Statistical Methods in
functional MRI
Lecture 3:
Signal and Noise in fMRI
04/09/13
Martin Lindquist
Department of Biostatistics
Johns Hopkins University
MRI
• MRI studies brain anatomy.
– Structural (T1) images
– High spatial resolution
– Can distinguish different types of tissue
Functional MRI
• An fMRI experiment consists of a sequence of
individual MR images, where one can study
oxygenation changes in the brain across time.
fMRI
• fMRI studies brain function.
–
–
–
–
Functional (T2*) images
Lower spatial resolution
Higher temporal resolution
Can relate changes in signal to an experimental manipulation
BOLD fMRI
• The most common approach towards fMRI uses the
Blood Oxygenation Level Dependent (BOLD)
contrast.
• BOLD fMRI allows us to measure the ratio of
oxygenated to deoxygenated hemoglobin in the
blood.
• In this lecture we will investigate the signal and noise
present in fMRI data.
• BOLD fMRI doesn’t measure neuronal activity
directly, instead it measures the metabolic demands
(oxygen consumption) of active neurons.
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BOLD Contrast
• Hemoglobin exists in two different states each of
which has different magnetic properties and produces
different local magnetic fields. (Pauling 1936)
The chart shows
physiological and
physical changes with
brain activation.
– Oxyhemoglobin (diamagnetic)
– Deoxyhemoglobin (paramagnetic)
• BOLD imaging takes advantage of the difference in
T2* between oxygenated and deoxygenated
hemoglobin.
Positive/negative
arrows indicate
positive/negative
correlations.
• Deoxyhemoglobin has the effect of suppressing the
MR signal. As the concentration of deoxy decreases
the signal increases.
BOLD Signal
BOLD Signal
• The change in the MR signal triggered by
instantaneous neuronal activity is known as the
hemodynamic response function.
• Initial increases in the concentration of
deoxyhemoglobin can lead to a decrease in
BOLD signal (“initial dip”; controversial).
• As neural activity increases, so does metabolic
demand for oxygen and nutrients.
• This is followed by an increase in signal, due to
an over-compensation in blood flow which dilutes
the concentration of deoxyhemoglobin and tips
the balance towards oxyhemoglobin.
• As oxygen is extracted from the blood, the
hemoglobin becomes paramagnetic which
creates distortions in the magnet field that cause
a T2* decrease (i.e. a faster decay of the signal).
• This increase in BOLD signal peaks about 4-6 s
following activation.
BOLD Signal
HRF
• After reaching its peak, the BOLD signal
decreases to an amplitude that lies below the
baseline level.
Positive rise
• This poststimulus undershoot is due to a
combination of reduced blood flow and increased
blood volume.
Initial dip
The strongest signal appears 5-6 seconds after activation.
2
HRF Properties
Temporal Resolution
• Decreasing the TR allows for a better estimate of the
HRF.
• Magnitude of signal changes is quite small
– 0.1 to 5%
– Hard to see in individual images
• Response is delayed and quite slow
– Extracting temporal information is tricky, but possible
– Even short events have a rather long response
• Exact shape of the response has been shown to
vary across subjects and regions.
From Huettel
Timing of Activity
Timing of Activity
• The hemodynamic response function is known to
differ across the brain and across subjects.
• It is possible to measure latency differences between
brain regions.
• Within a single brain region, vascular properties will
be similar across conditions, allowing accurate
estimation of timing differences.
• However, these differences may reflect changes in
vascular response and not directly reflect differences
in neuronal response.
• However, measuring sequential neural activity across
the brain is difficult due to differences in the
hemodynamic response function.
• To make inference about the relative timing,
researchers must find a way to selectively manipulate
one neuronal process while holding another constant.
Illustration
Interpretation
• How well does BOLD signal reflect increases in
neural firing?
Subjects moved a target
square across a display with a
joystick when it changed color.
• The BOLD signal corresponds relatively closely to the
local electrical field potential surrounding a group of
cells—which is itself likely to reflect changes in postsynaptic activity—under many conditions.
The relative latency between
BOLD activity in V1 and in
SMA increased roughly
linearly with reaction time,
while the latency between
SMA and M1 did not.
• Demonstrations have shown that high-field BOLD
activity closely tracks the position of neural firing and
local field potentials.
The variation in RT occurs
between V1 and motor
planning areas.
Menon at al. 1998
3
BOLD Response
• The evoked BOLD response in fMRI is a
complex, nonlinear function of the results of
neuronal and vascular changes.
• The shape of the response depends both on the
applied stimulus and the hemodynamic response
to neuronal events.
• There exist a number of methods for modeling
the BOLD response and the underlying HRF.
Logothetis 2002
Models
Balloon Model
• Nonlinear physiological-based models
– Consists of a set of ordinary differential equations that model
changes in blood volume, blood inflow, deoxyhemoglobin
and flow inducing signal and describe how these changes
impact the observed BOLD response.
• Linear time invariant (LTI) system
– Assumes that the neuronal activity (based on task
manipulations) constitutes the input, or impulse, and the
HRF is the impulse response function.
Balloon Model
• While the balloon model is more biophysically
plausible, it has a number of drawbacks.
– Requires the estimation of a large number of model
parameters.
– Does not always provide reliable estimates with noisy data.
– Doesn’t provide a direct framework for performing inference.
– Is not considered a feasible alternative for performing wholebrain multi-subject analysis of fMRI data.
• LTI models are more commonly used in practice.
LTI System
• The relationship between stimuli and the BOLD
response is often modeled using a linear time
invariant (LTI) system.
– Here the neuronal activity acts as the input or impulse
and the HRF acts as the impulse response function.
• In this framework the signal at time t, x(t), is
modeled as the convolution of a stimulus function
v(t) and the hemodynamic response h(t), that is,
x(t ) (v h)(t )
4
Convolution
Examples
Block Design
Event-Related
Experimental
Stimulus Function
Properties
Scaling – if the input is scaled by a factor b then
the BOLD response will also be scaled by b.
Superposition – the response to two different
stimuli applied together is equal to the sum of the
individual responses.
Hemodynamic
Response
Function
Time-invariance – if a stimulus is shifted by a time
t, then the response is shifted by t.
Predicted
Response
Properties
• The scaling principle is important as it implies
that the amplitude of the measured signal
corresponds to the amplitude of neuronal activity.
• The relative difference in amplitude in the signal
between two conditions can be used to infer that
the neuronal activity was similarly different.
• The superposition principle allows us to
differentiate between the response in various
brain regions to multiple closely spaced stimuli.
Non-linearity
• Studies have shown that the BOLD response is
roughly linear, with some departures from
linearity.
Comments
• A major shortfall when analyzing fMRI data is that
users typically assume a canonical HRF, which
leaves open the possibility for mismodeling the signal
in large portions of the brain.
• There is some evidence of refractory effects,
which are reductions in amplitude of a response
as a function of inter-stimulus intervals.
• There has therefore been a movement toward both
using more sophisticated models and enhanced
model diagnostics.
• There is evidence of non-linearity if the stimulus
are spaced closer than 5-6 s apart.
• Both of these areas fall squarely in the purview of
statisticians.
5
fMRI Noise
• The measured fMRI signal is corrupted by
random noise and various nuisance components
that arise due to hardware reasons and the
subjects themselves.
• Sources of noise:
‒ Thermal motion of free electrons in the system.
‒ Patient movement during the experiment.
‒ Physiological effects, such as the subject’s heartbeat
and respiration.
‒ Low frequency signal drift.
Handwerker 2004
fMRI Noise
fMRI Noise
• Some of these noise components can be
removed prior to analysis, while others can be
included as components in subsequent models.
• However, it is difficult to remove all sources of
noise and therefore significant autocorrelation will
be present in the signal.
• Characteristics:
– “1/f” in frequency domain
– Nearby time-points exhibit positive correlation
Excess Low
Freq.
variability
Power
Spectrum
• Scanner instabilities and not motion or
physiological noise may be the main cause of
the drift, as drift has been seen in cadavers.
• We need to include drift parameters in our
future models.
Autocorrelation
Function (ACF)
Issues
Drift
• Slow changes in voxel intensity over time
(low-frequency noise) is present in the fMRI
signal.
Nearby
time-points
positively
correlated
• Drift can have serious consequences:
– Experimental conditions that vary slowly may be confused
with drift.
– Experimental designs should use high frequencies (more
rapid alternations of stimulus on/off states).
• Bad Design:
2 min. rest
2 min. active
Typical Drift
Pattern & Magnitude
Typical Signal
Magnitude
... you’ll never detect this signal, due to the drift
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Motion
Physiological Noise
• Subject motion during the experiment can
also give rise to serious problems.
• Respiration and heart beat give rise to highfrequency noise.
• Typically motion correction is performed in
the pre-processing stages of the analysis.
• It can potentially be modeled, but if the TR is
too low there will be problems with aliasing.
• However, ‘spin-history’ artifacts may remain
that cannot be removed.
• Hence, this type of noise is difficult to
remove and is often left in the data giving
rise to temporal autocorrelations.
- This is caused by through-plane motion.
- Attempts to account for it are often made in the
modeling stage.
Nyquist Criteria
Time Series Analysis
• It is necessary to use a sampling rate at least
twice as big as the frequency of the curve
you seek to model.
•
A time series is an ordered sequence of
observations.
-
Interested in discrete equally spaced time series.
• Let {Xt} be a sequence of random variables with
E ( X t2 ) < ¥
Stationary Time Series
•
Autocorrelation
The mean function:
m X (t) = E(Xt )
•
Assume {Xt} is a stationary time series.
•
The autocovariance function (ACVF) lag h:
g X (h) = Cov(Xt+h, Xt )
The covariance function:
X
( r, s )
Cov( X r , X s )
E (Xr
•
•
X ( r ))( X s
•
X ( s ))
" r, s
The autocorrelation function (ACF) at lag h:
rX (h) = g X (h)
g X (0)
A time series is weakly stationary if the mean
and covariance function do not vary across time.
7
White Noise
AR Process
•
• A sequence of uncorrelated random
variables, each with mean 0 and variance
2.
{Xt} is a autoregressive process of order p,
AR(p), if
Xt
• We write {Zt}~ WN(0, 2).
1
Xt
1
2
Xt
2

where {Zt} ~ WN(0, 2) and
•
p
Xt
1 , p
Zt
p
are constants.
A new error (“innovation”, Z) added at each
time point; total signal (X) is propagated in time.
Gaussian White Noise
Ex. AR(1)
Xt
Xt
Zt
1
MA Process
0, 1, 
t
•
The ACF for an AR(1) process:
ì
ï
1,
r X (h) = í
|h|
ï j
î
Xt
if h = 0,
if h ¹ 0
q

1Z t 1
q Zt q
1 , p
are constants.
A new error (“innovation”, Z) added at each
time point; prior innovations (Z) are propagated
in time.
-1.0
-1.0
-0.5
0.0
0.5
0.5
q
•
1.0
1.0
=-0.7
0.0
Zt
where {Zt} ~ WN(0, 2) and
=0.7
-0.5
{Xt} is a moving-average process of order q,
MA(q), if
5
10
15
5
10
1:16
15
1:16
ARMA Process
Ex. MA(1)
Xt
Zt
Zt
0, 1, 
t
1
• {Xt} is a autoregressive moving-average
process, ARMA(p,q), if
The ACF for an MA(1) process:
ì 1,
if h = 0
ï
r X (h) = í q / (1+ q 2 ), if h = ±1
ï
if |h |> 2
î 0,
Xt
2
1Zt
Xt
2
1

1.0

p
Xt
q Zt
1 , p
p
q
,
1 , p
q
-0.5
-1.0
-1.0
-0.5
0.0
0.0
0.5
1.0
1
where {Zt} ~ WN(0, 2) and
are constants.
=-0.8
0.5
Xt
Zt
=0.8
q
1
2
4
6
1:11
8
10
2
4
6
8
10
1:11
8
Comments
Ex. ARMA(1,1)
Xt
• Different software packages use different models to
handle fMRI noise.
Xt
Zt
1
Zt
– White noise, AR or ARMA noise models.
1
The ACF for an ARMA(1,1) process is given by
1
)2
(
X
(
( h)
h 1
X
2
h
2
1
1
(1)
)2
2
0
|h| 1
2
• White noise is the easiest to work with, but is not
particularly realistic.
• The parameters of the AR model are easier to
estimate than those for ARMA parameters.
– Method of Moments vs. Maximum likelihood
|h| 1
Noise Models
• Autoregressive process
– AR(1): i = i-1 + i
• Software: fmristat, SPM99, SPM2, SPM5
Spatio-temporal Behavior
• The spatiotemporal behavior of these noise
processes is complex.
• AR + White Noise or ARMA(1,1)
– AR plus an independent WN series
• Software: SPM2b
• Arbitrary autocorrelation function
–
= corr( i, i-k )
• Software: FSL’s FEAT
k
Spatial maps of the model parameters from an AR(2)
model estimated for each voxel’s noise data.
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