Low Dimensional Embedding of fMRI datasets

Low Dimensional Embedding of fMRI datasets⋆ Xilin Shen, a Fran¸cois G. Meyer a,∗ aDepartment of Electrical Engineering, University of Colorado at Boulder Abstract We propose a novel method to embed a functional magnetic resonance imaging (fMRI) dataset in a low-dimensional space. The embedding optimally preserves the local functional coupling between fMRI time series and provides a low-dimensional coordinate system for detecting activated voxels. To compute the embedding, we build a graph of functionally connected voxels. We use the commute time, instead of the geodesic distance, to measure functional distances on the graph. Because the commute time can be computed directly from the eigenvectors of (a symmetric version) the graph probability transition matrix, we use these eigenvectors to embed the dataset in low dimensions. After clustering the datasets in low dimensions, coherent structures emerge that can be easily interpreted. We performed an extensive evaluation of our method comparing it to linear and nonlinear techniques using synthetic datasets and in vivo datasets. We analyzed datasets from the EBC competition obtained with subjects interacting in an urban virtual reality environment. Our exploratory approach is able to detect independently visual areas (V1/V2, V5/MT), auditory areas, and language areas. Our method can be used to analyze fMRI collected during “natural stimuli”. Key words: fMRI, Laplacian eigenmaps, embedding, natural stimuli. 1. Introduction At a microscopic level, a large number of internal variables associated with various physical and phys- iological phenomena contribute to dynamic changes in functional magnetic resonance imaging (fMRI) datasets. FMRI provides a large scale (as compared to the scale of neurons) measurement of neuronal activity, and we expect that many of these vari- ables will be coupled resulting in a low dimensional set for all possible configurations of the activated fMRI signal. We assume therefore that activated fMRI time series can be parametrized by a small number of variables. This assumption is consistent ⋆Submitted to Neuroimage, Sep. 2007. Revised Jan. 2008. ∗Corresponding author. Email address: fmeyer@Colorado.Edu (Fran¸cois G. Meyer). URL: ece.colorado.edu/∼fmeyer (Fran¸cois G. Meyer). with the usage of low dimensional parametric mod- els for detecting activated voxels (Petersson et al., 1999). This assumption is also consistent with the empirical findings obtained with principal compo- nents analysis (PCA) and independent components analysis (ICA) (B.B.Biswal and Ulmer, 1999; McK- eown et al., 2003), where a small number of compo- nents are sufficient to describe the variations of most activated temporal patterns. Both PCA and ICA make very strong assumptions about the compo- nents: orthogonality and statistical independence, respectively. Such constraints are convenient math- ematically but have no physiological justification, and complicate unnecessarily the interpretation of the components (Friston, 1998). A second limitation of PCA and ICA is that both methods only provide a linear decomposition of the data (Friston, 2005). There is no physiological reason why the fMRI signal should be a linear combination of eigen-images or Preprint submitted to Elsevier 22 October 2018 arXiv:0709.3121v2 [stat.ML] 16 Jan 2008 eigen-time series. In practice, the first components identified by PCA are often related to physiological artifacts (e.g. breathing), or coherent spontaneous fluctuations (Raichle and Mintun, 2006). These ar- tifacts can be responsible for most of the variability in the dataset. Stimulus triggered changes, which are much more subtle, rarely appear among the first components. The contribution of this paper is a novel ex- ploratory method to construct an optimal coordi- nate system that reduces the dimensionality of the dataset while preserving the functional connectiv- ity between voxels (Sporns et al., 2000). First, we define a distance between time series that quan- tifies the functional coupling (Fox et al., 2005), or connectivity between the corresponding voxels. We then construct an embedding that preserves this functional connectivity across the entire brain. After embedding the dataset in a lower dimen- sional space, time series are clustered into coherent groups. This new parametrization results in a clear separation of the time series into: (1) response to a stimulus, (2) coherent physiological signals, (3) ar- tifacts, and (4) background activity. We performed an extensive evaluation of our method comparing it to linear and nonlinear techniques using synthetic datasets and in vivo datasets. 2. Methods 2.1. Overview of our approach Our goal is to find a new parametrization of an fMRI dataset, effectively replacing the time series by a small set of features, or coordinates, that facili- tate the identification of task-related hemodynamic responses to the stimulus. The new coordinates will also be able to reveal the presence of physical or phys

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