Global variations of resting state fMRI (rfMRI) signal are considered nuisance effects that are commonly removed with global signal regression (GSR) [1]. However, GSR is controversial because it introduces anti-correlations in the resultant functional connectivity maps [2]. In this study, we describe a new method, sparse parameter global signal regression (SP-GSR), for removing global signal variations. We assume the global signal to be of low rank and the remaining signal can be split into orthogonal regressors with spatially sparse parameters. Under this assumption, SP-GSR is cast as an L-p minimization problem and solved with non-convex matrix decomposition and PCA.

SP-GSR is mathematically described as:

$$\begin{split}&min_{L,G,\beta}||\beta||_{p}\\s.t.\quad&Y=L+G\times \beta\\&rank(L)=1\\&0<p<1\end{split}$$

where Y denotes the input signal matrix, with T (total number of time points) rows and N (total number of voxels) columns; L is a T x N rank-1 matrix containing the global variation; G denotes design matrix containing orthogonal regressors for the remaining signal; is the corresponding sparse parameter matrix. The problem is non-convex and can be solved using non-convex matrix decomposition and PCA with the iterative approach in Fig1.

The simulated brain consists of 3 ROIs plus a common global variation (Fig. 2). The signal in voxel j at time t in a region m are assumed to be

$$y_{j,m}(t)=v_m(t)+g_b b(t)+g_e e_j(t)$$

$$$ v_m(t) $$$ is the time series shared in the region m, b(n) is the global variation assumed to follow an independent and identical Gaussian distribution (Gaussian i.i.d.), and $$$e_i(n) $$$ is Gaussian white noise in each voxel, also i.i.d. $$$g_b$$$ and $$$g_e$$$ are global variation and noise gain factors, respectively, that are constant within a group of subjects.

To ascertain the performance of SP-GSR in removing global signal variation, in the presence of noise, we performed a simulation first assuming only correlations within each region but no correlations between the regions plus global signal variation and noise. Correlation maps were derived from the simulated data using the 4 different approaches for global signal correction: (1) No correction (NC), (2) conventional global signal regression (GSR), (3) 1st-PC based Global Signal Regression (PC-GSR), where the 1st-PC element is used as the regressor to obtain global variant signal [3], and (4) SP-GSR. To study of the effect of inter-regional correlations, we also performed a simulation with correlations between the regions. The simulated were corrected for global signal variation with the 4 approaches listed above. Inter-regional correlations were calculated. In both cases, the results were compared with the ground truth.

We also tested our approach on data of 38 subjects downloaded from the Human Connectome Project (http://www.humanconnectomeproject.org/). We compared correlation maps obtained by seeding in posterior cingulate cortex using no correction, GSR, PC-GSR and SP-GSR.

Simulations

Fig. 3 shows the resultant correlation maps derived after the different correction schemes with seeds in region 1 and 2, respectively. With no correction, positive correlations are seen in all regions although they should only be seen in the region with the seed. With GSR, negative correlations are seen in regions other than the seeding region. The result of PC-GSR correction is almost the same as GSP, exhibiting significant erroneous negative correlations. In contrast, SP-GSR resulted in maps that are almost identical to the ground truth.

The results of inter-regional correlations are shown in Table 1. No correction led to significant positive correlations between the regions in the absence of inter-regional correlation and inflated positive correlations between the regions in the presence of inter-regional correlation. Both GSR and PC-GSR led to negative correlations between the regions in the absence of true inter-regional correlation and deflated positive correlations in the presence of inter-regional correlation. SP-GSR led to results that are almost the same as the ground truth.

Experimental Data

The result on experiment data (Fig. 4) exhibits wide spread positive correlation when global signal is not corrected. Correlation maps after GSR and PC-GSR showed similar results of the default mode network but with substantial wide spread anti-correlation (blue). SP-GSR identified a more prominent default mode network with anti-correlations much reduced and constrained in white matter.

In this work, we introduced a novel method, SP-GSR, for global signal removal. The method was demonstrated, by simulations, to be free of anti-correlation while preserving the true correlations. The analysis of experimental data with SP-GSR led to a exhibit more prominent and focused default mode network, with isolated negative correlations. The removal of the anti-correlated artifacts will increase the accuracy of the detected functional connectivity networks.