Clustering of resting state fMRI signals for extraction of functional brain networks has been showed to provide value in recent times [1, 2]. Independent component analysis (ICA) is the most commonly used technique to extract functional brain networks [3]. More recently non-negative matrix factorization (NMF) has been successfully utilized for identification of brain functional networks in single-subject resting state fMRI data [5, 6]. NMF may provide complementary information for analyzing resting state fMRI data [5, 6]. However, the technique has not been extended to provide group inferences. This is non-trivial, since the components obtained from single subject NMF are not ordered. Using temporal concatenation, similar to group ICA [7], we introduce a new framework for back reconstruction of individual subject from group analysis using NMF. This framework will make comparisons between groups possible for NMF.

NMF [4] is a low-rank approximation of a given feature space where, non-negativity constraint is imposed on the features. For non-negative $$$N\times T$$$ matrix, $$$X$$$ and a positive integer $$$k<\min(N,T)$$$ NMF finds non-negative matrices $$$W$$$ and $$$H$$$ of sizes $$$N\times k$$$ and $$$k\times T$$$ respectively,

$$X\approx{WH}$$

that minimize the following objective function,

$$D=\min_{W,H}{||X-WH||}^2$$

NMF is used for blind clustering of functional networks in the rs-fMRI BOLD signals. In context of brain functional networks in fMRI data, if $$$X$$$ matrix represents the rs-fMRI data for given subject then each columns of $$$W$$$ represents the functional maps and the rows of the matrix $$$H$$$ are the representative time courses of the corresponding functional maps.

In this work, a method to combine all subjects into an NMF analysis to estimate a single set of components is suggested. The NMF components of individual subjects are then back reconstructed from the combined coefficients matrix $$$H$$$. This approach enables the ordering the components in different subjects in the same way.

The proposed method can be illustrated using a two-subject case; consider single voxel data having two time points ($$$D_1 $$$,$$$D_2 $$$) from two subjects (subject $$$S_1$$$ and subject $$$S_2$$$ ),

$$S_1=\begin{bmatrix}D_1 & D_2 \end{bmatrix}$$

And

$$S_2=\begin{bmatrix}D_1 & D_2 \end{bmatrix}$$

The two subjects can be concatenated in to a single vector as,

$$X=\begin{bmatrix}S_1 & S_2 \end{bmatrix}$$

After applying NMF for two components,

$$\begin{bmatrix}S_1 & S_2 \end{bmatrix}=\begin{bmatrix}C_1 & C_2 \end{bmatrix}\begin{bmatrix}{Coff}_{1,1} & {Coff}_{1,2} & {Coff}_{1,3} &{Coff}_{1,4} \\{Coff}_{2,1} & {Coff}_{2,2} & {Coff}_{2,3} & {Coff}_{2,4} \end{bmatrix}$$

Where the elements of $$$W$$$ matrix $$$C_1$$$ and $$$C_2$$$ are the group components and $$$Coff$$$ are the elements of basis matrix $$$H$$$. Figure 1 shows group NMF block diagram. To back-reconstruct the individual subject components we multiply the inverse of partition of coefficients matrix $$$H$$$ corresponding to the desired subject’s data with the corresponding partition of concatenated subject matrix $$$X$$$.

$$\overbrace{S_1}=S_1\star\begin{bmatrix}{Coff}_{1,1} & {Coff}_{1,2} \\{Coff}_{2,1} & {Coff}_{2,2} \end{bmatrix}^{-1}$$

And

$$\overbrace{S_2}=S_2\star\begin{bmatrix}{Coff}_{1,3} & {Coff}_{1,4} \\{Coff}_{2,3} & {Coff}_{2,4} \end{bmatrix}^{-1}$$

Figure 2 gives the back reconstruction block diagram. In practical case to handle the scale of the fMRI data, before concatenation of individual subject data, principal component analysis (PCA) based dimensionality reduction on time dimension [7] is done for each subject. The reduction parameter is chosen such that, the original data should not be overly reduced to avoid losing important information.

Resting state fMRI scans were acquired on forty participants, each scanned on a 3 Tesla GE MRI scanner. Each scan consists of 6 minutes of BOLD sensitized 2D multi-band EPI acquisition with voxel dimensions of 2 x 2 x 3 mm with a sampling rate of 1.11 Hz (TR = 900 ms, TE = 30ms). Motion correction is applied on scan data followed by registration to T1-weighted image, followed by registration to atlas, removal of nuisances using aCompCor [8], spatial smoothing using 4 mm isotropic Gaussian window, band pass temporal filter with passband between 0.01 and 0.1 Hz. For group analysis, PCA based dimensionality reduction is applied in temporal direction to reduce time points with reduction ratio of 5. After PCA, data from all participants are concatenated in temporal direction followed by NMF to get group maps. Low-rank approximation using NMF up to 30 components is used, which gives 30 components some of which are neuronal and others are noise induced. For individual subject back reconstruction, the partition of the basis matrix corresponding to a given subject is projected back on the original subject data as given in equation (6). Figure 3 shows default mode network (DMN), Attention Network (AN) and Motor Network (MN) maps for the group and back reconstructed for the individual.In this paper, a framework for group inferences of individual subject from group NMF using temporal concatenation is introduced. The proposed framework can act as complementary technique for ICA based group inferences from fMRI data.