Alican Nalci^{1} and Thomas T. Liu^{1}

In resting-state fMRI, global signal regression (GSR), global signal subtraction (GSS) and global signal normalization (GSN) are widely used nuisance removal methods. So far these techniques have been treated as distinct operations and the relation between them has not been clearly described. In this paper, we mathematically and empirically show that GSS and GSN are nearly identical processes in resting-state fMRI. We further show that in terms of resting-state functional connectivity maps, GSS and hence GSN are similar processes to GSR when considering seed time courses that have a good fit to the global signal time course.

We denote a voxel time series as $$$\mathbf{x}=\tilde{\mathbf{x}}+\mu_{\mathbf{x}}$$$ where $$$\tilde{\mathbf{x}}$$$ is a zero-mean fluctuation term and $$$\mu_{\mathbf{x}}$$$ is a constant mean value. Similarly, the GS is represented as $$$\mathbf{g}=\tilde{\mathbf{g}}+\mu_{\mathbf{g}}$$$, and is computed as the mean time course over all voxels.

In GSN,
the computations are performed prior to removal of the temporal mean from the
voxel time series. Data are normalized at each time point by dividing the voxel
value by the corresponding GS value and then subtracting 1.0 from the result.^{2}
This approach is defined in line (1) of Figure 1. In lines (3) and (6) we
assume that the voxel and global signal means are identical, which can be achieved
without loss of generality by scaling the data such that all voxels have the
same mean value. In line (4), we assume that the mean component of the GS is
much larger than the zero-mean fluctuations, which is the case for most fMRI data.
The result (line 6) is simply the difference between the percent normalized signal
changes $$$\tilde{\mathbf{x}}[i]/ \mu_{\mathbf{x}}$$$ and $$$\tilde{\mathbf{g}}[i]/ \mu_{\mathbf{g}}$$$.

In GSS, the GS is subtracted from each voxel time series after percent change normalization of the signals. This can be expressed as $$$\mathbf{y}_{GSS}=\tilde{\mathbf{x}}/\mu_{\mathbf{x}}-\tilde{\mathbf{g}}/\mu_{\mathbf{g}}$$$, which is identical to the result derived above, hence $$$\mathbf{y}_{GSN}\approx \mathbf{y}_{GSS}$$$.

The process of GSR is described as $$$\mathbf{y}_{GSR}=\tilde{\mathbf{x}}-\alpha\tilde{\mathbf{g}}$$$ where the fit coefficient $$$\alpha = (\tilde{\mathbf{g}}^T\tilde{\mathbf{g}})^{-1}\tilde{\mathbf{g}}^T\tilde{\mathbf{x}}$$$. GSR and GSS are equivalent when $$$\alpha=1$$$ (normalization does not affect this equality). As shown in Figure 2, the mean value of alpha (over voxels) is equal to 1.0. For any given voxel, the relation between GSR and GSS will depend on how “close” $$$\alpha$$$ is to 1.0.

Figure 3 shows representative PCC and WM functional maps obtained after GSR, GSS, and GSN. The maps with GSS and GSN are nearly identical, consistent with the derivation in Figure 1. The mean (over all 68 scans) of the spatial correlations between the PCC and WM maps with GSS and GSN is 0.99 with a small standard deviation (0.003).

In contrast, spatial correlations between PCC maps for GSR and GSS range from 0.62 to 0.99 with a mean of 0.94. For WM maps, the spatial correlations range from 0.37 to 0.99 with a mean of 0.73. As indicated by the fit coefficients and correlation values listed under each column of Figure 3, the degree of similarity depends on the closeness of the fit coefficient to 1.0.

Figure 4 shows the PCC and WM seed fit coefficients versus the spatial correlations between the corresponding connectivity maps obtained after GSR and GSS. As the fit coefficient deviates from the ideal value of 1.0, the spatial correlations decrease sharply for the WM seed and more gradually for the PCC seed.

1. Murphy, K., Birn, R. M., Handwerker, D. A., Jones, T. B., Bandettini, P. A., Feb. 2009. The impact of global signal regression on resting state correlations: are anti-correlated networks introduced? NeuroImage 44 (3), 893–905.

2. Fox, M. D., Zhang, D., Snyder, A. Z., Raichle, M. E., Jun. 2009. The global signal and observed anticorrelated resting state brain networks. Journal of Neurophysiology 101 (6), 3270–3283.

3. Remes, J. J., Starck, T., Nikkinen, J., Ollila, E., Beckmann, C. F., Tervonen, O., Kiviniemi, V., Silven, O., May 2011. Effects of repeatability measures on results of fMRI sICA: a study on simulated and real resting-state effects. NeuroImage 56 (2), 554–569.

4. Power, J. D., Mitra, A., Laumann, T. O., Snyder, A. Z., Schlaggar, B. L., Petersen, S. E., Jan. 2014. Methods to detect, characterize, and remove motion artifact in resting state fMRI. NeuroImage 84, 320–341.

5. Fox, M. D., Snyder, A. Z., Vincent, J. L., Raichle, M. E., 2007. Intrinsic fluctuations within cortical systems account for intertrial variability in human behavior. Neuron 56 (1), 171–184.

Figure 1.
Approximate equality of
the GSN and the GSS processes. See text
for additional details.

Figure 2.
The derivation of the
result that the mean of the fit coefficients over all voxels is equal to 1.0.

Figure 3. Representative PCC and WM
functional connectivity maps obtained before (uncorrected) and after
application of GSR, GSS, and GSN. For GSN and GSS, the PCC and WM maps are
nearly identical, consistent with the theoretical arguments. The spatial
correlations between the GSR and GSS maps are indicated by the values listed at
the bottom. The GSR fit coefficients for
the seed time courses are also listed.

Figure 4.
Fit
coefficients for PCC (blue) and WM (red) time courses versus the spatial
correlations between the corresponding connectivity maps obtained after GSR and
GSS.
Spatial
correlations are very high $$$(r >.90)$$$ when the seed coefficients are
close to the ideal value of 1.0.
As
the seed coefficient deviates from the ideal value, the spatial correlations between the functional connectivity maps decrease for both seeds.
The
decrease is sharper for the WM seed and is more gradual for the PCC seed. The overall decrease is characterized by a
Gaussian fit $$$R^2=0.74$$$ that is shown with the black-dashed line.