Linear systems analysis of the fMRI signal.
ABSTRACT In 1995 when we began our investigations of the human visual system using fMRI, little was known about the temporal properties of the fMRI signal. Before we felt comfortable making quantitative estimates of neuronal responses with this new technique, we decided to first conduct a basic study of how the timecourse of the fMRI response varied with stimulus timing and strength. The results ended up showing strong evidence that to a first approximation the hemodynamic transformation was linear in time. This was both important and remarkable: important because nearly all fMRI data analysis techniques assume or require linearity, and remarkable because the physiological basis of the hemodynamic transformation is so complex that we still have a far from complete understanding of it. In this paper, we provide highlights of the results of our original paper supporting the linear transform hypothesis. A reanalysis of the original data provides some interesting new insights into the published results. We also provide a detailed appendix describing of the properties and predictions of a linear system in time in the context of the transformation between neuronal responses and the BOLD signal.

Article: A Mixed L2 Norm Regularized HRF Estimation Method for Rapid EventRelated fMRI Experiments.
[Show abstract] [Hide abstract]
ABSTRACT: Brain state decoding or "mind reading" via multivoxel pattern analysis (MVPA) has become a popular focus of functional magnetic resonance imaging (fMRI) studies. In brain decoding, stimulus presentation rate is increased as fast as possible to collect many training samples and obtain an effective and reliable classifier or computational model. However, for extremely rapid eventrelated experiments, the bloodoxygenleveldependent (BOLD) signals evoked by adjacent trials are heavily overlapped in the time domain. Thus, identifying trialspecific BOLD responses is difficult. In addition, voxelspecific hemodynamic response function (HRF), which is useful in MVPA, should be used in estimation to decrease the loss of weak information across voxels and obtain finegrained spatial information. Regularization methods have been widely used to increase the efficiency of HRF estimates. In this study, we propose a regularization framework called mixed L2 norm regularization. This framework involves Tikhonov regularization and an additional L2 norm regularization term to calculate reliable HRF estimates. This technique improves the accuracy of HRF estimates and significantly increases the classification accuracy of the brain decoding task when applied to a rapid eventrelated fourcategory object classification experiment. At last, some essential issues such as the impact of lowfrequency fluctuation (LFF) and the influence of smoothing are discussed for rapid eventrelated experiments.Computational and Mathematical Methods in Medicine 01/2013; 2013:643129. · 0.79 Impact Factor  SourceAvailable from: Karl J Friston[Show abstract] [Hide abstract]
ABSTRACT: In Kilner et al. [Kilner, J.M., Kiebel, S.J., Friston, K.J., 2005. Applications of random field theory to electrophysiology. Neurosci. Lett. 374, 174178.] we described a fairly general analysis of induced responsesin electromagnetic brain signalsusing the summary statistic approach and statistical parametric mapping. This involves localising induced responsesin peristimulus time and frequencyby testing for effects in timefrequency images that summarise the response of each subject to each trial type. Conventionally, these timefrequency summaries are estimated using posthoc averaging of epoched data. However, posthoc averaging of this sort fails when the induced responses overlap or when there are multiple response components that have variable timing within each trial (for example stimulus and response components associated with different reaction times). In these situations, it is advantageous to estimate response components using a convolution model of the sort that is standard in the analysis of fMRI time series. In this paper, we describe one such approach, based upon ordinary least squares deconvolution of induced responses to input functions encoding the onset of different components within each trial. There are a number of fundamental advantages to this approach: for example; (i) one can disambiguate induced responses to stimulus onsets and variably timed responses; (ii) one can test for the modulation of induced responsesover peristimulus time and frequencyby parametric experimental factors and (iii) one can gracefully handle confoundssuch as slow drifts in powerby including them in the model. In what follows, we consider optimal forms for convolution models of induced responses, in terms of impulse response basis function sets and illustrate the utility of deconvolution estimators using simulated and real MEG data.NeuroImage 09/2012; 64C:388398. · 6.25 Impact Factor
Page 1
Review
Linear systems analysis of the fMRI signal
Geoffrey M. Boyntona,⁎, Stephen A. Engelb,1, David J. Heegerc,2
aDepartment of Psychology, University of Washington, P.O. Box 351525, Seattle, WA 981951525, USA
bDepartment of Psychology, University of Minnesota, N218 Elliot Hall, 75 East River Road, Minneapolis, MN, USA
cDepartment of Psychology and Center for Neural Science, New York University, 6 Washington Place, New York, NY, 10003, USA
a b s t r a c ta r t i c l ei n f o
Article history:
Received 15 December 2011
Revised 12 January 2012
Accepted 16 January 2012
Available online xxxx
Keywords:
fMRI
Linearity
V1
Hemodynamics
In 1995 when we began our investigations of the human visual system using fMRI, little was known about the
temporal properties of the fMRI signal. Before we felt comfortable making quantitative estimates of neuronal
responses with this new technique, we decided to first conduct a basic study of how the timecourse of the
fMRI response varied with stimulus timing and strength. The results ended up showing strong evidence
that to a first approximation the hemodynamic transformation was linear in time. This was both important
and remarkable: important because nearly all fMRI data analysis techniques assume or require linearity,
and remarkable because the physiological basis of the hemodynamic transformation is so complex that we
still have a far from complete understanding of it. In this paper, we provide highlights of the results of our
original paper supporting the linear transform hypothesis. A reanalysis of the original data provides some in
teresting new insights into the published results. We also provide a detailed appendix describing of the prop
erties and predictions of a linear system in time in the context of the transformation between neuronal
responses and the BOLD signal.
© 2012 Elsevier Inc. All rights reserved.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Why primary visual cortex?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time domain study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameterized fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frequency domain study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameterized fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Impact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shift invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
Our paper “Linear Systems Analysis of Functional Magnetic Reso
nance Imaging in Human V1” published in the Journal of Neurosci
ence in 1996 (Boynton et al., 1996) began as a pilot study. Our
primary interest was to use the new technique of fMRI to study the
neuronal response properties in the human primary visual cortex
NeuroImage xxx (2012) xxx–xxx
⁎ Corresponding author. Fax: +1 206 685 3157.
Email addresses: gboynton@uw.edu (G.M. Boynton), engel@umn.edu (S.A. Engel),
david.heeger@nyu.edu (D.J. Heeger).
1Fax: +1 612 626 2079.
2Fax: +1 212 995 4981.
YNIMG09147; No. of pages: 10; 4C:
10538119/$ – see front matter © 2012 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2012.01.082
Contents lists available at SciVerse ScienceDirect
NeuroImage
journal homepage: www.elsevier.com/locate/ynimg
Please cite this article as: Boynton, G.M., et al., Linear systems analysis of the fMRI signal, NeuroImage (2012), doi:10.1016/
j.neuroimage.2012.01.082
Page 2
across a range of stimulus conditions. We were inspired by the work
of animal physiologists who make quantitative measurements of neu
ronal responses to visual stimuli that vary along dimensions such as
contrast, spatial frequency and orientation. However, at that time
there were very few fMRI studies that used parametric stimulus ma
nipulations to obtain quantitative measurements. Instead, fMRI stud
ies typically reported binary decisions about whether statistically
significant responses were found in various brain regions or voxels
(e.g. statistical parameter maps). It occurred to us that before we
could get started on a quantitative investigation of the human visual
system, we first had to conduct an investigation into the nature of
the fMRI signal itself.
What we were hoping for, and were generally able to support, was
a linear transform model for the timecourse of the hemodynamic re
sponse (Fig. 1). The linear transform model wasn't a new idea. In fact,
it had already been assumed in previous papers proposing methods
for analyzing fMRI data (Bandettini et al., 1993; Friston et al., 1995).
Surprisingly, however, no test of linearity had been conducted to
date.
The linear transform model posits that the timecourse of the fMRI
response can be predicted by convolving the timecourse of the neu
ronal response with a hemodynamic impulse response, followed by
additive Gaussian noise. For those less familiar with the definition
and properties of a linear system, we have provided a brief descrip
tion in Appendix A.
Why primary visual cortex?
We focused on the primary visual cortex (V1), for a few reasons.
First, since feasible applications of retinotopic mapping procedures
were still being worked out (in our lab and others), we needed a re
gion of interest that we could find based on structural information.
Fortunately for us, V1 lies consistently within the calcarine sulcus.
Knowing where to look was particularly important because at that
time we only had a single slice (!) of functional data.
Second, we wanted to study an area where we had some under
standing of what the neuronal responses might be to a given stimu
lus. A true test of the linear model should compare the relation
between the neuronal responses and the fMRI responses, but of
course we were not directly manipulating the neuronal response. Of
all areas in the mammalian brain, V1 is perhaps the most studied
and well understood. Most important for our purposes, it is well
known that nearly all V1 neurons increase their firing rate monoton
ically with stimulus contrast (Albrecht and Hamilton, 1982). We
could therefore be reasonably confident that the mean firing rate of
neurons in V1 increases monotonically with stimulus contrast. Note
that we did not assume that the visual system itself was linear: we in
cluded a nonlinear contrast–response function in the linear transform
model. We did, however, assume that when a stimulus was presented
continuously with a constant contrast that the neuronal responses
would likewise be constant over time. That is known to be incorrect
because of nonlinearities in neuronal activity including adaptation,
something which we dealt with posthoc in the original paper, and
which we discuss further below.
Third, we wanted robust reliable data, and we knew even then
that fMRI responses in V1 are stronger and more reliable than most
other brain areas — probably due to the relative homogeneity of the
neuronal population and to the experimenter's control over the
timecourse of the input to that area.
We designed our experiments to test the linear transform model
in both the time domain and in the frequency domain (See
Appendix A for an explanation of time and frequency domains). The
general procedure was to manipulate the intensity and duration of
the visual stimulus and measure the timecourses and amplitudes of
the fMRI responses in V1. We used two kinds of analyses to test the
linear model hypothesis. The first was a nonparametric investigation
of the data to see if the responses were consistent with some of the
properties of linearity. The second was a bruteforce parameterized
model fit to the data. Both of these tests were applied to two sets of
experiments — one emphasizing the time domain and the other em
phasizing the frequency domain.
Time domain study
We first tested linearity in the time domain using what is now
called a slow eventrelated design. We simply measured the time
course of the fMRI response to flickering checkerboards of different
contrasts and durations. We waited 30 s or so between repeated trials
to let the hemodynamics recover to baseline. After acquiring several
runs across several days, we averaged the timecourse of each re
sponse from voxels within a handdrawn region of interest, time
locked to the onset of each stimulus.
Scaling
If the hemodynamic transfer function is linear, then the time
courses of the fMRI response to stimuli for a given duration should
vary only by a scale factor with contrast. This follows from the scal
ability property of a linear system. Note that we are assuming that
the timecourse of the neuronal response follows the timecourse of
Fig. 1. The linear transform model.
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the stimulus (at least as far as the timescale of fMRI is concerned).
This means that even though the magnitude neuronal response
does not scale in proportion to stimulus contrast, the shape of the he
modynamic responses should remain the same across contrasts.
The results were surprisingly clear. The response magnitudes
scaled nicely across the three contrasts (25, 50 and 100%). Fig. 2
shows an example response from one of the subjects to a 12 s stimu
lus. Note how the shape of the timecourse did not vary — it only
scaled with contrast. (Note how these results compare to the predic
tions of the linear model shown in Appendix A, Fig. A1).
We called this lessthanperfect scaling time–contrast separabili
ty. From a statistical sense it can be thought of as time–contrast inde
pendence, in which there is no interaction between the effects of the
independent variables of time and contrast. Figs. 10 and 11 in the
original paper showed how scalability held up for the four durations
tested (3, 6, and 24 s) for the two subjects.
It is important to point out that we did not find (or expect to find)
that the BOLD signal would scale directly with stimulus contrast.
Most biological systems show compression for large inputs, so that
doubling the strength of the input produces much less than doubling
of the output. This is true for the sensory systems and for contrast in
particular. It is clearly visible in Fig. 2 (and Fig. A1): Doubling the con
trast from 50% to 100% produced a scaling of the output, but the scal
ing was much less than a factor of two. We know that this is true of
the underlying neuronal responses so it is not surprising that it is
inherited by the hemodynamic responses. But this was not a violation
of the linear transformation model, because it is still possible that the
BOLD signal is scaling with the underlying (compressed) neuronal re
sponse. Time–contrast separability was consistent with a linear sys
tem relating the neuronal and hemodynamic responses that was
preceded by a nonlinear relationship between stimulus strength and
neuronal response. This nonlinear component is termed the con
trast–response function, and was easily inferred from our data by
plotting the stimulus contrast against the magnitude of the corre
sponding fMRI response.
Time–contrast separability, or more generally time–stimulus
separability, is critical for allowing meaningful data analysis. Sup
pose, for example, that the vascular system reacted more quickly
to a stronger neuronal input, leading to a differently shaped hemo
dynamic response for high contrast stimuli. This would be a disas
ter for data analysis since we wouldn't be able to generate a single
template to correlate with our data. We'd be stuck in a circular di
lemma in which we'd need to know the strength of the neuronal
response to generate the appropriate template timecourse which
would then be used to detect and estimate the underlying
neuronal response.
Additivity
We next tested the property of additivity with the same time
domain data by, for example, treating a 12 second stimulus as
two successive 6 second stimuli. If additivity and shift invariance
hold, then we should be able to predict the response to the 12 sec
ond stimulus by adding two responses to the 6 second stimulus,
with the second response shifted by 6 s. Fig. 3(A) shows an exam
ple of this from one of the subjects. Shown in red is the average
time course to a 12second stimulus. The blue curve was generated
by adding the response to the 6econd stimulus to a shifted copy
of itself. The red and blue boxes on the timeline represent the
timecourses of the stimuli.
The curvesoverlap. This means thatthe sum of theresponses to two
successive6secondstimuliwasequaltotheresponsetothesumoftwo
successive 6second stimuli. That is, the results were consistent with
the principal of additivity (compare these results with the predictions
from the linear transform model shown in Appendix A, Fig. A2).
Since we used 3, 6, 12, and 24second stimuli we were able to
make a total of 6 predictions (e.g. a 24 second response was predicted
by shifting and adding six 3second responses, etc.). Most of the pre
dictions matched nicely, like the example in Fig. 3(A).
Fig. 3(B) shows what happened when a 3second responses was
used to predict the same 12second response. The fit wasn't as
good. The shifted and summed 3second prediction was too large
compared to actual response to the 12second stimulus. This violation
of the additivity property was also evident in our parametric fit to the
data, which will be described below.
Like scaling, it is not hard to imagine how additivity could have
failed for a complex phenomenon like the hemodynamic response,
especially considering that the vascular response provides an over
supply of oxygenated blood. Suppose, for example, that once a re
sponse to the first stimulus is underway, the vascular response to
the second stimulus is not needed, so little additional hemodynamic
response is added. The disproportionately large response to the 3
second stimulus might mean that this may actually be the case to
some extent. Or it might simply reflect a nonlinearity (such as adap
tation) in the neuronal response.
Parameterized fit
So far we have shown support for scaling and additivity using
parameterfree analyses of the data. Given this support, we went
ahead and fit the entire timedomain data sets with a parameterized
version of the linear transform model. The model had two separate
parts. The first was a parameterized contrast–response function,
and the second was a parameterization of the impulse response
function.
The form for the contrast response function was borrowed from
the singleunit electrophysiology (Albrecht and Hamilton, 1982)
and psychophysics (Legge and Foley, 1980) literature. The function
predicts the relation between stimulus contrast c, and neuronal re
sponse r as:
r c ð Þ ¼
acp
cpþ σ
ð1Þ
This function, called a hyperbolic ratio function or sometimes the
NakaRushton equation, predicts a decelerating or compressive re
sponse as a function of contrast, as is typically seen in singleunit
data from the macaque primary visual cortex (Albrecht and Geisler,
1991).
Fig. 2. Example of time–contrast separability. The shape of the response to a 12 second
stimulus does not vary across the three stimulus contrasts.
3
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Please cite this article as: Boynton, G.M., et al., Linear systems analysis of the fMRI signal, NeuroImage (2012), doi:10.1016/
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For the impulse response function (later called the hemodynamic
response function (Friston et al., 1998)) we chose the Gamma
distribution function (not to be confused with the regular Gamma
function, Γ, which is an extension of the factorial function for non
integers).
h t ð Þ ¼
1=τ
ðÞn−1
τ n−1
ð
ðÞe1=τ
Þ!
ðÞ
ð2Þ
The Gamma distribution function has two parameters, n (an in
teger) and a timeconstant, τ. Gamma distribution functions often
shows up in the context of linear systems. Perhaps the simplest
linear system is one with an exponential impulse response func
tion, which describes any selfequilibrating system that changes
over time at a rate proportional to the difference between the cur
rent state and the asymptotic state. An RC circuit is an example of
a linear system with an exponential impulse response function.
The Gamma distribution function is the impulse response function
for a cascade of n such exponential systems — that is a series of
exponential systems that feed in to the next. Gamma distribution
impulse response functions therefore show up in all sorts of
models, such as those describing fluid dynamics, heat conduction
and membrane potentials.
The first author became aware of the Gamma distribution func
tions in a linear stage of a model for the temporal properties of the
human visual system (Watson, 1986). The interested reader should
visit this chapter in the Handbook of Perception and Human Perfor
mance to see how closely our paper follows Watson's, 1986 working
model of the temporal sensitivity of the human visual system. We
essentially borrowed Watson's analysis and applied it at the scale of
seconds rather than milliseconds.
The full model predicts the fMRI response as the time course of the
stimulus convolved with the Gamma distribution function, scaled by
the contrast response function. Time–contrast separability is a simple
consequence of the two separate parameterizations for time and con
trast. The full parameterized model has a total of five free parameters:
two for contrast, two for the Gamma distribution function, and one
more describing a delay in the impulse response function. This five
parameter model was fit to the fMRI time courses for the three con
trasts and four durations using the nonlinear optimization algorithm
provided by MATLAB.
The model fit the data from the timedomain study well for both
subjects, except that it systematically underestimated the magnitude
of the fMRI response for the stimuli having shorter durations. This can
be seen in the original paper (Figs. 10 and 11) where the data points
exceed the predicted model curves for the 3second stimulus. This ap
parent nonlinearity for brief stimuli was noted above, and has since
been replicated in both the visual system (Vazquez and Noll, 1998;
Birn et al., 2001) and the auditory system (Robson et al., 1998). The
nonlinearity can be quite large; Birn et al. (2001) later found that
the fMRI response to a 250 ms visual stimuli can be 3–5 times larger
than what is expected from a longer stimulus. This has important im
plications for the interpretation and analysis of rapid eventrelated
fMRI studies that typically use brief stimulus events.
In the original paper we speculated about the possible causes of
the relatively large fMRI responses we found for short stimuli. One
possibility is a nonlinearity in the function relating neuronal activity
to hemodynamic response. But it could also be a purely neuronal ef
fect. For example, transient bursts of neuronal firing occur at the
Fig. 4. Reanalysis of the timedomain results. The amplitude parameters were allowed to vary for each of the four durations, showing how the fMRI responses to shorter stimuli are
relatively stronger than the responses to longer stimuli.
Fig. 3. Example of additivity. (A) Response to a 12 second stimulus predicted by the responses to two 6second stimuli. (B) Response to the same 12 second stimulus predicted by
the responses to four 3second stimuli.
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G.M. Boynton et al. / NeuroImage xxx (2012) xxx–xxx
Please cite this article as: Boynton, G.M., et al., Linear systems analysis of the fMRI signal, NeuroImage (2012), doi:10.1016/
j.neuroimage.2012.01.082
Page 5
onset and offset of neuronal responses (Albrecht et al., 1984; Maddess
et al., 1988; Bonds, 1991). Linear convolution with a typical hemody
namic impulse response function predicts that these transient bursts
would have a relatively large influence on brief stimuli because they
contribute relatively more to the average response over time.
In a subsequent paper, the first author teamed up with the Ban
dettini lab at NIH and tested the transient hypothesis by replicating
the original fMRI result using stimuli of durations 1, 3 and 6 s, but
also adding a gradual contrast ramp that is was intended to reduce
the strength of the onset and offset neuronal transients. A parallel
MEG study with the same subjects and stimuli showed that the
ramps did effectively reduce the transient neuronal activity. Howev
er, the nonlinearity in the BOLD signal remained (Tuan et al., 2008).
A related neuronal explanation is longterm adaptation. It is well
known that V1 neurons reduce their responses over time to long
duration stimuli (e.g. Bonds, 1991). Our flickering checkerboards,
lasting up to 24 s, are exactly the sort of stimuli used in adaptation ex
periments. In the original paper, we showed that a better fit was
obtained by incorporating a model of neuronal adaptation.
A reanalysis of our original data provides further support for this
idea. We have refit the original data with the linear transform
model, but allowed four different amplitudes (the parameter a in
Eq. (1) above) to vary for the four different stimulus durations
while the other parameters of the modified model were held fixed
to the original bestfitting parameters. Fig. 4 shows a plot of the
bestfitting amplitude parameters for each of the four stimulus dura
tions for the two subjects. Error bars represent bootstrapped esti
mates of parameter variability based on resampling using the
standard errors of the original data.
The reanalysis shows something not apparent in the original fig
ures; the bestfitting amplitude parameters decrease across the entire
range of stimulus durations. The effect seems to decay with duration
in an exponential fashion and a time constant of seconds, which is re
markably consistent with the timecourse of adaptation seen in elec
trophysiology (Bonds,1991)and
(Greenlee et al., 1991). For a similar analysis of the nonlinearity
using eventrelated designs, see (Heckman et al., 2007).
psychophysical experiments
Frequency domain study
As described in Appendix A, another way to investigate linearity is
in the frequency domain using sinusoidally modulating stimuli. Ideal
ly we would have liked to invoke a sinusoidal neuronal response, but
this was not possible without knowing the exact relation between
stimulus contrast and neuronal response. Instead we chose to use a
blockeddesign in which the stimuli turned on and off as a square
wave over time. The actual stimulus was a squarewave in space (ver
tically oriented bars of flickering checkerboards) that slowly drifted
horizontally at different rates. The timecourse of stimulation at any
given point in space therefore turned on and off at a rate related to
the speed of the drift. We chose this design over a fullfield on/off
stimulus because we were trying to minimize the effects of blood ves
sels that pool across large regions of cortex. The overall stimulation
across the visual field in our design remained roughly constant, unlike
the case for a fullfield stimulus. It is interesting to note that the ef
fects of draining veins on fMRI signals was a major concern back
then, but receives relatively little attention today — yet the problem
remains. Indeed, a recent study supports the idea that discrepancies
in the literature on retinotopic maps in visual area V4 may be due
an artifact caused by the transverse sinus that passes near V4 in
some subjects (Winawer et al., 2010).
In our study, we measured fMRI responses to drifting bars with
temporal periods of 10, 15, 30 and 45 s using five contrasts ranging
from about 3% to 100%. Our dependent measure was the amplitude
of the fMRI response at the fundamental frequency of modulation.
With a linear system, this measure should be proportional to the
on/offamplitude of theunderlying
Appendix A for an explanation). Not surprisingly, the amplitudes of
the responses decreased with decreasing temporal period (or increas
ing temporal frequency) due to the lowpass nature of the hemody
namic transformation.
More importantly, the results again showed strong support of
timecontrast separability, meaning that changing the temporal peri
od had only a scaling influence on the shape of the contrast–response
(see the original Fig. 6). Replotting the original results on a log
response axis emphasizes this point (Fig. 5). Curves that are multiple
scales of each other (as predicted by time–contrast separability)
should be parallel on a log–amplitude axis. The curves in Fig. 5 are
roughly parallel with no systematic deviations in shape across stimu
lus contrast.
neuronalresponse(See
Parameterized fit
We fit the same linear transform model to the data from the fre
quency domain experiment and found similar parameter values as
for the timedomain data. This was important because though the ex
perimental designs and dependent measures were quite different, the
same model should fit both data sets with roughly the same
parameters.
Noise
A final important result of our study related to the noise found in
fMRI data. In the linear transform model, noise is added after the con
volution stage (see Fig. 1). This makes specific predictions about the
fMRI response that can best be thought of in the frequency domain.
Fig. 5. Frequencydomain results plotted on a log–response axis to illustrate time–contrast separability.
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Truly additive and independent Gaussian noise (where an indepen
dent random number drawn from a normal distribution is added to
the response at every time point) contains sinusoids at all frequencies
with roughly equal amplitudes. If Gaussian noise is added after the
linear transform, then the Fourier transform of the output should
look like a signal (amplitudes at frequencies related to the input)
plus a flat spectrum of amplitudes across all frequencies. If, however,
noise was added before the linear transform, then the higher frequen
cy components of the noise should be attenuated by the lowpass
process of the hemodynamics.
We found that the noise spectrum of our frequencydomain study
was relatively flat (our original Fig. 9). This has important implica
tions for those using deconvolution to estimate the hemodynamic re
sponse functionfrom eventrelated
deconvolution is essentially linear regression which assumes additive
Gaussian noise. It also means filtering fMRI data with a lowpass tem
poral filter is a good idea, since the variability in the fMRI timecourse
at high temporal frequencies is likely due to stimulusindependent
noise factors.
A side notesubject gmb showed a splattering of noise around the
3–6 second temporal period, which is around the period of respira
tion (See Fig. 9B in the original publication). In retrospect, this respi
ratory artifact is probably not a hemodynamic issue but was probably
related to the physical stimulus. We projected our images from the
foot of the scanner table over the subjects' chest on to a screen just
at the subject's neck. The rising and falling of subject gmb's chest
may have periodically blocked the bottom of the image, leading to a
temporal modulation of the visual signal. Subject sae did not have
this problem.
experiments,because
Impact
Our results supported the linear transform model well enough
for us to confidently move on to quantitative studies of the
human visual system. As stated above, the results satisfied our
main concern that there might be an interaction between the
strength of the neuronal response and the shape of the hemody
namic response.
Our plans at the time were to use our knowledge of linearity to
analyze results from blockeddesign experiments. This is why
much of our paper focused on the frequencydomain experiment
and results. With linearity, we could be confident that the amplitude
of the fMRI signal in a blocked design experiment was proportional
to the amplitude of the underlying neuronal response. We then
went on and applied blockeddesigns in a variety of studies, using
the amplitude of the Fourier transform at the stimulus frequency as
our measure of neuronal response (e.g. Demb et al., 1997, 1998;
Boynton et al., 1999; Gandhi et al., 1999; Heeger et al., 1999;
Wandell et al., 1999).
What we did not anticipate was the impact that our paper would
have on the emerging method of rapid eventrelated designs (see
Huettel in this issue) in which brief stimuli are presented close
enough in time for their responses to overlap. In a typical event
related fMRI experiment, different types of events or conditions are
presented in a pseudo random order, resulting in a timevarying
fMRI signal that contains the sum of responses to individual events.
The hemodynamic response function for each event type is then
found that bestpredicts the fMRI results when convolved with the
timecourse of the stimulus (Dale, 1999). Clearly this deconvolution
process relies heavily on the assumption of linearity. Alternatively, a
canonical hemodynamic response function is assumed which, when
convolved with the stimulus timecourse, produces a predicted fMRI
timecourse that can be correlated with individual voxel time
courses. These correlations provide a statistical parameter map for lo
calizing eventrelated activity.
An anecdotal note: we actually tried to test the linear transform
model by measuring the fMRI response to a stimulus that followed
a complicated timecourse (ramping up and down in contrast, turn
ing on and off, etc.) In theory, we should have been able to use our
model and bestfitting parameters to the original data to predict the
resulting fMRI time course in V1. Unfortunately we didn't have
enough signaltonoise in our data set to provide a convincing figure
for the paper. Perhaps this is why we went on to using the blocked
design in future studies instead of pursuing similar overlapping de
signs, including eventrelated ones.
Note that brief stimulus presentations like those used in event
related fMRI appears to show the greatest violation of the linearity as
sumption. However, the reanalysis above (see Fig. 4) shows that an
other way to look at our results is that linearity fails for longer
stimulus durations (due perhaps to neuronal adaptation). The over
prediction of our original model in fits for short stimulus presenta
tions was probably simply because the model placed more weight
on the longerstimulus duration signals since they contained more
data points.
Overthepast15 yearsmuchefforthasbeenputintostudyingthebi
ologicalbasisofthehemodynamic couplingprocess.Asecondimpactof
our paper has been to provide a framework for some of these studies,
particularly for correlative studies that compare more direct measures
of neuronal activity to more direct measures of blood flow and oxygen
ation. Perhaps the most wellcited example of this is the study by
(Logothetis et al., 2001) that simultaneously measured the BOLD signal
and electrophysiological responses inthe monkey visualcortex.This ef
fortledtothesuggestionthattheBOLDsignalismorelinearlyrelatedto
localfield potentialsthanspikes,thoughbothare good predictorsofthe
BOLDsignal. A veryrecentexampleof a correlative study is (Kahn et al.,
2011)thatusedacombinationofoptogeneticstostimulatethesomato
sensory cortex in mice while measuring the BOLD signal and electro
physiological recordings.
Another recent example is a study by Sirotin and Das. They mea
sured simultaneously spiking activity and intrinsic optical signals.
They found that there was a component of the hemodynamic re
sponses that was timelocked to the stimulus presentations but not
differentially responsive to the different stimulus conditions. They
found likewise that there was such a baseline component in the spik
ing activity. The two baseline components (spiking and hemodynam
ic) were not related to one another (Sirotin and Das, Nature, 2009).
But after removing each of the two baseline response components,
the residual hemodynamic responses are very tightly linked with
spiking activity, such that the time courses of the hemodynamic re
sponses evoked by different stimulus contrasts are very well fit as a
shiftinvariant linear transform of the firing rates (A. Das, personal
communication). Temporal summation experiments, much like our
additivity ones, showed that once again, the hemodynamic coupling
process is remarkably linear.
Another unanticipated impact of our paper is that the bestfitting
parameters from our study are often used to model the hemody
namic response in a variety of studies. This is probably because cer
tain software packages, such as BrainVoyager have a model of the
hemodynamic response with parameters set to those in our paper.
Though the parameters are adjustable, many users just use the de
fault parameters. The two subjects (gmb and sae) were both au
thors, so it is amusing to note how the authors' brains contributed
to this paper in more than one way. But using these exact parame
ters is dangerous because it is well known that the hemodynamic
response function varies significantly across subjects and brain
areas e.g. (Aguirre et al., 1998). With our extremely small sample
size of two subjects, it doesn't seem wise to use our parameters
for estimates of the population. Indeed, given that the hemodynamic
response varies with age, (Kannurpatti et al., 2010) the parameters
from this paper are probably not even appropriate for our own,
now middleaged brains!
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Appendix A
In this appendix we define the properties of a linear system in the
context of the transformation between a neuronal response and the
BOLD signal.
A linear system satisfies the two properties of scaling and additiv
ity. For our model, we also assume a third property, shift invariance.
Scaling
For a system operating in the time domain, scaling means that the
timecourse of the output directly scales with the strength of the
input. For the case of a hemodynamic transform, L, we'll define the
input as a neuronal response over time, n(t), and the hemodynamic
response H(t) as the output of the system:
H t ð Þ ¼ L n t ð Þ½?ðA1Þ
For scaling to hold, if we multiply our input n(t) by a factor k,
then:
L kn t ð Þ½? ¼ kL n t ð Þ½? ¼ kH t ð ÞðA2Þ
The new output is equal to the original output multiplied by the
scale factor k.
In the case of the hemodynamic response, the slow, delayed time
course of the output is very different than the shape of the input, and
we are interested in whether a linear system relates the two. If this
system shows scaling, then when the input is scaled the shape of
the output will always remain the same except that it is scaled by
the same factor. Fig. A1 illustrates the scaling property for an example
hemodynamic response to three neuronal inputs identical in time,
but with different strengths. Shown are three neuronal responses
and the predicted hemodynamic responses over time, each scaled in
proportion to their corresponding neuronal response.
Additivity
The property of additivity states that given two inputs, the response
to the sum of the inputs is equal to the sum of the responses to each of
the inputs alone. For our system L, given two inputs n1(t) and n2(t):
L n1t ð Þ þ n2t ð Þ½? ¼ L n1t ð Þ½ ? þ L n2t ð Þ½?ðA3Þ
For the hemodynamic response, the additivity property is best il
lustrated for two neuronal responses that follow closely in time. If
they fall within a few seconds of each other, the output hemodynamic
response to the first input will still be ongoing when the
hemodynamic response to the second input begins. The additivity
principal says that this second hemodynamic response should simply
add to the first response, without interacting in any way.
Fig. A2 shows an example of how, for a linear shiftinvariant sys
tem, the hemodynamic responses to two successive 6second inputs
is identical to the sum of the responses to a 6second input added
to the same 6second input delayed by 6 s. That is, you can either
add the inputs and then measure the response, or you can measure
the responses to the individual stimuli first and then add them.
Shift invariance
The additional property of shift invariance is also assumed in the lin
ear transform model. Shift invariance means that the hemodynamic re
sponse to a later input neuronal response should look just like the
response to an earlier, identical input, but shifted in time by the delay
between the two inputs.
Convolution
The two properties of scaling and additivity combine in a very
powerful way. Consider an arbitrary neuronal timeseries n(t). We
can think of this timeseries a sequence of shifted brief impulses
δ(t), each scaled accordingly:
n t ð Þ ¼ ∑
i
kiδ t−i
ðÞðA4Þ
The response to an impulse input is called (naturally) the impulse
response function,
H t ð Þ ¼ L δ t ð Þ½?ðA5Þ
Then by the properties of additivity and scaling, the response to
the arbitrary input is:
L ∑
i
kiδ t−i
ðÞ
??
¼ ∑
i
kiL δ t−i
ðÞ½? ¼ ∑
i
kiH t−i
ðÞðA6Þ
That is, the response is the sum of shifted and scaled responses to
the impulse.
The above argument shows that the impulse response function
completely characterizes a linear system, because it can be used to pre
dict the response to any input. Of course, characterizing a timeseries
as a sum of discrete scaled impulses is only an approximation, but the
above argument holds for any size impulse, no matter how small. In
the limit, the impulse response function is defined to be the response
to an infinitely short input.
Fig. A3 shows a hemodynamic response to an arbitrary neuronal
timeseries defined as a sequence of scaled impulses. Each of these
brief impulses produces its own scaled response, and the response
to the entire time series is simply the sum of these individual scaled
responses.
This process of shifting and scaling a series of impulse responses to
predict a linear system's response to an arbitrary input is called con
volution. It is possible to go the other way and estimate the impulse
response given a linear system's input and measured output. The re
verse process is called deconvolution and is used in the analysis of
eventrelated fMRI experiments.
Frequency response
A second way to characterize a linear system is to work in the fre
quency domain, which is to think about how the system transforms
inputs that vary sinusoidally over time. Given a system that obeys
scaling and additivity, the response to a sinusoidal input is also a sinu
soid of the same frequency, though possibly shifted in phase.
Fig. A1. Example of the scaling property. Scaling the size of the input to a linear system
scales the size of the output without changing the shape of the response over time.
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Remarkably, this is only true for sinusoids (technically, it's true for all
complex exponentials which includes sinusoids), and is true for any
shaped impulse response function. Fig. A4 shows how a linear hemo
dynamic system should respond to sinusoidal inputs of varying fre
quency (temporal periods of 30, 15 and 10 s). Note how the
frequency of the output matches the frequency of the input.
According to the Fourier theorem, any arbitrary timeseries can be
described as a sum of sinusoids, each having a specific amplitude fre
quency and phase. Using an argument like the one above for convolu
tion, if we know how a linear system shifts and scales each of these
sinusoids then we can predict how the linear system will transform
any arbitrary input, since the output will simply be the sum of shifted
and scaled sinusoids that composed the input.
The way a linear system transforms a series of sinusoids is called
the frequency response function. Looking at Fig. A4, it can be seen
that the amplitude of the response is decreasing as the frequency of
the input increases (or as the period decreases). There is also a
specific phase delay for each frequency, but here we'll focus only on
amplitudes. This illustrates the typical lowpass property of the hemo
dynamic response; the higher the frequency of the input, the lower
the relative amplitude of the output of the system.
It is not hard to show that there is a direct relationship between
the impulse response function for a linear system and the corre
sponding frequency response function. This is the basis of the convo
lution theorem which states that convolution in the time domain is the
same is multiplying in the frequency domain. That is, the Fourier
transform of the output of a linear system is equal to the product of
the Fourier transforms of the input and the impulse response
function.
Fig. A5 illustrates the Fourier theorem for a blockeddesign exper
iment. The input to the system is an on/off blocked design with a
30 second period (period=1/frequency), plus some additive neuro
nal noise. It is assumed in this simulation that the input neuronal re
sponses are turned on and off following a square wave. The top row
shows the linear system operating in the time domain by convolving
the input neuronal response with the impulse response function. The
output is a smoothly varying, roughly sinusoidal modulation. Now
consider the Fourier transform of the neuronal input (in the bottom
left of Fig. A5). It shows that to build a squarewave signal, we need
to add discrete sinusoidal components of increasing frequency and
decreasing amplitude. If our temporal period is P seconds, then the
periods we need are P/n for n=1, 3, 5, 7,… and the corresponding
amplitudes are 4π/n. In our example of a square wave with a 30 sec
ond period, we need to add sinusoids of decreasing amplitudes start
ing with a sinusoid with a period of 30 s (called the fundamental
period), followed by sinusoids having periods of 30/3=10, 30/
5=6, 30/7=4.29, 30/9=3.33 s and so on.
We can think of a linear system as working on each of these
sinusoidal components independently. To see how the hemody
namics affect the input, we can look at the Fourier transform of
the impulse response function, shown in the middle of the bottom
row in Fig. A5. This graph shows how sinusoids are scaled by each
input frequency, and specifically, how the amplitude of the output
decreases with increasing frequency (decreasing temporal period).
To predict the output of the system to our squarewave input, we
simply multiply each of the amplitudes in the stimulus by their
corresponding value in the frequency response (multiplication is
actually done with complex numbers which takes into account
both amplitude and phase).
The Fourier transform of the resulting output is shown in the bot
tom right of Fig. A5. The already decreasing amplitudes in the input
decrease more rapidly in the output due to the lowpass nature of
the impulse response function. The higher frequencies in the input
serve to “squareoff” the fundamental 30second period sinusoid. At
tenuating these higher frequencies by the hemodynamic linear sys
tem causes the output to look less square and more sinusoidal. For
this example, the fMRI response to a 30 second period input has
only a small 10second period component and negligible higher fre
quency components. Nearly all of the response is at the fundamental
Fig. A2. Example of the additivity property. The response to the sum of two inputs is
equal to the sum of the responses to the inputs alone.
Fig. A3. Combining scaling and additivity. The response to any arbitrary stimulus can be predicted by the summed and shifted responses to an impulse input.
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frequency. This means that the analysis of results from a blocked
design experiment using a 30second period can focus on just the am
plitude of the 30second period Fourier component without throwing
out much information in the signal.
Note also that the property of scaling applies in the frequency
domain as well. Doubling the amplitude of a sinusoidal input dou
bles the amplitude of the sinusoidal output. In the early days of
fMRI, the majority of studies used a standard blockeddesign
paradigm in which two stimulus conditions alternated back and
forth throughout a scan at an even period. Using an on/off period
of 30 s or so, the resulting fMRI signal should modulate in a
roughly sinusoidal manner. With scaling, the amplitude of the si
nusoidal fMRI signal can serve as a measure of neuronal activity
(e.g. Boynton et al., 1999; Gandhi et al., 1999; Heeger et al.,
1999). That is, given a linear hemodynamic transform, the amplitude
of the sinusoidal output should be proportional to the amplitude of
Fig. A4. The response of a linear system to a sinusoidal input is equal to a sinusoid of the same frequency, only scaled in amplitude and delayed in time.
Fig. A5. The convolution theorem: For a linear system, convolution in the time domain is equivalent to multiplication in the frequency domain.
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the underlying neuronal response. A common application of the
blocked design is for phaseencoded retinotopic mapping experi
ments (See Engel, this issue). The analysis of these experiments
only relies on the response at the fundamental frequency of the
stimulus. In analyzing these experiments, a sinusoidal model of the
BOLD signal was sufficient due to the lowpass nature of the hemo
dynamic impulse response function (Engel et al., 1994).
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