Journal of Signal and Information Processing, 2012, 3, 469-480
http://dx.doi.org/10.4236/jsip.2012.34060 Published Online November 2012 (http://www.SciRP.org/journal/jsip)
Analysis of fMRI Single Subject Data in the Fourier
Domain Acquired Using a Multiple Input Stimulus
Experimental Design
Daniel Rio1, Robert Rawlings2*, Lawrence Woltz3, Jodi Gilman4, Daniel Hommer1
1Section of Brain Electrophysiology and Imaging, LCTS, National Institute on Alcohol Abuse and Alcoholism, National Institutes of
Health, Bethesda, USA; 2National Institute on Alcohol Abuse and Alcoholism, Columbia, USA; 3Synergy Research Inc., Monrovia,
USA; 4Laboratory of Neuroimaging and Genetics, Martinos Center for Biomedical Imaging, Massachusetts General Hospital,
Charlestown, USA.
Received July 28th, 2012; revised August 27th, 2012; accepted September 12th, 2012
ABSTRACT
Analysis of functional MRI (fMRI) blood oxygenation level dependent (BOLD) data is typically carried out in the time
domain where the data has a high temporal correlation. These analyses usually employ parametric models of the hemo-
dynamic response function (HRF) where either pre-whitening of the data is attempted or autoregressive (AR) models
are employed to model the noise. Statistical analysis then proceeds via regression of the convolution of the HRF with
the input stimuli. This approach has limitations when considering that the time series collected are embedded in a brain
image in which the AR model order may vary and pre-whitening techniques may be insufficient for handling faster
sampling times. However fMRI data can be analyzed in the Fourier domain where the assumptions made as to the
structure of the noise can be less restrictive and hypothesis tests are straightforward for single subject analysis, espe-
cially useful in a clinical setting. This allows for experiments that can have both fast temporal sampling and
event-related designs where stimuli can be closely spaced in time. Equally important, statistical analysis in the Fourier
domain focuses on hypothesis tests based on nonparametric estimates of the hemodynamic transfer function (HRF in
the frequency domain). This is especially important for experimental designs involving multiple states (drug or stimulus
induced) that may alter the form of the response function. In this context a univariate general linear model in the Fourier
domain has been applied to analyze BOLD data sampled at a rate of 400 ms from an experiment that used a two-way
ANOVA design for the deterministic stimulus inputs with inter-stimulus time intervals chosen from Poisson distribu-
tions of equal intensity.
Keywords: fMRI (BOLD); Time Series Analysis; Single Subject Analysis; Fourier Domain; Statistical Analysis;
Complex General Linear Model; Alcoholism Research
1. Introduction
Functional magnetic resonance imaging (fMRI) has be-
come a standard technique for investigating changes in
brain physiology over time. The most important tech-
nique developed for this purpose measures the blood
oxygen level dependent (BOLD) response to given stim-
uli. The BOLD signal arises from localized variations in
magnetic susceptibility caused by changes in blood oxy-
genation levels instigated by the underlying stimu-
lus-induced neuronal activation [1]. These changes can
be seen in T2* weighted MRI time series data [2,3] that
have a low signal-to-noise (SNR) and temporal SNR [4]
as measured in fMRI time series that also have high
temporal correlation [5].
Essentially from the very first attempts to analyze
fMRI BOLD time series data analyses have been carried
out in the time domain [6-8], most often in the context of
analyses using a general linear model applied to groups of
subjects [7,9-12]. The majority of these methodological
approaches require a preliminary attempt to model fMRI
time series data from single subjects with a few papers
looking at this problem more directly [13]. However there
are a number of concerns to deal with in analyses of fMRI
time series data carried out in the temporal domain.
Foremost among these is the problem associated with
temporal correlation as seen in fMRI BOLD time series
data [5] where a number of methods to control for this
problem [14,15] have been developed. These methods
*Retired.

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
470
attempt to remove the autocorrelation in single subject
data (a form of signal estimation) in order to ultimately
perform statistical inference or signal detection. One of
the most common approaches is pre-whitening of the
data [15]. Another common approach is to represent the
correlation in the data with a fixed order AR model [16]
that is often embedded in a restricted maximum likely-
hood (REML) estimation formulation [17]. However
pre-whitening of the fMRI BOLD data has been shown
to have limitations [15]. In particular it may be insuffi-
cient to remove the correlation across the temporal do-
main for rapidly sampled fMRI data or experimental de-
signs that include short inter-stimulus intervals. As well,
the application of fixed order AR models also have a
number of problems. Primarily, that in practice the cho-
sen order of the model is typically not verified for time
series at every spatial position or voxel acquired within
the brain. In fact the AR model order is typically not
even verified for a new experimental design and it would
be remarkable if such a constrained model were univer-
sally suitable. However if a variable order AR model
were fitted to every time series collected at every voxel
investigated then the modeled covariance structure would
change across these voxels and any resultant statistic
would be difficult to interpret across regions of the brain.
The use of parametric models for the hemodynamic
response function (HRF), somewhat separate from the
statistical analysis of fMRI data is another limitation in
temporal-based analysis of fMRI data [18,19]. Since the
general shape of the HRF is usually assumed before the
fMRI BOLD data is analyzed there is some concern with
experimental designs in which the shape is state-de-
pendent. This can for instance happen with the introduc-
tion of a vaso-active drug, such as alcohol, to a subject
during part of an experimental procedure [20]. Addition-
ally many additional parametric forms along with regres-
sion coefficients are typically included to correct for
other perceived confounds including those associated
with signal drift and head motion [21,22]. Furthermore
other procedures are used to correct e.g. for cardiac arti-
facts [23] and time shifts errors seen in multi-slice acqui-
sition of fMRI data [24]. Therefore the current standard
analysis of fMRI BOLD data in the temporal domain
necessitates the use of a number of empirically based ad
hoc analytical and/or statistical techniques. However Fou-
rier-domain-based analysis can in many instances provide
a more general mathematically suitable methodology.
The methodology in this paper focuses on the analysis
of fMRI time series data in the Fourier or frequency do-
main. While Fourier domain approaches have been ex-
tensively applied to time series in the general field of
signal processing they have had limited use for this ap-
plication. One of the earliest attempts at a Fourier-based
analysis of fMRI was that by Lange, et al. [25] that fo-
cused on the analysis of data obtained from a block ex-
perimental design. This approach used a parametric form
for the HRF and spatially averaged over adjacent voxels.
Another early paper that analyzed fMRI data in the fre-
quency domain was by Marchini and Ripley [26], how-
ever it was restricted to periodic stimuli. Neither of the
methodologies presented in these papers were developed
to carry out multivariate hypothesis testing for compli-
cated experimental designs. As such they were not based
on the statistical theory in the Fourier domain as devel-
oped by Brillinger [27,28] and first applied to investigate
fMRI time series data by Rio [29].
A more recent paper, also based on the work by Brill-
inger, is that by Bai et al. [30]. It focused on obtaining
unbiased estimates of the HRF using stochastic rather
than deterministic input stimuli (the usual design for
fMRI experiments). It uses a weighted estimate of the
transfer function and appropriate chi-square statistics to
analyze sample data from an fMRI experiment with a
“simple” design. In contrast our paper has deterministic
inputs or stimuli and an unweighted estimate of the
transfer function, an approach that provides estimates
with minimum mean square error. The development in
our paper therefore is toward a full “multivariate” ap-
proach for hypothesis testing to perform signal detection
in the spectral domain using an extension of the general
linear model methodology in the complex domain. Here
the use of stochastic inputs to model the stimuli and
weighted transfer function estimations would be inap-
propriate for our clinical studies. Thus the paper by Bai is
attempting to find the best estimate to the transfer function,
but not necessarily carrying out multivariate statistical hy-
potheses testing. Thus while these two papers are starting at
the same place they are going in different directions.
Focusing on the approach taken in this paper we pre-
sent a methodological approach developed for the Fou-
rier domain that allows us to analyze event-evoked time
series data similar to that collected in many fMRI ex-
periments. In this approach the noise model can be made
less restrictive and allows more direct analysis of fMRI
time series data for single subject [29,31,32]. In particu-
lar to demonstrate this methodology, fMRI BOLD data
[33] are taken from an experimental design that would
not typically be suited for some of the noise assumptions
made in analyzing these data in the time domain for sin-
gle subjects. Here the sampling rate for the experiment
was 400 ms and the experiment used a two-way ANOVA
design for the stimulus input. In addition, stimulus types
were constructed from Poisson processes of equal inten-
sity resulting in some inter-stimulus intervals (ISI)s being
very short. Thus the data for single subjects generated
from this experiment is well suited to analysis using the

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
471
Fourier-based methodology that will be presented in this
paper, in that the fast sampling rate and experimental
design are easily handled by this technique as compared
to temporal-based approaches.
2. Complex General Linear Model
2.1. The Model
The model in the time domain takes the usual form for
fMRI BOLD data [29,31]
 


s
tatrtt

   (1)
where the fMRI signal


s
t is considered to be a uni-
variate time series collected at discrete time points t and
at each spatial position

,,

x
xyz or voxel (implicit).
Here the fixed deterministic input stimuli


rt are fil-
tered by the hemodynamic response function (HRF),
, where the convolution operation is symbolically
represented by . Multiple (R) input stimuli require that
represents a 1 × R matrix and correspondingly the
response function represents a R × 1 matrix. Thus
each single-input stimulus time series has a correspond-
ing single response function represented by a corre-
sponding matrix entry in the response function

at

rt 

at


at.
The only assumption on the noise structure, represented
by the univariate time series


t

, is that it be stationary
with zero mean. Therefore the fMRI time series observed
at each spatial position is assumed to be described by a
linear time-invariant model, where the BOLD signal,

s
t


t
, is determined by a constant (with respect to time),
, plus a linear filter of several fixed determinis-
tic inputs and an additive error or noise term,

at

rt


. The analysis is then carried out in the Fourier do-
main, where the complex Fourier transform of Equation
1 can be written as
 
kkk
sar

k

 
  (2)
where k2πkT

, T represents the number of time
points and k represents the wave or frequency number.

k
a

, the response function’s representation in the
Fourier domain, is henceforth referred to as the hemody-
namic transfer function or HTF.
Periodograms are then constructed from the Fourier
transform of the input function,

k
r

, and the output
function,

k
s

, as follows

1
kk
2πT





IH
k

(3)
where and superscript H refers to the
Hermitian transpose. Estimates of cross-spectral density
functions are then constructed from these periodograms.
Thus corresponding to each of the periodogram functions
we construct a corresponding cross-spectral function [27]
that is

,,rs  
 
1m
k
km
ˆ2m 1
 






fI (4)
These estimates are constructed over disjoint fre-
quency bands, size k = −m to m with center frequency

and provide stable estimates of the cross-spectral
functions. Also, since these spectral functions will be
used in constructing hypothesis tests a Daniell or equal-
weighted averaging window is used to provide estimates
for these quantities. These cross-spectral functions take a
slightly different form [27] for the band centered at zero
frequency, however in this application to fMRI data this
band will be discarded due to artifacts (for example low
frequency motion drift) and we will not address it in this
paper. Band size is picked based on statistical power
considerations or on information as to the input power
spectrum. If however specific information is known as to
the spectral distribution of the input power (e.g. the
power of the input stimuli is concentrated in one or more
narrow bands) then smaller band sizes should be chosen.
Unbiased spectral-banded estimates of the power associ-
ated with the input stimuli are given by the cross-spectral
function


ˆrr

f and for the power of the output func-
tion or fMRI BOLD signal by

ˆss

f. Furthermore,
estimates can also be constructed for the error spectrum


ˆ


f represented by

g



(Equation 10 to be
presented in Section 2.3) and provide a measure of the
extent to which the output signal can be determined from
the input function for the given model.
2.2. Unbiased Non-Parametric Estimates of the
HTF
In Equation 2 we see that the HTF

k
a

 essentially
represents regression coefficients relating the stimulus
input function


k
r

 to the output fMRI BOLD signal


k
s

 in the Fourier domain. This represents a form of
the general linear model in the complex domain for mul-
tiple input and univariate output. Furthermore since the
noise


k


 in the Fourier domain is asymptotically
independent [27] it is possible to use the method of
maximum likelihood estimation as extended to the com-
plex domain to construct unbiased estimate of the HTF
[27,34] at the center frequencies using the cross-spectral
estimates as defined in Equation 4. That is
 
1
ˆˆ
ˆsr rr
Af f 




(5)
a R × 1 matrix (one entry for each stimulus input) at
every spatial position or voxel. This estimate of the HTF,
calculated at the center frequency for each band, is the
crucial quantity for which hypothesis tests are made in
this implementation of the complex linear model for
multiple inputs in the Fourier domain. Furthermore since
this estimated HTF is so important to our methodology,

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
472
some additional non-statistical quantitative functions (as
a prelude to inference tests) are constructed as follows to
investigate the general spatial pattern of the hemody-
namic response induced by the stimuli. Using the esti-
mated transfer function in the Fourier domain, the power
per band can be simply calculated by
 
ˆˆ
diag H pAA





(6)
and the total power (sum over all bands) for the rth transfer
function is given by

rrr
Pp

  (7)
where r = 1 to R. Finally we can construct the total power
for all transfer functions as

tot trPp







(8)
where tr refers to the trace (sum of diagonal entries). Thus
these functions give us a general sense of the hemody-
namic activity at every sampled voxel within the brain in
response to each individual input stimulus, , or for all
stimuli taken together, .
r
P
tot
P
2.3. Hypothesis Testing
Consider the test of the hypothesis


o
H: 0AB 

where B is the contrast matrix for multiple input and has
size R × b, where b = 1 to R depending on the particular
test being used. The contrast matrix B allows for testing
any individual or combination (contrast, marginal or in-
teraction tests) of the R stimulus input (functions). The
tests take the form of the following F-distribution
[27,31,32] at each spatial position and band (represented
by its center frequency

)


 

2b;2 2m1R
1
1H
TT
F
ˆ ˆ
2m 1
b
rr
ABBf BBA
g


ˆ













(9)
where

 

1
2m 1ˆˆˆˆ
2m 1 Rsssr rrrs
g
ffff




 





(10)
estimates the spectral error function,

ˆ


f, a measure
of the model’s fit to the data in the frequency domain.
A special case of general hypothesis test presented is
when the contrast matrix B is equal to the identity matrix.
Then the test is for the hypothesis

o
H: 0A 

and
Equation 9 takes the following form



 



H
2R;2 2m1R
ˆ ˆˆ
2m 1
FR
rr
Af A
g







 (11)
This statistic, which we will henceforth refer to as the
omnibus test, is related to the squared sample complex
multiple correlation coefficient or simply the

2
Rsr

statistic in the Fourier domain [27,35]. This statistic takes
the following form in our case
  



1
2ˆˆˆ
R1
ˆˆ
sr rrrs
sr
ss ss
ff fg
ff

 








(12)
ranges between 0 and 1, and measures in general how
well our statistical model fits the data. Essentially, either
spectral statistic conveys information as to how well the
fMRI signal is predicted by the stimulus inputs since
small values for the error term

g


lead to large


F

values for the omnibus test or equivalently

2
sr
R

values close to 1, in each associated band rep-
resented by the center frequency

.
3. Application
3.1. Experimental Design
Selected data were taken from the following more exten-
sive experiment [33], previously analyzed using a stan-
dard temporal fMRI group based analysis as imple-
mented in AFNI [8]. Four types of visual inputs were
presented to control and alcoholic subjects using a split
screen format. They consisted of the following paired
presentation in a 2-way ANOVA experimental design: 1)
Alcoholic beverage and positive image; 2) Non-alcoholic
beverage and positive image; 3) Alcoholic beverage and
negative image; 4) Non-alcoholic beverage and negative
image. Positive and negative visual images were taken
from the International Affective Picture System (IAPS)
[36] and paired (simultaneously, side-by-side) with al-
coholic or non-alcoholic beverages (i.e. milk, orange
juice). There were 55 presentations for each type of input
stimulus combination. The duration of a single stimulus
image presentation was 800 ms and the ISIs between the
paired inputs were sampled from an exponential distribu-
tion. The average ISI in seconds for each input stimulus
presented individually was: Positive and alcohol—8.58
seconds, positive and non-alcohol—9.61 seconds, nega-
tive and alcohol—9.24 seconds and negative and non-
alcohol—9.77 seconds. The average ISI for all stimulus
presentations (without differentiation as to the type of
stimulus) was 3.15 seconds. Control and alcoholic sub-
jects viewed these images or a neutral background and
the entire input stimuli time sequence consisted of 1400
time points and all subjects received the same sequence
of input stimuli.

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
473
The sampling rate for these 1400 time points, typically
referred to as the time to repetition (TR) to acquire each
volume of MRI data, was 400 ms making the experi-
mental scan time 9 min 20 sec for each subject. Note that
by sampling at a rate of 400 ms, significantly longer time
series were acquired than in typical fMRI BOLD ex-
periments. This enabled us to both increase statistical
power and filter out cardiac artifacts at lower frequency
bands due to aliasing, where the spectral power of the
stimulus inputs was generally focused. Furthermore, de-
signing the stimulus input functions to place spectral
power into many frequency bands in the Fourier domain
allowed us to maximize both hypothesis testing and es-
timation [37] by providing better estimates of the transfer
function. A segment of the stimulus presentation se-
quence timing and sample stimuli, along with the distri-
bution of ISIs can be seen in Figure 1.
3.2. Experimental Scanning Parameters
Each scan consisted of approximately 1400 T2
*-weighted
echo-planer MR volumes (composed of 5 contiguous
slices—128 × 128 voxels with spatial dimensions 1.875
mm × 1.875 mm × 6 mm) acquired at a sampling rate of
400 ms (or as previously stated at TR = 0.4 seconds) with
a sixteen-channel head coil. Brain structural information
was collected using MPRAGE T1-weighted 3D volumet-
ric images (256 × 256 × 120 voxels with dimensions
0.856 mm × 0.856 mm × 1.2 mm).
3.3. Image Preprocessing
Preprocessing of the acquired functional images con-
sisted of the following steps. First, co-registration of the
1400 functional images to align these images over time
and then structural to functional registration (all within
subject) using the AFNI programs 3dvolreg and
3dAllineate respectively. At the same time the AFNI
program 3dautomask was used to construct a binary
masks for the functional image. These masks were used
to restrict all further analyses to only data (or corre-
spondingly voxels) collected within the brain. Next, spa-
tial filtering with a Gaussian spatial filter, 4 mm full
width half maximum (FWHM), and application of a low
pass frequency filter, cutoff at 0.9 Hz to mediate possible
cardiac contributions to the BOLD signal that could be
aliased into lower frequencies, were applied in the pro-
gram SRView [unpublished]. The SRView program was
Figure 1. (a) A segment of the input stimulu s sequence showing the inte rleaved sequence of pr esentation for the four stimuli
long with a color code for each; (b) The distribution of ISIs for all four stimuli taken collectively. a

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
474
also the developmental platform for all the Fourier-based
analytic techniques presented in this paper. Note impor-
tantly that no other preprocessing of the data was made
(and none was required in contrast to standard preproc-
essing of fMRI data in the time domain [21-24]).
3.4. Analyses
Prior to the collecting of fMRI data, the power associated
with the stimuli input timing were calculated. Band size
(and the associated center frequencies) was chosen based
on the observed spectral power distributions and from
previously analyzed data results [29,31,32]. Thus a band
size of 15 ( in Equation 7) frequencies was taken.
This also corresponds to the number of subjects in a rea-
sonably sized group or equivalently in this application,
the number of asymptotically independent frequencies in
each band. Also note that once a uniform band size was
chosen the partitioning of the Fourier frequencies into
m7
bands and associated center frequencies was set. Fur-
thermore since a low-pass filter was used in preprocess-
ing the fMRI data, tests were restricted to bands below. 9
Hz (which was less than the Nyquist frequency) and ef-
fectively only 35 disjoint bands (or center frequencies)
were used. Also the band centered at zero frequency


0 

was not used because it contains a number of
low frequency artifacts including those often associated
with motion or possibly signal drift. More importantly,
an additional motivation for not using the band centered
at zero frequency was that there was little power supplied
by the stimulus inputs at the very low frequencies (see
results, Section 4.1, Figure 2(a)) within this band and
therefore no corresponding response would be expected
for the linear model assumed. Once the fMRI data were
collected estimates of the HTF were constructed (Equa-
tion (5) and the associated spectral power of these func-
tions was investigated (Equations (6)-(8)).
Figure 2. (a) Plots of frr(λ), the banded spectral power for each of the four input stimuli; (b) Three images showing the spatial
extent of the total power associated with the omnibus, negative with alcoholic, and negative with non-alcoholic HTFs on an
alcoholic subject. In the green region of interest (ROI) placed in the brain insula the total power of the associated HTFs av-
eraged over this region for the negative with alcoholic stimuli is 57% less than that for the negative with non-alcoholic stimu-
us HTF. l

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
475
Model goodness of fit (Equation (12)) was explored
using the omnibus hypothesis test

o
H: 0A 

using
the F-test statistic (Equation (11)). These tests were car-
ried out at every voxel with the binary functional brain
mask (see Section 3.3) and indicated whether any of the
stimuli produced a significant response at any center
frequency. In essence the F-test statistic image was con-
structed for every band (or center frequency), then a
p-value was selected and an associated F-value threshold
was used to test for signal response in these images. Al-
ternatively, one can consider looking at one spatial posi-
tion or voxel, setting a threshold across all bands calcu-
lated for that voxel (see Figure 3(a) Section 4.2), and
then combining the voxel results at each center frequency
to produce spatial images for each band. In either case
this produced frequency-specific spatial patterns associ-
ated with signal response to the input stimuli (see Figure
3(b) Section 4.2). These resultant spatial patterns are
presented using multiple p-level mask whose corre-
sponding F-statistics with appropriate degrees of freedom
(dof) were used as threshold values for the F-test statistic
image produced. A color look-up-table (LUT) is used to
present these threshold values. The algorithm is:
Input: Select multiple p-values p0, ···, pn and associated
color values in LUT
Calculate corresponding F-value, F(p1), ···,F(pn) such
that dof (2b; 2(2 m + 1-R))
Loop over voxels within brain mask
Extract F-values at voxel, indexed by band numbers
Loop over band numbers
Get F for band
if (F < F(p
0)) maskPixel = 0
if (F >= F(p
0) and F < F(p1)) maskPixel = 1
···
if (F >= F(p
n – 1) and F < F(pn)) maskPixel = n
if (F >= F(p
n)) maskPixel = n+1
End loop over band numbers
End loop over image voxels
Output: Multi-value mask and associated color LUT
Next, as a simple method to control for multiple tests
(or limit the number of false positives) when looking at
every voxel originally sampled within the full brain mask
a “voxel limiting” spatial mask was produced as follows.
The omnibus F-test images (related to

2
Rsr

, a
measure of model fit—see Section 2.3) at each center
frequency were strictly threshold at p = 0.0001 to pro-
duce binary masks for each band. Then the masks were
combined to produce one spatial binary mask using a
Boolean or operation. This mask enabled us to limit the
number of voxels looked at with the specific inference
tests for interaction, main and simple effects for the
ANOVA design presented. These specific hypothesis tests
were selected by using appropriate values for the contrast
Figure 3. Representative omnibus multilevel F-test (and
associated p-values) results showing some low frequency
bands where significant activation occurred in the occipital
regions of the brain for four alcoholic subjects. Each row
compares two different alcoholic subjects at a particular
band or center frequency.
matrix B (see Table 1 Section 4.2) in Equation (9). Results
for these tests are presented at multiple p-value levels
(ranging from 0.001 to 0.01) and overlaid on the regis-
tered structural image for the individual subject. These
tests were structured by a scheme not typically used in
the neuro-imaging field but closer to traditional multi-
variate hypothesis testing procedures. First, hypothesis
tests for interactions were strictly applied at a p-value
threshold of 0.004 and a Boolean OR binary mask pro-
duced (similar to that for the omnibus hypothesis tests)
that included only those voxels that failed the interaction
hypothesis tests (that is an image mask composed of
voxels in which no interaction was seen). Only then were
main effect hypothesis tests (e.g. contrast of positive
versus negative input stimuli images regardless of the
beverage currently shown) performed at those voxels for

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
476
which the interaction hypothesis tests were not signifi-
cant. On the other hand, in order to better investigate the
spatial pattern of the simple effects (e.g. the positive with
alcohol combined stimulus image) hypothesis tests asso-
ciated with single-input combined stimuli (as seen in
Figure 1(a)) were not restricted to voxels in which the
interaction hypothesis tests were accepted as would be
typically done in rigorous statistically-based multivariate
presentations. This may introduce a few possible false
positive voxels in the masks for simple effects but was a
concession to the importance of presenting their full spa-
tial activation pattern (as is typically done in the neuro-
imaging field). In fact this is the usual procedure for re-
porting fMRI analysis as performed in the temporal do-
main. Alternatively it would be completely wrong to in-
vestigate voxels with hypothesis tests for main effects
where hypothesis test for interactions are affirmative.
While the hypothesis-testing logistics seem at first view
complicated, Figure 4 (presented in the results Section
4.2) easily clarifies the process. Generally we feel that
this type of presentation or a similar approach should be
taken when investigating more complicated experimental
designs such as presented in this paper, however they
often are not [33].
Figure 4. Chart showing available hypothesis tests for in-
teraction, marginal and simple effects for one alcoholic
subject at one particular slice.
4. Results
Only a selected sample of relatively interesting results is
presented to demonstrate the utility of analyzing fMRI
data in the Fourier domain on single subjects. Note that
all functional related results are presented as mask on the
subject’s co-registered structural image.
4.1. Spectral Power of the Input Stimuli and
Some Selected Examples of the Spatial
Patterns of the Power of the HTF and
Temporal Shape of the HRF
For ideal Poisson processes, the input functions temporal
distribution would distribute power equally at all fre-
quencies in the power spectrum. Of course in this case,
where the stimulus is finite and the sampling is discrete
(at a sampling rate of 400 ms), the shape of the power
spectra is compromised from this ideal case as can be
seen in Figure 2(a) but is much more uniform than for
typical fMRI stimulus input designs. While the power
spectrum is not uniform from frequency to frequency and
decreases in power at higher frequencies, these features
would subside for longer-duration experiments and the
high frequency fall off would decrease for increasing
sampling rates with point stimulations.
Focusing further on the specific difference seen in an
alcoholic subject we look at the total transfer power for
both the omnibus case and the total transfer power asso-
ciated for the negative with alcoholic and negative with
non-alcoholic image stimuli. As mentioned before, these
functions can give us a sense of the general response of the
system (in this case the brain) to the input stimuli at every
voxel. These results are presented in Figure 2(b). The
color scheme in this omnibus mask is unscaled with
higher power represented by red to lower power repre-
sented by orange. However individual stimuli power
masks (again the color scheme simply represents larger
power if the color is darker) were normalized between-
stimuli by virtue of having the inputs represented by bi-
nary sequences. Thus it is possible to compare values
across stimuli where a region of interest (ROI) was sam-
pled in the insula region for this alcoholic subject. Here
we see a 57% decrease in power between the negative
stimuli with a non-alcoholic beverage to that with the
alcoholic beverage for the region (green box) shown. This
can be compared to the often-reported signal change in the
temporal domain for the BOLD response of typically a
few percent [38]. Using these results it might be possible
to conclude that the BOLD response to a negative image
in association with an alcohol image is diminished as
compared to the BOLD response seen when a negative
images is combined with a non-alcoholic image for this
alcoholic subject.

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
477
Looking now at the banded (or frequency) power dis-
tribution of the transfer function for selected brain regions
in the alcoholic subject (Figure 5) shows an interesting
result in that the occipital region of this subject shows
dominant low frequency responses while the insula region
response is generally at much higher frequencies. A
neighborhood of selected voxels where the power has
been plotted as a function of bands from these two regions
shows this quite dramatically. It generally appears than
that the occipital region response is slower and more ex-
tended in the temporal domain than that for the insula,
which shows a more peaked response due to the higher-
frequency components.
4.2. A Sampling of Hypothesis Testing Results
A synopsis of all hypothesis tests for this experimental
design is presented in Table 1, for which a more com-
plete description of these tests follows. Sample images of
spatial masks associated with the hypothesis tests are
Figure 5. A look at 3 × 3 voxel ROI neighborhoods in the occipital and insula regions of the brain from an alcoholic subject
showing the frequency-banded power of the omnibus HTF (Nyquist cut off at band 35 has been applied) along with some
slices showing the spatial extent of frequency bands that showed relatively high power (voxels having the maximum value
represented in red). Note especially the difference in the transfer function frequency dependence between the occipital and
insula regions of the brain.
Table 1. Synopsis of hypothesis tests.
Test Type Mask Used Contrast Matrix (B)
Omnibus whole brain Identity
Interaction omnibus mask BT=[1/2 –1/2 –1/2 1/2]
Main Effects
Emotional Valence non-interaction voxels only BT=[1/2 1/2 –1/2 –1/2]
Beverage Type non-interaction voxels only BT=[1/2 –1/2 1/2 –1/2
Simple Effects—Individual Stimuli
(Positive, Alcohol) omnibus mask BT = [1 0 0 0]
(Positive, NonAlc) omnibus mask BT = [0 1 0 0]
(Negative, Alcohol) omnibus mask BT = [0 0 1 0]
(Negative, NonAlc) omnibus mask BT = [0 0 0 1]
Simple Effects—Contrast
(Pos, Alc vs. NonAlc) omnibus mask BT = [1 –1 0 0]
(Neg, Alc vs. NonAlc) omnibus mask BT = [0 0 1 –1]
(Pos vs. Neg, Alc) omnibus mask BT = [1 0 –1 0]
(Pos vs. Neg, NonAlc) omnibus mask BT = [0 1 0 –1]

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
478
presented in Figures 3 and 4 for selected subjects and
slices. Testing of the HTF for signal detection

o
H: 0A   was performed using the omnibus F-sta-
tistic (Equation (11)) at all center frequencies as previ-
ously detailed (see Section 3.4). Examples of the thresh-
old being applied at six adjacent voxels in the occipital
brain region of the brain for a sample subject are pre-
sented in Figure 3(a). In addition, some selected low
frequency bands are presented for four alcoholic subjects
in Figure 3(b) using multilevel p-values (see Section
3.4). These were chosen to demonstrate the spatial extent
of the low-frequency activation (generally restricted to
gray matter) among a number of subjects. Note that there
is no expectation that the activations will occur in exactly
the same bands across every subject since each subject’s
transfer function can have a slightly different shape. But
it is of interest to note that activation in the occipital re-
gions for all four subjects shown have a major low fre-
quency response. Also note the red regions (p < 0.0001)
that were used to spatially restrict the more detailed hy-
pothesis tests that follow (see Section 3.4).
Next, hypothesis tests of the form


o
H: 0AB 

for interaction, main and simple effects for the experi-
mental design presented were systematically performed
using Equations (9) and (10) with the appropriate values
for the B contrast matrix. A list of all hypothesis test,
spatial mask applied and associated B contrast matrix
values used for these tests can be seen in Table 1. Sam-
ple results for these hypothesis tests can be seen in Fig-
ure 4 for a sample slice that included the insula region of
the brain. These hypothesis tests were performed in the
following systematic order. First, hypothesis tests for
interactions between the input stimuli for emotional con-
tent and beverage were performed. Results from these
tests for interaction (threshold of p < 0.004) were used
for voxel selection where a binary mask was created by
selecting voxels in which no interactions were seen (light
green, p > 0.004). Tests for main effects were then re-
stricted to the voxels within this mask. This prevents
combining difference in the emotional states with the
alcohol beverage input image and those same differences
with the non-alcohol beverage input image when they
have different slopes. Hypothesis tests for main effects
were then applied and the results are presented in the top
right hand side and bottom left hand side of Figure 4.
Finally hypo thesis tests for simple effects associated
with the four stimuli (see Section 3.1) were performed
(see Ta b le 1) and the results are also presented in Figure
4 (center of figure along with a color key). Hypothesis
tests for simple contrast can be easily found by scanning
either vertically or horizontally for the desired test result.
Note that color keys were not provided for all hypothesis
tests results presented in Figure 4, however all color
keys were generally the same (having the same range of
p-values, dark 0.001 to light 0.02), similar to that shown
for the individual stimuli.
There is a main effect of emotional valence (positive
versus negative) in the insula for this alcoholic subject
(Figure 4, bottom left of figure). We also see a main
effect of beverage type (alcohol versus non-alcohol), but
it is somewhat muted. This is also clarified by looking at
the simple effects hypothesis tests results (Figure 4, four
images in the center). Here we see the hypothesis test
results mask for the effect of the positive-alcohol stimu-
lus is much stronger than the hypothesis test results mask
for the negative-alcohol stimulus. This is further con-
firmed by looking at the hypothesis test results for the
simple contrast of positive versus negative emotion stim-
uli with an alcoholic beverage image (presented Figure 4,
bottom row, center). Finally it should be noted that in
general both the occipital and insula regions of the brain
for this alcoholic subject’s chosen brain slice show a
reasonable activation (or hemodynamic response) spatial
pattern for the hypothesis tests presented [33].
5. Conclusions
In conclusion we have presented a number of analytic
techniques in the Fourier domain that provides a some-
what new and different perspective into the analysis of
fMRI BOLD data as compared to what have become the
standard temporal-domain-based approaches. The meth-
odology has a number of advantages that can be espe-
cially useful for single subject fMRI BOLD data analysis
obtained from complicated event-driven experimental
designs. This has allowed us to analyze fMRI BOLD
time series data for single subjects obtained at a sampling
rate of 400 ms without resorting to unrealistic restrictions
on the structure of the noise and to implement a full set
of statistical hypothesis tests based on a complex general
linear model framework. This is critically important for
those designs that include relatively short inter-stimulus
intervals and fast acquisition rates as presented in this
paper, but could also be a potential problem in the analy-
sis of data obtained from any fMRI experiment. This has
enabled us to analyze fMRI time series data with a large
number of sampled points (collected in a short time) and
thereby increase statistical power for single-subject ana-
lyses.
Finally we have shown that it is possible to use tradi-
tional signal processing and statistical techniques to in-
vestigate and detect signal changes in fMRI BOLD time
series data acquired from a relatively complicated evoked
response experimental design. Toward this end we have
implemented a number of investigative and statistical
procedures and shown that in particular a Fourier-based

Analysis of fMRI Single Subject Data in the Fourier Domain Acquired Using a
Multiple Input Stimulus Experimental Design
479
approach can be implemented for single subjects analysis
of fMRI BOLD time series data with fast acquisition
times. This gave robust results even for an experimental
design that is challenging to analyze in the temporal do-
main using a multistage group analysis approach [33].
We have also shown how to look at the analysis of these
data in a systematic manner using a complex univariate
multiple-regression methodology. Furthermore, this me-
thodology approach to fMRI data analysis is still rela-
tively novel and a number of extensions are in the proc-
ess of being developed. These include techniques to bet-
ter handle the multiple comparison problem (beyond
what was implemented in the paper to restrict the hypo-
thesis testing of main and simple effects by the general
omnibus hypothesis test produced mask) and to more
fully implement the complex general linear model in the
Fourier domain to handle multivariate inputs (needed to
analyze multiple fMRI runs data for an individual subject)
and analyses for subject groups.
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