Group NMF Analysis for Resting State fMRI

Bhushan Patil^{1}, Mahesh Panicker^{1}, Radhika Madhavan^{1}, and Suresh Joel^{1}

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.

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2. Van Dijk, Koene RA, et al. "Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization." Journal of neurophysiology103.1 (2010): 297-321.

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5. Lee, Jong-Hwan, et al. "Investigation of spectrally coherent resting-state networks using non-negative matrix factorization for functional MRI data." International Journal of Imaging Systems and Technology 21.2 (2011): 211-222.

6. Mahesh Panicker, Bhushan Patil, Ek Tsoon Tan, Suresh Joel, ” Blind Functional Clustering of Resting State fMRI using Non-Negative Matrix Factorization”’ 21st Annual Meeting of the Organization for Human Brain Mapping, OHBM 2015.

7. Karhunen, Juha, et al. "On neural blind separation with noise suppression and redundancy reduction." International Journal of Neural Systems 8.02 (1997): 219-237.

8. Behzadi, Yashar, et al.”A component based noise correction method (CompCor) for BOLD and perfusion based fMRI.” Neuroimage 37.1 (2007): 90-101.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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