{"title": "Locality and low-dimensions in the prediction of natural experience from fMRI", "book": "Advances in Neural Information Processing Systems", "page_first": 1001, "page_last": 1008, "abstract": "Functional Magnetic Resonance Imaging (fMRI) provides an unprecedented window into the complex functioning of the human brain, typically detailing the activity of thousands of voxels during hundreds of sequential time points. Unfortunately, the interpretation of fMRI is complicated due both to the relatively unknown connection between the hemodynamic response and neural activity and the unknown spatiotemporal characteristics of the cognitive patterns themselves. Here, we use data from the Experience Based Cognition competition to compare global and local methods of prediction applying both linear and nonlinear techniques of dimensionality reduction. We build global low dimensional representations of an fMRI dataset, using linear and nonlinear methods. We learn a set of time series that are implicit functions of the fMRI data, and predict the values of these times series in the future from the knowledge of the fMRI data only. We find effective, low-dimensional models based on the principal components of cognitive activity in classically-defined anatomical regions, the Brodmann Areas. Furthermore for some of the stimuli, the top predictive regions were stable across subjects and episodes, including Wernicke\u00d5s area for verbal instructions, visual cortex for facial and body features, and visual-temporal regions (Brodmann Area 7) for velocity. These interpretations and the relative simplicity of our approach provide a transparent and conceptual basis upon which to build more sophisticated techniques for fMRI decoding. To our knowledge, this is the first time that classical areas have been used in fMRI for an effective prediction of complex natural experience.", "full_text": "Locality and low-dimensions in the prediction of\n\nnatural experience from fMRI\n\nFranc\u00b8ois G. Meyer\n\nCenter for the Study of Brain, Mind and Behavior,\nProgram in Applied and Computational Mathematics\n\nPrinceton University\n\nfmeyer@colorado.edu\n\nGreg J. Stephens\n\nCenter for the Study of Brain, Mind and Behavior,\n\nDepartment of Physics\nPrinceton University\n\ngstephen@princeton.edu\n\nBoth authors contributed equally to this work.\n\nAbstract\n\nFunctional Magnetic Resonance Imaging (fMRI) provides dynamical access into\nthe complex functioning of the human brain, detailing the hemodynamic activ-\nity of thousands of voxels during hundreds of sequential time points. One ap-\nproach towards illuminating the connection between fMRI and cognitive function\nis through decoding; how do the time series of voxel activities combine to provide\ninformation about internal and external experience? Here we seek models of fMRI\ndecoding which are balanced between the simplicity of their interpretation and the\neffectiveness of their prediction. We use signals from a subject immersed in vir-\ntual reality to compare global and local methods of prediction applying both linear\nand nonlinear techniques of dimensionality reduction. We \ufb01nd that the prediction\nof complex stimuli is remarkably low-dimensional, saturating with less than 100\nfeatures. In particular, we build effective models based on the decorrelated com-\nponents of cognitive activity in the classically-de\ufb01ned Brodmann areas. For some\nof the stimuli, the top predictive areas were surprisingly transparent, including\nWernicke\u2019s area for verbal instructions, visual cortex for facial and body features,\nand visual-temporal regions for velocity. Direct sensory experience resulted in\nthe most robust predictions, with the highest correlation (c \u223c 0.8) between the\npredicted and experienced time series of verbal instructions. Techniques based on\nnon-linear dimensionality reduction (Laplacian eigenmaps) performed similarly.\nThe interpretability and relative simplicity of our approach provides a conceptual\nbasis upon which to build more sophisticated techniques for fMRI decoding and\noffers a window into cognitive function during dynamic, natural experience.\n\n1 Introduction\n\nFunctional Magnetic Resonance Imaging (fMRI) is a non-invasive imaging technique that can quan-\ntify changes in cerebral venous oxygen concentration. Changes in the fMRI signal that occur during\nbrain activation are very small (1-5%) and are often contaminated by noise (created by the imaging\n\n\fsystem hardware or physiological processes). Statistical techniques that handle the stochastic nature\nof the data are commonly used for the detection of activated voxels. Traditional methods of analy-\nsis \u2013 which are designed to test the hypothesis that a simple cognitive or sensory stimulus creates\nchanges in a speci\ufb01c brain area \u2013 are unable to analyze fMRI datasets collected in \u201cnatural stimuli\u201d\nwhere the subjects are bombarded with a multitude of uncontrolled stimuli that cannot always be\nquanti\ufb01ed [1, 2].\nThe Experience Based Cognition competition (EBC) [3] offers an opportunity to study complex re-\nsponses to natural environments, and to test new ideas and new methods for the analysis of fMRI\ncollected in natural environments. The EBC competition provides fMRI data of three human sub-\njects in three 20-minute segments (704 scanned samples in each segment) in an urban virtual reality\nenvironment along with quantitative time series of natural stimuli or features (25 in total) ranging\nfrom objective features such as the presence of faces to self-reported, subjective cognitive states\nsuch as the experience of fear. During each session, subjects were audibly instructed to complete\nthree search tasks in the environment: looking for weapons (but not tools) taking pictures of people\nwith piercings (but not others), or picking up fruits (but not vegetables). The data was collected with\na 3T EPI scanner and typically consists of the activity of 35000 volume elements (voxels) within\nthe head. The feature time series was provided for only the \ufb01rst two segments (1408 time samples)\nand competitive entries are judged on their ability to predict the feature on the third segment (704\ntime samples, see Fig. 1). At a microscopic level, a large number of internal variables associated\n\nFigure 1: We study the variation of the set of features fk(t), k = 1,\u00b7\u00b7\u00b7 , K as a function of the\ndynamical changes in the fMRI signal X(t) = [x1(t),\u00b7\u00b7\u00b7 , xN (t)] during natural experience. The\nfeatures represent both external stimuli such as the presence of faces and internal emotional states\nencountered during the exploration of a virtual urban environment (left and right images). We predict\nthe feature functions fk for t = Tl+1,\u00b7\u00b7\u00b7 T , from the knowledge of the entire fMRI dataset X , and\nthe partial knowledge of fk(t) for t = 1,\u00b7\u00b7\u00b7 , Tl. The \u201ctoy\u201d activation patterns (middle diagram)\nillustrate the changes in \u201cbrain states\u201d occurring as a function of time.\n\nwith various physical and physiological phenomena contribute to the dynamic changes in the fMRI\nsignal. Because the fMRI signal is a large scale (as compared to the scale of neurons) measurement\nof neuronal activity, we expect that many of these variables will be coupled resulting in a low di-\nmensional set for all possible con\ufb01gurations of the activated fMRI signal. In this work we seek a\nlow dimensional representation of the entire fMRI dataset that provides a new set of \u2018voxel-free\u201d\ncoordinates to study cognitive and sensory features.\nWe denote a three-dimensional volumes of fMRI composed of a total of N voxels by X(t) =\n[x1(t),\u00b7\u00b7\u00b7 , xN (t)]. We have access to T such volumes. We can stack the spatio-temporal fMRI\ndataset into a N \u00d7 T matrix,\n\n(1)\n\n\uf8ee\uf8ef\uf8f0 x1(1)\n\n...\n\nxN (1)\n\nX =\n\n\uf8f9\uf8fa\uf8fb ,\n\nx1(T )\n\n\u00b7\u00b7\u00b7\n...\n\u00b7\u00b7\u00b7 xN (T )\n\n...\n\nwhere each row n represents a time series xn generated from voxel n and each column j represents\na scan acquired at time tj. We call the set of features to be predicted fk, k = 1, ,\u00b7\u00b7\u00b7 , K. We are\ninterested in studying the variation of the set of features fk(t), k = 1,\u00b7\u00b7\u00b7 , K describing the subject\n\n?tjTttjti0tTlkf(t)tit0\fexperience as a function of the dynamical changes of the brain, as measured by X(t). Formally, we\nneed to build predictions of fk(t) for t = Tl+1,\u00b7\u00b7\u00b7 T , from the knowledge of the entire fMRI dataset\nX , and the partial knowledge of fk(t) for the training time samples t = 1,\u00b7\u00b7\u00b7 , Tl (see Fig. 1).\n\nFigure 2: Low-dimensional parametrization of the set of \u201cbrain states\u201d. The parametrization is\nconstructed from the samples provided by the fMRI data at different times, and in different states.\n\n2 A voxel-free parametrization of brain states\n\nWe use here the global information provided by the dynamical evolution of X(t) over time, both\nduring the training times and the test times. We would like to effectively replace each fMRI dataset\nX(t) by a small set of features that facilitates the identi\ufb01cation of the brain states, and make the\nprediction of the features easier. Formally, our goal is to construct a map \u03c6 from the voxel space to\nlow dimensional space.\n\n\u03c6 : RN (cid:55)\u2192 D \u2282 RL\n\nX(t) = [x1(t),\u00b7\u00b7\u00b7 , xN (t)]T (cid:55)\u2192 (y1(t),\u00b7\u00b7\u00b7 , yL(t)),\n\n(2)\n(3)\nwhere L (cid:28) N. As t varies over the training and the test sets, we hope that we explore most of\nthe possible brain con\ufb01gurations that are useful for predicting the features. The map \u03c6 provides a\nparametrization of the brain states. Figure 2 provides a pictorial rendition of the map \u03c6. The range\nD, represented in Fig. 2 as a smooth surface, is the set of parameters y1,\u00b7\u00b7\u00b7 , yL that characterize\nthe brain dynamics. Different values of the parameters produce different \u201cbrain states\u201d, associated\nwith different patterns of activation. Note that time does not play any role on D, and neighboring\npoints on D correspond to similar brain states. Equipped with this re-parametrization of the dataset\nX , the goal is to learn the evolution of the feature time series as a function of the new coordinates\n[y1(t),\u00b7\u00b7\u00b7 , yL(t)]T . Each feature function is an implicit function of the brain state measured by\n[y1(t),\u00b7\u00b7\u00b7 , yL(t)]. For a given feature fk, the training data provide us with samples of fk at cer-\ntain locations in D. The map \u03c6 is build by globally computing a new parametrization of the set\n{X(1),\u00b7\u00b7\u00b7 , X(T )}. This parametrization is built into two stages. First, we construct a graph that is\na proxy for the entire set of fMRI data {X(1),\u00b7\u00b7\u00b7 , X(T )}. Second, we compute some eigenfunc-\ntions \u03c6k de\ufb01ned on the graph. Each eigenfunctions provides one speci\ufb01c coordinate for each node\nof the graph.\n\n2.1 The graph of brain states\n\nWe represent the fMRI dataset for the training times and test times by a graph. Each vertex i\ncorresponds to a time sample ti, and we compute the distance between two vertices i and j by\nmeasuring a distance between X(ti) and X(tj). Global changes in the signal due to residual head\nmotion, or global blood \ufb02ow changes were removed by computing a a principal components analysis\n(PCA) of the dataset X and removing a small number components. We then used the l2 distance\nbetween the fMRI volumes (unrolled as N \u00d71 vectors). This distance compares all the voxels (white\nand gray matter, as well as CSF) inside the brain.\n\nDttt\u03c6j0t\u03c6i\f2.2 Embedding of the dataset\n\nOnce the network of connected brain states is created, we need a distance to distinguish between\nstrongly connected states (the two fMRI data are in the same cognitive state) and weakly connected\nstates (the fMRI data are similar, but do not correspond to the same brain states). The Euclidean\ndistance used to construct the graph is only useful locally: we can use it to compare brain states\nthat are very similar, but is unfortunately very sensitive to short-circuits created by the noise in the\ndata. A standard alternative to the geodesic (shortest distance) is provided by the average commute\ntime, \u03ba(i, j), that quanti\ufb01es the expected path length between i and j for a random walk started at i.\nFormally, \u03ba(i, j) = H(j, i) + H(i, j), where H(i, j) is the hitting time,\n\nH(i, j) = Ei[Tj] with Tj = min{n \u2265 0; Zn = j},\n\ndi = Di,i =(cid:80)\n\nfor a random walk Zn on the graph with transition probability P, de\ufb01ned by Pi,j = wi,j/di, and\nj wi,j is the degree of the vertex i. The commute time can be conveniently computed\nfrom the eigenfunctions \u03c61,\u00b7\u00b7\u00b7 , \u03c6N of N = D 1\n2 , with the eigenvalues \u22121 \u2264 \u03bbN \u00b7\u00b7\u00b7 \u2264 \u03bb2 <\n\u03bb1 = 1. Indeed, we have\n\n2 PD\u2212 1\n\nN(cid:88)\n\n1\n\n(cid:32)\n\n(cid:33)2\n\n.\n\n\u2212 \u03c6k(j)(cid:112)dj\n\n\u03c6k(i)\u221a\ndi\n\n1 \u2212 \u03bbk\nAs proposed in [4, 5, 6], we de\ufb01ne an embedding\n\n\u03ba(i, j) =\n\nk=2\n\ni (cid:55)\u2192 Ik(i) =\n\n1\n\n1 \u2212 \u03bbk\nBecause \u22121 \u2264 \u03bbN \u00b7\u00b7\u00b7 \u2264 \u03bb2 < \u03bb1 = 1, we have\n. We can therefore\nneglect \u03c6k(j)\u221a\nfor large k, and reduce the dimensionality of the embedding by using only the \ufb01rst\n1\u2212\u03bbk\nK coordinates in (4). The spectral gap measures the difference between the \ufb01rst two eigenvalues,\n\u03bb1 \u2212 \u03bb2 = 1 \u2212 \u03bb2. A large spectral gap indicates that the low dimensional will provide a good\napproximation. The algorithm for the construction of the embedding is summarized in Fig. 3.\n\n> 1\u221a\n\n1\u2212\u03bbN\n\n1\u2212\u03bb3\n\n1\u2212\u03bb2\n\n1\u221a\n\nk = 2,\u00b7\u00b7\u00b7 , N\n> \u00b7\u00b7\u00b7\n\n,\n\n\u03c6k(i)\u221a\ndi\n1\u221a\n\n(4)\n\nAlgorithm 1: Construction of the embedding\n\nInput:\n\n\u2013 X(t), t = 1,\u00b7\u00b7\u00b7 , T , K: number of eigenfunctions.\n\nAlgorithm:\n\n1. construct the graph de\ufb01ned by the nn nearest neighbors\n2. \ufb01nd the \ufb01rst K eigenfunctions, \u03c6k, of N\n\n\u2022 Output: For ti = 1 : T\n\n\u2013 new co-ordinates of X(ti): yk(ti) = 1\u221a\n\u03c0i\n\n\u03c6k(i)\u221a\n1\u2212\u03bbk\n\nk = 2,\u00b7\u00b7\u00b7 , K + 1\n\nFigure 3: Construction of the embedding\n\nA parameter of the embedding (Fig. 3) is K, the number of coordinates. K can be optimized\nby minimizing the prediction error. We expect that for small values of K the embedding will not\ndescribe the data with enough precision, and the prediction will be inaccurate. If K is too large, some\nof the new coordinates will be describing the noise in the dataset, and the algorithm will over\ufb01t the\ntraining data. Fig. 4-(a) illustrates the effect of K on the performance of the nonlinear dimension\nreduction. The quality of the prediction for the features: faces, instruction and velocity is plotted\nagainst K. Instructions elicits a strong response in the auditory cortex that can be decoded with as\nfew as 20 coordinates. Faces requires more (about 50) dimensions to become optimal. As expected\nthe performance eventually drops when additional coordinates are used to describe variability that\nis not related to the features to be decoded. This con\ufb01rms our hypothesis that we can replace about\n15,000 voxels with 50 appropriately chosen coordinates.\n\n\f2.3 Semi-supervised learning of the features\n\nThe problem of predicting a feature fk at an unknown time tu is formulated as kernel ridge regres-\nsion problem. The training set {fk(t) for t = 1,\u00b7\u00b7\u00b7 , Tl} is used to estimate the optimal choice of\nweights in the following model,\n\nTl(cid:88)\n\n\u02c6f(tu) =\n\n\u02c6\u03b1(t)K(y(tu), y(t)),\n\nwhere K is a kernel and tu is a time point where we need to predict.\n\nt=1\n\n2.4 Results\n\nWe compared the nonlinear embedding approach (referred to as global Laplacian) to dimension\nreduction obtained with a PCA of X . Here the principal components are principal volumes, and for\neach time t we can expand X(t) onto the principal components.\nThe 1408 training data were divided into two subsets of 704 time samples. We use fk(t) in a subset\nto predict fk(t) in the other subset. In order to quantify the stability of the prediction we randomly\nselected 85 % of the training set (\ufb01rst subset), and predicted 85 % of the testing set (other subset).\nThe role, training or testing, of each subset of 704 time samples was also chosen randomly. We\ngenerated 20 experiments for each value of K, the number of predictors. The performance was\nquanti\ufb01ed with the normalized correlation between the model prediction and the real value of fk,\n\nr = (cid:104)\u03b4f est\n\nk (t), \u03b4fk(t)(cid:105)/\n\n(5)\nwhere \u03b4fk = fk(t)\u2212(cid:104)fk(cid:105). Finally, r was averaged over the 20 experiments. Fig. 4-(a) and (b) show\nthe performance of the nonlinear method and linear method as a function of K. The approach based\non the nonlinear embedding yields very stable results, with low variance. For both global methods\nthe optimal performance is reached with less than 50 coordinates. Fig. 5 shows the correlation\ncoef\ufb01cients for 11 features, using K = 33 coordinates. For most features, the nonlinear embedding\nperformed better than global PCA.\n\nk )2(cid:105)(cid:104)\u03b4f 2\nk(cid:105),\n\n(cid:113)(cid:104)\u03b4(f est\n\n3 From global to local\n\nWhile models based on global features leverage predictive components from across the brain, cog-\nnitive function is often localized within speci\ufb01c regions. Here we explore whether simple models\nbased on classical Brodmann regions provide an effective decoder of natural experience. The Brod-\nmann areas were de\ufb01ned almost a century ago (see e.g [7]) and divide the cortex into approximately\n50 regions, based on the structure and arrangement of neurons within each region. While the ar-\neas are characterized structurally many also have distinct functional roles and we use these roles to\nprovide useful interpretations of our predictive models. Though the partitioning of cortical regions\nremains an open and challenging problem, the Brodmann areas represent a transparent compromise\nbetween dimensionality, function and structure.\nUsing data supplied by the competition, we warp each brain into standard Talairach coordinates and\nlocate the Brodmann area corresponding to each voxel. Within each Brodmann region, differing in\nsize from tens to thousands of elements, we build the covariance matrix of voxel time series using\nall three virtual reality episodes. We then project the voxel time series onto the eigenvectors of the\ncovariance matrix (principal components) and build a simple, linear stimulus decoding model using\nthe top n modes ranked by their eigenvalues,\n\nn(cid:88)\n\nk (t) =\nf est\n\nwk\n\ni mk\n\ni (t).\n\n(6)\n\ni (t)} are the\nwhere k indexes the different Brodmann areas, {wk\nmode time series in each region. The weights are chosen to minimize the RMS error on the training\ni (t)(cid:105).\nset and have a particularly simple form here as the modes are decorrelated, wk\nPerformance is measured as the normalized correlation r (Eq. 5) between the model prediction and\n\ni } are the linear weights and {mk\n\ni = (cid:104)S(t)mk\n\ni=1\n\n\fFigure 4: Performance of the prediction of natural experience for three features, faces, instructions\nand velocity as a function of the model dimension. (a) nonlinear embedding, (b) global principal\ncomponents, (c) local (Brodmann area) principal components. In all cases we \ufb01nd that the predic-\ntion is remarkably low-dimensional, saturating with less than 100 features. (d) stability and inter-\npretability of the optimal Brodmann areas used for decoding the presence of faces. All three areas\nare functionally associated with visual processing. Brodmann area 22 (Wernicke\u2019s area) is the best\npredictor of instructions (not shown). The connections between anatomy, function and prediction\nadd an important measure of interpretability to our decoding models.\n\nthe real stimulus averaged over the two virtual reality episodes and we use the region with the lowest\ntraining error to make the prediction. In principle, we could use a large number of modes to make a\nprediction with n limited only by the number of training samples. In practice the predictive power\nof our linear model saturates for a remarkably low number of modes in each region. In Fig 4(c) we\ndemonstrate the performance of the model on the number of local modes for three stimuli that are\npredicted rather well (faces, instructions and velocity).\nFor many of the well-predicted stimuli, the best Brodmann areas were also stable across subjects and\nepisodes offering important interpretability. For example, in the prediction of instructions (which\nthe subjects received through headphones), the top region was Brodmann Area 22, Wernicke\u2019s area,\nwhich has long been associated with the processing of human language. For the prediction of the\nface stimulus the best region was usually visual cortex (Brodmann Areas 17 and 19) and for the\nprediction of velocity it was Brodmann Area 7, known to be important for the coordination of visual\nand motor activity. Using modes derived from Laplacian eigenmaps we were also able to predict an\nemotional state, the self-reporting of fear and anxiety. Interestingly, in this case the best predictions\ncame from higher cognitive areas in frontal cortex, Brodmann Area 11.\nWhile the above discussion highlights the usefulness of classical anatomical location, many aspects\nof cognitive experience are not likely to be so simple. Given the reasonable results above it\u2019s natural\n\n1601200local eigenmodesglobal eigenmodes\u0bc0r\u0bc10.90Best Area (faces)(c)(b)00.930100 facesinstructionsvelocityBrodmann37Brodmann19Brodmann21(d)\u0bc0r\u0bc1local eigenmodes16030148facesinstructionsvelocity 12001000\u0bc0r\u0bc10.9(a)global Laplacianfacesinstructionsvelocity\fFigure 5: Performance of the prediction of natural experience for eleven features, using three differ-\nent methods. Local decoders do well on stimuli related to objects while nonlinear global methods\nbetter capture stimuli related to emotion.\n\nto look for ways of combining the intuition derived from single classical location with more global\nmethods that are likely to do better in prediction. As a step in this direction, we modify our model\nto include multiple Brodmann areas\n\nk (t) =(cid:88)\n\nf est\n\nn(cid:88)\n\nwl\n\niml\n\ni(t),\n\n(7)\n\nl\u2208A\n\ni=1\n\nwhere A represents a collection of areas. To make a prediction using the modi\ufb01ed model we \ufb01nd the\ntop three Brodmann areas as before (ranked by their training correlation with the stimulus) and then\nincorporate all of the modes in these areas (nA in total) in the linear model of Eq 7. The weights\n{wl\ni} are chosen to minimize RMS error on the training data. The combined model leverages both\nthe interpretive power of single areas and also some of the interactions between them. The results\nof this combined predictor are shown in Fig. 5 (black) and are generally signi\ufb01cantly better than\nthe single region predictions. For ease of comparison, we also show the best global results (both\nnonlinear Laplacian and global principal components). For many (but not all) of the stimuli, the\nlocal, low-dimensional linear model is signi\ufb01cantly better than both linear and nonlinear global\nmethods.\n\n4 Discussion\n\nIncorporating the knowledge of functional, cortical regions, we used fMRI to build low-dimensional\nmodels of natural experience that performed surprisingly well at predicting many of the complex\nstimuli in the EBC competition. In addition, the regional basis of our models allows for transparent\ncognitive interpretation, such as the emergence of Wernicke\u2019s area for the prediction of auditory\ninstructions in the virtual environment. Other well-predicted experiences include the presence of\nbody parts and faces, both of which were decoded by areas in visual cortex. In future work, it will\nbe interesting to examine whether there is a well-de\ufb01ned cognitive difference between stimuli that\ncan be decoded with local brain function and those that appear to require more global techniques.\n\narousalbodydoginterior/exteriorfacesfearful/anxiousfruits/veggiehitsinstructionsweapons/toolsvelocity0.90\u0bc0r\u0bc1 global eigenbrainglobal laplacianlocal eigenbrain\fWe also learned in this work that nonlinear methods for embedding datasets, inspired by manifold\nlearning methods [4, 5, 6], outperform linear techniques in their ability to capture the complex\ndynamics of fMRI. Finally, our particular use of Brodmann areas and linear methods represent only\na \ufb01rst step towards combining prior knowledge of broad regional brain function with the construction\nof models for the decoding of natural experience. Despite the relative simplicity, an entry based on\nthis approach scored within the top 5 of the EBC2007 competition [3].\n\nAcknowledgments\n\nGJS was supported in part by National Institutes of Health Grant T32 MH065214 and by the Swartz\nFoundation. FGM was partially supported by the Center for the Study of Brain, Mind and Behavior,\nPrinceton University. The authors are very grateful to all the members of the center for their support\nand insightful discussions.\n\nReferences\n[1] Y. Golland, S. Bentin, H. Gelbard, Y. Benjamini, R. Heller, and Y. Nir et al. Extrinsic and\nintrinsic systems in the posterior cortex of the human brain revealed during natural sensory\nstimulation. Cerebral Cortex, 17:766\u2013777, 2007.\n\n[2] S. Malinen, Y. Hlushchuk, and R. Hari. Towards natural stimulation in fMRI\u2013issues of data\n\nanalysis. NeuroImage, 35:131\u2013139, 2007.\n\n[3] http://www.ebc.pitt.edu.\n[4] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data represen-\n\ntation. Neural Computations, 15:1373\u20131396, 2003.\n\n[5] P. B\u00b4erard, G. Besson, and S. Gallot. Embeddings Riemannian manifolds by their heat kernel.\n\nGeometric and Functional Analysis, 4(4):373\u2013398, 1994.\n\n[6] R.R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis,\n\n21:5\u201330, 2006.\n\n[7] E.R. Kandel, J.H. Schwartz, and T.M. Jessell. Principles of Neural Science. McGraw-Hill, New\n\nYork, 2000.\n\n\f", "award": [], "sourceid": 908, "authors": [{"given_name": "Francois", "family_name": "Meyer", "institution": null}, {"given_name": "Greg", "family_name": "Stephens", "institution": null}]}