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GLM for fMRI. Emily Falk, Ph.D. University of Pennsylvania Thanks to Thad Polk and Elliot Berkman. Review of preprocessing. (De-noise). Realign. Slice Timing Correct. Smooth. Predictors. Acquire Functionals. Y. X. y = X β + ε. Template. 1 st level (Subject) GLM.

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Glm for fmri

GLM for fMRI

Emily Falk, Ph.D.

University of Pennsylvania

Thanks to Thad Polk and Elliot Berkman


Review of preprocessing

Review of preprocessing


Glm for fmri

(De-noise)

Realign

Slice Timing Correct

Smooth

Predictors

AcquireFunctionals

Y

X

y = Xβ + ε

Template

1st level (Subject) GLM

Determine Scanning Parameters

β

Co-Register

βhow - βwhy

Normalize

Acquire Structurals (T1)

Contrast

All subjects

2nd level (Group)

GLM

Threshold


Glm for fmri

Determine Scanning Parameters


Determine scanning parameters

Determine Scanning parameters

Temporal Resolution

Signal-Noise Ratio (SNR)

-

-

-

-

Coverage/ Field of View

Spatial Resolution

-


Glm for fmri

(De-noise)

Slice Timing Correct

Acquire Functionals

Determine Scanning Parameters

Acquire Structurals (T1)


Glm for fmri

Slice 4

Slice

Slice 3

Slice 2

Slice 1

TR #1

TR #2

TR #3

TR #4

Time


Glm for fmri

(De-noise)

Realign

Slice Timing Correct

Acquire Functionals

Determine Scanning Parameters

Acquire Structurals (T1)


Realignment

Realignment

Minimize sum of squared diff

reslice


Glm for fmri

(De-noise)

Realign

Slice Timing Correct

AcquireFunctionals

Template

Determine Scanning Parameters

Co-Register

Normalize

Acquire Structurals (T1)


Co registration

Co-registration

Can’t use minimized squared difference on different image types (different tissue -> signal intensity mapping)

Instead use mutual information (maximize)

reference

moved image


Normalization

normalization

12 parameter affine transformation

Trans x Pitch x Roll x Yawx Zoom x Sheer

Sheer

Zoom

Even better with segmentation!


Smoothing

Smoothing


Glm for fmri

(De-noise)

Realign

Slice Timing Correct

Smooth

Predictors

AcquireFunctionals

Y

X

Template

Determine Scanning Parameters

Co-Register

Normalize

Acquire Structurals (T1)


Forming predictors

Forming predictors

How

Why

X

=

Events

Basis Function

Predictors (X)


Glm for fmri

(De-noise)

Realign

Slice Timing Correct

Smooth

Predictors

AcquireFunctionals

Y

X

y = Xβ + ε

Template

1st level (Subject) GLM

Determine Scanning Parameters

β

Co-Register

βhow - βwhy

Normalize

Acquire Structurals (T1)

Contrast

All subjects

2nd level (Group)

GLM

Threshold


Glm for fmri

(De-noise)

Realign

Slice Timing Correct

Smooth

Predictors

AcquireFunctionals

Y

X

y = Xβ + ε

Template

1st level (Subject) GLM

Determine Scanning Parameters

β

Co-Register

βhow - βwhy

Normalize

Acquire Structurals (T1)

Contrast

All subjects

2nd level (Group)

GLM

Threshold


Lecture outline

Lecture Outline

  • Hypothesis Testing (covered Sunday)

    Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons

  • General Linear Model

    Regression, multiple regression, model fitting, matrix notation, design matrix, example, issues

  • Overview of fMRI data analysis

    Build design matrix, fit model to get betas, contrasts, statistical parametric maps, threshold for significance (correcting for multiple comparisons)


Statistical model

Statistical Model

In fMRI, we experimentally manipulate various independent variables (e.g., task, stimulus) while scanning

We are interested in constructing a model of the predicted brain activity that can be used to explain the observed fMRI data in terms of the independent variables.

In fitting the model to the data, we obtain parameter estimates and make inferences about their consistency with the null hypothesis.


General linear model

General Linear Model

The General linear model (GLM) approach treats the data as a linear combination of predictor variables plus noise (error).

The predictors are assumed to have known shapes, but their amplitudes are unknown and need to be estimated.

The GLM framework encompasses many of the commonly used techniques in fMRI data analysis (and data analysis more generally).


The glm family

The GLM Family

DV

Predictors

Analysis

Regression

Continuous

One predictor

Multiple Regression

Continuous

Two+ preds

One continuous

2-sample t-test

Categorical

1 pred., 2 levels

General Linear Model

One-way ANOVA

Categorical

1 p., 3+ levels

Factorial ANOVA

Categorical

2+ predictors

Two measures, one factor

Paired t-test

Repeated measures

More than two measures

Repeated measures ANOVA


A simple linear model with one predictor a made up non fmri example

A simple linear model with one predictor (a made up, non-fMRI example)

  • Fit a straight line to the data, the “best fit”

  • This line is a simplification, a model with two parameters: intercept and slope

  • Can use the model to make predictions

  • Can test the slope parameter against a null hypothesis of zero, and make inferences about whether there is a statistically significant relationship

Blushing

(blood flow to cheeks)

Attractiveness


The regression model

The Regression Model

  • Vector notation: y, x, and e are vectors of N values corresponding to N observations

Outcome

(DV)

Intercept

(constant)

Error (residual)

Predictor value

slope

For point i:


Fitting and residual variance

Fitting and Residual Variance

  • The line is being pulled by vertical rubber bands attached to each point. Vector of red lines is e

  • Minimize squares of vertical distance to lines. Minimize:

Blushing

(blood flow to cheeks)

Attractiveness


The multiple regression model basic model for the glm

The Multiple Regression ModelBasic Model for the GLM

  • Structural Model for Regression

DV

Pred1

Pred2

Predk

Variables

Parameters

Slope 1

Slope k

intercept

Slope 2

Error

Matrix notation

  • solve for beta vector

  • minimize sum of squared residuals


Matrix notation

Matrix Notation

  • Alternatively, we can write

as

Design matrix

Residuals

Observed Data

Model parameters


Matrix notation1

Matrix Notation

Design matrix

Residuals

Observed Data

Model parameters

Is this same as


Design matrix

Design Matrix

In fMRI the design matrix specifies how the factors of the model change over time.

The design matrix is an np matrix where n is the number of observations over time and p is the number of model parameters


Another made up non fmri example

Another Made-up, Non-fMRI Example

  • Does exercise predict life-span?

  • Control for other variables that might be important. i.e., gender (M/F)

Females

Males


A non fmri example

A Non-fMRI Example

Outcome Data

Design matrix

Model parameters

Residuals

=

+

X


Multicollinearity

Multicollinearity

  • Coefficients (betas) for individual predictors test for variance uniquely explained by that predictor

  • When predictors are intercorrelated, interpreting the betas can become very tricky!

    • Changes in sign of one beta when you add others

    • Changes in significance of one beta when you add others

  • This is because the predictors are attempting to explain the SAME variance

  • This is called multicollinearity.


Implications for fmri

Implications for fMRI

  • You have at least some control over the design matrix, X, because you manipulate the stimulation.

  • Avoid multicollinearity and complex issues with good experimental design!

  • Do not use a design with many or highly correlated predictors and expect the modeling to sort everything out


Lecture outline1

Lecture Outline

  • Hypothesis Testing (covered last night)

    Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons

  • General Linear Model

    Regression, multiple regression, model fitting, matrix notation, design matrix, example, issues

    QUESTIONS?


Lecture outline2

Lecture Outline

  • Hypothesis Testing (covered last night)

    Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons

  • General Linear Model

    Regression, multiple regression, model fitting, matrix notation, design matrix, example, issues

  • Overview of fMRI data analysis

    Model specification, parameter estimation, contrasts, statistical parametric maps, threshold for significance (correcting for multiple comparisons)


A simple research question

A simple research question?

  • What are the neural correlates of positive valuation (see recent meta analysis by Bartra et al.)


Fmri example

fMRI: Example


Glm for fmri

On

X1

Off

On

X2

Off

Time


Glm for fmri

β2

β2

β1

β1

β2

β2

β1

β1

β2

β2

β1

β2

β1

β1

β2

β1

β2

β2

β1

β2

β2

β1

β1

β1

β2

β1

β2

β2

β1

β2

β2

β1

β1

β1

β2

β2

β1

β1

β2

β1

β1

β2

β1

β2

β2

β1


Glm for fmri

-


The spm way of plotting the variables

The SPM way of Plotting the Variables

X

y

e

+

=


T test contrasts spm t

contrast ofestimatedparameters

T =

varianceestimate

T-test – contrasts – SPM{t}

Question:

Difference between seeing loved one and random face > 0 ?

=

b1 – b2 = cTb> 0 ?

cT = 1 -1 0 0 0 0 0 0

b1b2b3b4b5 ...

H0: cTb=0

Null hypothesis:

Test statistic:


T test another simple example

1

10

20

30

40

50

60

70

80

0.5

1

1.5

2

2.5

Design matrix

T-test: Another Simple Example

Whyversus rest

Q: activationduringattribution?

cT = [ 1 0 ]

Null hypothesis: 1=0


More specifically

More specifically…


Isolating mental processes

Isolating Mental Processes

  • Tasks are typically designed to isolate mental processes of interest by comparing parameter estimates (betas)

    • Viewing hot vs. neutral faces

    • Why vs. how

  • The comparison involves a linear combination of the parameter estimates

    • Note: investigating a single parameter estimate is also a contrast and sometimes termed a contrast against the implicit baseline


Voxel wise time series analysis

Model

specification

Parameter

estimation

Hypothesis

Statistic

Voxel-wise Time Series Analysis

Time

Time

BOLD signal

single voxel

time series

SPM


Fmri example one voxel

fMRI Example: One Voxel

amplitude

time

Source: J-B. Poline

Temporal series fMRI

voxel time course


Model specification building the design matrix

Model Specification:Building The Design Matrix

fMRI Data

Design matrix

Residuals

Model

parameters

=

+

X

Predicted task

response

intercept


Parameter estimation model fitting

Parameter Estimation/Model Fitting

Find  values that produce

best fit to observed data

y

=

 0

+

 1

+

ERROR


The spm way of plotting the variables1

The SPM Way of Plotting the Variables

y

X

e

+

=


Glm for fmri

What SPM Computes

  • F-contrast

    • spm_ess_####.img

      • Extra sum-of-squares image

    • spm_F_####.img

      • F-statistic image

  • ResMS.img

    • Residual sum-of-squares (σ2)

  • beta_####.img

    • Parameter estimates (β)

  • T-contrast

    • spm_con_####.img

      • Contrast image

      • If you take the appropriate beta_####.img’s and linearly combine them, it will be identical to the spm_con_####.img

    • spm_T_####.img

      • T-statistic image


Statistical parametric maps

spmT_???? image

SPM{t}

Statistical Parametric Maps

beta_???? images

con_???? image


T test one dimensional contrasts spm t

contrast ofestimatedparameters

T =

varianceestimate

T-test –One Dimensional Contrasts – SPM{t}

box-car amplitude > 0 ?

=

b1 = cTb> 0 ?

Question:

cT = 1 0 0 0 0 0 0 0

H0: cTb=0

b1b2b3b4b5 ...

Null hypothesis:

Test statistic:


Model specification building the design matrix adding predictors

Model Specification:Building The Design Matrix, Adding Predictors

fMRI Data

Design matrix

Residuals

Model

parameters

=

+

X

Predicted task

response

intercept


The spm way of plotting the variables2

The SPM Way of Plotting the Variables

y

X

e

+

=


T test one dimensional contrasts spm t1

contrast ofestimatedparameters

T =

varianceestimate

T-test –One Dimensional Contrasts – SPM{t}

box-car amplitude > 0 ?

=

b1 = cTb> 0 ?

Question:

cT = 1 -1 0 0 0 0 0 0

H0: cTb=0

b1b2b3b4b5 ...

Null hypothesis:

Test statistic:


T test a simple example

1

10

20

30

40

50

60

70

80

0.5

1

1.5

2

2.5

Design matrix

T-test: A Simple Example

Why vs. How

Q: Greateractivationduringwhythanhow?

cT = [ 1 -1 ]

Null hypothesis: 1-2=0


T test a slightly more complicated example

T-test: A slightly more complicated example

Whathappens in thebrainduringattribution?

A

C

B

D


Another slightly more complicated design

Another slightly more complicated design

  • Do peer ratings change underlying assessments of the attractiveness of faces or merely public reporting (see work by Klucharev et al. and Zaki et al.)

  • Manipulate peer ratings (higher, lower, same, no rating)


T contrasts

t-contrasts

  • We can estimate the full design matrix, and then apply contrasts to betas after model fitting to estimate effects of interest and make inferences.

Design Matrix (X)

B

C

D

A

Covariates

fit

apply C

Time

Courtesy of Tor Wager


T contrasts1

t-contrasts

  • Statistical contrasts typically sum to zero so that E(C’b) = 0. Makes testing easy.

  • Scaling doesn’t matter for most inference, but good to get into habit of using fractions if you need to average.

    • [1 -1] and [.5 -.5] give same statistical result (though can affect other calculations like percent signal change).

Design Matrix (X)

C’b is effect magnitude estimate for (A + B) – (C + D), equivalent to mean(A,B) - mean(C,D) or WHY> HOW

B

C

D

A

-1

-1

Time

C’b is beta for A only, or WHY_faces > rest

C’b is sum (average) beta for A and B, or WHY > rest


Two kinds of contrasts

Two Kinds of Contrasts

  • T-contrast

    • Test of a specific combination of predictors

      • Linear combination differs from zero (e.g., why – how)?

    • Signed: effect can be positive or negative

      • Some areas might be higher for why and some higher for how

  • F-contrast

    • Test on a set of predictors

      • Variance explained by a set of predictors greater than zero?

    • Unsigned: amount of variance explained can only be positive

    • Any t-contrast can be specified as a F-contrast (without the sign)


F contrast

F-Contrast

  • Test of set of linear contrasts

    • 1) Does a set of regressors explain a significant amount of variance?

      • E.g., model with and without temporal derivatives

    • 2) Set of differences

      • E.g., an overall effect of time (T1-T2; T2-T3)


Summary

Summary

  • Contrasts can be a linear combination of parameter estimates (t-contrast) or a set of linear combinations of parameter estimates (F-contrast)

  • Inference on these contrasts can be performed at the single subject level where degrees of freedom are dictated by number of scans and number parameters modeled


Questions

Questions?


Note multiple runs

Note: Multiple Runs

Put participant in scanner

Localizer

Functional Run 1

Block 1

Trial 1

Trial 2

Trial n

Block 2

Block n

Functional Run 2

Functional Run n

Structural

Take participant out of scanner

  • Each run is estimated by a separate GLM

    • Run: one set of scans (typically 5-10 min)

    • SPM calls this a “Session” although in typical parlance, a “Session” refers everything that occurs in-between putting a subject in the scanner and taking them out

  • Due to

    • Mean signal varies on a run-by-run basis

      • Different intercept term for each run

      • SPM scales images on a per run basis so that mean signal across a run is 100

    • Low frequency drift and temporal autocorrelation are defined within a run

Block

Run

Session


Design matrix in spm one run

Design Matrix in SPM: One Run


Design matrix in spm multiple runs

Design Matrix in SPM: Multiple Runs

Run 1

Predictors of interest

Intercept


Design matrix in spm multiple runs1

Design Matrix in SPM: Multiple Runs

Predictors of interest

Run 2

Intercept


Note re contrasts and multiple runs

Note re: contrasts and multiple runs

  • If your runs all have the same conditions in the same order (aim for this):

    • You can specify the contrast weights for one run and SPM will assume same for other runs

  • If your runs don’t conform this way:

    • You can specify custom contrast across all run/conditions


Example

Example

Input: 0 0 1

Input: 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1


Questions1

Questions?


Parametric modulation

Parametric modulation


What if you want to model a continuous variable

What if you want to model a continuous variable?

e.g., making mental state attributions for people rated more or less similar to the participant


Parametric modulation1

Parametric Modulation

  • Modeling changes in brain activity as a function of performance or other continuous variables across multiple levels can provide strong inference

    • Load, confidence, RT, etc.

Goal (purchase) value of an item

Courtesy of Tor Wager


Examples

Examples

  • Effect of attribution (Why > How) modulated by:

    • Identification with person in the photo

    • Attractiveness of person in the photo

    • Accuracy of attribution (i.e., consensus with other raters)

  • Parametric effect on face viewing of:

    • Ppt. attractiveness rating of face

    • Peer rating of face

    • Peer rating of face, controlling for initial rating of face


Glm for fmri

Parametric Psychological Events

Principle Predictor: Captures Average Response

Parametric Modulator: Captures Modulations

Idealized Data

Fitted Response

Courtesy of Tor Wager


Parametric modulation details

Parametric Modulation: Details

  • Modulators are orthogonalizedw.r.t. principle regressor and each other

    • Order matters

    • Modulators will capture residual activation not captured by preceding regressors

  • Example:

    • Why, Why*Similarity, Why*Attractiveness

    • Why, Why*Attractiveness, Why*Similarity


Parametric modulation details1

Average response

Parametric

modulator

Parametric Modulation: Details

  • Modulators model differences in amplitude of response only

    • Not duration

  • Suggested to vary duration of event rather than use parametric modulation in these cases (“variable epoch” model)

(parametric modulator framework)


Glm for fmri

Visual cortex: Contrast-modulation (red) vs. duration-modulation (blue)

Grinband et al., 2008


Parametric modulation contrasts with pms

Parametric Modulation: Contrasts with PMs

y

X

e

+

=

AA*pm

Contrast: 10= main effect of condition A

Contrast: 01= effect of pm


Parametric modulation summary

Parametric Modulation: Summary

  • Include modulator(s) can be a good idea to capture parametric effects

  • Understand that modulator(s) are orthogonalized to principle regressor and each other when making inferences

    • Can be adjusted in SPM code by removing calls to “spm_orth”

  • Amplitude modulators may not capture appropriate variance if modulation affects both amplitude and duration of response

    • Including “variable epochs” and orthogonalizing them w.r.t. principle regressor may be more appropriate


Covariates nuisances

Covariates/ Nuisances


Nuisances in fmri

Nuisances in fMRI

  • fMRI data contains large contributions from factors of non-interest (nuisances)

    • E.g. Scanner instability

  • A few such nuisances are:

    • Low frequency drift

    • Temporal autocorrelation

    • Motion


Low frequency drift

Low Frequency Drift

Frequency domain

fMRI Noise: Time domain

Unfiltered

Lot’s of low frequency power!


Effect of low frequency drift

blue= data

black = mean + low-frequency drift

green= predicted response, taking into account low-frequency drift

red= predicted response, not taking into account low-frequency drift

Effect of Low Frequency Drift


Accounting for low frequency drift

Accounting for Low Frequency Drift

High-pass filter:

High frequency elements are allowed to pass

Default in SPM is 128

In SPM, filter is applied to both data and design


High pass filtering

60 s

HP filter

1/60 =

.016 Hz

High-Pass Filtering

Frequency domain

fMRI Noise: Time domain

Unfiltered

Filtered

Courtesy of Tor Wager


Low frequency drift considerations

Low Frequency Drift Considerations

  • Avoid long epochs or conditions that repeat at a very low frequency

  • Check if filter removes power from predictors of interest and adjust if necessary

  • Rule of thumb: 2 x the longest periodicity in the design

Predictor

Filtered frequencies

Power


Nuisances in fmri1

Nuisances in fMRI

  • fMRI data contains large contributions from factors of non-interest (nuisances)

    • E.g. Scanner instability

  • A few such nuisances are:

    • Low frequency drift

    • Temporal autocorrelation

    • Motion


Temporal autocorrelation

Temporal Autocorrelation

  • Nearby time-points exhibit positive correlation

    • Physiological and scanner instability

r = 1

Lag 0


Temporal autocorrelation1

Temporal Autocorrelation

  • Nearby time-points exhibit positive correlation

    • Physiological and scanner instability

r = 0.6

Lag 1


Temporal autocorrelation2

Temporal Autocorrelation

  • Nearby time-points exhibit positive correlation

    • Physiological and scanner instability

r = 0.3

Lag 2


Temporal autocorrelation3

Temporal Autocorrelation

  • Nearby time-points exhibit positive correlation

    • Physiological and scanner instability

r = 0.1

Lag 3


Temporal autocorrelation4

Temporal Autocorrelation

  • Nearby time-points exhibit positive correlation

    • Physiological and scanner instability

  • GLM assumes that errors are independent

    • Correlated errors violate this assumption (changes degrees of freedom)

Autocorrelation function

Autocorrelated

Errors

Random errors


Temporal autocorrelation solution

Temporal Autocorrelation: Solution

  • Model correlation structure, V

    • Model data ignoring temporal autocorrelation

    • Use residuals to estimate temporal autocorrelation

      • Assume autoregressive (AR) model

        • Signal at time t depends on signal at previous time points plus noise

  • Use estimated autocorrelation to filter the data and design matrix


Autocorrelation case study

“Model serial correlations: none”

“Model serial correlations: AR(1)”

Autocorrelation: Case Study

Courtesy of Tor Wager


Temporal autocorrelation take home

Temporal Autocorrelation: Take Home

  • Understand that autocorrelation exists and that there are algorithms to correct it

    • Specific algorithm will vary from package to package

  • Understand that it impacts inference on individuals (1st level) if not accounted for

    • Residual variance will be biased

  • Parameter estimates will be unbiased if autocorrelation is not taken into account

  • Parameter estimates will be less precise if autocorrelation is not taken into account


Nuisances in fmri2

Nuisances in fMRI

  • fMRI data contains large contributions from factors of non-interest (nuisances)

    • E.g. Scanner instability

  • A few such nuisances are:

    • Low frequency drift

    • Temporal autocorrelation

    • Motion


Motion confounds

Motion Confounds

  • Realignment spatially corrects images for motion

  • However, motion has profound affects on signal that realignment does not correct

Signal Spikes

Motion

Power et al., 2012, NeuroImage


Motion regression

Motion Regression

  • Realignment procedures output motion parameters

    • 6 per scan (x, y, z, pitch, roll, yaw)

    • Including these parameters as nuisance regressors can capture motion-related signal changes and improve model fits


Motion regression improvements

Motion Regression Improvements

  • Motion parameters capture absolute displacement over time

    • i.e. motion drift

  • Motion-related signal artifacts are typically reflected in scan-to-scan changes

    • i.e. displacement relative to last scan

  • Good idea to include differential motion to capture scan-to-scan changes

    • But remove sign

  • Recommended approach is to include 24 total motion regressors(Lund et al., 2005, NeuroImage):

    • Displacement, Differential, Squared Displacement, Squared Differential


Motion regression caveats

Motion Regression: Caveats

  • Motion can sometimes be synched to task

    • Button press

    • Shifting during ITI

    • Motion regressors can be confounded with task-related regressors in these cases

  • May be a good idea to set a motion tolerance criterion and exclude motion regressors from low motion subjects


Summary1

Summary

  • GLM is fit to each voxel, estimating linear contribution of each predictor

  • Parameter estimates (β’s) estimate the amplitude of each predictor

  • Estimates are affected by presence and absence of other predictors, so it is important to model the data as accurately as possible

  • Nuisance confounds are addressed in both the data and design

    • High-pass filter: removed low frequency drift

    • Pre-whiten: remove temporal autocorrelation

  • Nuisance confounds and methods to address them limit the kinds of predictors that can be effectively included in the GLM

  • Multiple runs (called “Session” in SPM) are each estimated by a separate GLM


Additional notes

Additional notes


Hemodynamic response

Hemodynamic Response

  • Power of predictors depends on the appropriateness of the basis function

  • Thus far, assumed a “canonical” hemodynamic response function (HRF)

    • Double-gamma function

    • Peak 5s

    • Undershoot 15s (1/6 height of peak)

    • ~20s to return to baseline

  • True HRF varies between individuals AND regions


Hrfs vary across brain regions and tasks problem for canonical model

HRFs Vary Across Brain Regions and TasksProblem for Canonical Model!

Checkerboard, n = 10

Thermal pain, n = 23

  • HRF shape depends on:

    • Vasculature

    • Time course of neural activity

Stimulus On

Aversive picture, n = 30

Aversive anticipation

Courtesy of Tor Wager

See Aguirre et al. 1998, NeuroImage


Hrf and bias

HRF and Bias

  • Typically, we are interested in the amplitude of the HRF

  • If true HRF peaks earlier or later than expected, amplitude estimates will be systematically biased

    • True HRFs often peak earlier than canonical HRF (c.f. Handwerker et al., 2004, NeuroImage; Steffener et al., 2010, NeuroImage)

  • May be especially problematic for closely spaced events

    • Unaccounted for variance can be shifted to earlier or later events


Basis sets hrf derivatives

Basis Sets: HRF + Derivatives

  • One popular approach is to use derivatives of the canonical HRF to capture timing and dispersion related variance

Canonical only

Canonical + derivs

2nd derivative of the HRF is the “dispersion” derivative: captures narrower or wider responses than expected

1st derivative of the HRF is the “temporal” derivative: captures peaks that occur earlier or later than expected


Basis sets hrf derivatives1

Basis Sets: HRF + Derivatives

Note, including derivative, but basing tests only on the non-deriv does not help!

Adding derivative can account for delay

Calhoun et al., 2004, NeuroImage

Calhoun et al., 2004, NeuroImage


Basis sets finite impulse response

Basis Sets: Finite Impulse Response

  • Another approach is to make no assumptions about the shape of the HRF

  • Instead, a set of shifted delta functions is used to capture any response: Finite Impulse Response (FIR) set


Glm for fmri

  • Fit is sometimes better with multiple basis sets, inference is harder

  • Most common approach is to use the SPM canonical HRF


Outline

Outline

  • Providing a better model fit

    • Basis functions

  • Modeling continuous variation

    • Parametric modulators

  • Improved confound reduction

    • Motion regression


Take home

Take-Home

  • Model fits can be made more accurate with the use of basis sets

    • Can help if canonical HRF is not appropriate

    • BUT, cost in more complicated inference or methods to extract amplitude estimates from set of fitted parameters

  • Parametric modulators can capture changes across several levels

    • Can lead to stronger inferences

    • BUT, note that duration modulation may require a variable epoch approach

  • Motion regression can substantially reduce variability

    • Can remove severe signal spikes

    • BUT, may be a good idea to set a motion tolerance criterion


Spm listserv

SPM Listserv

  • The SPM listserv is a great, active resource for questions about SPM and SPM-related analyses

  • Subscribe

    • Search “spm listserv”, click on first link (SPM LIST at JISCMAIL.AC.UK), then click on subscribe

  • If you have a question

    • First search the archives to see if it’s already been asked and answered

    • If it hasn’t, email your question to

      • [email protected]


Lecture outline3

Lecture Outline

  • Hypothesis Testing (covered last night)

    Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons

  • General Linear Model

    Regression, multiple regression, model fitting, matrix notation, design matrix, example, issues

  • Overview of fMRI data analysis

    Model specification, parameter estimation, contrasts, statistical parametric maps, threshold for significance (correcting for multiple comparisons)

    QUESTIONS?


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