Signal Processing for Functional Brain Imaging: General ...
Transcript of Signal Processing for Functional Brain Imaging: General ...
Signal Processing for Functional Brain Imaging:General Linear Model
Melissa SaenzAuditory Neuroscience Group, CHUV/[email protected]
Check the website (by this evening)
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n Exercise Set 1 (run GLM in MATLAB)- reinforce lectures, not graded but good practice for final exam- solutions posted the Tuesday night before the next class- question period at beginning of the next class
n Review - matrix algebra
n Class Slides
n Journal club dates: 28.03, 11.04, 18.04, 02.05, 16.05, 30.05- assignments given next week
Overviewn Basics: the fMRI BOLD signaln GLM method (part 1, today 28.02.13)
n intuitive explanationn matrix algebra explanation
n model generationn parameter estimationn hypothesis testing
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n hypothesis testing continuedn t-test, f-test, contrasts
n enriching the modeln accounting for imaging artifacts, physiological noise
n from single-subject to group-level analysis
n GLM method (part 2, next week 07.03.13 )
MRI versus functional MRI (fMRI)
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MRI fMRI
n Series of 3D volumesn 3x3x3 mmn every 2-4 sec n during 5-10 minutes
high resolution(1 mm)
low resolution(~3 mm)
…n Single 3D volume
n 1x1x1 mmn takes couple of minutes
The idea behind fMRI is simple: we measure a series of MRI images over time (i.e. a movie) and we test for small changes in signal intensity over time that are related to brain activity.
MRI versus functional MRI (fMRI)
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fMRI
…
The idea behind fMRI is simple: we measure a series of MRI images over time (i.e. a movie) and we test for small changes in signal intensity over time that are related to neural activity.
n Series of 3D volumesn 3x3x3 mmn every 2-4 sec n during 5-10 minutes
voxel (in this case, 3x3x3 mm)
voxel timecourseor timeseries
some vocabulary:
What is the BOLD signal?
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n Blood Oxygenation Level Dependent (BOLD) signal - measure of the ratio of oxygenated to deoxygenated hemoglobin
n Hemoglobin carries oxygen in the blood (6 binding sites per molecule)
n Oxygenation vs. deoxygenated hemoglobin have different magnetic properties. Higher ratio of oxy vs. deoxy yields higher local MRI signal
n Following neural activity, metabolic demand rises, blood flow locally increases bringing oxygenated hemoglobin --> the BOLD signal rises
Red blood cells carry hemoglobin
What is the BOLD signal?
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n Blood Oxygenation Level Dependent (BOLD) signal - measure of the ratio of oxygenated to deoxygenated hemoglobin
BOLD impulse response following brief neural eventn Hemoglobin carries oxygen in the blood (6 binding sites per molecule)
n Oxygenation vs. deoxygenated hemoglobin have different magnetic properties. Higher ratio of oxy vs. deoxy yields higher local MRI signal
n Following neural activity, metabolic demand rises, blood flow locally increases bringing oxygenated hemoglobin --> the BOLD signal rises
What is the BOLD signal?
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n Blood Oxygenation Level Dependent (BOLD) signal - measure of the ratio of oxygenated to deoxygenated hemoglobin
…
BOLD response: voxel time course
BOLD is an indirect measure of neural activity. Not perfect, but allows non-invasive, repeatable measures of human brain activity at a reasonable spatial resolution.
n Hemoglobin carries oxygen in the blood (6 binding sites per molecule)
n Oxygenation vs. deoxygenated hemoglobin have different magnetic properties. Higher ratio of oxy vs. deoxy yields higher local MRI signal
n Following neural activity, metabolic demand rises, blood flow locally increases bringing oxygenated hemoglobin --> the BOLD signal rises
Hemodynamic Response Function (HRF)
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n the BOLD impulse response function
Boynton et al., 1996
Canonical hrf model (gamma function)Empirical hemodynamic responses to brief events
Lindquist et al., 2008
time to peak4–6 sec
return to baseline~20 seconset time
~2 sec
n Most analyses assume the HRF constant across brain areas and individuals,n for now let’s assume that’s a valid assumption (although some differences do exist!)n often modeled by a gamma function (or difference between two gamma functions)
Time (seconds)
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n Aim of the gamen FMRI brain map
n Task positive and task negative voxels
n Contrasting conditions n High-dimensional data
n 100’000 intracranial voxels
n 100-1’000 timepointsn 10-100 trialsn 10-100 subjects
GLM
General linear modeln The most commonly used tool in fMRI data analysis
n hypothesize the BOLD response for a given task (predictors or regressors)
n look for evidence in the measure data (confirmatory analysis)
n Parametric model-based hypothesis testingn model-based vs. blindn univariate vs. multivariate
n Applies to data a single voxel at a time. Thus if we measure the whole brain, the analysis is repeated ~ 100,000 times (mass-univariate).
n parametric vs. non-parametricn assume null distribution, estimate its parameters
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the GLM equation
n Three conditions
Tutorial fMRI experiment
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Time
faces objects
+
Zzz
baseline
Is there a face-selective brain region?
[inspired from SPM course on GLM]
Tutorial fMRI experiment
Time
facesobjectsbaseline
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Is there a face-selective brain region?
n For a given voxel (time-series), we try to figure out just what type that is by “modelling” it as a linear combination of the hypothetical time-series
Model: a set of hypothetical time-series
measured
≈
“known”regressors
β1 · + β2 · + β3 ·
unknownparameters 14
faces objects baseline
n For a given voxel (time-series), we try to figure out just what type that is by “modelling” it as a linear combination of the hypothetical time-series
Estimation: find “best” parameters
≈ β1 · + β2 · + β3 ·
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2 3 4 0 1 2 0 1 2 0 1 2
faces objects baseline
n For a given voxel (time-series), we try to figure out just what type that is by “modelling” it as a linear combination of the hypothetical time-series
Estimation: find “best” parameters
≈ β1 · + β2 · + β3 ·
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2 3 4 0 1 2 0 1 2 0 1 2
Not brilliant! 0 0 3
faces objects baseline
n For a given voxel (time-series), we try to figure out just what type that is by “modelling” it as a linear combination of the hypothetical time-series
Estimation: find “best” parameters
≈ β1 · + β2 · + β3 ·
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2 3 4 0 1 2 0 1 2 0 1 2
Neither this 1 0 4
faces objects baseline
n For a given voxel (time-series), we try to figure out just what type that is by “modelling” it as a linear combination of the hypothetical time-series
Estimation: find “best” parameters
≈ β1 · + β2 · + β3 ·
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2 3 4 0 1 2 0 1 2 0 1 2
Nice Fit! 0.83 0.16 2.98
faces objects baseline
n And the same model can be fitted to each time-series, with different parameter values of course
Estimation: find “best” parameters
≈ β1 · + β2 · + β3 ·
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1 2 3 0 1 2 0 1 2 0 1 2
another voxel time course with more weight given to the
second predictor
0.68 0.82 2.17
faces objects baseline
n And the same model can be fitted to each time-series, with different parameter values of course
Estimation: find “best” parameters
≈ β1 · + β2 · + β3 ·
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1 2 3 0 1 2 0 1 2 0 1 2
Doesn’t care 0.03 0.06 2.04
faces objects baseline
Time
��� =
�
⇧⇤0.830.162.98
⇥
⌃⌅
beta_0001.img
beta_0002.img
beta_0003.img
n Same model for all voxels, different parameter values
Estimation: putting things together
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��� =
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⇧⇤0.030.062.04
⇥
⌃⌅
��� =
�
⇧⇤0.680.822.17
⇥
⌃⌅
n What is a “best” fit?
Parameter estimation
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datasome fit
residual error
��� =
�
⇧⇤00
3.31
⇥
⌃⌅ ei�
i ei = 0�
i e2i = 17.16
databest fit
��� =
�
⇧⇤0.830.162.98
⇥
⌃⌅�
i ei = 0�
i e2i = 9.47
residual error
Parameter estimation aims to reduce the sum of the squared errors (ordinary least squares method)
n General linear model (GLM): the “design matrix” fashion
Model revisited
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≈
�
⇧⇤�1
�2
�3
⇥
⌃⌅ �1 + �2 + �3 �1 + �2 + �3
or, in matrix notation: y � X���
FMRI statistical analysis
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Voxel-wise General Linear Model (GLM)
y = X� + e
y : N � 1 — measured timecourse
X : N � L — columns contain L regressors
� : L� 1 — unknown parameters
e : N � 1 — estimation error
Model specification: setup “design matrix” X
Parameter estimation: find “best” �
Hypothesis testing and inference
Model specificationn From stimuli to modeled BOLD responsen Convolve stimulus time course with HRF
n blocks (epochs)
n events
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⊗ =
⊗ =
Parameter Estimation
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Equivalent to solving for using simple matrix algebra:
n For X to be invertible, the columns of X must have full rank; no column is a linear combination of any other column
n If X is rank deficient, there is not a unique solution for the parameter estimates.
MATLAB code to implement eq. 3 is simply:beta=inv(X’ * X) * X’ * y
Assumes that exists!(eq. 1)
(eq. 2)
(eq. 3)
(eq. 1)
Parameter Estimationn Let’s make some assumptions about the noise ( )
n is normally distributed , such that mean = 0, variance = n elements of the vector are uncorrelated n typically written as
n columns of X are independent of e
n Then we meet the conditions of the Gauss-Markov theorem and we are guaranteed that the OLS estimator
produces a minimum variance linear unbiased estimate (MVUE) of
[Worsley et al., 1996]
Nice! MVUE is the gold standard in the parameterestimation world!
will use forhypothesis testing
Contrasts
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- contrast to extract the difference between the firsttwo parameters for hypothesis testing
- contrast to extract the first parameter for hypothesis testing
0.67
Hypothesis test of the contrast
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Standard t-statistic:t=extracted parameter value/sqrt(variance)
Consider contrast cT� = 0 (null hypothesis)
we know � is asymptotically normal N (�,�2(XTX)�1)
then cT � is asymptotically normal N (cT�,�2cT (XTX)�1c)
test statistic (effect size / uncertainty of effect size)
t =cT �
�qcT (XTX)�1c
,
where we estimate (reminder: RTR = R2 = R and e = Ry)
�2 =eT e
tr(R), since E[eT e] = E[eTRTRe] = �2
tr(R)
and t follows Student t-distribution with tr(R) = rank(R) = N �L degrees of
freedom
var( )
The full picture
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get data
model
get error
get effect size
��� =
�
⇧⇤0.830.162.98
⇥
⌃⌅
t = = 6.42
contrast
cT = [1 0 0]
Overviewn Basics: the fMRI BOLD responsen GLM method (part 1, today 28.02.13)
n intuitive explanationn matrix algebra explanation
n model generationn parameter estimationn hypothesis testing
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n hypothesis testing continuedn t-test, f-test, contrasts
n enriching the modeln accounting for imaging artifacts, physiological noise
n from single-subject to group-level analysis
n GLM method (part 2, next week 07.03.13 )
Generality of GLM
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From Wiki page