Caltech's Economic fMRI Decision Analysis Group - Bayesian Blind Source Separation

In an fMRI experiment, Participants are commonly presented two stimuli, A and B of equal time duration. There are several rounds of these stimuli repeated as ABABAB...AB that make up a scan session. Imaging takes place while the participants are responding either actively or passively to the presented stimuli. In our experiments, tasks A and B were both 32 seconds in length and 8 rounds were presented as ABABABABABABABAB to make up a total of 512 seconds for the scan session. It was decided to take 24 axial slices, each being 64x64 voxels. The dimensions of the voxels were 3x3x5 (in mm). Observations were taken in each of the p=98304 voxels every 4 seconds so that there are n=128 observations in each.

The Methods

Data is collected on subjects in the form of MRIs while performing Economic tasks. This data consists of n=128 entire brain volume scans of dimension p=98304. There are p=98304 voxels and each has time courses of length n=128. In a typical analysis of fMRI data, the time courses for each of the p voxels are assumed to be independent as well as the n observations that make up the time courses. Typically the time course in each voxel is fit to a multiple linear Regression model of the form

(x_i| a,b,

,s_i)

a+bi+

s_i +

where
x_i is the observed voxel value at time i,
a and b capture any mean and linear trend,

is the mixing or regression coefficient,
s_i is the value of the "known" reference function (i.e. square, sine, triangular, trapesoidal, ... wave possibly shifted) at time point i,
and

_i is the i^th random error term.

This can be generalized to fit the multiple regression model

(x_i| a,b,

₁,...,

_m, s_i1,...,s_im)

a+bi+

_k=1^m

_k s_ik +

where the additional subscript k is for the m "known" reference wave functions. For example corresponding to the several simple tasks that make up a more complex experimental task, the wave form of "known" higher frequency signal, the participants EKG or respiration. There can be several reference wave functions but the first will be assigned to correspond to the experimental stimulus.

After the reference function(s) is (are) chosen and the regression coefficients determined, the observed time course is detrended by subtracting out the linear trend (and any other reference functions contribution) and scaled by the coefficient of the reference function. Then the correlation coefficient between the reference function of interest and the data y after detrending and subtraction of all other underlying source component reference functions is computed and used to determine significance. A colormap is selected and voxels colored according to their correlation level.

For example, if the first reference function was the one of interest, then the detrended time course is

y_i = (x_i-a-bi- Sum

_k=2^m

_k s_ik)/

₁

and the correlation coefficient between the y_i's and the s_i1's is computed.

But how well are these reference wave functions really known. Which one(s) do we use, and is it exactly the same for all time, or does it change slightly? There is currently debate as to which wave function to use and what lag? Two researchers could have the same data but analyze it differently accourding to their subjectively selected reference functions and arrive at differing conclusions. This is why we adopt the Bayesian paradigm. Available prior knowlege regarding the reference function (and other parameters) is quantified using prior distributions. Uncertainty in the reference functions is assessed complete with a mean and variance. Further, uncertainty as to their amplitudes and the observation error covariance matrix is quantified in the form of prior distributions. The prior knowlege in the form prior distributions is combined with the current data and the reference function along with the other parameters is determined.

The fMRI problem is exactly the problem of source separation. Source separation refers to the experimental situation where mixed signals are observed, not the original unmixed source signals. The original source signals are unobserved. The problem is to unmix and obtain the underlying source signals so they can be analyzed. The observed time course in the voxles is made up of unobserved contributions from several sources. We are looking for the one (or more) source(s) associated with the response of the participant to the presented stimmulus.

We will be taking a Bayesian statistical approach to source separation. In Bayesian source separation and more generally the Bayesian approach to statistical inference, available prior information either from subjective expert experience or prior experiments is incorporated into the inferences. This prior information yields progressively less influence in the final results as the sample size increases, thus allowing the data to "speak the truth." The Bayesian source separation statistical model has several advantages over the currently used methods. Among the advantages are that the observed signals are measured with nonzero error, the source components are not constrained to be statistically independent, and the number of sources is not confined to be the same as the number of observed signals. It can be greater or smaller.

We believe that the assumption of statistically independent source components is unrealistic for use in separating cerebral activity. The observed activation in each voxel is composed of several periodic reference functions that may correspond to stimuli such as several simple tasks that make up a more complex experimental task, an EKG, and respiration which may be correlated. Further, In the brain, the responses to stimuli activate regions which act synergistically or in concert. These regions which are activating together are obviously correlated.

Once the source signals for the cognitive activity are separated, we can concentrate on those sources that correspond to the responses to the stimuli.

Some fMRI Results

In analyzing the results of functional magnetic resonance imaging, the identification of significant activation in voxels is a crucial task. In computing the activation level, a standard method is to select an assumed to be known reference function and perform a multiple regression of the time courses on it and a linear trend. Once the linear trend is found, the correlation between the assumed to be known reference function and the detrended observed time-course in each voxel is computed and voxels colored according to their correlation. But the most important question is: How do we choose the reference function? We developed a Bayesian statistical approch to determining the underlying source reference function based on Bayesian source separation, and used it on both sumulated and real fMRI data. This underlying reference function is the unobserved response due the presentation of the experimental stimulus.

The detrended time course _ _ of a voxel in the highlighted area, the prior square ._ , and the Bayesian reference function __ .
Note the similarity between the detrended time course and the Bayesian reference function. The Bayesian reference function captures the dynamically changing hemodynamic response of the subject to the presentation of the stimulus.

Activation with prior square reference function.
If the threshold is raised, the area starts to disappear.
Note that the activations are low and buried in the noise.
Square

Activations with Bayesian reference function.
If the threshold is raised, the area starts to slowly disappear.
Note that the activations are higher stand out above the noise.

Recent Work: Rowe, D. B. (2001). Bayesian Source Separation of Functional Sources. [.ps .pdf] Forthcoming in the Journal of Interdisciplinary Mathematics. (Describes Bayesain BSS when sources can be functionals like sinusoids for independent vectors. Simulation example to mimic fMRI.)

Rowe, D. B. (2001). A Bayesian Approach to Blind Source Separation. Forthcoming in the Journal of Interdisciplinary Mathematics. [.ps, .pdf] (Reviews PCA, ICA, and FA then uses Bayesian BSS to unmix correlated source components.)

Rowe, D. B.(2001). Bayesian Source Separation of fMRI Signals. [.ps] [.pdf] Bayesian Source Separation of fMRI Signals. In MaxEnt 2000: Twentieth International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, CNRS, Gif sur Yvette, France, July 8-13, 2000. (Develops Bayesian Source Separation for fMRI activation. Applies it in a simulation then a real example.)

Rowe, D. B.(2001). Bayesian Source Separation for Reference Function Determination in fMRI. Volume 45, Issue 5. Magnetic Resonance in Medicine. (Develops Bayesian Source Separation to determine the reference function for fMRI instead of assuming it is known.)

Rowe, D. B.(March 1999). A General Bayesian Approach to Blind Source Separation with Correlation. Unpublished manuscript. Last revised March 22, 1999. [.ps, .pdf] (Unmixes normalized source components with correlated vectors.)

Rowe, D. B. (March 1999). Bayesian Blind Source Separation. In submission. (Unmixes normalized source components.)

Work in Progress

Rowe, D. B.(April 2000). Bayesian Blind Source Separation of fMRI Signals. [.ps,.pdf] Working Manuscript not publically available. (Develops Bayesian BSS for fMRI signals and applies to both simulated and real fMRI data. Great results so far.)

Rowe, D. B.(November 1999). The General Blind Source Separation Model And A Bayesian Approach With Correlation. [.ps .pdf] Working Manuscript. (Described the general BSS problem including delayed mixing and unmixes correlated source components with correlated source and error vectors.)

Updated November 1, 2001 by Daniel Rowe.