Projects:BrainConnectivity

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Exploration of Functional Connectivity via Clustering

Generative Model for Functional Connectivity

In the classical functional connectivity analysis, networks of interest are defined based on correlation with the mean time course of a user-selected `seed' region. Further, the user has to also specify a subject-specific threshold at which correlation values are deemed significant. In this project, we simultaneously estimate the optimal representative time courses that summarize the fMRI data well and the partition of the volume into a set of disjoint regions that are best explained by these representative time courses. This approach to functional connectivity analysis offers two advantages. First, is removes the sensitivity of the analysis to the details of the seed selection. Second, it substantially simplifies group analysis by eliminating the need for the subject-specific threshold. Our experimental results indicate that the functional segmentation provides a robust, anatomically meaningful and consistent model for functional connectivity in fMRI.

We formulate the problem of characterizing connectivity as a partition of voxels into subsets that are well characterized by a certain number of representative hypotheses, or time courses, based on the similarity of their time courses to each hypothesis. We model the fMRI signal at each voxel as generated by a mixture of Gaussian distributions whose centers are the desired representative time courses. Using the EM algorithm to solve the corresponding model-fitting problem, we alternatively estimate the representative time courses and cluster assignments to improve our random initialization.

Experimental Results

We used data from 7 subjects with a diverse set of visual experiments including localizer, morphing, rest, internal tasks, and movie. The functional scans were pre-processed for motion artifacts, manually aligned into the Talairach coordinate system, detrended (removing linear trends in the baseline activation) and smoothed (8mm kernel).

Fig. 1 shows the 2-system partition extracted in each subject independently of all others. It also displays the boundaries of the intrinsic system determined through the traditional seed selection, showing good agreement between the two partitions. Fig. 2 presents the results of further clustering the stimulus-driven cluster into two clusters independently for each subject.

Fig 1. 2-System Parcelation. Results for all 7 subjects.	Fig 2. 3-System Parcelation. Results for all 7 subjects.

Fig.3 presents the group average of the subject-specific 2-system maps. Color shading shows the proportion of subjects whose clustering agreed with the majority label. Fig. 4 shows the group average of a further parcelation of the intrinsic system, i.e., one of two clusters associated with the non-stimulus-driven regions. In order to present a validation of the method, we compare these results with the conventional scheme for detection of visually responsive areas. In Fig. 5, color shows the statistical parametric map while solid lines indicate the boundaries of the visual system obtained through clustering. The result illustrate the agreement between the two methods.

Fig 3. 2-System Parcellation. Group-wise result.	Fig 4. Validation: Parcelation of the intrinsic system.

Fig 5. Validation: Visual system.

Comparison between Different Clustering Schemes

As a continuation to the above experiments, we apply two distinct clustering algorithms to functional connectivity analysis: K-Means clustering and Spectral Clustering. The K-Means algorithm assumes that each voxel time course is drawn independently from one of k multivariate Gaussian distributions with unique means and spherical covariances. In contrast, Spectral Clustering does not presume any parametric form for the data. Rather it captures the underlying signal geometry by inducing a low-dimensional representation based on a pairwise affinity matrix constructed from the data. Without placing any a priori constraints, both clustering methods yield partitions that are associated with brain systems traditionally identified via seed-based correlation analysis. Our empirical results suggest that clustering provides a valuable tool for functional connectivity analysis.

One downside of Spectral Clustering is that it relies on the eigen-decomposition of an NxN affinity matrix, where N is the number of voxels in the whole brain. Since N is on the order of ~200,000 voxels, it is infeasible to compute the full eigen-decomposition given realistic memory and time constraints. To solve this problem, we approximate the leading eigenvalues and eigenvectors of the affinity matrix via the Nystrom Method. This is done by selecting a random subset of "Nystrom Samples" from the data. The affinity matrix and spectral decomposition is computed only for this subset, and the results are projected onto the remaining data points.

Experimental Results

We validate these algorithms on resting state data collected from 45 healthy young adults (mean age 21.5, 26 female). Four 2mm isotropic functional runs were acquired from each subject. Each scan lasted for 6m20s with TR = 5s. The first 4 time points in each run were discarded, yielding 72 time samples per run. The entire brain volume is partitioned into an increasing number of clusters. We perform standard preprocessing on each of the four runs, including motion correction by rigid body alignment of the volumes, slice timing correction and registration to the MNI atlas space. The data is spatially smoothed with a 6mm 3D Gaussian filter, temporally low-pass filtered using a 0.08Hz cutoff, and motion corrected via linear regression. Next, we estimate and remove contributions from the white matter, ventricle and whole brain regions (assuming a linear signal model). We mask the data to include only brain voxels and normalized the time courses to have zero mean and unit variance. Finally, we concatenate the four runs into a single time course for analysis.

We first study the robustness of Spectral Clustering to the number of random samples. In this experiment, we start with a 4,000-sample Nystrom set, which is the computational limit of our machine. We then iteratively remove 500 samples and examine the effect on clustering performance. After matching the resulting clusters to those estimated with 4,000 samples, we compute the percentage of mismatched voxels between each trial and the 4,000-sample template. This procedure is repeated twice for each participant.

Fig 6. Varying the number of Nystrom Samples	Fig 7. Nystrom consistency for 2,000 random samples

Fig 6. depicts the median clustering difference when varying the number of Nystrom samples. Values represent the percentage of mismatched voxels w.r.t. the 4,000-sample template. Error bars delineate the 10th-90 percentile region. The median clustering difference is less than 1% for 1,000 or more Nystrom samples, and the 90th percentile difference is less than 1% for 1,500 or more samples. This experiment suggests that Nystrom-based SC converges to a stable clustering pattern as the number of samples increases. Based on these results, we chose to use 2,000 Nystrom samples for the remainder of this work. At this sample size, less than 5% of the runs for 2,4,5 clusters and approximately 8% of the runs for 3 clusters differed by more than 5% from the 4,000-sample template.

The box plot in Fig 7. summarizes the consistency of Nystrom-based Spectral Clustering across different random samplings. The red lines indicate median values, the box corresponds to the upper and lower quartiles, and error bars denote the 10th and 90th percentiles. Here, we perform SC 10 times on each participant using 2,000 Nystrom samples. We then align the cluster maps and compute the percentage of mismatched voxels between each unique pair of runs. This yields a total of 45 comparisons per participant. In all cases, the median clustering difference is less than 1%, and the 90th percentile value is less than 2.1%. Empirically, we find that Nystrom SC predictably converges to a second or third cluster pattern in only a handful of participants. This experiment suggests that we can obtain consistent clusterings with only 2,000 Nystrom samples.

Fig 8. Clustering results across participants. The brain is partitioned into 5 clusters using Spectral Clustering/K-Means, and various seed are selected for Seed-Based Analysis. The color indicates the proportion of participants for whom the voxel was included in the detected system.

Spectral Clustering	K-Means	Seed-Based
Cluster 1, Slice 37	Cluster 1, Slice 37	PCC, Slice 37
Cluster 1, Slice 55	Cluster 1, Slice 55	vACC, Slice 55
Cluster 2, Slice 55	Cluster 2, Slice 55	Visual, Slice 55
Cluster 3, Slice 31	Cluster 3, Slice 31	Motor, Slice 31
Cluster 4, Slice 31	Cluster 4, Slice 31	IPS, Slice 31
Cluster 5, Slice 37	Cluster 5, Slice 37

Fig 8. shows clearly that both Spectral Clustering and K-Means can identify well-known structures such as the default network, the visual cortex, the motor cortex, and the dorsal attention system. Spectral Clustering and K-Means also identified white matter. In general, one would not attempt to delineate this region using seed-based correlation analysis because we regress out the white matter signal during the preprocessing. In our experiments Spectral Clustering and K-Means achieve similar clustering results across participants. Furthermore, both methods identify the same functional systems as seed-based analysis without requiring a priori knowledge about the brain and without significant computation. Thus, clustering algorithms offer a viable alternative to standard functional connectivity analysis techniques.

Comparison of ICA and Clustering for the Identification of Functional Connectivity in fMRI

Although ICA and clustering rely on very different assumptions on the underlying distributions, they produce surprisingly similar results for signals with large variation. Our main goal is to evaluate and compare the performance of ICA and clustering based on Gaussian mixture model (GMM) for identification of functional connectivity. Using the synthetic data with artificial activations and artifacts under various levels of length of the time course and signal-to-noise ratio of the data, we compare both spatial maps and their associated time courses estimated by ICA and GMM to each other and to the ground truth. We choose the number of sources via the model selection scheme, and compare all of the resulting components of GMM and ICA, not just the task-related components, after we match them component-wise using the Hungarian algorithm. This comparison scheme is verified in a high level visual cortex fMRI study. We find that ICA requires a smaller number of total components to extract the task-related components, but also needs a large number of total components to describe the entire data. We are currently applying ICA and clustering methods to connectivity analysis of schizophrenia patients.