first-level fmri modelingmumford.fmripower.org/2010_summercourse/lev1_2010.pdf · 2010-07-19 ·...

First-level fMRI modeling

UCLA Advanced NeuroImagingSummer School, 2010

Goal in fMRI analysisTask on

Find voxels with BOLD time series that look like this

Voxelwithsignal

Voxelnosignal

Voxelsignaland drift

Delay of BOLD response

Voxelwithsignal

Voxelnosignal


Voxelwithsignal

Voxelnosignal


Starts off high

BOLD issues

• BOLD response is delayed– Convolution– FIR modeling

• BOLD time series suffer from low frequency noise– Highpass filtering– Prewhitening– Precoloring

• Scaling the data– Grand mean scaling– Intensity normalization

Recall the GLM

Single voxel timeseries

How to make a good model

• Explain as much variability in the dataas possible– If you miss something it will go into the

residual error, e

Big residuals ⇒Big variance⇒Small t stat

Understanding the data

• Time series drifts down in beginning• BOLD response is delayed

Simplest Model

=

Simplest Model

Modeling the delay

• Hemodynamic response function– Real data was used to find good models for

the hemodynamic response

Stimulus

HRF

(double gamma)

Convolution

• Combine HRF and expected neuralresponse

Typically model derivative of convolved HRF to adjust for smalldifferences in onset (<1s)

Different HRF’s

Too symmetric

Basic shape okay,but no post stimulusundershoot

Includes poststimulusundershoot

Assumptions of canonicalHRF

• BOLD increases linearly

Dale & Buckner, 1997

Assumptions of canonicalHRF

• The width, heightand delay arecorrect

• Lindquist & Wager(2007)– From what I’ve

seen only lookslike it would workwith 1 task

Finite impulse response model• FIR

– Make no assumption about the shape ofthe HRF

Constrained basis set

• Lower the number of regressors in themodel by using a basis set

• Constrained to shapes that arereasonable for HRF shapes

Constrained basis set

Basis set HRF possibilities

FLOBS

• fMRIB Linear OptimalBasis Sets– Generates a set of

basis sets to modelsignal

– Specify ranges fordifferent portions of thehrf

Comparison

More thoughts aboutcanonical HRF

• Advantages:– Simpler analysis– Easily interpretable outcome– Simplifies group analysis

• Disadvantages– Biased if canonical HRF is incorrect

Unbiased basis sets

• Advantages– Not biased towards a particular shape– Allows testing of hypotheses about specific

HRF parameters• Disadvantages

– Less powerful– Makes group analysis more difficult– Tend to overfit the data (i.e., fit noise)

We can make a design matrix!

• Start with task blocks or delta functions• Convolve• Estimate the GLM and carry out

hypothesis!

Convolved Boxcar

The Noise

• White noise– All frequencies have similar power– Not a problem for OLS

More Noise• Colored noise

– Has structure– OLS needs help!

What about the drift?

• Sources– Head motion– Cardiac noise– Respiratory noise– Scanner noise

What the noise looks like

Power spectra of noise data(Zarahn, Aguirre, D’Esposito, NI, 1997)

1/fstructure

More noise

Average spectrum of principal components(Mitra & Pesaran, Biophysical Journal, 1999)

More noise


Low frequencynoise

More noise


Breathing

More noise


Cardiac

The 1-2 punch

• Punch 1: Highpass filtering– FSL uses gaussian weighted running line smoother– SPM fits a DCT basis set

• Punch 2: Prewhitening– We’ll get to that later

• There’s also a thing called lowpass filtering(precoloring), but generally it isn’t so great andnobody uses it

Highpass filtering

• Simply hack off the low frequency noise– SPM: Adds a discrete cosine transform

basis set to design matrix

Highpass filtering• FSL: Gaussian-weighted running line

smoother– Step 1: Fit a Gaussian weighted running

line

Highpass filtering• FSL: Gaussian-weighted running line

smoother– Step 1: Fit a Gaussian weighted running

line

Fit at time t is a weightedaverage of data around t

Highpass filtering

– Step 2: Subtract Gaussian weightedrunning line fit

– IMPORTANT: Must apply filter to both thedata and the design.

• FSL has ‘apply temporal filter’ box in designsetup. Leave it checked!

Highpass filtered design (FSL)

If it wasn’tfiltered, thistrend wouldn’t behere.

Highpass filtering

Filter below .01 Hz

Filter cutoff

• High, but not higher than paradigmfrequency– Look at power spectrum of your design and

base cutoff on that– Block design: Longer than 1 task

cycle…usually twice the task cycle– Event related design: Larger than 66 s

(based on the power spectrum of acanonical HRF of a single response)

High-pass Filtering• Removes the worst of the low frequency trends

High-pass

From S. Smith

Highpass filtering

• What does it do to the signal??

Signal Powerspectrum

Lowpassfilter

Highpassfilter

Woolrich et al, NI 2001

Highpass filtering

• What does it do to the signal??

Signal Powerspectrum

Lowpassfilter

Highpassfilter

Filtering conclusions

• Lowpass filtering– Idea is to swamp out high frequency noise– Easily removes important signal in ER designs– Choose cutoff to remove noise, but avoid your signal

• Highpass filtering– Removes low frequency drift– We typically avoid designs with low frequencies, so highpass

filtering is always used

Bandpass filtering

• High and lowpass filtering– Common in functional connectivity analysis

since it allows you to focus on a specificfrequency

– Not typically used in standard fMRIanalyses

Model with HP filter

=

Parameter ofinterest

Model with HP filter

=

Parameter ofinterest

Use contrast

c=(1 0 0 0 0 0 0 0 )

Convolution & HP filter

Punch 2: Prewhitening

• Highpass filtering– Analogous to using a roller to paint a

wall…you can’t get the edges very neatly• Prewhitening

– More precise estimate of correlation…likeusing a brush for the edges

Prewhitening

• Remember Gauss Markov?– If our errors are distributed with mean 0,

constant variance and not temporallyautocorrelated then our estimates areunbiased and have the smallest variance ofall unbiased estimators.

– Uh oh,

(even after highpass filtering)

Prewhitening

• Find K such that

• Premultiply GLM by K

Prewhitening

• Find K such that

• Premultiply GLM by KAwesome! G-Mholds for our*new* model

Whitening

• OLS can be used on whitened model–

–

Prewhitening

• Step 1: Fit the linear regressionignoring temporal autocorrelation

• Step 2: Use residuals from firstregression to estimate temporalautocorrelation to obtain K

• Step 3: Create prewhitened model andestimate in usual way

Estimating V

• We don’t know V, so we estimate it

• There’s a bias problem….– SPM uses a global covariance estimate to

help with this– FSL uses a local estimate, but smoothes it

fMRI noise

• Tends to follow 1/ftrend

• Autoregressive (AR)models fit it well–

Whitening FSL

• Estimates V locally• Step 1: Estimate raw autocorrelations

Whitening FSL


Lag 1

[e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12]

[e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12]

Take products and average

Whitening FSL


Lag 2

[e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12]

[e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12]


Whitening FSL


Lag 7

[e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12][e1 e2 e3 e4 e5 e6 e7…]


Whitening FSL

• Estimates V locally• Step 1: Estimate raw autocorrelations• Step 2: Smooth spatially• Step 3: Within voxel, smooth

correlation estimates (Tukey taper)– Correlation estimates at high lags aren’t

estimated well, so they are down-weighted

Whitening FSL

Woolrich et al., 2001, NI

Time Domain Spectral Domain

Raw estimate

Whitening FSL

Woolrich et al., 2001, NI

Time Domain Spectral Domain

Tukey Taper

Whitening SPM

• Globally estimates correlation– Correlation of time series averaged over

voxels• Structured correlation estimate

– Scaled AR(1) with correlation 0.2 pluswhite noise

Whitening SPM

Only 2 parameters are estimated

Comparisons

• Bandpass (solid), Prewhitening withoutbias correction (dot-dash), highpass(dashed)

PW

HP

BP

Friston, NI 2000

PW

HP

BP

Summary of filters

• Although bandpass (and lowpass) hasthe best looking bias, it is less efficient

• Highpass filtering doesn’t remove all ofthe structured noise

• Prewhitening with bias correction withhighpass filtering is the traditionalcombination

Convolution, HP filter, Whitening

Scaling• Grand Mean Scaling

– Removes intersession variance in globalsignal due to changes in gain of scanneramplification

– Allows us to combine data across subjects– Whole 4D data set is scaled by a single

number– Automatically done in software packages– Doesn’t change variability between time

points

Scaling

• Proportional scaling– Forces each volume of 4D dataset to have

the same mean– Also done by modeling the global signal– Idea is to remove background activity– Problems can occur if true activation is

wide spread– Negative activations may result

Intensity Normalizaton

without

with

Signal is lost and negative activation artifacts

Junghofer et al, 2004, NI

Other modeling considerations

• Adding the derivative of the HRF

• Adding motion parameters to the model

Model HRF & Derivative

0 5 10 15-0.2

0

0.2

0.4

0.6

0.8

1

1.2

“Shifted” HRF

0 5 10 15-0.2

0

0.2

0.4

0.6

0.8

1

1.2 Blue line is sum ofHRF and its derivative.

Temporal derivative

• We model the derivative, but don’t studyinferences of it– Linquist, et al (NI, 2008) suggest this is a

bad idea…may lead to bias• Generally I’d say we don’t worry about

the canonical HRF too much

Collinearity

• When designing your study, you wantyour tasks to be uncorrelated

• Correlation between regressors lowersthe efficiency of the parameterestimation

• Parameter estimates are highly variable– Can even flip signs

Why is it a problem?

Why is it a problem?

• There are an infinite # of solutions forand

…etc

Collinearity illustration

X1

X2

Cor=0.96

X1

X2

Cor= -0.2

Correlated RegressorsIntercept X1 X2

Highly variableoverexperiments

Inflated forcorrelatedcase (green)

T statistic

Bias can go ineither direction

Residuals don’t change

• The designs explain the same amountof variability

Collinearity• You can’t fix it after the data have been

collected• You can’t tell from the t statistic if you

had collinearity– FSL has some diagnostics

Absolutevalue ofcorrelation

Bad=whiteoff diagonal

Eigenvaluesfrom SVD

Bad=near 0

You are now first levelmodeling experts!

• You know why we convolve, highpassfilter and prewhiten

• If you don’t trust the canonical HRFthere are other options using basisfunctions

• Precoloring (lowpass filtering) typicallyisn’t used because it removes signal

Other things you learned

• Prewhitening was orginally viewed as toobiased, but methods have been developed toalleviate this problem

• Grand mean scaling is necessary in groupanalyses

• Proportional scaling is generally frownedupon (although popular in resting statefunctional connectivity studies)

Last things you learned

• Add derivative as well as motionparameters (unconvolved) to soak upextra variance

• Check for collinearity!– You should think about it before you collect

your data.

first-level fmri modelingmumford.fmripower.org/2010_summercourse/lev1_2010.pdf · 2010-07-19 ·...

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