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


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BOLD impulse response following brief neural eventn Hemoglobin carries oxygen in the blood (6 binding sites per molecule)




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…

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)



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


Estimation: find “best” parameters

≈ β1 · + β2 · + β3 ·

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2 3 4 0 1 2 0 1 2 0 1 2




≈ β1 · + β2 · + β3 ·

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2 3 4 0 1 2 0 1 2 0 1 2

Not brilliant! 0 0 3




≈ β1 · + β2 · + β3 ·

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2 3 4 0 1 2 0 1 2 0 1 2

Neither this 1 0 4




≈ 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


n And the same model can be fitted to each time-series, with different parameter values of course


≈ 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


n And the same model can be fitted to each time-series, with different parameter values of course


≈ 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


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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�� =

�

⇧⇤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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signal processing for functional brain imaging: general

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