1 Introduction
The sensitivity of the statistical data analysis of fMRI data is limited by amount of data available. Hence there is an increasing interest in gathering data from multiple sessions and/or multiple subjects. It is hoped that this will increase statistical sensitivity in testing hypotheses about human cognition [1, 2]. Here we focus on the problem of analyzing fMRI data across multiple subjects all experiencing the same time synchronized stimulus. The raw fMRI data is thus temporally aligned, but is neither anatomically nor functionally aligned across subjects. Traditional methods of addressing this via anatomical registration, [3, 4, 5], ignore the inherent functional variability across subjects. More recent approaches attempt to use anatomical and functional data to align, or register, functional structure [6, 7, 8]. A distinct approach learns a factor model that jointly models the functional responses. Such factor methods have shown promising results and are attracting increased attention, e.g., [9, 10, 11, 12, 13, 14].
To make things more precise let denote the data from subject , , where are the number of voxels in the 3D volume along the axes, and is number of time samples (TRs) in the experiment. Typically one rearranges into a by matrix, where . If one is only interested in voxels within a given region of interest (ROI), then is simply the submatrix formed by restriction to those voxels. In this setting, the basic goal is to jointly factorize the matrices into a product so as to minimize some constrained or regularized objective function. Here is the number of factors, , and . The “mixing matrices” are subject specific (acknowledging the distinct functional topographies of the subjects), while the “source matrix” provides a set of shared elicited time responses. For this reason we call such models shared response models.
Form by stacking the along the spatial dimension, and similarly form . Then where is the shared response and is the model error. and are then determined by minimizing a regularized cost function of the terms in the decomposition subject to constraints on and .
As a specific example, PCA selects and to minimize . The resulting has orthonormal columns and is partitioned into submatrices , . Since , the sources are formed as linear combinations of potentially all voxel time courses. Similarly, using a compact SVD: , , and is the first columns of . So the columns of are linear combinations of potentially all columns of .
In general, in a shared response factor model can be formed as a linear combination of all columns in and can be formed using a linear combination of all voxel time courses. In this sense the factor model has no assurance of preserving spatial locality. Such methods are usually applied to large predefined ROIs, such as PMC [14] or ventral temporal cortex (VT) [13], to investigate the relationship between the ROI and a specific stimulus or cognitive state.
In this paper we specifically set out to preserve spatial locality in forming the shared response factor model. This is important in exploratory analyses where one seeks to detect and study small local regions in the brain where shared information is present. There are two possible approaches: a searchlight analysis [15, 16, 17, 18], or adding spatial regularization to the factorization cost function. Both approaches are worthy of study. Here we focus on the searchlight approach.
A searchlight uses a fixed number of neighboring voxels to conduct analysis for each voxel location, and the same analysis is conducted on all locations. This idea can be used to extend any ROIbased factor model to overlapping searchlights. We combine factor models of the SRM form and searchlight analysis to enable localized analysis in the whole brain multisubject fMRI functional alignment. In detail, a fixed sized searchlight centering at voxel is used to scan over the whole brain. For each searchlight location, a factor model is used to functionally align across subjects, and an analysis is performed based on the results of the alignment. Statistics from the analysis (e.g., classification accuracy) is assigned to the center voxel of that specific searchlight. We report the accuracy of each searchlight on a given classification task. This helps identify locations in the brain in which information is shared across subjects for a specific cognitive task.
For multisubject neuroscience datasets and experiments, we provide an effective method for locating where the shared information is over the whole brain while keeping the quality of the found shared information. So our method can serve as a first step in multisubject fMRI analysis to help identify regions that worth further investigation in a neuroscience experiment. In some factor models, the number of latent factors can be prespecified. In this case, we record the k value that gives the best analysis result for each searchlight. We report both the accuracy and the best value on a brain map as a proxy for the presence and richness of a shared cognitive state across subjects. We also developed two new variants of the ICA and group ICA factor models. These variants show good performance in the functional alignment task.
2 Some Specific Factor Models
Examples of (probabilistic and deterministic) factor methods that have been used in multisubject fMRI analysis include PCA [9], ICA [10, 11], Group ICA [12], hyperalignment (HA) [13], and shared response model (SRM) [14, 19].
The desirable factor models to be combined with searchlight analysis in multisubject functional alignment should have the following properties: 1) has an adjustable number of factors ; 2) shows good performance in large area multisubject functional alignment. Below, we discuss and compare several candidates using a consistent framework.
PCA: Standard PCA is a deterministic model. We have already outlined how to obtain the and in this case.
(1) 
subject to . This is done in a graphical model framework that provides Bayesian regularization. This SRM can be considered as a variant of pPCA in a multisource setting [20, 14]. The shared response of some held out data for subject can be computed as .
Hyperalignment (HA) [13]: HA is a deterministic model that learns orthogonal to minimize (1). It does not have a selectable factor dimension . For , the nonprobabilistic version of the SRM in [14] yields HA. So HA can be considered a special case of the SRM in [14]. A searchlight application of HA has been explored in [17]. For these reasons we do not explore HA further here.
Independent Component Analysis (ICA):
ICA learns statistically independent signals as measured by kurtosis or negentropy
[10, 11]. We use the FastICA algorithm, an efficient probabilistic method [10] optimizing negentropy of the shared response . This is formulated as(2) 
where is a nonquadratic function, e.g. , and
is a standard normal random variable. This yields
. is then partitioned into submatrices , .SRICA: We also study a new algorithm we call shared response ICA (SRICA). Motivated by the framework in [14], we modify the FastICA algorithm. In SRICA, the block structure of the subject data is preserved by spatial concatenation in both and . The key difference is that instead of learning a joint matrix , we iteratively learn to ensure blockwise structure in . This is summarized in Algorithm 1^{1}^{1}1We acknowledge the help of Jacob Simon in coding SRICA.. Here we follow the convention of working with unmixing matrices instead of . The function is the derivative of in (2) [21].
Group ICA (GICA): Group ICA, an algorithm for making group inferences, uses two applications of PCA and an application of ICA [12]. The original algorithm first performs subject specific PCA along the temporal dimension. Then the projected data matrices for all subjects are concatenated to form a joint data matrix. A second PCA is then performed on the joint data matrix. Lastly, an ICA is performed on the projected data matrix after the second PCA. We apply GICA along the spatial dimension to learn a low dimensional shared response space. See Algorithm 2.
Dataset  TRs (s/TR) 

audiobook (narrated story) [22]  449(2) 
sherlockmovie (audiovisual movie) [23]  1976(2) 
sherlockrecall (movie freerecall) [23]  4371009(2) 
3 Experiments and Results
Three datasets (Table 1), collected using a 3T Siemens Skyra scanner with different preprocessing pipelines, are used to test and compare the searchlight factor models. The sherlockmovie dataset was collected while the subjects watched a minute BBC episode ”Sherlock”. The sherlockrecall dataset was collected while the same subjects verbally reiterated the ”Sherlock” episode without any hints. The audiobook dataset was collected while a different set of subjects listened to a minute narrated story.
In all experiments, we use searchlights on data downsampled by 2. For each searchlight, we try and report the highest accuracy and the corresponding value. To ensure a fair comparison, for SRGICA, we set to be the number of voxels in the searchlight and . We tried other values, but this resulted in less accuracy. Note that there are relatively few voxels per searchlight to begin with. The accuracies are computed based on the projected shared response of held out data using learned subjectspecific maps .
Time Segment Matching: This experiment is designed to test if the shared response we learned can be generalized to new data. That is, what is the quality of the shared information extracted. We use the audiobook and sherlockmovie datasets. The fMRI data are split into two halves along the temporal axis, one for training and the other for testing, and the roles reversed and the results averaged. In this experiment, we first use training data of all 9 subjects for learning the shared response. Then, a random 9 TR time segment from the testing subject (1 of the 9 subjects), called test segment, projected to the shared response space. The other 8 subjects’ testing data is projected to the shared response space and averaged. We then locate this time segment by maximizing Pearson correlation between the average response and response from the test segment. Segments overlapping with the test segment are excluded in matching. Assuming independent time segments, chance accuracy is for audiobook and for sherlockmovie. The accuracy and k brain maps for different searchlight factor models are shown in in Fig. 1 and Fig. 2. Note that we threshold the accuracy to give a more clear visualization of the most informative area. We also compute a single number accuracy by aggregating the local shared response from all searchlights. This accuracy is compared with accuracy from whole brain factor models with features ( for SRGICA). The results are shown in Fig. 4.
Scene Recall matching: This experiment is designed to test if brain functional patterns are similar when subjects are recalling the same scene. We use the sherlockmovie for training and sherlockrecall for testing. In sherlockrecall, TRs collected when the subject was recalling the same scene are averaged and projected to the shared response space using
learned from the training data. The projected recall data along with the corresponding scene labels are used to train a SVM classifier. The projected recall data from a testing subject is used to test the classifier. Chance accuracy is
. The accuracy and k brain maps for a subset of models are shown in Fig. 3. Accuracies for searchlight factor models and whole brain factor models are computed the same way as time segment matching experiment and are shown in Fig. 4.4 Discussion and Conclusion
We have investigated how well various factor models can locate informative regions in a searchlight based analysis of multisubject fMRI data. This approach highlights local brain regions that are most informative of the cognitive state of interest using both accuracy and brain maps. Early auditory cortex (EAC) and early visual cortex (EVC) are the most informative regions in time segment matching experiment for audiobook and sherlockmovie dataset, respectively. This matches the type of stimulus in these datasets. Scene recall is a more complex task. In this case a higher level cognitive region, PMC, is more informative. The results demonstrate that the approach can effectively locate meaningful informative local regions. In some neuroscience experiments, it is not clear which regions will be most relevant to the stimulus and/or task. The searchlight factor model approach helps locate regions worthy of further exploration. Moreover, since the searchlight approach preserves spatial locality, we expected the overall accuracy to drop as a consequence of the searchlight constraint. In fact, as shown in Fig. 4, the overall accuracy does not drop in most cases, and sometimes even significantly increases. The brain maps also help reveal the effectiveness of the various factor models. For example, consider the brain maps of SRICA and SRM. While the accuracy maps of these methods are very close to each other, SRICA uses a smaller to achieve this accuracy. This suggests that each factor in SRICA has an improved representation capability. Overall the SRICA is showing strong performance across the three experiments.
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