Independent Component Analysis of fmri Data

Transcription

1 Independent Component Analysis of fmri Data Denise Miller April 2005 Introduction Techniques employed to analyze functional magnetic resonance imaging (fmri) data typically use some form of univariate data analysis to determine regions of task-related activity. Changes in blood oxygen level dependent (BOLD) signal at each voxel can successfully be analyzed using univariate techniques. This type of analysis can identify voxels where the BOLD signal changes are significantly correlated to an expected time course for an experiment. However, univariate analysis is not able to effectively determine correlation, or lack of it, between the voxels identified as active. Two voxels may both be significantly correlated to the experiment s time course but uncorrelated with each other, implying potentially different causes of signal changes in the areas represented by these voxels. Multivariate analysis of fmri data would be ideal, but application of multivariate techniques is complicated by the fact that the number of datapoints (voxel values) greatly outnumbers the timepoints in a typical fmri dataset. Independent component analysis (ICA) is one type of multivariate data analysis which has been used to successfully analyze fmri data. The standard application of ICA is blind source separation. Observed data is assumed to be a linear combination of a number of components, where each component corresponds to a signal which may be either one of the original source signals or noise. Since source signals are often statistically independent, ICA attempts to unmix the observed data into its source signals by finding the linear transformation which provides the set of components which are as independent as possible. Signals within fmri data, for example, would include task-related activity signals, physiological signals (due to heartbeat and breathing), motion-related signals and signals due to the MRI equipment. BOLD signal changes in an fmri experiment are typically very small, and physiological signals are thought to be a major contributing factor to the measured signals. With ICA, it is hoped that signals from various sources can be successfully separated from one another. In this project, the FastICA algorithm [4] was applied to fmri data from the synaethesia dataset. Once independent components within the fmri data were identified, the set of components that correspond to task-related activation was determined by comparing the components time courses with the expected time course for the experiment. An activation map was created for each of these components. Areas of task-related activity identified by ICA were also compared to the active brain regions identified using BOLDfold, and to those identified by creating an Afni fico map. Method Independent Component Analysis Independent component analysis is based on the assumption that the observed data, X, is a linear combination of a set of independent components, C, representing the source signals. For fmri data, X is an nxv matrix where n is the number of time points, and v is the number of voxels within each image. With noise terms included, the relationship between the observed data and the source signals can be expressed as X = MC + N However, noise is often not specifically modeled in ICA algorithms. In this case, it is assumed that ICA will find noise as separate components. Omitting noise terms, X = MC 1

2 In other words, the observed fmri signals are assumed to be the linear sum of contributions of individual components at each voxel. Each row of the matrix C represents one component. The mixing matrix, M, indicates the relative contribution of each component at each time point to the observed data. The ICA algorithm iteratively determines the unmixing matrix, W, such that C = WX W is not necessarily the exact inverse of M; it may be permuted and/or linearly scaled. Data could then be reconstructed from X = W -1 C In the case where W -1 is equal to M, the data could be perfectly reconstructed. When applied to fmri data, the components identified by ICA consist of a set of voxel values and associated time courses. Either independence of the spatial patterns or of the time courses can be used to separate the signals. The choice of spatial or temporal independence when analyzing fmri data is somewhat controversial [2], although spatial independence appears to be the most common assumption in published fmri analyses using ICA. For this project, spatial independence was assumed. Each row of C provides the map voxel values of one component. The corresponding column in the W -1 matrix gives the time course for that component (i.e. column 1 of W -1 gives the time course for the component represented by row 1 of C). As the fmri data changes with time, ICA assumes that the changes are the result of changes in the relative contributions of the components to the observed data, not due to changes in the component maps themselves. Component maps do not change during the course of an experiment. Voxel values of a component map represent the contribution of the individual voxels to that component. To determine the voxels which contribute significantly to a particular component map, the map values can be scaled using z-scores (the number of standard deviations of the voxel value from the mean voxel value in the map). Voxels with absolute z-score greater than a certain threshold can be considered active voxels for that component. ICA can extract a number of independent components from the fmri data up to the number of time points in that data. The sum of all task-related components provides the full picture of task-related brain activity. The order of the components determined by ICA is not significant. A fundamental assumption of ICA is that the components do not have gaussian distributions. In fact, FastICA finds the components within the observed data by maximizing nongaussianity [5]. McKeown et al. [9] examined both the assumption of non-gaussianity of source signals of fmri data, and the assumption that the fmri components were spatially independent, and found both to be valid. The FastICA Algorithm For this project, FastICA was used to find independent components within the fmri data. FastICA code is freely available from FastICA provides two approaches to calculate the independent components within observed data. In the Deflation technique, independent components are estimated one by one using hierarchical decorrelation. In the Symmetric technique, all independent components are estimated at once using symmetric decorrelation. Deflation is the default. FastICA performs two pre-processing steps. In the first step, the data is centered by subtracting the mean. This step is performed to simplify ICA computations. The mean is then added back to the data after the signals have been estimated. In the second pre-processing step, principal component analysis (PCA) is used in order to whiten the data. The observed data is transformed so that the covariance matrix of the transformed data equals unity. Although this step is not required by all ICA algorithms, all ICA algorithms converge better with whitened data. PCA may also reduce the dimension of the data, and determine the number of independent components. 2

3 The FastICA algorithm makes some key assumptions about the data. As with all ICA algorithms, FastICA assumes that the independent components are non-gaussian. Also, FastICA assumes that the number of observed mixtures (n) is at least as large as the number of independent components. This assumption is not strictly required by ICA, but is necessary in order to identify the individual components and not just the mixing matrix, M. FastICA makes a further simplification by assuming that the dimension of the observed data equals the number of independent components. Full details of the FastICA algorithm are described by Hyvarinen et al. [5]. Code Developed for this Project For this project, ICA was used to analyze the synaethesia fmri dataset to determine regions of the brain which had task-related activity. The input data for the analysis was a series of Afni time series images. Afni Matlab functions were used to read and write fmri maps in Afni format. FastICA routines were used to perform independent component analysis. In addition, a number of custom developed routines were created. In addition to running the appropriate Afni Matlab and FastICA functions, these routines convert data into the expected formats, separate the individual components identified by FastICA, determine the correlation of the time course of each of these components to the expected time course of the experiment, and convert the component maps for correlated components into Afni maps. The custom developed code is included in the following Matlab routines. convertafni.m separatesignals.m converttoafni.m convertfunctoafni. m runica.m Reads the Afni time series data using Afni Matlab functions. Converts data into an nxv matrix for input to the FastICA routine. Compares the time course of each component found by FastICA to the expected time course for the experiment. Any time points which fall within the stabilization period at the beginning of the experiment are excluded from the comparison. For each component which has a correlation value greater than or equal to the specified cutoff value, a data file is created to store the component map for that component (the corresponding row of the C matrix). If requested, a time series image data file is also created for each significantly correlated component. The time series data is constructed by multiplying the appropriate column of the W -1 matrix with the component row from the C matrix. Converts data from the component map data file (created by separatesignals.m) into the required format. Creates a raw component map from this data, in Afni format. This map will have voxel values which indicate the relative contribution of each voxel to the component. Also creates an Afni intensity map which indicates only those voxels which contribute significantly to the component, as determined by their z-score. Z-score is computed for each voxel, and for each voxel with a z-score above the specified threshold, the voxel value in the intensity map is set to its z-score. Values for all insignificant voxels are set to zero in this map. These intensity maps can be used to identify the regions of task-related brain activity. Converts data from the time series data files created by separtesignals.m into Afni time series images. Convenience routine which runs convertafni.m, FastICA and separatesignals.m All code developed for this project is available on io.usask.ca. Full instructions for running the project code are provided in the Independent Component Analysis of fmri Data User Guide. 3

4 ICA Analysis of the Synaethesia fmri Dataset The Deflation (default) technique of FastICA was used for most of the analysis of synaethesia data. The Symmetric technique was used in only a limited number of cases for comparison purposes. A square wave was used as the comparison function for the experiment. The comparison function identified that the first 5 time points should be ignored when correlating the time course of each component with the expected time course, as these initial time points fall within the stabilization period. Results The ICA project code was applied to the data for several of the subjects from the synaethesia fmri study. In general, ICA tended to identify multiple components which had time series that were correlated to the expected time course for each of the experiments. Correlation values of component time courses to the expected time course were generally low. As a result, a value of 0.3 was specified as the cutoff point. Data for all components with a correlation value greater than or equal to 0.3 was saved. Data for the components with the highest correlation values above this threshold were then converted to Afni maps in subsequent steps. Typically experiments had between 2 and 6 components with a correlation value above this threshold. Each analysis also produced a number of components which were negatively correlated to the task. In creating the intensity maps, a z-score cutoff of 4 seemed to be appropriate in most cases. Using a z- score value below this level tended to leave a number of marked voxels which appeared to be noise, such as isolated active voxels, and, in some cases, voxels outside the head. Use of a z-score cutoff higher than 4 eliminated a number of regions which had been identified as active. When the regions identified by ICA as active during the task were compared to the active regions identified by BOLDfold or by Afni fico maps, there was typically some amount of overlap in addition to active regions identified uniquely by ICA or the other techniques. It is likely that the amount of overlap would increase if contributions of additional components were added to the comparison. However, this is also likely to increase the number of voxels identified as active by ICA alone. The figures below show the results of using the ICA project code on the test results from subject E11 in the arithn test, in which the subject was shown simple equations and asked to solve them. For this case, ICA identified only two components with a correlation value above 0.3: component 70 had the highest correlation value at , and component 18 was second at The next highest correlation value of any of the components for this case was only Figures 1 and 3 show the regions identified as active by these components individually. In Figure 4, the active regions identified by both task-related components are combined. The active areas identified by these components are overlaid with the active areas identified by BOLDfold analysis of this case. Active areas identified only by the components, those identified only by BOLDfold, and those areas identified by both ICA and BOLDfold are indicated in different colours. Figure 2 shows the time course for voxels which contribute significantly to component 70. 4

5 Figure 1: Component 70. All pixels with a z-score greater than or equal to 3.0 are shown. Correlation coefficient for component 70 is Figure 2: Sample time course for voxels which contribute significantly (high z-score) to component 70. 5

6 Figure 3: Component 18. All pixels with z-score greater than or equal to 3.0 are shown. Correlation coefficient for component 18 is

7 In the figure below, the two components which are most task-related are combined. Voxels identified as active by either component 70 or component 18 are shown in red. The results of using BOLDfold to identify task-related brain activity are also shown on these images, with these voxels marked in blue. The overlap of active areas identified by ICA and BOLDfold is shown in yellow. Figure 4: Comparison of areas of task-related activity as identified by the two more task correlated components, and by BOLDfold. Voxels shown in red indicate areas of activity identified by the components only; voxels shown in blue indicate areas of activity identified by BOLDfold only; voxels shown in yellow indicate areas of activity identified by both methods. Discussion One of the problems of ICA is that it gives a large number of components, in no particular order. Kiviniemi et al. [6] attempt to alleviate this problem by specifying that the number of principal components chosen during the PCA pre-processing step of FastICA be 40. The number 40 was selected in order to still provide 99.99% coverage of signal variance. However, it is not clear how it was determined that 40 was the required number of components to provide this coverage. It would be interesting to see the results of limiting the number of components in the analysis of the synaethesia dataset. This was not attempted. A very minor change to the runica.m routine would be required in order to supply the maximum number of components as an input parameter. However, the challenge with reducing the number of components this way is that the unmixing matrix would no longer be square, and so no longer invertible. It is the corresponding column of W -1, the inverse of the unmixing matrix, which provides the time course for each component. So, by forcing the number of components to be smaller than the number of timepoints, there is no longer a way to determine which of the components has a time course which is correlated to the expected time course of the experiment. Voxel scores within component maps may have either positive or negative values. Negative z-scores of 7

8 voxels within a component map indicate voxels where the fmri signal is modulated opposite to the time course of that component. Voxels with negative z-scores were not examined in this project, although this would be interesting for future work. In this project, a correlation cutoff of 0.3 and a z-score cutoff of 4 seemed to be most appropriate. In the ICA analysis of fmri data performed by McKeown et al. [8], a correlation cutoff of 0.4, and a z-score cutoff of 2 were used. Kiviniemi et al. [6] performed ICA analysis of fmri data also using the FastICA routine. As was done in the synaethesia data analysis of this project, Kiviniemi et al. used FastICA s deflation approach. For their work, they found that a z-score cutoff of 6 was appropriate. For the synaethesia dataset, ICA identified a number of components which had time courses that were significantly correlated to the expected time course of the experiment. The typical number of these taskrelated components was between 2 and 6 of the 143 components identified. In McKeown s study [8], for each experiment exactly one ICA component was found which had a time course which was significantly correlated (correlation coefficient of 0.64 to 0.94) with the reference function for the experiment. Arfanakis et al. [1] report results which are closer to the results of this project. In their analysis of fmri data, Arfanakis et al. found that in most cases there was more than one ICA component with a time course that corresponded to the task activation pattern. The maximum number of such task-related components was found to be 7 of 144 components. In the work of McKeown et al. [8], regions of maximum activity in task-related components overlapped active regions found by the standard correlation method. However, their ICA analysis also found additional regions of task-related activation which were not identified by the conventional method. McKeown et al. state that, in general, ICA, PCA and correlation analysis methods for finding active voxels will usually not give identical results, although there may be overlap [8]. This finding is consistent with the results of the ICA analysis of synaethesia data when compared to results from BOLDfold and Afni (correlation) analysis of the same data. For the majority of the ICA analysis of synaethesia data, FastICA s default deflation technique was used. This approach is consistent with that described by Kiviniemi et al. [6] in their ICA analysis of fmri data. However, Hyvaarinen states that the deflation technique is useful for exploratory data analysis [5]. It is not clear whether this statement is meant to imply that deflation should only be used for an initial exploratory analysis. When the symmetric approach was used to analyze synaethesia data, it was found that this technique identified a different pattern of components than deflation. That is, there was a noticeable difference in the number of components with time courses significantly correlated to the expected time course when each of these techniques was applied to the same input data. Additional FastICA analysis using the symmetric technique should be run against the synaethesia data, in order to determine the extent of the differences in components identified. For this project, it was assumed that the spatial patterns of the underlying components of observed fmri data were independent. The spatial independence assumption seems to be consistent with most published ICA analyses of fmri data, including that of McKeown et al. [8]. However, the debate between selection of spatial independence vs. temporal independence when analyzing fmri data appears to be ongoing. Calhoun et al. [2] recommend that both spatial ICA and temporal ICA be applied to fmri data, and the authors cite examples where both approaches have been used in combination for analysis of fmri data. It would be worthwhile to examine the results of an ICA analysis of the synaethesia data where temporal independence is assumed. The comparison function used for all ICA analysis in this project was a standard square wave. Although a square wave is an acceptable comparison function for block fmri experiment design, a better comparison function would be a square wave convolved with the expected hemodynamic response function. If a more accurate comparison function were used, it would be interesting to see whether additional or different components were identified as task-related, or whether the same pattern of components was identified, but just with higher correlation coefficients. Calhoun et al. [2] identify signals of interest within fmri data as including: 8

9 Task-related signals: Signals which are correlated to reference waveform for the experiment Transiently task-related signals: The responses of brain are not necessarily regular. Signals may, for example, die out before end of a stimulus, resulting in a transiently task-related pattern of activation. Function-related signals: Similarities between voxels within a functional domain of the brain (e.g. motor cortex) As the code is currently written, the separatesignals.m routine will only detect and process task-related signals. However, the information for all source signals is available in the output from the FastICA routine. It would be interesting to expand the existing code in order to separate out the other signals of interest. McKeown et al. [9] identify a pattern to regions of the brain where ICA models fit better or less well. According to their findings, white matter is better modeled than the cortex or regions around blood vessels. The authors state that this disparity may be explained by the fact that the number of spatially independent components differs between the more active gray matter and white matter. They speculate that the actual mixture of components may be nonlinear. McKeown et al. suggest that advances in the application of ICA to fmri data may result from separate analyses of different subsets of the brain. 9

10 References [ 1 ] Arfanakis, Konstantinos, Cordes, Dietmar, Haughton, Victor M., Moritz, Chad H., Quigley, Michelle A., and Meyerand, Mary E., Combining independent component analysis and correlation analysis to probe interregional connectivity in fmri task activation datasets, Magnetic Resonance Imaging, Volume 18, pp , [ 2 ] Calhoun, V.D., Adali, T., Hansen, L.K., Larsen, J., and Pekar, J.J., ICA of Functional MRI Data: An Overview, 4 th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA 2003), April 2003, Available from: Accessed: [ 3 ] Calhoun, V.D., Adali, T., Pearlson, G.D., and Pekar, J.J., Spatial and Temporal Independent Component Analysis of Functional MRI Data Containing a Pair of Task-Related Waveforms, Human Brain Mapping, Volume 13, pp , [ 4 ] Hyvarinen, Aapo, Survey on Independent Component Analysis, Available from: Accessed: [ 5 ] Hyvarinen, Aapo, and Oja, Erkki, Independent Component Analysis: Algorithms and Applications, Neural Networks, Volume 13, No. 4, pp , [ 6 ] Kiviniemi, Vesa, Kantola, Juha-Heikki, Jauhiainen, Jukka, Hyvarinen, Aapo, and Tervonen, Osmo, Independent component analysis of nondeterministic fmri signal sources, NeuroImage, Volume 19, pp , [ 7 ] McKeown, Martin J., Detection of Consistently Task-Related Activations in fmri Data with Hybrid Independent Component Analysis, NeuroImage, Volume 11, pp , [ 8 ] McKeown, Martin J., Makeig, Scott, Brown, Greg G., Jung, Tzyy-Ping, Kinderman, Sandra S., Bell, Anthony J., and Sejnowski, Terrence J., Analysis of fmri Data by Blind Separation Into Independent Spatial Components, Human Brain Mapping, Volume 6, pp , [ 9 ] McKeown, Martin J., and Sejnowski, Terrence J., Independent Component Analysis of fmri Data: Examining the Assumptions, Human Brain Mapping, Volume 6, pp , [ 10 ] Sarty, Gordon E., Computing Brain Activity Maps from fmri Time-Series Images. 10