Enhancing the network specific individual characteristics in rs-fMRI functional connectivity by dictionary learning

Abstract

Most fMRI inferences are based on analyzing the scans of a cohort. Thus, the individual variability of a subject is often overlooked in these studies. Recently, there has been a growing interest in individual differences in brain connectivity also known as individual connectome. Various studies have demonstrated the individual specific component of functional connectivity (FC), which has enormous potential to identify participants across consecutive testing sessions. Many machine learning and dictionary learning-based approaches have been used to extract these subject-specific components either from the blood oxygen level dependent (BOLD) signal or from the FC. In addition, several studies have reported that some resting-state networks have more individual-specific information than others. This study compares four different dictionary-learning algorithms that compute the individual variability from the network-specific FC computed from resting-state functional Magnetic Resonance Imaging (rs-fMRI) data having 10 scans per subject. The study also compares the effect of two FC normalization techniques, namely, Fisher Z normalization and degree normalization on the extracted subject-specific components. To quantitatively evaluate the extracted subject-specific component, a metric named $\text{Overlap}$ is proposed, and it is used in combination with the existing differential identifiability $\left(I_{\text{diff}}\right)$ metric. It is based on the hypothesis that the subject-specific FC vectors should be similar within the same subject and different across different subjects. Results indicate that Fisher Z transformed subject-specific fronto-parietal and default mode network extracted using Common Orthogonal Basis Extraction (COBE) dictionary learning have the best features to identify a participant.

Keywords: Fisher Z transform; brain atlas; degree normalization; dictionary learning; fMRI; functional connectivity; individual connectome; resting-state networks.

FIGURE 1

Overview of preprocessing, FC computation, and the Train‐Test Split. The 4‐dimensional rs‐fMRI data are preprocessed in MATLAB 2019b using the SPM 12 toolbox, further segmented into ROIs using a predefined Brain Atlas. The BOLD time courses within an ROI are averaged to get one time course for an ROI. Moreover, the time courses belonging to each of the seven networks (Yeo et al., 2011) are chosen one by one, and an FC matrix specific to the rs‐network is formed using Pearson Correlation. Since this correlation matrix is symmetric, only the upper‐triangular elements are vectorized, and a data matrix is formed by concatenating the upper triangular vectors from each scan. The train‐test split is further performed by placing five sessions per subject for training and the other five for test over the data matrix, the Fisher Z transformed, and the Degree Normalized data matrix. These matrices are given as input to the dictionary learning algorithms.

FIGURE 2

The Training data matrix $\mathit{Y}$ is given as input to the PCA algorithm, which gives m principal components (PCs). First $\tilde{m}$ PCs are chosen, and the data are reconstructed to get the subject‐specific components. During testing, the test data $\tilde{\mathit{Y}}$ are projected over the PCs computed while training, and the reconstructed matrix is the subject‐specific data matrix.

FIGURE 3

RPCA training and testing overview. The data matrix $\mathit{Y}$ goes through the RPCA algorithm during the training phase, which decomposes $\mathit{Y}$ into a low‐rank matrix ${\mathit{L}}_{\mathit{\text{rpca}}}$ and a sparse matrix ${\mathit{S}}_{\mathit{\text{rpca}}}$. ${\mathit{S}}_{\mathit{\text{rpca}}}$ having the individual differences is further decomposed by Online Dictionary Learning (ODL) which finds out the basis ${\mathit{D}}_{\mathit{\text{rpca}}}$ whose linear combination can construct the individual components to any data matrix. This gives the subject‐specific components for training. During the test, least squares are used to get the coefficients ${\tilde{\mathit{\alpha }}}_{\mathit{\text{rpca}}}$ for the test matrix $\tilde{\mathit{Y}}$ which on reconstruction gives the subject‐specific component.

FIGURE 4

K‐SVD training and testing overview. K‐SVD algorithm decomposes the data matrix $\mathit{Y}$ into a dictionary ${\mathit{D}}_{\mathit{\text{ksvd}}}$ and sparse coefficients $\mathit{X}$. The product ${\mathit{D}}_{\mathit{\text{ksvd}}}{\mathit{X}}_{\mathit{\text{ksvd}}}$ represents the common or shared information. This, when subtracted with the data matrix $\mathit{Y}$ gives the subject‐specific component. During testing, least squares are used to find the coefficients for the Dictionary obtained during the test. This, when subtracted from the Test data matrix $\tilde{\mathit{Y}}$, gives us the Subject‐specific matrix for the test.

FIGURE 5

COBE training and testing overview. Here instead of the Global data matrix $\mathit{Y}$ the subject data matrix ${\mathit{Y}}_i$ is given individually to the COBE algorithm which decomposes the matrix into a dictionary ${\mathit{D}}_{\mathit{\text{cobe}}}$ and coefficients matrix ${\mathit{X}}_{\mathit{\text{cobe}}}^i$ for every subject. The product ${\mathit{D}}_{\mathit{\text{cobe}}}{\mathit{X}}_{\mathit{\text{cobe}}}^i$ is subtracted from ${\mathit{Y}}_i$ for every subject which gives the subject‐specific components. During test, the least‐squares algorithm is used to get the coefficients for the dictionary ${\mathit{D}}_{\mathit{\text{cobe}}}$ obtained during training. Finally, the subject‐wise subtraction of the reconstructed matrix with the test data matrix ${\tilde{Y}}_i$ gives the subject‐specific matrix during test.

FIGURE 6

(a) $I_{\mathit{\text{diff}}}$ computation: The subject‐specific data matrix consisting of $i^{\mathit{th}}$ and the $j^{\mathit{th}}$ sessions of all the subjects. Further ${\mathit{C}}_{i,j}$ is computed by taking Pearson correlation. $I_{\mathit{\text{self}}}$ is the mean of diagonal elements and $I_{\mathit{\text{others}}}$ is the mean of non‐diagonal elements of the correlation matrix obtained in b. $I_{\mathit{\text{diff}}}^{i,j}$ is computed using Equation (16). Finally, $I_{\mathit{\text{diff}}}$ is computed as mean of the $I_{\mathit{\text{diff}}}$ values computed taking every ${\left(i,j\right)}^{\mathit{th}}$ pair. (b) $\mathit{\text{Overlap}}$ computation: A Pearson correlation matrix is computed from the subject‐specific data matrix. This correlation matrix was partitioned into $p^2$ sub‐matrices each of size $s\times s$, then the histogram of the diagonal blocks would represent within‐subject similarity (WSS) values and the histogram of non‐diagonal blocks would represent between‐subject similarity (BSS) values. Using these histograms, a threshold is computed such that any value above the threshold is considered within‐subject and below the threshold is considered between‐subject. The threshold that minimizes the error is taken and stored in memory during the training phase. During the testing phase, the threshold computed during training is used. The total number of values in error is called $\mathit{\text{Overlap}}$. (c) Ratio of $I_{\mathit{\text{diff}}}$ to $\mathit{\text{Overlap}}$: Plot shows the variation of the ratio to with the variation of $I_{\mathit{\text{diff}}}$ and $\mathit{\text{Overlap}}$.

FIGURE 7

Number of Principle components (PC) used versus the ratio of $I_{\mathit{\text{diff}}}$ to $\mathit{\text{Overlap}}$. Plots show how the ratio changes as we change the number of PCs used to reconstruct the train data matrix $\mathit{Y}$. The number of PCs varied from 1 to 189 in steps of 5. (a–c) The effect of the different normalization methods on the ratio.

FIGURE 8

Effect of changing the number of atoms ($k$) and the $\lambda$ parameter of the dictionary computed by the Online dictionary learning algorithm on the Sparse connectivity Traits matrix obtained by the R‐PCA algorithm on the training data matrix $\mathit{Y}$. The number of dictionary atoms k is varied from 25 to 70 in steps of 5, and the $\lambda$ parameter is varied from 0.1 to 0.5 in steps of 0.05.

FIGURE 9

Variation of number of atoms ($k$) in the K‐SVD dictionary and the sparsity ($s_0$) in the corresponding coefficient matrix. $k$ and $s_0$ is varied from 2 to 8, across all the rs‐networks and normalization methods.

FIGURE 10

Variation of number of common components $C$ in COBE algorithm. We can observe the change in the ratio of $I_{\mathit{\text{diff}}}$ to $\mathit{\text{Overlap}}$ with respect to the number of common components chosen in the COBE algorithm. The number of common components was varied from 2 to 5 for all networks across all normalization methods (a–c).

FIGURE 11

(a–c) Comparison between the dictionary learning algorithms during training. Following parameter check, we choose the parameters which maximize the ratio of $I_{\mathit{\text{diff}}}$ to $\mathit{\text{Overlap}}$ and present the results here. (d–f) Comparison between the dictionary learning algorithms during test. Using the parameters and the thresholds for $\mathit{\text{Overlap}}$ computation during training, these are the results obtained during the testing phase. Results were obtained using the Schaefer 400 atlas.

FIGURE 12

Effect of varying the number of time points in the BOLD signal on the ratio of $I_{\mathit{\text{diff}}}$ to $\mathit{\text{Overlap}}$ computed using the (a) the original functional connectivity (b) the subject‐specific functional connectivity extracted by DL algorithms during the training phase. The time points were varied as 1, 3, 5, 7, and 8.5 min. Results were obtained using the Schaefer 400 atlas.

FIGURE 13

Effect of variation of brain atlas on the ratio of $I_{\mathit{\text{diff}}}$ to $\mathit{\text{Overlap}}$ during the training phase. On the x‐axis brain atlas are arranged in decreasing order of average number of voxels per ROI.

FIGURE 14

Results were repeated 252 $({}^{10}{C}_5$) times, each time taking a different set of training and testing data for the different dictionary learning algorithms. (a–c) The standard deviation of the ratio computed at each iteration across the different networks and the different normalization methods during training. (d–f) The results during the test.