Full correlation matrix analysis of fMRI data

Full correlation matrix analysis of fMRI data
Full correlation matrix analysis of
fMRI data
Yida Wang∗ 1 , Jonathan D. Cohen2,3 , Kai Li1 , Nicholas B. Turk-Browne2,3
1 Department
of Computer Science, Princeton University
of Psychology, Princeton University
3 Princeton Neuroscience Institute, Princeton University
2 Department
Abstract
Functional brain imaging produces huge amounts of data, of which only a fraction are analyzed. Existing
univariate and multivariate analyses of brain activity ignore interactions between regions, and analyses
of interactions (functional connectivity) are typically biased toward regions of interest chosen based on
their activity profile. This technical report provides a provisional description of an unbiased technique for
functional connectivity, full correlation matrix analysis (FCMA). This technique calculates and analyzes
all pairwise relationships between voxels over multiple time windows by leveraging advances in parallel
computing and machine learning. FCMA enables the identification of neural mechanisms that support
cognitive processes but may be invisible to activity-based methods.
I.
Introduction
can distinguish between tasks[5]. For example,
consider a brain region that is equally active
for two conditions, but interacts differentially
with other regions depending upon the condition; such a region would not be identified by
analyses that contrast these conditions or classify their patterns of activity. However, insofar
as this region’s interactions with other regions
differ according to condition, it should show
condition-specific patterns of correlations.
Second, although the analysis of correlations in fMRI data, or functional connectivity[6, 7],
has become prevalent, this approach has other limitations. Such analyses typically involve
first identifying a small set of "seed" regions
that show task-related activity, and then examining correlations between these seeds and the
rest of the brain. However, by choosing seeds based on activity, this procedure is saddled
with the limitations of activity-based methods,
and may fail to identify the region in the example above (since its activity is not selective).
Thus, correlations may be needed at the outset
to identify regions that exhibit task-related in-
Functional magnetic resonance imaging (fMRI)
studies often seek to associate regions of neural
activity with specific cognitive processes[1, 2].
However, this univariate, sometimes phrenological approach has been challenged by multivariate pattern analysis (MVPA) methods[3, 4].
Findings from MVPA have been offered as evidence that representations in the brain, rather
than being strictly localized, are distributed
over multiple regions. However, the neural
mechanisms supporting cognitive processing
may be even more widely distributed than previously thought. The failure to recognize this
reflects biases in existing analysis methods.
First, most applications of MVPA have focused on neural activity. Although activity can
reveal important information about neural representations, it may fail to reveal interactions
between brain regions that support neural processes. That is, even if neurons show identical mean firing rates across behavioral tasks,
when and how they interact with each other
∗ Please
address correspondence to: [email protected]
1
teractions. Furthermore, the use of seeds itself,
regardless of how they are chosen, is subject to
the limitation of focusing on localized rather
than distributed processes.
Here we describe initial technical details of
a new method, full correlation matrix analysis
(FCMA), that surmounts these limitations by
performing unbiased multivariate analyses of
whole-brain functional connectivity. The currency of this technique is the full correlation
matrix: the temporal correlation in fMRI activity of every voxel in the brain with every other
voxel. A separate matrix is computed for each
time epoch of interest in a task, such as trial or
block, just as would be done for mean activity
or activity patterns. These matrices are then
labeled with the task condition for that epoch,
and they are submitted to MVPA. Critically, the
input is now correlation patterns, rather than
activity patterns. This analysis determines, in
an exhaustive manner, which patterns of correlations distinguish between conditions.
This analysis may sound straightforward,
but it is intractable using a naive approach such
as calculating the Pearson correlation pair-bypair, storing these values, and then analyzing
the full set of correlations. Consider a relatively small fMRI dataset with 18 subjects, 12
epochs per subject over two experimental conditions, and 34,470 voxels in the brain. On a
machine with 2 4-core 2.6 Ghz Xeon CPUs, running Matlab corr function in batch mode, the
computation of all 594 million pairwise correlations for all epochs above takes 2.5 hours and
requires at least 475.2 GB of disk space at the
end (and much more memory at intermediate
stages). Reducing the Pearson correlation to
matrix multiplication and rewriting the computation in C++ using highly tuned linear algebra
packages shortens the total running time to 348
seconds on the same machine. Thus, the correlation computation alone is not an impediment.
The real problem arises when this massive
amount of data needs to be analyzed. To make
sense of the data, machine learning techniques
such as MVPA are needed. The correlations
would first need to be preprocessed, including
normalizing each coefficient with the Fisher
transform and then z-scoring all coefficients
across the matrix within subject. To classify
two conditions, we would divide these correlation matrices into training and test sets, such as
by leaving out one subject at a time for randomeffects cross-validation. In the example above,
the classifier would be trained on 204 matrices
to find a boundary separating the conditions
in a 594 million dimensional hyperspace and
tested on the remaining 12 matrices to obtain
a classification accuracy. Using nested feature
selection for data-driven dimensionality reduction, training and testing a basic linear classifier
on the correlation matrices in C++ takes 35.91
days on the same machine described above.
For all 18 subjects, this process must repeated
17 more times for a total of 646.31 days.
To address these challenges, we have been
developing a software toolbox for FCMA that
exploits advances in parallel computing and
machine learning. Using this toolbox on a cluster with 66 of the aforementioned machines,
we show that the time required for the analysis
outlined above is reduced from two years to
about an hour.
II.
FCMA Toolbox
Our toolbox exploits hardware and software
advances to render FCMA tractable, all in a
user environment that is readily accessible to
neuroscientists. It does so by incorporating efficient algorithms for correlation computation,
massive parallelization to manage the scale of
the data, and on-demand analysis without the
need for storage space or time. The toolbox
runs on a compute cluster with any number of
nodes, uses controller-worker architecture, and
takes standard NIFTI format images as its input. A dataset consisting of fMRI activity over
time in a set of voxels is divided in multiple
time epochs and is stored on a disk accessible
to each node. The (relatively small) data are
copied into the memory of each worker and
the controller dynamically allocates a subset
of voxels whenever a node becomes available.
A vector is computed for every epoch of each
voxel using matrix multiplication, reflecting
2
its correlation with every other voxel in that
epoch. To handle the high dimensionality without imposing biases, these epoch vectors are
submitted to MVPA and the resulting crossvalidation accuracy is assigned to the voxel.
This can be used for nested feature selection,
leading to a final round of MVPA over the correlation matrices of the automatically selected
voxels.
Optimization of correlation computation.
The Pearson correlation between several pairs
of variables can be reduced to a matrix multiplication by normalizing the data[8]. Specifically,
the timeseries for a given voxel and epoch is
normalized by subtracting the mean and then
dividing each value by the root sum of squares
of the mean-centered data. These are O(n) operations and thus computation time scales linearly and poses little burden. The derivation
of how this normalization turns the Pearson
correlation for a set of variables into matrix
multiplication is provided in the Appendix.
Matrix multiplication is significantly faster
than other approaches for computing correlations. In addition, it can benefit from generally
available technological advances in modern CPUs, such as the single instruction, multiple
data (SIMD) set. The toolbox implements advanced linear algebra algorithms from the GotoBlas library[9] to exploit this hardware. At
peak performance, using SSE instructions over
128-bit XMM registers, these algorithms allow
for eight single-precision floating point operations (four additions plus four multiplications)
in one CPU cycle.
Parallelization of correlation computation.
The FCMA toolbox was written in C++ to run
on a compute cluster with modern commercial machines and X86 architecture, each with
reasonable memory size (e.g., 16 GB). Because
of this, it can leverage large-scale computing
techniques to further accelerate computation
and analysis. A controller/worker model is
used (Fig. 1), in which a controller process coordinates numerous worker processes running
on multiple machines with the Message Passing Interface (MPI). The controller allocates
computation and analysis tasks to the workers;
typically, one process is assigned to each node
in order to fully utilize its resources. Each process consists of multiple threads to compute
and analyze multiple voxels simultaneously
within one node.
The first step in the toolbox is to read in preprocessed fMRI data (e.g., corrected for head
motion and other sources of noise), as well as
text files specifying the experimental design.
Based on this design, the data are partitioned
into epochs. For every voxel and epoch, the
data are normalized as described above. The
controller then directs all available worker processes to load the full data from the storage
device into memory and dynamically assigns
each a subset of voxels to analyze. In other
words, the full correlation matrix is automatically divided into groups of rows, and they are
spread across worker processes. When analysis
in a worker finishes, the controller collects the
results, stores them in memory, and assigns
another group of voxels.
By distributing the full correlation computation in this way over a 66-node cluster, and
by using data normalization and optimized
matrix multiplication algorithms in GotoBlas,
total correlation computation time for our example dataset is reduced from 2.5 hours to 0.73
seconds.
Parallelization of voxelwise classifier analysis. To avoid the burden of storing full
correlation matrices to disk (and associated
write/read time), analysis is performed online
within the nodes, immediately after correlation computation. Specifically, after a worker
process has computed a correlation matrix of
the assigned subset of voxels with the rest of
the voxels for each epoch, the same row of all
matrices is extracted. Each of these rows comprises a vector of the correlations between a
given voxel and all other voxels in the brain for
one epoch. These correlation vectors are then
labeled with the condition of the experimental design to which the epoch was assigned,
and submitted to MVPA as training (or test)
patterns. Each vector reflects a point in a highdimensional space, and the goal of MVPA is
to determine how accurately the points with
3
Hard drive
Controller CPU
Worker CPUs
RAM
# nodes W
1
1
Dynamically assign
subset of voxels (S)
1
Compute correlation
vectors
Epoch 1 1
Voxel 1
Epoch 5
E
RAM
TE
TE
Train and test
MVPA classifier
V
1
V
Labels Acc
=
%
=
%
=
%
=
%
Epoch E
2
Voxel 2
RAM
# voxels
3
# TRs/
epoch
Voxel 3
RAM
1 TE
Voxel S
1 2
# epochs
E
W
V
Data
V
RAM
1
T
#TRs
Result
Figure 1: FCMA correlation computation workflow. The toolbox runs using a controller/worker model, in which each
worker process first loads the full data into its memory. The controller process does the following: assigns
a subset of voxels to each worker; instructs the worker to compute the correlation between each of these
voxels and the rest of the brain in each epoch using matrix multiplication; instructs the worker to analyze
the correlation vectors for each voxel across epochs with MVPA and supplied condition labels; collects the
analysis result (i.e., cross-validation accuracy) for each voxel and loads it into memory; and returns to the
first step to assign another subset of voxels until there are none left. Finally, the controller writes the results
to disk.
different labels can be separated. MVPA is run
with multiple threads, with each thread processing the correlation vectors across different
epochs for one voxel.
The FCMA toolbox uses a linear Support
Vector Machine (SVM) classifier based on
LibSVM[10]. On a server with 2 4-core 2.6Ghz
Xeon CPUs, the standard single-threaded implementation of LibSVM takes about 90 seconds to perform cross-validation on the 204
correlation vectors from one voxel in our example dataset. To speed up this process, we
accelerated the original LibSVM algorithm by
pre-computing the linear kernel matrices with
GotoBlas. This reduced the running time to 2
seconds per voxel in a single thread.
MVPA of correlation patterns. Classification
of the correlation vectors results in a crossvalidation accuracy for each voxel, in terms
of how informative its patterns of correlations
with the rest of the brain are about the task
conditions (Fig. 2). FCMA can stop here, providing an unbiased, whole-brain "map" of the
extent to which different brain regions exhibit
task-related changes in functional connectivity.
This map is, for voxelwise correlations, analogous to the map generated by a voxelwise
(univariate) GLM analysis of brain activity. By
extension, just as MVPA can identify patterns
of activity across voxels that differentiate between task conditions, so too can it be applied
to identify patterns of correlation that differentiate between conditions.
The results of the voxelwise classifier analysis described above can also be used as feature
selection for MVPA, by ranking voxels according to the accuracy with which their individual patterns of correlation differentiate condi4
tions and then selecting a subset that exceeds some threshold (e.g., percentage of voxels,
absolute accuracy level, statistical significance
over cross-validation folds, etc.) for the final
round of MVPA. The FCMA toolbox implements this step as well: A chosen subset of
voxels can be submitted to the toolbox again to
obtain and analyze with MVPA the correlation
matrix for all of these voxels. Critically, this requires that the original data be partitioned into
training and testing sets, with the feature selection step only applied to the training set and
the overall classification accuracy derived from
the held-out testing set. The training and test
sets are cycled to obtain overall cross-validation
performance. Using the FCMA toolbox on our
cluster, the complete procedure was reduced
from 646 days to 72.3 mins.
tion and machine learning algorithms and by
leveraging multi-thread parallelism within nodes and multi-node parallelism across nodes.
The speedups we achieved with a test dataset
are listed in Table 1.
III.
To the best of our knowledge, FCMA is the
first toolbox in the neuroscience community
that takes advantage of high-performance computing to efficiently analyze the full correlation matrix of fMRI data. FCMA is extremely
powerful because it deals gracefully with large
amounts data, optimally splitting and scheduling problems based on the latest techniques in
parallel computing and minimizing the need
for slow data transfer by managing memory
intelligently during online analysis. Beyond
flexible parallelization, FCMA accelerates discovery by improving standard algorithms for
correlation and classification. FCMA can be
run on a compute cluster with any number
of nodes, limited only by the computation resources of the hardware.
For performance benchmarking, we used a
compute cluster consisting of 66 nodes. Each
node was equipped with two Intel Xeon E5430
processors, 16 GB memory, and 4 TB local
disk, and could run 8 threads simultaneously
at peak. The toolbox has been ported to other
systems as well, including a 128-node cluster at
the Princeton Institute for Computational Science and Engineering (PICSciE). Construction
MVPA of activity patterns. For comparison
purposes, the toolbox also incorporates standard MVPA on the pattern of averaged activity
over time for every voxel. This uses exactly the
same process as for correlation patterns (Fig.
2), except taking activity instead of correlation
as inputs, and generates an unbiased, wholebrain "map" of the extent to which different
brain regions exhibit task-related changes in
averaged activity over blocks. This is the same
approach as in the standard MVPA toolbox,
but we include it in FCMA toolbox to allow
easy comparisons.
Analysis benchmarks. In summary, the FCMA
toolbox accelerates the analysis of unbiased
functional connectivity by optimizing correla-
Additional analysis component
Discussion
Elapsed time in minutes (days)
Speedup (x Baseline)
930686 (646.31)
21152 (14.69)
3620 (2.51)
72 (0.05)
1.0
44.0
257.1
12855.0
Baseline
Improved linear SVM
Multi-core parallelism (8 cores)
Multi-node parallelism (66 nodes)
Table 1: The speedups FCMA achieved using our 66-node cluster by running the example application of 34,470
voxels and 216 epochs. The baseline code was written in C++ using Gotoblas library and LibSVM in
single-thread mode. Improved linear SVM code pre-computed the linear kernel matrices used in LibSVM.
Multi-core parallelism code used OpenMP to launch 8 shared memory threads in one cluster node to run the
program in parallel. Multi-node parallelism code runs MPI to coordinate 66 cluster nodes to work together in
controller-worker mode. Long elapsed times were estimated by extrapolating from a portion of the dataset.
5
is underway of a dedicated 50-node cluster for
FCMA, featuring two Intel E5-2670 processors
and two Intel Xeon Phi boards per node (total
of 800 processor cores and 6,000 coprocessor
cores).
of covariance and standard deviations:
n
∑ ( xi − x̄ )(yi − ȳ)
i =1
s
s
corr ( X, Y ) =
n
n
∑ x2j −n x̄
( n − 1)
Though applied to voxel pairs in fMRI,
this technique could help uncover relationships between variables across a range of
applications[11]. Even within cognitive neuroscience, our case study involved the autocorrelation of one dataset, but two or more datasets
(e.g., from different phase offsets, regions, or
even brains) could be submitted without modification of the toolbox for cross-correlation.
By making all code publicly available, we
hope that researchers will explore these exciting avenues (http://princetonuniversity.
github.io/fcma-toolbox).
n
=
∑( s
∑ y2j −nȳ
j =1
j =1
n −1
n −1
( xi − x̄ )
n
i =1
∑
j =1
(yi − ȳ)
s
x2j
n
− n x̄
∑
j =1
y2j
)
− nȳ
(2)
Applying the mean subtraction and root
sum-of-squares division, let:
xi0 = s
( xi − x̄ )
n
(3)
∑ x2j − n x̄2
j =1
The formula above then becomes:
In summary, our toolbox enables the possibility to examine interactions between brain
areas in an unbiased, exhaustive manner. Examining such interactions can reveal the engagement of neural systems transparent to traditional, activity-based analysis methods.
corr ( X, Y ) = X 0 · Y 0
(4)
This can be expanded from vectors to matrices:
X 0 = ( x10 , x20 , · · · , xn0 )
(5)
and
A = ( X10 , X20 , · · · , Xn0 ) T
IV.
Appendix
Then the correlation matrix, where every
voxel is represented by a row and column, is
given by:
C = AA T
(7)
The correlations among several voxel timecourses were computed via matrix multiplication. The mean activity was first subtracted
from each timecourse, and this mean-centered
timecourse was then divided by its root sum of
squares. The Pearson correlation between two
of the resulting normalized timecourses is their
pointwise product, which for an arbitrary number of timecourses becomes a dot product[9].
A derivation of this is provided below:
V.
cov( X, Y )
σX σY
Acknowledgements
This work was made possible by grants from
the J. Insley Blair Pyne fund at Princeton
University, the John Templeton Foundation,
the National Science Foundation MRI BCS1229597, and the National Institutes of Health
R01 EY021755. Intel and Fusion-io provided significant in-kind donations of computer hardware. The opinions expressed in
this publication are those of the authors
and do not necessarily reflect the views
of these agencies or corporations. Code
available at: http://princetonuniversity.
github.io/fcma-toolbox.
The formula for the population Pearson
product moment correlation is:
corr ( X, Y ) =
(6)
(1)
This can be re-written based on estimates
6
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Figure 2: Classification procedure. (a) The preprocessed fMRI data set contains n subjects, each represented with a k
voxels by t epochs matrix. (b) For standard MVPA of activity patterns, vectors are defined for each subject
and epoch as the average fMRI signal over time in every voxel (µi ). For MVPA of correlation patterns,
vectors are defined for each subject and epoch as the pairwise correlation of the fMRI signal over time between
every voxel and every other voxel (rij ). (c) The same nested cross-validation pipeline can be applied to activity
and correlation patterns. The inner loop serves to select features (voxels) for classification: A training set (Si )
is divided into m pieces to do an m-fold cross-validation that identifies the l (e.g., 2000) voxels with highest
performance. (d) The outer loop is n-fold, with each fold leveraging the selected voxels to train a model on Si
and test it on the left-out test set (Ti ). This results in a classification accuracy (Pi ), which is then averaged
across folds (P) to quantify overall performance. In addition, a map of the proportion of the folds in which a
voxel was feature selected can be output as an image.
8
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