Supplemental Data
Feature-reduction and semi-simulated data in functional connectivity-based cortical parcellation
Xiaoguang Tian2,*, Cirong Liu2,5,*, Tianzi Jiang4,5, Joshua Rizak2, Yuanye Ma1,2,3,6, Xintian Hu1,2,3,6
1Yunnan Key Lab of primate Biomedical Research, China
2Kunming Institute of Zoology, Chinese Academy of Sciences, Kunming, Yunnan, China
3State Key Laboratory of Brain and Cognitive Science, Institute of Biophysics, Chinese Academy of Sciences, Beijing, China
4LIAMA Center for Computational Medicine, National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing, China
5The University of Queensland, Queensland Brain Institute, QLD 4072, Australia
6Kunming Bio-International
Functional connectivity maps by different feature reduction approaches
Based on Principal Component Analysis (PCA):
The feature reduction approach by PCA differed from the approach by Affinity Propagation (AP) as it was directly done on the raw functional connectivity maps. For example, suppose we let B= n × p represent seed regions’ functional connectivity maps of n samples (seed regions’ voxels) under p variables (features of each one seed region voxel), as:
(1)
The first step required a subtraction of the mean of the variables under each of the sample.
(2)
Where, the mean of variables
Thus, the whole adjusted matrix was represented as:
(3)
Secondly, the above adjusted matrix was transferred into a covariance matrix
(4)
Then, the eigenvalues and eigenvectors from the covariance matrix were obtained by singular value decomposition (SVD). However, the number of eigenvectors corresponding to the non-zero eigenvalues determined by the SVD was the number of samples n subtracted by 1, because the experimental data for the number of samples (voxels) n was less than the number of variables p (features). Lastly, if we select r satisfied eigenvectors, the functional connectivity maps after PCA would be created as:
(5)
where B’ was the final functional connectivity maps and E was eigenvector matrix.
In this experiment, we averaged 25 subjects’ raw seed region functional connectivity maps to obtain the group functional connectivity maps and then performed the above feature reduction by PCA on the group functional connectivity maps. We select all principal components (PCA-all) and principal components covering 95% cumulative contribution (PCA-95) to build new functional connectivity maps.
Based on Affinity Propagation:
As illustrated in Fig. 1, the feature reduction based on Affinity Propagation first applied a gross parcellation of all voxels within the whole brain. We used the standard Affinity Propagation algorithm implemented in Matlab and freely available through http://www.psi.toronto.edu/index.php?q=affinity%20propagation [1]. The functional connectivity maps of the whole brain were transformed into a collection of negative-valued similarities in order to satisfy a special inputting format needed for Affinity Propagation algorithm. Briefly, there were two elements ,. The negative-valued similarity between them was given by. This transformation allowed Affinity Propagation to automatically receive the best number of clusters (K) setting by a certain preference, rather than by pre-specifying an arbitrary number of clusters. Nonetheless, this preference needed to be set to a common value. The value was defined as the median of the input similarities that resulted in a moderate number of clusters as described by Frey and Dueck [1]. Then Affinity Propagation used a process of iteration under fixed maximal numbers (max=2000), where two kinds of message were exchanged between data points. The first type of message, termed “Responsibility” reflected the accumulated evidence for how well-suited a point was to serve as the exemplar for point , while taking into account all other potential exemplars for point . The message was updated following the rule:
(6)
Another message, termed “Availability” , was sent from candidate exemplar point to point , and reflected the accumulated evidence for how appropriate it would be for point to choose point as its exemplar, taking into account the support from all other points that point should be an exemplar. The second message was also updated as follows:
(7)
Meanwhile, the self-availability which reflected the accumulated evidence that point was an exemplar for point , was based on the positive responsibilities , sent to candidate exemplar from all other points. The self-availability was also updated as follows:
(8)
When updating these messages, there was a damping factor between 0 and 1 used to avoid numerical oscillations, with a default damping value . This message-passing procedure was set to terminate after fixed maximal number of iterations or changes in the messages fell below a convergence threshold (ncov=25).
After the gross parcellation of the whole brain, an average of the time series of the voxels in the same resulting clusters was taken. Finally, new functional connectivity maps of seed region were generated by the correlation representing the relationship between raw seed region time series and these new whole brain time series.
In this experiment, we first averaged 25 subjects’ functional connectivity maps of the whole brain to obtain the group whole brain functional connectivity maps and then performed the gross clustering of the whole brain described above. By averaging the time courses of voxels in the same clusters, we obtained new whole brain time courses in each subjects. Then the seed region functional connectivity maps were calculated from the correlation between seed region time courses and the new whole brain courses in each subject. Finally, by averaging the seed region functional connectivity maps of the 25 subjects, we obtained the group seed region functional connectivity maps with the AP-based feature reduction.
Fig. S1 Flowchart of proposed method.
Fig. S2. Illustration of examples of resulting clusters of R-SMA (K = 2 and 3). PCA-all: resulting clusters using functional connectivity maps with PCA-based feature reduction selecting all components; Raw: resulting clusters using functional connectivity maps without feature reduction; AP: resulting clusters using functional connectivity maps with AP-based feature reduction; K: the number of clusters defined in K-means. The coordinate of the slice is x = 5.
Fig. S3. Illustration of examples of resulting clusters of the cingulate cortex (K = 4 and 6). PCA-all: resulting clusters using functional connectivity maps with PCA-based feature reduction selecting all components; Raw: resulting clusters using functional connectivity maps without feature reduction; AP: resulting clusters using functional connectivity maps with AP-based feature reduction; K: the number of clusters defined in K-means. The coordinate of the slice is x = 5.
Table S1. One-way repeated measures ANOVA and post hoc test on the three Eigenmaps.
P value / < 0.0001
P value summary / ***
Are means signif. different? (P < 0.05) / Yes
Number of groups / 3
F / 18.13
R squared / 0.0004013
Was the pairing significantly effective?
R squared / 0.7353
F / 5.559
P value / < 0.0001
P value summary / ***
Is there significant matching? (P < 0.01) / Yes
Bonferroni's Multiple Comparison Test / Mean Diff. / t / Significant? P < 0.01? / Summary
R-SMA vs Cingulate Cortex / -0.00005387 / 4.505 / Yes / ***
R-SMA vs R-PFC / -0.00006833 / 5.713 / Yes / ***
Cingulate Cortex vs R-PFC / -0.00001445 / 1.209 / No / ns
References
[1] Frey BJ, Dueck D. Clustering by passing messages between data points. Science 2007, 315: 972-976.