data for 45 subjects. The comparisons were performed before and after global signal regression.

Resting-state data from all 45 subjects were preprocessed, then stripped of neural signals, structured noise signals, and global signal with fsl_regfilt, regressing out of the data all of the time courses derived from single-subject ICA, followed by GSR, as in Kelly Jr. et al., 2010. The artificial network took the shape of a cross whose beams were 5 × 5 voxels wide and 45 voxels long, oriented parallel to the x and y axes respectively and centered on the point [0, −24, 6]. Artificial signal (Fig. 2, top, left) was added to each voxel in the cross, with amplitude 1% (peak-to-peak) of MR signal intensity. We chose this time course as the signal of interest because its smoothness made it easier to identify signal distortions due to noise and other factors. MNI brain space was divided into octants with origin at [0, −24, 6]. To represent localized sources of structured noise, one sawtooth time course at 2% MR signal intensity was added to each octant, excluding the xy-, yz-, and xz-planes, with frequencies (within the typical resting-state range-of-interest) of 1/8, 1/12, 1/20, 1/28, 1/44, 1/52, 1/68, and 1/76 Hz (Fig. 2, right column). Global noise was represented by adding a sawtooth time course at 2% MR signal intensity and frequency 1/92 Hz to all brain voxels (Fig. 2, bottom, left). For the GSR cases, a second set of 45 fMRI data sessions was created by applying GSR at this point (removing most, but not all of the added artificial global noise signal).

Fig. 2.

Signals added to fMRI data after regressing out structured noise and neural signals identified with ICA. Left column: Time course of artificial signal of interest, and sawtooth signal (1/92 Hz) added globally to all voxels. Right column: One sawtooth signal simulating a local noise source added to each octant, in frequencies 1/8, 1/12, 1/20, 1/28, 1/44, 1/52, 1/68, and 1/76 Hz.

SCA was derived using a small, 7.5-mm-radius spherical “ball” seed composed of 33 voxels, as in Biswal et al., 2010. To create misalignments of this spatial regressor with the target network, as would be expected in natural fMRI data (Duncan et al., 2009), for each subject the seed was displaced from the center by a random distance (Gaussian variable with μ = 0 and σ = 6 mm) in each of the x, y, and z directions, rounded to the nearest voxel (For convenience and purposes of illustration, we varied the spatial locations of the seeds/spatial regressors rather than the locations of the artificial neural network, as would ordinarily occur in natural fMRI data). SDR was derived under identical circumstances as for SCA, using spatial regressors consisting of binary spatial maps of voxels in seeds used for SCA.

Our primary hypothesis was that without GSR, SDR would outperform SCA, resulting in higher mean PD over the 45 subjects, using the cross map as the SOC. Wilcoxon’s signed-rank test, two-tailed, with α = 0.05, was used to test our primary hypothesis against the null hypothesis, as well as to test the remaining pair-wise comparisons among the four methods (SCA or SDR, with or without GSR), uncorrected for multiple comparisons. To illustrate the global signal filtering effect of SDR, we examined correlations between derived time courses and the mean global signal for brain voxels across volumes. Pair-wise comparisons of the correlations from each of the four methods were tested for statistical significance against the null hypothesis with α = 0.05, using Wilcoxon’s signed-rank test, two-tailed, uncorrected for multiple comparisons.

3.4. Experiment 2

To simulate a more natural distribution of noise, we repeated Experiment 1 with “physiological noise” concentrated in “gray matter.” We designated as gray matter a brain region containing the cross voxels and bounded by a rectangular prism 53 × 53 × 14 voxels in size, constituting approximately 38.5% of brain voxels, which was the average percentage of gray matter in our fMRI data as determined by segmentation with FSL FAST (Smith, 2002). To facilitate comparison with Experiment 1, rather than increasing the amplitude of artificial noise in “gray matter,” we reduced its amplitude in the surrounding brain areas: The synthetic fMRI data for “gray matter” voxels were identical to those in Experiment 1, while artificial noise sources for the other brain voxels were reduced in amplitude to 1% of MR signal intensity.

3.5. Experiment 3

3.5.1. Participants and selection of downloaded data

Publicly available task-fMRI data were downloaded through The fMRI Data Center (Haaland et al., 2003; Van Horn et al., 2001) from a study of 14 healthy, right-handed adults (Haaland et al., 2004).

3.5.2. Experimental designs for data collection

FMRI data were used from a study of the association between brain hemispheric lateralization patterns and motor task complexity (Haaland et al., 2004), which involved pressing buttons in response to visual cues, in a complex or simple pattern. We used data from the first run of the complex condition with the right hand, as described in Kelly et al., 2010.

3.5.3. Data acquisition

Images were acquired using a 1.5T GE Signa scanner. Echo-planar (EP) images were collected using a single-shot, blipped, gradient-echo EP pulse sequence with TR = 4 s, TE = 40 ms, 22 contiguous sagittal 6-mm thick slices, FOV = 24 × 24 cm, 64 × 64 matrix, and voxel size = 3.75 × 3.75 × 6 mm. High-resolution 3-D spoiled gradient-recalled at steady-state anatomic images were collected with TR = 24 ms, TE = 5 ms, flip angle = 40 degrees, number of excitations = 1, slice thickness = 1.2 mm, FOV = 24 cm, resolution = 256 × 128.

3.5.4. Image preprocessing

FMRI image preprocessing was performed using FSL (version 5.0, on Ubuntu 12.04), FMRIB’s Software Library (www.fmrib.ox.ac.uk/fsl) (Jenkinson et al., 2012; Smith et al., 2004; Woolrich et al., 2009), involving non-brain removal (Smith, 2002); motion correction (Jenkinson et al., 2002); high-pass temporal filtering using Gaussian-weighted least-squares straight line fitting with σ = 100 seconds; spatial smoothing using a Gaussian kernel with full-width half-maximum 8 mm; co-registration to high-resolution T1-weighted images; and normalization of all images to standard space (MNI, Montreal Neurological Institute atlas, resolution 4 × 4 × 4 mm) with affine registration (Jenkinson et al., 2002; Jenkinson and Smith, 2001). To allow for T1 equilibrium, the first two volumes of fMRI data were discarded.

3.5.5. GLM processing with a priori time courses

Z-statistic spatial maps were derived from task-fMRI data by regressing a priori time courses onto fMRI data using FSL FEAT with FILM local autocorrelation correction (Woolrich et al., 2001). The a priori time courses were derived from a block function representing task on/off intervals, convolved with a Gaussian function with peak lag = 5 seconds and σ = 2.8 seconds. Group-level task-related spatial maps were derived from multilevel linear modeling with FSL FLAME (FMRIB’s Local Analysis of Mixed Effects) (Woolrich et al., 2004) using individual-subject, task-related GLM results (i.e., a “random-effects” analysis).

3.5.6. Method comparison

SCA and SDR were compared against binary SOC maps to evaluate their ability to detect FC in natural fMRI data. The SOC maps were derived by converting the z-statistic spatial map for each subject to a map of probabilities for each voxel of obtaining a higher z-score by chance, based on the cumulative normal distribution; then thresholding resulting probability maps with false discovery rate q = 0.05 (Benjamini and Hochberg, 1995). We presumed that our block-design experiment would elicit synchronized brain activity in a constellation of brain regions that could be mapped using GLM or SCA: We have previously shown that for our natural experiment, GLM-derived maps correlated highly (~0.7 or greater) with SCA maps derived using seeds chosen from voxels with highest z-scores in the corresponding GLM-derived maps (Kelly et al., 2010). We chose a seed of comparable volume as for Experiments 1 and 2 in the shape of a 19-voxel spherical “ball” created by omitting the corners of a 3 × 3 × 3 voxel cube, centered on MNI coordinates [−46, −30, 48], the location that yielded the highest average seed-voxel z-score (5.52) from the group-GLM map within a cuboid region containing the visual and motor cortices, with vertices [38, −102, −32] and [−50, 10, 68].

Our primary hypothesis was that without GSR, SDR would outperform SCA, resulting in higher mean PD over the 14 subjects. Wilcoxon’s signed-rank test, two-tailed, with α = 0.05, was used to test our primary hypothesis against the null hypothesis, as well as to test the remaining pairwise comparisons among the four methods (SCA or SDR, with or without GSR), uncorrected for multiple comparisons. To illustrate the global signal filtering effect of SDR, we examined the correlations between time courses derived and global signal. Pairwise comparisons of the correlations from each of the four methods were tested for statistical significance against the null hypothesis with α = 0.05, using Wilcoxon’s signed-rank test, two-tailed, uncorrected for multiple comparisons.

3.6. Illustrative Example

To illustrate possible effects of global signal filtering inherent with dual regression, for a selected axial slice in one subject, we show maps of voxel correlations with global signal prior to fMRI data preprocessing, as well as after preprocessing with global signal subtraction (GSS), corresponding to the effect of SDR, and after global signal regression (GSR). For SCA and SDR, 1) without global signal filtering, 2) with GSS, and 3) with GSR, we show FC maps that result from placing a seed within the motor cortex, where the seed time course remains positively correlated with global signal after global signal subtraction; and from placing a seed within CSF, where the seed time course becomes anticorrelated with global signal after global signal subtraction.

4. Results

Fig. 3 shows an example of FC maps derived in Experiments 1 and 2. For all three experiments: Without GSR (GSR−), PD scores were significantly higher for SDR compared with SCA (Fig. 4, left column). The seed time courses for SCA GSR− showed significantly higher correlations with global signal compared with SDR GSR−, and compared with cases using GSR (GSR+, Fig. 4, right column). After GSR, FC maps and PD detection accuracy scores for SCA and SDR became identical, as expected from theory. GSR significantly improved PD scores for SCA, but not for SDR.

Fig. 3.

Detection of functional connectivity (FC) in Experiments 1 and 2 for a selected subject. Top row: Cross containing artificial signal (yellow) and seed (red) randomly deviated from cross center. Bottom rows: Correctly detected FC (yellow) for Experiments 1 (uniform noise distribution, left column) and 2 (“natural” noise distribution, right column) for SCA (second row), SDR (third row), and either method after global signal regression, GSR (fourth row). White rectangles for Experiment 2 show borders of the cuboid region defining “gray matter,” outside of which artificial noise amplitudes in brain voxels were cut in half compared with Experiment 1. Shown are axial, coronal, and transverse slices of the subject’s high resolution image normalized to standard space, intersecting at seed center, MNI coordinates [−6, −24, 3].

Fig. 4.

Compared with SCA without GSR (GSR−), PD accuracy scores were higher for SDR, as well as for both methods after GSR (GSR+), in all three experiments (left side, top, middle, and bottom). Shown above each of the three right-most bars is the statistical significance of difference in PD-score compared with the left-most bar (** for p < 0.01 and *** for p < 0.001, using two-tailed Wilcoxon signed-rank tests, uncorrected for multiple comparisons); and differences among the right-most bars were non-significant. The improved PD scores corresponded to significant (p < 0.001 compared with SCA GSR− case) reductions in correlations between derived signal of interest and mean global signal in all three experiments: Experiment 1, uniform global noise, top right; Experiment 2, global noise concentrated in and around brain regions containing signal of interest, middle right; and Experiment 3, natural data, bottom right. Box plots show quartiles and ranges of correlations.

4.1. Illustrative Example

Widespread positive correlations with global signal were seen throughout the brain (Fig. 5, left top corner), being generally higher in gray matter. Subtraction of the global signal from voxel time courses reduced these correlations, diminishing positive correlations throughout most of gray matter, while making time courses in other regions anticorrelated with the global signal (Fig. 5, left, second and third rows). GSR, unsurprisingly, rendered all voxels uncorrelated with the global signal (Fig. 5, left lower corner). For SCA, FC z-scores for seeds located in the motor cortex (Fig. 5, middle, top row) and CSF (Fig. 5, right top corner) were positive throughout the brain, generally higher for the motor cortex seed than the CSF seed. For SDR, FC z-scores were generally positive for the motor cortex seed (Fig. 5, middle, second row) and negative for the CSF seed (Fig. 5, right, second row). After GSS or GSR, SCA and SDR resulted in identical maps showing a balance of positive and negative values in derived FC z-score maps (Fig. 5, middle and right columns, third and fourth rows).

Fig. 5.

Impact of global signal denoising on derived functional connectivity for a selected subject in Experiment 3 (showing axial slice at MNI coordinates x = −18). The first column (r_GS) shows for each voxel the correlation between its time course and global signal in the absence of global signal filtering (SCA) and after preprocessing with GSS or GSR; and SDR effectively subtracts out the global signal from each time course, equivalent to GSS. For SCA without denoising (top row), nearly all brain voxels’ time courses were highly correlated with the global signal, biasing FC toward high positive z-scores, whether derived from a seed in the motor cortex (FC_seed, middle column, seed not in slice shown) or in the lateral ventricle’s cerebrospinal fluid (FCCSF, last column, seed outlined with black square). For SDR (second row), subtraction of the global signal (i.e., mean-centering brain voxel values at each volume) prior to deriving seed time courses reduced this positive FC bias for the motor cortex seed, but resulted in a negative bias for the CSF seed. After preprocessing with GSS or GSR (bottom two rows), SCA and SDR resulted in identical FC for both methods, with a balance of positive and negative z-scores. White voxels in the middle column show where FC was detected in voxels that were part of the standard-of-comparison map derived from a random-effects analysis based on the experimental paradigm.

In summary, like GSS and GSR, SDR reduced the bias in FC z-scores due to global signal, compared with SCA; but unlike GSS and GSR, SDR produced widespread negative correlations when the seed was located in a region containing relatively less global signal than other regions, as for our CSF seed. Choosing seeds that lie within gray matter can usually prevent such negative correlations, but not always: In our natural data we found isolated regions of gray matter, including in the brain stem and right superior frontal gyrus that contained lower-than-average global signal, resulting in SDR time courses that were negatively correlated with the global signal, leading to a negative bias in FC derived from SDR for seeds placed in those regions (work not shown).

5. Discussion

SDR can derive FC in place of SCA, using a binary spatial map of the seed used for SCA as the spatial regressor for dual regression. Results for SDR become identical to those derived with SCA when global signal is filtered from fMRI data with GSR or other denoising that fixes the mean brain voxel intensity across volumes. However, if the global signal is not already filtered from fMRI data, SDR may detect FC more accurately than SCA by the intrinsic filtering of the spatial regression. SDR’s filtering of global signal can in some cases spawn spurious negative correlations between seed and other voxels. These negative correlations can negatively bias FC derived from SDR, which should be considered in evaluating results from SDR, as well as results from other variants of dual regression, because it is a byproduct of spatial regression.

These properties of SDR, derived from theory, were confirmed empirically in Experiments 1–3. Experiment 1 showed that when artificial global signal was added uniformly across all voxels, time courses derived from SCA were highly correlated with the global signal, while those derived from SDR were nearly uncorrelated with the global signal; and the accuracy of FC detection was significantly lower for SCA compared with SDR. Experiment 2 showed that when the added artificial global signal amplitude was twice as high in brain regions containing the artificial signal of interest compared with other brain regions, then time courses derived from SCA were again highly correlated with the global signal, while those derived from SDR were significantly less correlated with the global signal; and the accuracy of FC detection was significantly lower for SCA compared with SDR. Results for Experiment 3, our natural condition, were comparable to those of Experiment 2. For all three experiments, the application of GSR in preprocessing caused FC maps derived from SCA to become identical to those derived from SDR, resulting in a significant improvement in FC detection accuracy for SCA, but not for SDR.

The largest components of fMRI global signal frequently consist of noise from subject motion, cardiac fluctuations, and respiratory activity (Liu et al., 2017). Removing noise from resting-state fMRI data is commonly performed with GSR (Liu et al., 2017), which can reduce the typically positive bias in correlations among voxel time courses (Caballero-Gaudes and Reynolds, 2017; Fox et al., 2009; Murphy et al., 2009; Murphy and Fox, 2017) and in some cases can filter from fMRI data noise that reduces sensitivity of FC estimates (Aguirre et al., 1997; Liu et al., 2017; Zarahn et al., 1997) and systematically biases results of FC group comparison (Ciric et al., 2017; Power et al., 2012, 2014, 2017a, 2017b; Satterthwaite et al., 2012, 2013; Tyszka et al., 2014; van Dijk et al., 2012). However, GSR has been the subject of considerable discussion (Caballero-Gaudes and Reynolds, 2017; Chai et al., 2012; Chen et al., 2012; Ciric et al., 2017; Fox et al., 2009; Hampson et al., 2010; He and Liu, 2012; Li et al., 2019; Liu et al., 2017; Lydon-Staley et al., 2019; Murphy et al., 2009; Murphy and Fox, 2017; Muschelli et al., 2014; Parkes et al., 2018; Power et al., 2015, 2017a, 2017b; Weissenbacher et al., 2009; Yan et al., 2013b, 2013c; Zhu et al., 2015) related to concerns that in some cases it may artificially introduce noise (Jo et al., 2013; Liu et al., 2017; Saad et al., 2012), filter neural signal from the data (Keilholz et al., 2017; Schölvinck et al., 2010), potentially bias FC group comparison (Gotts et al., 2013; Hahamy et al., 2014; Saad et al., 2012; Song et al., 2012), and introduce spurious anticorrelations among voxels, as is mathematically mandated (Murphy et al., 2009). The optimal choice between using GSR or no filtering of global signal appears to vary depending upon the study in question; and the lack of consensus concerning whether it is preferable to use GSR or not has prompted some authors to propose that SCA studies be analyzed with and without GSR, so that results from both methods can be presented if they differ (Bijsterbosch et al., 2020).

However, other methods for filtering global signal may also be considered (Liu et al., 2017; Murphy et al., 2013), including SDR or the equivalent SCAx, neither of which have been systematically investigated in the literature, to our knowledge. Some methods circumvent GSR’s “mathematical mandate”, which suggests that they might be less likely to cause spurious anticorrelations than GSR, but even these methods will necessarily alter the correlation structure of the fMRI data, which must be considered in the interpretation of FC results (Murphy and Fox, 2017). Instead of limiting the derivation of FC from SCA to be performed with and without GSR, a “multiverse analysis” could include results from multiple viable methods for filtering global signal, to openly address the potential variability in results that depends upon method choice (Botvinik-Nezer et al., 2019; Steegen et al., 2016). Understanding the nature of signal filtering by each of these FC methods would be key to evaluating results.

The pros and cons of filtering global signal with SDR, compared with GSR or no filtering, stem from SDR’s less aggressive form of global signal filtering. When performing SCA after preprocessing with GSR, global signal is completely filtered from both the time course for the seed and the time course for every voxel with which it is to be correlated. Filtering global signal from only one of these time courses, as with SDR, suffices to suppress a positive FC bias due to global signal, while reducing the risk of spurious anticorrelations. For example, compared with global signal subtraction (Equation 8) SDR does not normally create spurious anticorrelated “networks” consisting of regions that are low in global signal (Jezzard et al., 2001), such as CSF (e.g., compare periphery of images for SDR before and after GSS, Fig. 5, middle column, second and third rows), because global signal is only subtracted from the seed time course (Equation 7), not from all voxel time courses in the fMRI data. In this sense, SDR can be considered a “partial” filtering compared with GSR or GSS. SDR also involves “partial” filtering in the sense that for voxels where the global signal is relatively high, unlike GSR, SDR does not completely eliminate the global signal (See Fig 4., correlations with global signal for SDR in Experiments 2 and 3, right column, bottom two rows), which partly represents neural signals (Caballero-Gaudes and Reynolds, 2017; Keilholz et al., 2017; Liu, 2016; Liu et al., 2017; Schölvinck et al., 2010; Yan et al., 2013a).

For some applications, SDR may not remove noise as thoroughly as GSR, and widespread negative correlations between seed and other brain voxel time courses would occur in cases where the seed falls on a region containing relatively less global signal than other brain regions. Such cases could indicate misalignments between seed location and intended brain target, which would become apparent in FC maps derived from SDR (Fig. 5, right, second row, showing predominately negative z-scores), but not in those derived from SCA (Fig. 5, right, top, showing predominately positive z-scores) or after application of GSS or GSR (Fig. 5, right, bottom two rows, showing a balance of positive and negative z-scores). Thus, SDR can facilitate identification of some cases of misalignment between seeds and their target brain regions, via inspection of FC maps as part of quality control procedures, or by testing SDR time courses for negative correlations with global signal.

Cases where time courses derived from spatial regression become anticorrelated with the global signal should be relatively uncommon because spatial regressors representing neural signals of interest would usually correlate spatially with regions where global signal is higher than average, such as in gray matter (Krüger and Glover, 2001). None of our 14 natural fMRI data cases from Experiment 3 resulted in SDR time courses that were anticorrelated with the global signal. However, after examining 56 cases from our natural fMRI data (using runs 1 and 2 for both left and right hands from our task-fMRI experiment) from a separate study comparing standard dual regression with single-map dual regression (Kelly Jr. et al., 2019), we found a single case where spatial regression produced a time course that was negatively correlated with the global signal (r = −0.1 for both methods). Upon inspection it did not appear that the spatial regressor in this case was grossly misaligned with the intended visuomotor network, and FC z-scores in visuomotor regions were comparable to those from the other fMRI sessions, so discarding this case from group comparison did not seem necessary, as might normally be considered; but generally, applying GSR prior to dual regression might be considered in order to fix to zero the correlation with global signal from each time course derived from spatial regression, thereby eliminating a potential source of variability in resulting FC maps.

The question of whether to perform GSR when using dual regression goes beyond the scope of the current paper, but in some cases the choice is obvious. Typically, GSR has not been performed prior to standard dual regression (Baggio et al., 2015; Chahine et al., 2017; Filippini et al., 2009; Geissmann et al., 2018; Leech et al., 2011; Pannekoek et al., 2015), considered controversial and unnecessary because noise from head motion and physiological signal fluctuations is filtered via temporal nuisance regressors included in the temporal regression (Smith et al., 2014). However, global signal is filtered from these temporal nuisance regressors in spatial regression, so even for standard dual regression, GSR may provide more complete filtering of global signal. GSR should be considered when comparing FC results from dual regression with SCA, as otherwise results may become biased in favor of dual regression due to partial filtering of global signal. For example, Smith et al. (2014) found superior FC results for standard dual regression compared with a variant of SCA that did not include GSR but did include partial regression of standard dual regression’s temporal nuisance regressors. A fairer comparison might involve preprocessing with GSR and omitting dual regression’s temporal nuisance regressors that can reduce the accuracy of derived FC maps (Kelly Jr. et al., 2019).

Understanding the global signal filtering inherent with SDR can facilitate interpretation of results derived with SDR and provide a window to understanding more complex applications of dual regression. For researchers who empirically prefer SDR to SCA, the current paper elucidates the mechanism responsible for improved FC results in some cases—a partial filtering of global signal, equivalent to subtracting the global signal from the seed time course (SCAx). The paper also confirms that FC results from SDR can be equivalent to SCA, as indicated on an FSL webpage (FMRIB Software Library, 2017), but only for cases where global signal is removed from fMRI data in preprocessing. In such cases, the choice of which method to use becomes a matter of convenience, depending upon programs and data available in the data processing pipeline. The larger point is that dual regression does not circumvent the controversy concerning appropriate filtering of global signal, because deriving FC with dual regression inherently employs a form of global signal filtering that can be considered a partial application of conventional global signal subtraction.

6. Conclusion

SDR can replace SCA in deriving FC. When fMRI data are preprocessed with GSR, or any method that completely removes mean global signal, FC results derived with SDR become identical to those derived with SCA. Otherwise, SDR differs from SCA in that the mean global signal is subtracted out of the derived seed time course via SDR’s spatial regression. This filtering of global signal, inherent with all forms of dual regression, provides denoising that often may improve accuracy of detection of FC, but in some cases can induce spurious negative correlations between the seed time course and time courses of other brain voxels, thereby negatively biasing r-values or z-scores in FC maps. These spurious anti-correlations should therefore be considered when using dual regression, in the same way that they are considered when applying GSR.

Highlights:

Seed-based dual regression (SDR) can replace seed-based connectivity analysis (SCA)

SDR provides more accurate functional connectivity (FC) than SCA in some cases

FC maps from SDR and SCA become identical after global signal regression

Dual regression can result in spurious negative correlations that bias FC maps

Pronounced spurious negative FC bias can easily be identified and corrected

Appendix A

In this appendix we elucidate and quantify SDR’s partial filtering of global signal by demonstrating that FC derived from SDR is identical to that derived from SCAx, a variant of SCA where the mean global signal for brain voxels across volumes is subtracted from the seed time course prior to calculating correlations with voxel time courses.

A.1. Nomenclature

By convention, matrices and vectors are respectively denoted by capital letters and lowercase letters in bold. Let $\mathit{Y}={\left[{\mathit{y}}_1,{\mathit{y}}_2,\dots ,{\mathit{y}}_N\right]}^{\top }\in {ℝ}^{N\times n}$ be the matrix representing the fMRI data from all voxels in the brain, where N is the number of voxels and n is the number of time points. The vector of the time course of the i^th voxel is given by ${\mathit{y}}_i\in {ℝ}^{n\times 1}$, and the volume of Y at time point j given by ${\mathit{v}}_j\in {ℝ}^{N\times 1}$. We denote S = {1, 2, …, N_S} as the indices for N_S voxels in the seed region, ${\mathit{Y}}_S={\left[{\mathit{s}}_1,{\mathit{s}}_2,\dots ,{\mathit{s}}_{N_S}\right]}^{\top }\in {ℝ}^{N_S\times n}$ as the matrix of time courses from the seed region voxels, and $\mathit{S}=\frac{1}{N_{\mathit{S}}}{\sum }_{i=1}^{N_S}{s}_i$ as the vector of their mean time course. Similarly, we denote Q = {1, 2, …, N_Q} as the indices for voxels in the brain that do not include the seed region and ${\mathit{Y}}_Q={\left[{\mathit{q}}_1,{\mathit{q}}_2,\dots ,{\mathit{q}}_{N_Q}\right]}^{\top }\in {ℝ}^{N_Q\times n}$ as the matrix of their times courses (N_Q = N − N_S). g is the global mean time course defined as $\mathit{g}=\frac{1}{N}{\sum }_{i=1}^N{\mathit{y}}_i$, and s* = s − g is the seed time course for SCAx.

A.2. Proof

Equation 1, seed-based FC expressed as a Pearson correlation coefficient (r_si) for each voxel, becomes

$$r_{si}=Corr\left(\mathit{s},{\mathit{y}}_i\right)=E\left[\frac{(\mathit{s}-E[\mathit{s}])}{{\sigma }_{\mathit{s}}}\cdot \frac{\left({\mathit{y}}_i-E\left[{\mathit{y}}_i\right]\right)}{{\sigma }_{{\mathit{y}}_i}}\right],$$ (A.1)

and FC from SCAx $\left(r_{s^{*}i}\right)$ is given by replacing s with s*:

$$r_{s^{*}i}=Corr\left(\mathit{s}-\mathit{g},{\mathit{y}}_i\right).$$ (A.2)

FC for SDR is derived with dual regression by first regressing the binary seed map onto the fMRI data (Kelly Jr. et al., 2019; Kelly et al., 2010):

$${\mathit{v}}_j=d_j\mathit{b}+c_j+{\mathit{e}}_j$$ (A.3)

where vector v_j represents the spatial map of voxel values at time point j, vector b represents the binary spatial map with voxel value 1 for seed voxels and 0 for non-seed voxels, and d_j and constant c_j are chosen to minimize the sum of squares of residuals in e_j. The resulting time course $\mathit{d}={\left[d_1,d_2,\dots ,d_n\right]}^{\top }\in {ℝ}^{n\times 1}$ can then be regressed onto the fMRI data to yield parameter estimates to be converted to an FC z-score map; or alternatively, FC for SDR can be derived in terms of r-values via

$$r_{di}=Corr\left(\mathit{d},{\mathit{y}}_i\right).$$ (A.4)

The elements of d, d_j, derived from regression of the binary seed map onto the fMRI data can be related to r_bj, the point-biserial correlation coefficient between v_j and b, by

$$d_j=\left({\sigma }_{{\mathit{v}}_j}/{\sigma }_{\mathit{b}}\right)\cdot r_{bj}$$ (A.5)

and r_bj is also given by

$$r_{bj}=\left(1/{\sigma }_{v_j}\right)\sqrt{p(1-p)\cdot N/(N-1)}\left({\overline{s}}_j-{\overline{q}}_j\right)$$ (A.6)

where σ_b and ${\sigma }_{v_j}$ are the sample standard deviations of b and v_j, p is the proportion of brain voxels in the seed, ${\overline{s}}_j$ is the mean value of seed voxels and ${\overline{q}}_j$ is the mean value of non-seed brain voxels at time point j (Cohen 1988; Cohen et al., 2003). From Equations A.5 and A.6,

$$d_j=\left(1/{\sigma }_{\mathit{b}}\right)\cdot \sqrt{p(1-p)\cdot N/(N-1)}\left({\overline{s}}_j-{\overline{q}}_j\right).$$ (A.7)

The mean value of all brain voxels $\left({\overline{v}}_j\right)$ at time point j is

$${\overline{v}}_j=\left(N_S{\overline{s}}_j+N_Q{\overline{q}}_j\right)/\left(N_S+N_Q\right).$$ (A.8)

The mean value of seed voxels for SCAx at time point j is

$${\overline{s^{*}}}_j={\overline{s}}_j-{\overline{v}}_j={\overline{s}}_j-\left(N_S{\overline{s}}_j+N_Q{\overline{q}}_j\right)/\left(N_S+N_Q\right),$$ (A.9)

which is equivalent to

$${\overline{s^{*}}}_j=\left(N_Q/\left(N_s+N_Q\right)\right)\left({\overline{s}}_j-{\overline{q}}_j\right).$$ (A.10)

From Equations A.7 and A.10, noting that b, p, N, N_s, and N_Q do not vary with time (j),

$$d_j=k{\overline{s^{*}}}_j$$ (A.11)

where constant $k=\left(1/{\sigma }_{\mathit{b}}\right)\sqrt{p(1-p)\cdot N/(N-1)}/\left(N_Q/\left(N_s+N_Q\right)\right)>0.$. For the case where b is a binary map, its sample standard deviation is given by ${\sigma }_{\mathit{b}}=\sqrt{p(1-p)\cdot N/(N-1)}$, which leads to the simplification k = (N_s + N_Q)/N_Q = 1/(1 − p). By definition,

$$\mathit{d}={\left[d_1,d_2,\dots ,d_n\right]}^{\top }={\left[k{\overline{s^{*}}}_1,k{\overline{s^{*}}}_2,\dots ,k{\overline{s^{*}}}_n\right]}^{\top }=k{\mathit{s}}^{*},\text{so}$$ (A.12)

$$r_{di}=Corr\left(\mathit{d},{\mathit{y}}_i\right)=Corr\left(k{\mathit{s}}^{*},{\mathit{y}}_i\right)=Corr\left({\mathit{s}}^{*},{\mathit{y}}_i\right)=r_{s^{*}i}.$$ (A.13)

Thus, FC derived from SDR is identical to that derived from SCAx.

Appendix B

This appendix supports the statement that preprocessing of fMRI data with global signal normalization, global signal subtraction, or global signal regression fixes to a constant value the mean brain voxel intensity for each volume. This statement is true by definition for global signal normalization, is derived directly from the definition of the mean for global signal subtraction, and is proven below for global signal regression.

Let g be the mean global signal for brain voxels in an fMRI session having n time points; and let $\overline{g}$ be the mean of g, and g_m be the vector with every element equal to $\overline{g}\left({\mathit{g}}_m=\overline{g}1\in {ℝ}^{n\times \mathbf{1}}\right)$. Let g₀ be the mean-centered global signal, defined as g₀ = g − g_m. Let g_GSR be the mean global signal after global signal regression. We prove our statement by showing that g_GSR = g_m.

Let y_i be the n × 1 vector of time course for the i^th brain voxel. From the definition of the mean

$$\mathit{g}=\frac{1}{N}{\sum }_{i=1}^N{\mathit{y}}_i,$$ (B.1)

where N is the number of brain voxels.

We represent all time course vectors using an orthogonal basis v₁, v₂, …, v_n, having v₁ = g₀:

$${\mathit{y}}_i={\sum }_{i=1}^n{b}_{ij}{\mathit{v}}_j$$ (B.2)

for appropriate choices of scalars b_ij. Combining Equations B.1 and B.2:

$$\mathit{g}=\frac{1}{N}{\sum }_{i=1}^N{\mathit{y}}_i=\frac{1}{N}{\sum }_{i=1}^N{\sum }_{i=1}^n{b}_{ij}{\mathit{v}}_j=\frac{1}{N}{\sum }_{i=1}^N{b}_{i1}{\mathit{g}}_{\mathbf{0}}+\frac{1}{N}{\sum }_{i=1}^N{\sum }_{j=2}^n{b}_{ij}{\mathit{v}}_j.$$ (B.3)

From the definition of g₀,

$$\mathit{g}={\mathit{g}}_0+{\mathit{g}}_{\mathit{m}}$$ (B.4)

and noting that g₀ and g_m are orthogonal, then from Equations B.3 and B.4

$${\mathit{g}}_{\mathit{m}}=\frac{1}{N}{\sum }_{i=1}^N{\sum }_{j=2}^n{b}_{ij}{\mathit{v}}_j.$$ (B.5)

Global signal regression sets the coefficients for g₀ to zero in Equation B.3, without changing the coefficients for v₂, v₃, …, v_n, because these latter vectors are orthogonal to the global signal. Thus,

$${\mathit{g}}_{\mathit{G}\mathit{S}\mathit{R}}=\frac{1}{N}{\sum }_{i=1}^N{\sum }_{j=2}^n{b}_{ij}{\mathit{v}}_j={\mathit{g}}_{\mathit{m}}.$$ (B.6)