Locality and low-dimensions in the prediction of natural experience from fMRI Advances in Neural Information Processing Systems 20, Schölkopf B., Platt J. and Hofmann T. (Editors), ©  MIT Press, 2008

Locality and low-dimensions in the prediction of natural experience from fMRI thanks: Advances in Neural Information Processing Systems 20, Schölkopf B., Platt J. and Hofmann T. (Editors), ©  MIT Press, 2008

François G. Meyer
Center for the Study of Brain, Mind and Behavior,
Program in Applied and Computational Mathematics
Princeton University
fmeyer@colorado.edu &Greg J. Stephens
Center for the Study of Brain, Mind and Behavior,
Department of Physics
Princeton University
gstephen@princeton.edu &
Both authors contributed equally to this work.
Abstract

Functional Magnetic Resonance Imaging (fMRI) provides dynamical access into the complex functioning of the human brain, detailing the hemodynamic activity of thousands of voxels during hundreds of sequential time points. One approach towards illuminating the connection between fMRI and cognitive function is through decoding; how do the time series of voxel activities combine to provide information about internal and external experience? Here we seek models of fMRI decoding which are balanced between the simplicity of their interpretation and the effectiveness of their prediction. We use signals from a subject immersed in virtual reality to compare global and local methods of prediction applying both linear and nonlinear techniques of dimensionality reduction. We find that the prediction of complex stimuli is remarkably low-dimensional, saturating with less than 100 features. In particular, we build effective models based on the decorrelated components of cognitive activity in the classically-defined Brodmann areas. For some of the stimuli, the top predictive areas were surprisingly transparent, including Wernicke’s area for verbal instructions, visual cortex for facial and body features, and visual-temporal regions for velocity. Direct sensory experience resulted in the most robust predictions, with the highest correlation () between the predicted and experienced time series of verbal instructions. Techniques based on non-linear dimensionality reduction (Laplacian eigenmaps) performed similarly. The interpretability and relative simplicity of our approach provides a conceptual basis upon which to build more sophisticated techniques for fMRI decoding and offers a window into cognitive function during dynamic, natural experience.

 

Locality and low-dimensions in the prediction of natural experience from fMRI thanks: Advances in Neural Information Processing Systems 20, Schölkopf B., Platt J. and Hofmann T. (Editors), ©  MIT Press, 2008


  François G. Meyer Center for the Study of Brain, Mind and Behavior, Program in Applied and Computational Mathematics Princeton University fmeyer@colorado.edu Greg J. Stephens Center for the Study of Brain, Mind and Behavior, Department of Physics Princeton University gstephen@princeton.edu Both authors contributed equally to this work.

1 Introduction

Functional Magnetic Resonance Imaging (fMRI) is a non-invasive imaging technique that can quantify changes in cerebral venous oxygen concentration. Changes in the fMRI signal that occur during brain activation are very small (1-5%) and are often contaminated by noise (created by the imaging system hardware or physiological processes). Statistical techniques that handle the stochastic nature of the data are commonly used for the detection of activated voxels. Traditional methods of analysis – which are designed to test the hypothesis that a simple cognitive or sensory stimulus creates changes in a specific brain area – are unable to analyze fMRI datasets collected in “natural stimuli” where the subjects are bombarded with a multitude of uncontrolled stimuli that cannot always be quantified [1, 2].
The Experience Based Cognition competition (EBC) [3] offers an opportunity to study complex responses to natural environments, and to test new ideas and new methods for the analysis of fMRI collected in natural environments. The EBC competition provides fMRI data of three human subjects in three 20-minute segments (704 scanned samples in each segment) in an urban virtual reality environment along with quantitative time series of natural stimuli or features (25 in total) ranging from objective features such as the presence of faces to self-reported, subjective cognitive states such as the experience of fear. During each session, subjects were audibly instructed to complete three search tasks in the environment: looking for weapons (but not tools) taking pictures of people with piercings (but not others), or picking up fruits (but not vegetables). The data was collected with a 3T EPI scanner and typically consists of the activity of 35000 volume elements (voxels) within the head. The feature time series was provided for only the first two segments (1408 time samples) and competitive entries are judged on their ability to predict the feature on the third segment (704 time samples, see Fig. 1).

Figure 1: We study the variation of the set of features as a function of the dynamical changes in the fMRI signal during natural experience. The features represent both external stimuli such as the presence of faces and internal emotional states encountered during the exploration of a virtual urban environment (left and right images). We predict the feature functions for , from the knowledge of the entire fMRI dataset , and the partial knowledge of for . The “toy” activation patterns (middle diagram) illustrate the changes in “brain states” occurring as a function of time.

At a microscopic level, a large number of internal variables associated with various physical and physiological phenomena contribute to the dynamic changes in the fMRI signal. Because the fMRI signal is a large scale (as compared to the scale of neurons) measurement of neuronal activity, we expect that many of these variables will be coupled resulting in a low dimensional set for all possible configurations of the activated fMRI signal. In this work we seek a low dimensional representation of the entire fMRI dataset that provides a new set of ‘voxel-free” coordinates to study cognitive and sensory features.

We denote a three-dimensional volumes of fMRI composed of a total of voxels by . We have access to such volumes. We can stack the spatio-temporal fMRI dataset into a matrix,

(1)

where each row represents a time series generated from voxel and each column represents a scan acquired at time . We call the set of features to be predicted . We are interested in studying the variation of the set of features describing the subject experience as a function of the dynamical changes of the brain, as measured by . Formally, we need to build predictions of for , from the knowledge of the entire fMRI dataset , and the partial knowledge of for the training time samples (see Fig. 1).

Figure 2: Low-dimensional parametrization of the set of “brain states”. The parametrization is constructed from the samples provided by the fMRI data at different times, and in different states.

2 A voxel-free parametrization of brain states

We use here the global information provided by the dynamical evolution of over time, both during the training times and the test times. We would like to effectively replace each fMRI dataset by a small set of features that facilitates the identification of the brain states, and make the prediction of the features easier. Formally, our goal is to construct a map from the voxel space to low dimensional space.

(2)
(3)

where . As varies over the training and the test sets, we hope that we explore most of the possible brain configurations that are useful for predicting the features. The map provides a parametrization of the brain states. Figure 2 provides a pictorial rendition of the map . The range , represented in Fig. 2 as a smooth surface, is the set of parameters that characterize the brain dynamics. Different values of the parameters produce different “brain states”, associated with different patterns of activation. Note that time does not play any role on , and neighboring points on correspond to similar brain states. Equipped with this re-parametrization of the dataset , the goal is to learn the evolution of the feature time series as a function of the new coordinates . Each feature function is an implicit function of the brain state measured by . For a given feature , the training data provide us with samples of at certain locations in . The map is build by globally computing a new parametrization of the set . This parametrization is built into two stages. First, we construct a graph that is a proxy for the entire set of fMRI data . Second, we compute some eigenfunctions defined on the graph. Each eigenfunctions provides one specific coordinate for each node of the graph.

2.1 The graph of brain states

We represent the fMRI dataset for the training times and test times by a graph. Each vertex corresponds to a time sample , and we compute the distance between two vertices and by measuring a distance between and . Global changes in the signal due to residual head motion, or global blood flow changes were removed by computing a a principal components analysis (PCA) of the dataset and removing a small number components. We then used the distance between the fMRI volumes (unrolled as vectors). This distance compares all the voxels (white and gray matter, as well as CSF) inside the brain.

2.2 Embedding of the dataset

Once the network of connected brain states is created, we need a distance to distinguish between strongly connected states (the two fMRI data are in the same cognitive state) and weakly connected states (the fMRI data are similar, but do not correspond to the same brain states). The Euclidean distance used to construct the graph is only useful locally: we can use it to compare brain states that are very similar, but is unfortunately very sensitive to short-circuits created by the noise in the data. A standard alternative to the geodesic (shortest distance) is provided by the average commute time, , that quantifies the expected path length between and for a random walk started at . Formally, , where is the hitting time,

for a random walk on the graph with transition probability , defined by , and is the degree of the vertex . The commute time can be conveniently computed from the eigenfunctions of , with the eigenvalues . Indeed, we have

As proposed in [4, 5, 6], we define an embedding

(4)

Because , we have . We can therefore neglect for large , and reduce the dimensionality of the embedding by using only the first coordinates in (4). The spectral gap measures the difference between the first two eigenvalues, . A large spectral gap indicates that the low dimensional will provide a good approximation. The algorithm for the construction of the embedding is summarized in Fig. 3.

Algorithm 1: Construction of the embedding
 

  • Input:

    • , : number of eigenfunctions.

  • Algorithm:

    1. construct the graph defined by the nearest neighbors

    2. find the first eigenfunctions, , of

  • Output: For

    • new co-ordinates of :

 

Figure 3: Construction of the embedding

A parameter of the embedding (Fig. 3) is , the number of coordinates. can be optimized by minimizing the prediction error. We expect that for small values of the embedding will not describe the data with enough precision, and the prediction will be inaccurate. If is too large, some of the new coordinates will be describing the noise in the dataset, and the algorithm will overfit the training data. Fig. 4-(a) illustrates the effect of on the performance of the nonlinear dimension reduction. The quality of the prediction for the features: faces, instruction and velocity is plotted against . Instructions elicits a strong response in the auditory cortex that can be decoded with as few as 20 coordinates. Faces requires more (about 50) dimensions to become optimal. As expected the performance eventually drops when additional coordinates are used to describe variability that is not related to the features to be decoded. This confirms our hypothesis that we can replace about 15,000 voxels with 50 appropriately chosen coordinates.

2.3 Semi-supervised learning of the features

The problem of predicting a feature at an unknown time is formulated as kernel ridge regression problem. The training set for is used to estimate the optimal choice of weights in the following model,

where is a kernel and is a time point where we need to predict.

2.4 Results

We compared the nonlinear embedding approach (referred to as global Laplacian) to dimension reduction obtained with a PCA of . Here the principal components are principal volumes, and for each time we can expand onto the principal components.

The 1408 training data were divided into two subsets of time samples. We use in a subset to predict in the other subset. In order to quantify the stability of the prediction we randomly selected 85 % of the training set (first subset), and predicted 85 % of the testing set (other subset). The role, training or testing, of each subset of time samples was also chosen randomly. We generated 20 experiments for each value of , the number of predictors. The performance was quantified with the normalized correlation between the model prediction and the real value of ,

(5)

where . Finally, was averaged over the 20 experiments. Fig. 4-(a) and (b) show the performance of the nonlinear method and linear method as a function of . The approach based on the nonlinear embedding yields very stable results, with low variance. For both global methods the optimal performance is reached with less than 50 coordinates. Fig. 5 shows the correlation coefficients for 11 features, using coordinates. For most features, the nonlinear embedding performed better than global PCA.

3 From global to local

While models based on global features leverage predictive components from across the brain, cognitive function is often localized within specific regions. Here we explore whether simple models based on classical Brodmann regions provide an effective decoder of natural experience. The Brodmann areas were defined almost a century ago (see e.g [7]) and divide the cortex into approximately 50 regions, based on the structure and arrangement of neurons within each region. While the areas are characterized structurally many also have distinct functional roles and we use these roles to provide useful interpretations of our predictive models. Though the partitioning of cortical regions remains an open and challenging problem, the Brodmann areas represent a transparent compromise between dimensionality, function and structure.

Using data supplied by the competition, we warp each brain into standard Talairach coordinates and locate the Brodmann area corresponding to each voxel. Within each Brodmann region, differing in size from tens to thousands of elements, we build the covariance matrix of voxel time series using all three virtual reality episodes. We then project the voxel time series onto the eigenvectors of the covariance matrix (principal components) and build a simple, linear stimulus decoding model using the top modes ranked by their eigenvalues,

(6)

where indexes the different Brodmann areas, are the linear weights and are the mode time series in each region. The weights are chosen to minimize the RMS error on the training set and have a particularly simple form here as the modes are decorrelated, . Performance is measured as the normalized correlation (Eq. 5) between the model prediction and

Figure 4: Performance of the prediction of natural experience for three features, faces, instructions and velocity as a function of the model dimension. (a) nonlinear embedding, (b) global principal components, (c) local (Brodmann area) principal components. In all cases we find that the prediction is remarkably low-dimensional, saturating with less than 100 features. (d) stability and interpretability of the optimal Brodmann areas used for decoding the presence of faces. All three areas are functionally associated with visual processing. Brodmann area 22 (Wernicke’s area) is the best predictor of instructions (not shown). The connections between anatomy, function and prediction add an important measure of interpretability to our decoding models.

the real stimulus averaged over the two virtual reality episodes and we use the region with the lowest training error to make the prediction. In principle, we could use a large number of modes to make a prediction with limited only by the number of training samples. In practice the predictive power of our linear model saturates for a remarkably low number of modes in each region. In Fig 4(c) we demonstrate the performance of the model on the number of local modes for three stimuli that are predicted rather well (faces, instructions and velocity).

For many of the well-predicted stimuli, the best Brodmann areas were also stable across subjects and episodes offering important interpretability. For example, in the prediction of instructions (which the subjects received through headphones), the top region was Brodmann Area 22, Wernicke’s area, which has long been associated with the processing of human language. For the prediction of the face stimulus the best region was usually visual cortex (Brodmann Areas 17 and 19) and for the prediction of velocity it was Brodmann Area 7, known to be important for the coordination of visual and motor activity. Using modes derived from Laplacian eigenmaps we were also able to predict an emotional state, the self-reporting of fear and anxiety. Interestingly, in this case the best predictions came from higher cognitive areas in frontal cortex, Brodmann Area 11.

Figure 5: Performance of the prediction of natural experience for eleven features, using three different methods. Local decoders do well on stimuli related to objects while nonlinear global methods better capture stimuli related to emotion.

While the above discussion highlights the usefulness of classical anatomical location, many aspects of cognitive experience are not likely to be so simple. Given the reasonable results above it’s natural to look for ways of combining the intuition derived from single classical location with more global methods that are likely to do better in prediction. As a step in this direction, we modify our model to include multiple Brodmann areas

(7)

where A represents a collection of areas. To make a prediction using the modified model we find the top three Brodmann areas as before (ranked by their training correlation with the stimulus) and then incorporate all of the modes in these areas (nA in total) in the linear model of Eq 7. The weights are chosen to minimize RMS error on the training data. The combined model leverages both the interpretive power of single areas and also some of the interactions between them. The results of this combined predictor are shown in Fig. 5 (black) and are generally significantly better than the single region predictions. For ease of comparison, we also show the best global results (both nonlinear Laplacian and global principal components). For many (but not all) of the stimuli, the local, low-dimensional linear model is significantly better than both linear and nonlinear global methods.

4 Discussion

Incorporating the knowledge of functional, cortical regions, we used fMRI to build low-dimensional models of natural experience that performed surprisingly well at predicting many of the complex stimuli in the EBC competition. In addition, the regional basis of our models allows for transparent cognitive interpretation, such as the emergence of Wernicke’s area for the prediction of auditory instructions in the virtual environment. Other well-predicted experiences include the presence of body parts and faces, both of which were decoded by areas in visual cortex. In future work, it will be interesting to examine whether there is a well-defined cognitive difference between stimuli that can be decoded with local brain function and those that appear to require more global techniques. We also learned in this work that nonlinear methods for embedding datasets, inspired by manifold learning methods [4, 5, 6], outperform linear techniques in their ability to capture the complex dynamics of fMRI. Finally, our particular use of Brodmann areas and linear methods represent only a first step towards combining prior knowledge of broad regional brain function with the construction of models for the decoding of natural experience. Despite the relative simplicity, an entry based on this approach scored within the top 5 of the EBC2007 competition [3].

Acknowledgments

GJS was supported in part by National Institutes of Health Grant T32 MH065214 and by the Swartz Foundation. FGM was partially supported by the Center for the Study of Brain, Mind and Behavior, Princeton University. The authors are very grateful to all the members of the center for their support and insightful discussions.

References

  • [1] Y. Golland, S. Bentin, H. Gelbard, Y. Benjamini, R. Heller, and Y. Nir et al. Extrinsic and intrinsic systems in the posterior cortex of the human brain revealed during natural sensory stimulation. Cerebral Cortex, 17:766–777, 2007.
  • [2] S. Malinen, Y. Hlushchuk, and R. Hari. Towards natural stimulation in fMRI–issues of data analysis. NeuroImage, 35:131–139, 2007.
  • [3] http://www.ebc.pitt.edu.
  • [4] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computations, 15:1373–1396, 2003.
  • [5] P. Bérard, G. Besson, and S. Gallot. Embeddings Riemannian manifolds by their heat kernel. Geometric and Functional Analysis, 4(4):373–398, 1994.
  • [6] R.R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21:5–30, 2006.
  • [7] E.R. Kandel, J.H. Schwartz, and T.M. Jessell. Principles of Neural Science. McGraw-Hill, New York, 2000.
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