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Separating Processes within a Trial in Event-Related Functional MRI

II. Analysis

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

P ress

INTRODUCTION

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: jmo@npg.wustl.edu.

NeuroImage 13, 218–229 (2001)

doi:10.1006/nimg.2000.0711, available online at http://www.idealibrary.com on

218

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Copyright © 2001 by Academic Press

All rights of reproduction in any form reserved.

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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

experimental design?

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

data?

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

Statistical

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-

lation method.

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.

METHODS

Experimental Methods

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.

Simulation Methods

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

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SEPARATING PROCESSES WITHIN A TRIAL, II

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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

coefficients.

Statistical Analysis

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-

tion.

distributed

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

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OLLINGER, CORBETTA, AND SHULMAN

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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.

RESULTS

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-

ficient.

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

decaying

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-

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SEPARATING PROCESSES WITHIN A TRIAL, II

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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

response.

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

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

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OLLINGER, CORBETTA, AND SHULMAN

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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.

Statistical Analysis

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-

tion method.

Models for the Signal and Noise

Signal Model

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:

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SEPARATING PROCESSES WITHIN A TRIAL, II

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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.

Noise Model

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

cardiac signal.

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.

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OLLINGER, CORBETTA, AND SHULMAN

Page 8

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-

affected.

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-

tered.

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-

225

SEPARATING PROCESSES WITHIN A TRIAL, II

Page 9

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-

dershoot.

DISCUSSION

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

minimum ITIs.

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–

9.4 s).

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).

Statistical Analysis

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

226

OLLINGER, CORBETTA, AND SHULMAN

Page 10

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

power.

High-Pass Filtering

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.

227

SEPARATING PROCESSES WITHIN A TRIAL, II

Page 11

APPENDIX

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???

(1)

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

elsewhere,0,

(2)

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

h?t? ??

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

(3)

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?

(4)

where ?n(t) is the gamma function defined by

?n?t? ?

1

??n?tn?1e?t. (5)

Convolving with a boxcar function as before yields the

result that

h?t? ??

0, t ? 0

?

0

?

0

t

?6?t ? u?du ?

1

6?

1

0

t

?16?t ? u?du, 0 ? t ? T

T

?6?t ? u?du ?

6?

0

T

?16?t ? u?du, T ? t

,

(6)

where the integral is given by

?

0

t

?n?t ? u?du ?

e?t

??h???

i?0

n

n!tn?i

?n ? i?!??

t

t?T

.(7)

ACKNOWLEDGMENTS

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-

tions.

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