Separating Processes within a Trial in Event-Related Functional MRI
J . M. Ollinger,*,†,1M. Corbetta,*,† and G. L. Shulman*,†
*Department of Radiology and †Department of Neurology, Washington University, St. Louis, Missouri 63110
Received April 17, 2000
Many cognitive processes occur on time scales that
can significantly affect the shape of the blood oxy-
genation level-dependent (BOL D) response in event-
related functional MR I. T his shape can be estimated
from event related designs, even if these processes
occur in a fixed temporal sequence (J . M. Ollinger,
G. L . Shulman, and M. Corbetta. 2001. NeuroImage
13: 210–217). Several important considerations come
into play when interpreting these time courses.
F irst, in single subjects, correlations among neigh-
boring time points give the noise a smooth appear-
ance that can be confused with changes in the BOL D
response. Second, the variance and degree of corre-
lation among estimated time courses are strongly
influenced by the timing of the experimental design.
Simulations show that optimal results are obtained
if the intertrial intervals are as short as possible, if
they follow an exponential distribution with at least
three distinct values, and if 40% of the trials are
partial trials. T hese results are not particularly sen-
sitive to the fraction of partial trials, so accurate
estimation of time courses can be obtained with
lower percentages of partial trials (20–25%). T hird,
statistical maps can be formed from F statistics com-
puted with the extra sum of square principle or by t
statistics computed from the cross-correlation of the
time courses with a model for the hemodynamic re-
sponse. T he latter method relies on an accurate
model for the hemodynamic response. T he most ro-
bust model among those tested was a single gamma
function. F inally, the power spectrum of the mea-
sured BOL D signal in rapid event-related paradigms
is similar to that of the noise. Nevertheless, high-
pass filtering is desirable if the appropriate model
for the hemodynamic response is used.
© 2001 Academic
A major goal of studies using blood oxygenation
level-dependent (BOLD) weighted functional MRI
(FMRI) (Ogawa et al., 1990; Kwong et al., 1992) has
been to identify regions of the brain that are activated
while performing specific cognitive tasks. The most
widely used approaches to this problem begin by com-
puting a map of t statistics that test at each voxel the
null hypothesis of noactivation (Friston et al., 1995a,b;
Worsley et al., 1995). A threshold is then applied to
isolate statistically significant voxels or regions (Wors-
ley et al., 1992, 1995; Friston et al., 1994). Computation
of these statistical maps requires two quantities: the
magnitude of the response and its variance at each
voxel. These parameters are usually estimated from
linear models of the data (Friston et al., 1995a,b).
These models are usually constructed by postulating a
functional form for the BOLD response and using this
function as a regressor in the model. The BOLD re-
sponse is easy tocharacterize for block designs because
their long task and control periods can be modeled by a
simple square wave. Modeling the hemodynamic re-
sponse as a linear system improves this model by more
accurately modeling the rise time, delay, and fall time
of the BOLD response (Boynton et al., 1996).
Event-related designs (Buckner et al., 1996) are
more difficult to analyze because the task period is
short with respect to the hemodynamic response.
Therefore, the shape of the BOLD response is domi-
nated by the shape of the hemodynamic response,
which is known to vary across subjects and regions of
the brain (Lee et al., 1995; Buckner et al., 1996, 1998;
Kim et al., 1997; Schacter et al., 1997; Aguirre et al.,
1998). Moreover, the shape of the responses may de-
pend on the experimental paradigm. For example, the
BOLD response in areas that maintain information
over a delay period (such as in match-to-sample tasks)
1To whom correspondence and reprint requests should be ad-
dressed at Washington University School of Medicine, Neuroimaging
Laboratory, Campus Box 8225, St. Louis, MO 63110. Fax: 314-362-
6110. E-mail: email@example.com.
NeuroImage 13, 218–229 (2001)
doi:10.1006/nimg.2000.0711, available online at http://www.idealibrary.com on
Copyright © 2001 by Academic Press
All rights of reproduction in any form reserved.
will depend on the duration of the interval. The shape
of the hemodynamic response therefore carries impor-
tant information about the cognitive processes related
to the observed activation.
In the companion paper (Ollinger et al., 2001), we
introduced and validated a method for separately esti-
mating the time course of two successive BOLD re-
sponses occurring at a fixed interval. This method
makes noassumptions about the shape of the hemody-
namic response. It only requires that the sequence of
trials during the experiment includes partial trials, in
which only the first BOLD response is present, ran-
domly intermixed with trials in which both BOLD re-
sponses are present. Moreover, the interval between
trials (the intertrial interval or ITI) must be randomly
varied. Here we consider four issues related to the
analysis of these time course data:
1. What is the effect of correlations among points in
the estimated time courses?
2. What are the optimal values of parameters of the
3. How can the time courses derived from this
method be statistically analyzed?
4. How is the estimation and statistical analysis of
these time courses affected by high-pass filtering the
Effect of correlations. Individual points in time
courses estimated with the linear model will in general
be correlated. These correlations result primarily from
the overlap of nearby responses in rapidly presented
event-related studies. We discuss how these correla-
tions affect the interpretability of the data.
Optimal experimental design.
those that yield the largest estimated magnitudes with
the best statistical properties while satisfying the be-
havioral constraints of the experiment. We define the
best statistical properties as low variance of the esti-
mated effects, equal variance across effects, and mini-
mum correlation among effects. We present the results
of Monte Carlo simulations used to determine the dis-
tribution of ITIs and percentage of partial trials that
yield the best statistical properties.
Statistical analysis of time courses.
maps must be generated from the time courses in order
to identify activated regions. We evaluate several dif-
ferent methods for computing these maps. The first
method computes an F statistic using the extra sum-
of-squares approach (Beck and Arnold, 1977). An in-
teresting aspect of this technique is that significance
levels are determined without making any assumption
about the shape of the hemodynamic response. Statis-
tical maps can also be generated by cross-correlating
the estimated time courses with a model for the hemo-
dynamic response and then forming t statistics. Since
the resulting magnitude is strongly affected by the
Optimal designs are
shape and timing of the model function, variations in
the shape of the observed BOLD response with condi-
tion can produce spurious changes in magnitude.
Therefore, it is important that the models be as robust
as possible. In this paper, we evaluate the robustness
of two models that have been used in the literature,
both as regressors and as kernels for the cross-corre-
High-pass filtering.High-pass filtering has been
used toreduce the effects of low-frequency noise, which
can adversely affect the analysis of time courses. For
example, low-frequency noise introduces correlations
in the sampled magnetic resonance (MR) signal that
affect the validity of statistical techniques that assume
independent sampling. While filtering can be benefi-
cial, its effects depend on the nature of the experimen-
tal paradigm. Rapidly presented event-related designs
have very different frequency-domain characteristics
than either block designs or widely spaced event-re-
lated designs. In this paper, we analyze the effects of
high-pass filtering in different experimental para-
digms by presenting models for the noise and signal
components of the data.
As described in Ollinger et al. (2001), data were
acquired from four subjects. The same stimuli and
image acquisition protocol were used here. The high-
contrast flickering checkerboard was presented in a
rapid event-related design with the ITIs uniformly dis-
tributed across intervals of 2.4, 4.7, and 7.1 s.
Monte Carlosimulations were used todetermine the
optimal values of the minimum and maximum ITIs
and the optimum fraction of partial trials. We define
optimum values to be those that minimize the mean
variance of points in the time courses, that minimize
the root-mean-square (RMS) variation of the variance,
and that minimize the RMS value of the correlation
coefficients. Thesemetrics werecomputed for each sim-
ulated data set and then averaged across data sets. An
experiment with threestimuli combined in twotypes of
trials was simulated. Each trial consisted of the first
stimulus (which we call the cue) followed after a two-
frame interval by either the second or third stimulus
(which we call the targets). Randomly generated de-
sign matrices for measured data consisting of 2048
points of data were created for paradigms with a range
of partial trial fractions and ITIs. The ITI is defined as
the interval between the end of one trial and the be-
ginning of the next. In the first simulation run, the
SEPARATING PROCESSES WITHIN A TRIAL, II
minimum ITI (ITImin) was varied from zero to five TRs
while the maximum ITI was varied from ITImin? 1 to
7 TRs with a partial-trial fraction of 0.25. Note that the
data are discretely sampled and that we are consider-
ing data measured under thenull hypothesis. Sincethe
sampling rate only affects the representation of the
hemodynamic response, which is absent under the null
hypothesis, therelevant unit of timeis a singlesample,
i.e., a single TR. Therefore, the statistical results do
not depend on the duration of the TR, although calcu-
lations of the magnitude of the response obviously do.
The first simulation run was repeated twice: once
with a uniform distribution of ITIs and again with an
exponential distribution. The exponential distribution
was formed by assigning one-half of the ITIs to the
minimum ITI (ITImin), one-fourth to ITImin ? 1, one-
eighth toITImin? 2 and soforth until themaximum ITI
was reached. All remaining trials were assigned to the
maximum ITI. Note that this method forces the expo-
nential and uniform distributions to be the same if
there are only two ITIs. The second simulation run
varied the partial-trial fraction from 0.1 to 0.8 for ITI
ranges of 0–3, 1–4, and 2–5 for both uniform and
exponential ITI distributions. The first run was then
repeated for the partial-trial fraction that yielded the
best statistical properties. All simulations were re-
peated for time course lengths of 8 and 16 TRs. For a
TR of 2.5 s, the 8 TR time courses would sample the
first 20 s of the BOLD response. The 16 TR time
courses would sample the first 40 s. A pseudo-random
number generator was used togenerate 25 realizations
of stimuli meeting each of these specifications. The
design matrix A was computed for each realization.
As in all linear models with nonorthogonal regres-
sors, the covariance matrix is strongly affected by the
experimental design. In the case of rapidly presented
FMRI studies, this source of covariance is dominant
(Ollinger and McAvoy, 2000). Since the general linear
model assumes that the noise in the data is white and
Gaussian, the covariance matrix of the estimates bˆis
proportional toKall? (ATA)?1(Beck and Arnold, 1977).
The covariance matrix of the effects of interest is given
by a submatrix K of Kall. The variances of the estimates
are given by the diagonal terms kiiand the correlation
coefficients are given by the normalized off-diagonal
terms, i.e., by kij/?kiikjj. The metrics described above
were computed from these variances and correlation
Statistical maps were generated by cross-correlating
the time courses with models for the hemodynamic
response and then forming t statistics. Several models
for the hemodynamic response have been proposed.
One is a gamma function (Boynton et al., 1996), which
can be modified by adding a variable delay (Dale and
Buckner, 1997). An alternative model uses a Gaussian
function (Clark et al., 1997; Zarahn et al., 1997). A
more complex model uses a set of three gamma func-
tions and their derivatives toaccount for undershoot at
the tail of the function and to desensitize the model to
slight changes in temporal delay (Friston et al., 1998).
This model is perhaps the most accurate, but is of
limited usefulness in random effects models because it
yields six magnitudes rather than one, and these can-
not be combined to form a single, normally distributed
value that represents the magnitude of the activation.
A simplified version, referred to in the SPM package
(Wellcome Department of Cognitive Neurology, Lon-
don: http://www.fil.ion.ucl.ac.uk/spm) as the “canoni-
cal” model and referred tohere as the “SPM canonical”
model, is the difference of twogamma functions: one to
model the peak of the response and the other to model
the postpeak undershoot.
The sensitivity of the statistical maps to the model
for the hemodynamic response was characterized by
computing statistical maps using the two most
widely used methods: The SPM canonical model and
the delayed gamma function model (Dale and Buck-
ner, 1997). Both were convolved with a boxcar func-
tion to model the duration of the stimulus. Details
are given in the Appendix. E ach model was used two
ways: as a regressor in the design matrix and as a
kernel in the cross-correlation of the model function
with the time courses estimated as described in
Ollinger et al. (2001). All models included a high-
pass filter with a cutoff frequency of 0.014 Hz (i.e.,
the high-pass filter removes the lowest four frequen-
cies in the data). This filter was implemented by
including sine and cosine functions at each frequency
below the cutoff frequency as regressors in the de-
sign matrix. The variance was estimated from the
residuals (Friston et al., 1995a, Friston et al., 1995b;
Worsley and Friston, 1995), and t statistics were
formed as the ratio of the magnitude to its standard
deviation. F statistics were computed using the
extra sum-of-squares approach (Beck and Arnold,
1977). These t and F statistics were then “Gaussian-
ized,”i.e., transformed to normally
statistics with the same significance probabilities.
The unsmoothed statistical maps were corrected
for multiple comparisons using a Bonferroni correc-
Noise and Signal Power Spectrum
The noise power spectrum was determined by scan-
ning a subject under two conditions. In the first, 1280
volumes were collected in a fixation state with a TR of
2.5 s and a voxel size of 3 ? 3 ? 3 mm. In the second,
the subject viewed a flickering checkerboard in a block
OLLINGER, CORBETTA, AND SHULMAN
paradigm. Statistical maps were computed as de-
scribed above for the activation data. A Bonferroni
threshold was then applied to yield a map of activated
voxels in visual cortex. The power spectra were then
computed at each voxel for each run of the fixation
data. These were then averaged across runs to yield
average power spectra at each voxel and then averaged
again across activated voxels to yield a low-noise esti-
mate of the noise power spectra. This power spectrum
is representative of regions that are predominantly
gray matter in visual cortex at rest. We assume that
this is representative of the power spectrum of the
noise during a task-induced activation.
Correlations in the Estimated Time Courses
Correlations among points in the estimated time
courses arisefrom twosources: correlations in themea-
sured data and correlations induced by the design. We
consider correlations induced by the experimental de-
sign here. Note that these correlations quantify the
relationship between the noise at a point in one esti-
mated time course and the noise at points in the same
or other estimated time courses. They do not describe
correlations among the signals. Under the assump-
tions of the linear model, the estimated time courses
are always equal to the true time course corrupted by
this correlated noise.
The structure of these correlations can be seen in the
profile through a covariance matrix for the simulated
experiment shown at the top of Fig. 1. Each trial of the
compound trial experiment (open circles) involved a
cue (effect 1) that was followed after two TRs by one of
two targets (effect 2 or 3). Each trial of the rapid
event-related experiment contained a single stimulus
that could be of three different types (effect 1, 2, or 3).
This profile is taken through row 8, which is the eighth
point (t ? 8) of the first effect in the compound trials.
The value at column eight is equal to 1 since any point
is perfectly correlated with itself. There are two prom-
inent features. First, in the compound trial experi-
ment, there are negative peaks at rows corresponding
to t ? 6 of each target effect and positive peaks at t ?
8 of each target effect. The negative peaks are ex-
plained by noting that the eighth point of the cue
overlays the sixth point of each target response, since
the target effect begins two TRs after the cue effect.
Any noise that is attributed to the eighth point of the
cue must be taken away from these points in the target
response, which results in a negative correlation coef-
The positive peaks in both the compound trial and
event-related experiments in Fig. 1 are more difficult
to explain. They can be understood by considering the
simpler case of a rapidly presented event-related de-
sign with three effects. If the stimuli are widely spaced
and presented in such a way that each point in thedata
contributes to one and only one effect, the columns
corresponding to the estimated effects will be collinear
with the columns used to estimate the mean. Such a
design does not yield a unique solution, but it can be
constrained by forcing the regressors for the effects of
interest to sum to 0 at each row of the design matrix.
This yields unique time courses but with an added
constant. This constant is of no concern if a contrast
(which must sum to 0) is used to generate statistical
maps because the constant term will cancel. However,
this constant represents a significant, constant corre-
lation among time points. As the trials become more
closely spaced, this correlation is nolonger constant, as
is shown in the bottom panel of Fig. 1. Instead, the
correlation coefficients become smoothly
functions with peaks at the same point in each effect
(i.e., the cross-correlations with frame 3 for effect 1 are
F IG. 1.
sponding to the eighth point of the first effect. Each plot shows the
correlation coefficient of this point of the estimated time course with
each point of every other effect. The graph at the top shows the
correlations for the compound event-related study (open squares)
and a rapidly presented event-related study. The minimum ITI in
each case was three TRs. The graph at the bottom shows the corre-
lations for a rapidly presented paradigm for four values of the min-
imum ITI. This demonstrates how reducing the minimum ITI decor-
relates the estimates.
Profiles through the covariance matrix at the row corre-
SEPARATING PROCESSES WITHIN A TRIAL, II
at a maximum for frame 3 of effects 2 and 3). These
correlations will not cancel when a contrast is applied
and may alter theshapeof thetimecoursein ways that
are difficult topredict. Moreover, the value of the noise
at adjacent points should change smoothly. Referring
to the plots in Fig. 5 of Ollinger et al. (2001), we see
that there is very little spike-like noise in the tails of
the estimated BOLD responses. For any given subject,
the tails look smooth and interpretable despite the fact
that the variance is appreciable. Therefore, noise could
be interpreted as a smooth change in the shape of the
One might think that removing the mean from the
data prior tothe estimation step rather than as part of
it would eliminate some of this correlation. We com-
pared precorrecting the data with correcting as part of
the estimation procedure by comparing time courses
for the high-contrast study. The maximum difference
in the percentage change was 0.0264 for a time course
with a peak value of 1.36. This difference is negligible.
Simulation results for mean variance, RMS varia-
tion of the variance, and RMS correlation coefficient
are shown in Fig. 2. The results were the same for
partial-trial fractions of 0.25 and 0.4. Several conclu-
sions can be drawn. First, exponentially distributed
ITIs yield lower variance. This is not surprising, since
the exponential distribution concentrates trials at
shorter ITIs, which yields more trials per study. Sec-
ond, the mean and RMS variation of the variance are
relatively insensitive tothe maximum ITI for exponen-
tially distributed ITIs. Again, this is not surprising
since most trials with the exponential distribution are
at very short ITIs regardless of the maximum ITI. The
variance increases as the maximum ITI increases for
uniform ITIs because more trials have a long ITI,
thereby decreasing thetotal number of trials. TheRMS
correlation coefficients are roughly equivalent for uni-
form and exponential distributions. In both cases, they
are near their minimum values if there are at least
three distinct ITIs.
A plot of these same statistics against the fraction of
partial trials is shown in Fig. 3. This plot used ITIs
exponentially distributed over the range of one to four
TRs. The curves all had the same shape for all ITIs
tested (zero to three, one to four, and two to five, both
uniformly and exponentially distributed). It can becon-
cluded that if variance were the only consideration, the
optimum fraction of partial trials would be 0.4. Behav-
ioral considerations usually require a lower fraction of
partial trials and Fig. 3 indicates that the variance
functions begin to bottom out for fractions of 0.2–0.3.
Previous studies have found reliable separation of the
components of a compound trial using fractions of 0.25
(Shulman et al., 1999) and 0.20 (Corbetta et al., 2000).
These simulation results were computed from esti-
mates of 8-point time courses. Repeating the simula-
F IG. 2.
RMS correlation coefficient vs maximum ITI. The minimum ITI was
varied from zerotofive TRs while the maximum ITI was varied from
ITImin ? 1 to 7 TRs. The light lines with open symbols represent
studies with uniformly distributed ITIs and the heavy lines with
closed symbols represent studies with exponentially distributed ITIs.
Plots of mean variance, RMS error of the variance, and
F IG. 3.
correlation coefficient vs fraction of partial trials for an exponential
distribution of ITIs of zero to three TRs.
Mean variance, RMS variation of the variance, and RMS
OLLINGER, CORBETTA, AND SHULMAN
tions for 16-point time courses yields curves with the
sameshapes. However, themean varianceincreases by
13%, the RMS variation of the variance decreases by
11%, and the RMS correlation coefficient decreases by
25%. Therefore, estimating 16 points would decrease z
scores by an average of 5.9%, but would yield estimates
with more homogeneous variances.
The sensitivity of the statistical maps to the model
for the hemodynamic response was quantified by z
statistics averaged over V1 for the high-contrast study.
The SPM canonical model and the delayed gamma
function model were evaluated at four different delays:
the nominal value of 2 s and shifts from this value of
?1, 1, and 2 s. The means across subjects are shown in
Fig. 4. If thedelay is known exactly, theSPM canonical
model yields the highest z statistics. The performance
of the canonical model falls off rapidly, however, as the
delay of the true BOLD response deviates from that of
the model response. Both models yield lower z statis-
tics when they are formed by cross-correlating the
model response with the estimated time course of the
BOLD response rather than by using the model as a
regressor in the design matrix. Note, however, that
this disadvantage decreases and may actually reverse
if the delay is not optimal. With the cross-correlation
technique, the gamma function model yields higher z
statistics and is more robust to time shifts than the
canonical model. The F statistic approach, which does
not makeany shapeassumptions, yields z scores equiv-
alent to the highest obtained with either cross-correla-
Models for the Signal and Noise
A frequency domain representation for the signal is
easy to develop for paradigms with regularly spaced
stimuli, such as block designs and widely spaced event-
related studies. Any periodic function can be repre-
sented by the sum of sinusoids at frequencies that are
a multiple of the paradigm frequency. For example, a
block design with task/control blocks of 40 s each would
be represented by the weighted sum of sinusoids at
frequencies of 0.0125, 0.0250, 0.0375 Hz, etc., where
the weights decrease rapidly with frequency. Simi-
larly, a widely spaced event-related study with stimuli
presented every eight TRs (20 s) would be represented
by sinusoids at frequencies of 0.05, 0.10, 0.15 Hz, etc. A
representation for the paradigms proposed here can be
inferred by considering a rapidly presented event-re-
lated study and then assuming that the power spec-
trum for a compound event-related study will be simi-
lar. We model the stimulus as a train of impulses, s(t),
and assume that brain hemodynamics are well mod-
eled by a linear system with an impulse response, h(t),
the hemodynamic response function. The BOLD re-
sponse is therefore given by s(t) * h(t), where “*” de-
notes convolution. Moreover, if the ITIs follow an ex-
ponential distribution, i.e., half the ITIs are equal to0,
one-fourth of the ITIs are equal to 1 TR, one-eighth of
the ITIs are equal to 2 TRs, etc., then the stimulus s(t)
is well modeled as a homogeneous Poisson process
(Snyder and Miller, 1991). By Campbell’s theorem
(Snyder and Miller, 1991), the BOLD response has a
power density spectrum given by P(?) ? ??H(?)?2,
where ? is the intensity of the Poisson process, H(?) is
the Fourier transform of h(t), and ? ? ? denotes magni-
tude. If h(t) is modeled by the gamma function h(t) ?
Atexp(??t) (Boynton et al., 1996), the power spectrum
F IG. 5.
and the computed signal power (closed circles) vs temporal fre-
quency. The noise power was measured during a fixation task. The
signal power was computed analytically for a rapid event-related
design with an exponential distribution of ITIs and minimum ITI of
one TR. The signal power for a widely spaced design (open triangles)
and the rapidly presented design used in the experimental study
presented here (closed squares) are shown in the inset.
Magnitude of the measured noise power (open squares)
F IG. 4.
Modeling the hemodynamic response in the design matrix with a
gamma function (Gamma); modeling the hemodynamic response in
the design matrix with the SPM canonical model (SPM Canonical);
estimating the response time course and then cross-correlating with
a gamma function (TC then Gamma); estimating the response time
course and then cross-correlating with the SPM canonical model (TC
then “SPM”); and finally, computing the time courses and their
Gaussianized F statistic (TC F statistic).
Mean estimated Z scores vs time shift for five cases:
SEPARATING PROCESSES WITHIN A TRIAL, II
is given by P( f ) ? ??1/(?2? ?2)?2, where ? ? 2?f. A
typical valueof ? is 0.8 s (Daleand Buckner, 1997). The
resulting power spectrum in Fig. 5 shows that there is
significant signal power at all frequencies, including
very low ones. For example, the summed responses to
overlapped trials do not return to zero and will there-
forehavea nonzeromean, sothereis even signal power
at a frequency of zero.
There are twoshortcomings tothis analysis. First, it
does not include the postpeak undershoot of the hemo-
dynamic response. This causes an underestimateof the
signal power at very low frequencies. Second, it as-
sumes that the hemodynamic response has infinite
support. This leads to a slight overestimate of the
signal power at very low frequencies. Nevertheless, the
general shape is representative of the power spectrum
of the measured BOLD response.
The evolution of the noise from the neuronal level to
the received signal is modeled by a linear system as
shown in Fig. 6. The inputs to the system at the neu-
ronal level arethepresented stimuli and low-frequency
fluctuations in neuronal firing that may arise from
functional connectivity (Biswal et al., 1995). Coupling
between neuronal firing and hemodynamics can be
modeled as the output of a linear system whose kernel
is the hemodynamic response function. This implies
that a short burst of neuronal firing causes the tran-
sient increase in oxygenated blood described by the
hemodynamic response. In the presence of the mag-
netic field, this HRF induces the BOLD signal. At this
stage, noise due to bulk subject motion, cardiac pulsa-
tility, respiratory motion, and various other noise
sources in the MR signal is introduced. The BOLD
response is observed through the coil and receiver,
which add thermal noise to the signal.
Two of these noise sources are easy to deal with.
First, thenoiseduetobaselinefluctuations in neuronal
firing is quite small since it is of the same order of
magnitude as the firing changes due to functional con-
nectivity, and these are small (Biswal et al., 1995;
Xiong et al., 1999) and can be neglected. Second, ther-
mal noise added to the RF signal in the coil and re-
ceiver is uncorrelated, stationary, and Gaussian to a
very good approximation in regions of the image with a
mean much greater than zero and therefore satisfies
the assumptions of the linear model. Noise introduced
into the BOLD signal (as distinguished from the two
previous noisesources) is moredifficult tocharacterize.
Noise due to respiratory motion occurs at a frequency
of approximately 0.1 Hz. It is not usually modeled.
Noise due to cardiac pulsatility is usually sampled at
rates toolow toadequately represent it. For example, if
thepulserateis 60 bpm, theNyquist sampling interval
will be 0.5 s. A typical TR of 2.5 s undersamples this
signal by a factor of five. The effect will be to alias the
An empirically determined noise power spectrum
(see Methods) is shown in Fig. 5. As has been observed
by others (Zarahn et al., 1997), the spectrum can be
modeled as the sum of a 1/f decay and a constant. The
constant term is assumed tocorrespond tothe thermal
noise, and the low-frequency noise is caused by the
other effects mentioned above (Zarahn et al., 1997).
Since uncorrelated, normally distributed data yield a
uniform power spectrum, the deviation of the spectrum
shown in Fig. 5 from a uniform distribution can be
interpreted as correlations among points in the mea-
sured data. This deviation stems from thephysiological
and MR noise which is concentrated at low frequencies
(Zarahn et al., 1997).
Several approaches have been suggested for dealing
with correlated noise. Optimal filtering can be used to
prewhiten the data (Buonocore and Maddock, 1997),
but this requires additional data collection in order to
define the filter. Another approach appeals to the
matched-filter theorem to justify smoothing with the
HRF (Friston et al., 1995a,b). This, however, only deals
with the baseline noise. A third approach is to model
the correlations as a first-order autoregressive process
(Burock, 1998; Purdon and Weisskoff, 1998). The most
widespread approach is to high-pass filter the data.
The goal of filtering is to remove noise at frequencies
that do not contain the signal. Since the variance is
computed from the residuals of the linear model, re-
moving these frequencies decreases the variance,
thereby increasing the t statistics estimated from the
model. Filtering does not reduce the incidence of false
positives because it does not remove noise at the par-
adigm frequency. False positives can be reduced by
designing experiments sothat the signal energy occurs
predominantly at frequencies that contain relatively
small amounts of noise.
Effects of High-Pass Filtering
As described above, block paradigms often concen-
tratemost of their energy at frequencies on theorder of
0.0125 Hz. Figure 5 shows that this frequency lies near
the 1/f portion of the noise spectrum. This explains the
F IG. 6. Noise sources in a typical FMRI study.
OLLINGER, CORBETTA, AND SHULMAN
susceptibility of block designs to physiological noise
(Zarahn et al., 1997). As shown in the inset to Fig. 5,
widely spaced event-related designs would typically
have the most energy at roughly 0.055 Hz. This fre-
quency lies outside the low-frequency noise peak,
which explains thereduced susceptibility of thesestud-
ies to physiological noise. In both cases, one would
expect high-pass filtering to improve statistical power
by reducing the variance since it should decrease the
estimated variance while leaving the BOLD signal un-
The situation is more complex for rapidly presented
event-related designs. These designs spread the signal
energy across the entire spectrum as shown in Fig. 5
for the analytical model. This has two effects. First, it
deemphasizes low-frequency noise so the estimation is
relatively less sensitive to these frequencies; but sec-
ond, it places some signal energy at the noisiest fre-
quencies. Therefore, high-pass filtering will remove
part of the BOLD signal. The power spectrum for the
experimental design used here is shown in the inset to
Fig. 5. It is neither as smooth nor as flat as that of the
analytical model. This results from the finite length of
the experiment and from the nonexponential distribu-
tion of the ITIs.
The effect on the estimates can be understood by
considering the postpeak undershoot. This undershoot
has a duration of roughly 90 s (Fransson et al., 1998;
Mandeville et al., 1998) and is therefore represented
predominantly by frequency components at approxi-
mately 0.01 Hz. One would expect it to be strongly
affected by the high-pass filter. This hypothesis was
tested by analyzing the data both with and without a
high-pass filter with a cutoff at 0.014 Hz. As shown in
Fig. 7, the high-pass filter eliminates the undershoot
from the estimated time course. This implies that mod-
els for the hemodynamic response function should not
include the undershoot if the data are high-pass fil-
The merit of the high-pass filter was investigated by
analyzing how filtering affected z scores for the high-
contrast studies for five methods of computing statis-
tical maps. For the first two methods, Z scores were
obtained by coding the SPM canonical and gamma
function regressors into the design matrix as regres-
sors. The other three methods estimated time courses
and their associated F statistics for these conditions:
(1) rapidly presented with 8 estimated points, (2) rap-
idly presented with 16 estimated points, and (3) widely
spaced with 8 estimated points. The differences be-
tween z statistics computed with and without high-
pass filtering are shown in Fig. 8. Three of the methods
do not model the undershoot as part of the hemody-
namic response: the gamma function regressor and the
two methods that estimate 8-point time courses (these
truncate the undershoot). In each of these cases, the z
scores increased for each subject when the high-pass
filter was included. The SPM canonical model accounts
for the undershoot twice: as part of the hemodynamic
response model and in the high-pass filter. Again,
there were consistent improvements with the high-
pass filter for three of four subjects, but only a very
small improvement for thefourth. Wespeculatethat in
this subject, theundershoot matched theundershoot in
the model well enough that the high-pass filter made
little difference. The case of the 16-point time course
was more interesting. For one subject, high-pass filter-
ing reduced the z statistic, for another there was a
small improvement, and for the other two it increased
the z statistic. Interestingly, the twosubjects for whom
thez statistic increased (subjects 2 and 4) werealsothe
subjects that appeared to have more motion artifact.
We infer from this that high-pass filtering the data can
degrade sensitivity in cooperative subjects under rela-
tively noise-free conditions, but can improve perfor-
mance under noisy conditions. The changes in z statis-
tics were small, typically on the order of 0.25, and the
mean z score was between 5 and 5.5 depending on the
F IG. 8.
and without a high-pass filter for four subjects, s1 through s4, and
five processing methods. The “SPM” and “Gamma” used the SPM
canonical model and a gamma function, respectively, as regressors.
The other three approaches all computed time courses and their
associated Gaussianized F statistics. The first two estimated 8- and
16-point time courses from the rapidly presented high-contrast
study, while the third estimated an 8-point time course from the
widely spaced high-contrast study.
The difference between statistical maps computed with
F IG. 7.
mated from the high-contrast experiment using two different mod-
els. The first incorporated a high-pass filter that modeled the lowest
four frequency components while the other did not.
Mean time course of the high-contrast response esti-
SEPARATING PROCESSES WITHIN A TRIAL, II
computational method. This amounts to a 5% change
in the estimated magnitude, which is unlikely tomake
an appreciable difference in most studies.
We draw the following conclusions about high-pass
filtering. For block designs and widely spaced event-
related designs, it removes correlations from the data
and reduces variance estimates at the cost of a reduced
number of degrees of freedom (Friston et al., 1995a,b;
Worsley and Friston, 1995). Since most FMRI experi-
ments have hundreds to thousands of degrees of free-
dom, this reduction does not havea significant effect on
sensitivity. Filtering may also be useful for rapidly
presented event-related designs. It again removes cor-
relations from the data and reduces variance esti-
mates, although it alsoremoves some signal. In partic-
ular, since filtering removes the postpeak undershoot,
it is best done in conjunction with a model for the
hemodynamic response that does not include that un-
Correlations in the Estimated Time Courses
One might think that time courses estimated with
FMRI could be analyzed in the same way as those
acquired with single-unit recordings. This, however, is
not the case. In single-unit recordings, the noise is
uncorrelated across time points and has approximately
equal variances at each time point. This gives the noise
component of the data a spike-like appearance, which
can be easily recognized. In FMRI, however, the noise
across time points in the estimated time courses shows
a high degree of correlation. This follows from the
simulation results as well as the empirical evidence in
the time courses. These correlations imply that the
noise in estimated time courses will be smooth rather
than impulsive and could be interpreted as features of
the BOLD response. As one would expect, this smooth
noise is greatly reduced in across-subject averages.
This suggests that estimated time courses should be
analyzed with caution in single subjects. Monte Carlo
results show that correlations between estimated time
points of the same effect are minimized by using short
Optimal Experiment Design
Optimal designs are those that yield the largest es-
timated magnitudes with thebest statistical properties
while satisfying the behavioral constraints of the ex-
periment. We define the best statistical properties as
low variance, equal variance across effects, and mini-
mum correlation among effects. The simulations show
that the first two properties are best achieved with
short ITIs that follow an exponential distribution,
probably because these designs maximize the number
of trials. The overall magnitude of correlation is mini-
mized by using at least three distinct ITIs. Although
the correlation among effects is relatively independent
of the choice of ITIs, shorter minimum ITIs reduce
correlations among points within a given time course,
as noted above. Short minimum ITIs yield good statis-
tical properties, but they may also produce nonlinear
interactions among effects. In particular, the presence
of modest nonlinearities (Miezin et al., 2000) suggests
that exponential distributions which produce a succes-
sion of trials involving short ITIs may be undesirable.
Since response magnitude increases for longer mini-
mum ITIs (Miezin et al., 2000), the optimum range of
ITIs is likely to be a minimum of one to two TRs
(2.4–4.7 s) and a maximum of three to four TRs (7.1–
Low variance, equal variance across effects, and
minimum correlation among effects are all minimized
by using 40% partial trials. Behavioral constraints,
however, will generally require a lower fraction (Shul-
man et al., 1999; Corbetta et al., 2000). A fraction of
0.25 yielded reliable separation of the low-contrast and
high-contrast responses in the study presented here,
and good results have also been obtained in previous
studies with fractions of 0.25 (Shulman et al., 1999)
and 0.2 (Corbetta et al., 2000).
When the model for the hemodynamic response was
accurate, coding the model function into the design
matrix as a regressor yielded a higher z score than
estimating a time course and cross-correlating that
time course with the model function. This advantage
decreased, however, and sometimes reversed, when the
delay time was inaccurate. Both model functions
yielded results that wereaffected by errors in thedelay
time. This has the undesirable effect of confounding
thedelay of theresponsewith its measured magnitude.
The delayed gamma function model was more robust
than the SPM canonical model in this respect.
Therelevanceof this sensitivity tothedelay timecan
be assessed by considering the standard deviation of
the time-to-peak of the hemodynamic response, which
is approximately 1.1 s (Aguirre et al., 1998) for a
500-ms visual stimulus. This implies that the range of
values tested here is likely to occur in many experi-
ments. Moreover, the stimuli presented here had a
much longer duration of 4.7 s, which leads to much
wider hemodynamic responses. Magnitudes computed
from these wider hemodynamic responses would be
expected tobemorerobust with respect tovariations in
timing than the relatively narrow hemodynamic re-
sponses in Aguirre et al. (1998). Therefore, we would
expect experiments using brief stimuli tobe even more
OLLINGER, CORBETTA, AND SHULMAN
sensitive to changes in the delay parameter than the
data presented here suggest.
These results suggest that the model functions
should only be used as regressors in the design matrix
if they are known to accurately model the form of the
BOLD responses in all regions of interest. In the ab-
sence of this detailed prior information, response mag-
nitude will be confounded with the timing and width of
the response. Estimating the time courses reduces the
possibility of missing significant activations that de-
part from the standard shape or delay and prevents
biased analyses that favor conditions that yield hemo-
dynamic responses that match the model function.
Results for the hemodynamic response in V1 to a
passive visual stimulus may not be strictly represen-
tative of the response of other areas or the response
during demanding cognitive tasks. This seems partic-
ularly likely if the task has an extended duration and
engages cognitive processes of different durations at
different time points during the task. It might there-
fore make more sense to estimate the time course at
each voxel and then conduct a statistical analysis on
the time course functions. Otherwise, the analysis is
biased toward those areas and conditions that yield
canonical time courses. The use of strong shape as-
sumptions may only be advisable for areas and tasks,
which have already been well characterized (i.e., V1
responses during passive sensory stimulation). Al-
though the F statistic, which does not make any as-
sumption about shape, yielded lower z statistics than
those obtained by coding the model functions into the
design matrix as regressors, their equivalent z scores
were equal to the highest values yielded by the corre-
lation of time courses with model functions. Moreover,
when the delay of the hemodynamic response was off
by as littleas 1 s, theF statistics wereequivalent tothe
highest obtained using model functions as regressors
to obtain t statistics.
This analysis holds for data analyzed at the level of
a single subject. At this level, any modeling choice can
affect either the magnitude or the variance of an effect.
Most experiments, however, are analyzed across sub-
jects. In these analyses, magnitudes of activation are
extracted from each subject and then compared in sec-
ond-level analyses using paired t tests or analysis of
variance (ANOVA) tests. These second-level analyses
are based on the across-subject variance, which may
not be dominated by the variance at the single-subject
level. Therefore, any aspect of the model that affects
the magnitude of the response should affect the group
and individual analyses equally while any aspect that
only affects the within-subject variance term may or
may not have a strong impact on the group analyses.
Since the choice of model functions primarily affects
the magnitude of the estimated response, we expect
these results to carry over to group analyses.
The need for a model for the hemodynamic response
can be eliminated if the time courses are analyzed with
ANOVA methods (Shulman et al., 1999; Corbetta et al.,
2000). This is done by treating time as a main effect
with levels corresponding to each estimated point of
the BOLD responses. If the experiment has one task
with two levels, the resulting ANOVA would have two
main effects (task and time) and one interaction term.
The main effect of time would test the hypothesis that
estimated time courses averaged across tasks do not
significantly vary from zero. The interaction term
would test whether the time course varied significantly
across the two tasks.
A fundamental assumption of ANOVAs is that the
variance is homogeneous, i.e., that the variances are
equal and that data points are uncorrelated. The
time courses discussed here violate this assumption.
Therefore, ANOVAs can only be used if the degrees
of freedom of the ANOVA are adjusted appropriately
(Box, 1954; Ollinger and McAvoy, 2000). This incurs
a loss of statistical power that becomes more severe
as the variance becomes less homogeneous. There-
fore, proper choice of the timing parameters of a
study can have a significant effect on its statistical
The power spectrum of the data changes signifi-
cantly with the type of design and the timing of the
stimuli. Block designs concentrate the signal power at
low frequencies; widely spaced event-related designs
concentrate it at much higher frequencies; and rapidly
presented event-related designs spread it throughout
the power spectrum. Since the noise power spectrum
peaks at low frequencies (Fig. 5), the event-related
designs are less sensitive to artifacts due to low-fre-
quency noise than block designs. Therefore, these de-
signs may be preferable for studies where low-fre-
quency noise due to patient motion is expected to be
high, such as studies in patient populations and stud-
ies that require speaking in the scanner.
High-pass filtering is beneficial for block designs and
widely spaced event-related designs, since it removes
1/f noise from the data, thereby reducing the variance
estimates and whitening the data. For similar reasons,
filtering can alsobe useful for rapidly presented event-
related designs. In addition, however, the signal in
rapid event-related designs has power at low frequen-
cies, so filtering affects the estimated time course. As
shown in Fig. 7, the contribution to the signal due to
the postpeak undershoot is largely removed. This leads
to better fits by models that do not accurately account
for the undershoot.
SEPARATING PROCESSES WITHIN A TRIAL, II
Twomodels for the hemodynamic response toa brief
stimulus were used: a single gamma function (Boynton
et al., 1996) and the difference of twogamma functions
(Friston et al., 1999). These models were modified to
account for the relatively long duration of the stimuli
used in this paper by convolving them analytically with
a boxcar function. Details are given below.
Gamma Function Model
The hemodynamic response is modeled as the convo-
lution h(t) ? g(t) *b(t), whereg(t) is thedelayed gamma
function given by
g?t? ? ??t ? ??e?a?t???
The variable ?, which determines the width of the
hemodynamic response, is fixed at 1.25 s (Dale and
Buckner, 1997). The variable ?, which represents the
delay of the response, defaults to a value of 2 s. The
second function in the convolution, b(t), is the boxcar
function given by
r?t? ??1/T ,0 ? t ? T
where the variable T is the assumed duration of neu-
ronal firing. For studies where the stimulus is sus-
tained, such as the flickering checkerboard used in the
studies presented here, T is the duration of the stimu-
lus presentation. In most studies, T is the duration of
neuronal firing and is on the order of tens or hundreds
of milliseconds. In those cases, we assume that T is
small compared with the TR and model h(t) by an
impulse function. When T is not small, the convolution
is given by
0, t ? 0
e??t??2t2? 2?t ? 2?, 0 ? t ? T
e???t?T???2?T ? t?2? 2??T ? t? ? 2?
? e??t??2t2? 2?t ? 2?, T ? t
SPM Canonical Model
The canonical model used in the SPM package (Fris-
ton et al., 1999) is defined as
g?t? ? ?6?t? ? 1/6?16?t?
where ?n(t) is the gamma function defined by
Convolving with a boxcar function as before yields the
0, t ? 0
?6?t ? u?du ?
?16?t ? u?du, 0 ? t ? T
?6?t ? u?du ?
?16?t ? u?du, T ? t
where the integral is given by
?n?t ? u?du ?
?n ? i?!??
This work was supported by Grants EY00379 and EY12148 from
the National Institutes of Health and by a grant from the McDonnell
Center for Higher Brain Function. Theauthors thank Francis Miezin
for the data used tocompute the noise power spectrum, Tom Conturo
and Erbil Akbudak for theMRI pulsesequencedevelopment, and Avi
Snyder for the development of the preprocessing software. We also
thank Randy Buckner and Mark McAvoy for a careful reading of the
manuscript. Finally, we thank the referees for their useful sugges-
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