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FMRI Time Series Analysis. Mark Woolrich & Steve Smith Oxford University Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB). FMRI Time Series Analysis Overview. Noise modelling (autocorrelation) Signal modelling: Complex parameterised HRF model

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fmri time series analysis

FMRI Time Series Analysis

Mark Woolrich & Steve Smith

Oxford University Centre for

Functional Magnetic Resonance Imaging of the Brain

(FMRIB)

fmri time series analysis overview

FMRI Time Series AnalysisOverview

  • Noise modelling (autocorrelation)
  • Signal modelling:
    • Complex parameterised HRF model
    • Optimised basis functions for HRF modelling
fmri noise
FMRI Noise
  • Time series from each voxel contains low frequency drifts and high frequency noise
  • Drifts are scanner-related and physiological (cardiac cycle, breathing etc)
  • Both high and low frequency noise hide activation

Power Spectral Density

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

High-pass

high frequency noise
High-Frequency Noise
  • Unless high-frequency noise is modelled or corrected for:
    • Incorrect stats (probably false positives)
    • Inefficient stats (false negatives)
slide8

Temporal Filtering and the GLM

S is a matrix for temporal filtering

Prewhitening

(Bullmore et al ‘96)

S = K-1

Best Linear Unbiased Estimator

Precolouring

(Worsley et al ‘95)

S = L

(L is a low pass filter)

Smothers intrinsic autocorrelation

different regressors
Different Regressors

Fixed ISI single-event with jitter

Randomized ISI single-event

Boxcar

Regressor

Low-pass

Low-pass

Low-pass

FFT

fmrib s improved linear modelling film
FMRIB's Improved Linear Modelling (FILM)

Performs prewhitening LOCALLY:

  • Fit the GLM and estimate the raw autocorrelation on the residuals
  • Spectrally and spatially smooth autocorrelation estimate
  • Construct prewhitening filter to "undo" autocorrelation
  • Use filter on data and design matrix and refit
fmrib s improved linear modelling film1
FMRIB's Improved Linear Modelling (FILM)

Performs prewhitening LOCALLY:

  • Fit the GLM and estimate the raw autocorrelation on the residuals
  • Spectrally and spatially smooth autocorrelation estimate
  • Construct prewhitening filter to "undo" autocorrelation
  • Use filter on data and design matrix and refit
spectral smoothing
Spectral Smoothing

Autocorrelation estimate

Raw autocorrelation

IFFT

FFT

Power Spectral Density

Tukey taper smoothed

non linear spatial smoothing
Non-linear Spatial Smoothing

Gaussian spatial smoother with weights:

Ii= EPI signal intensity

t = brightness threshold

unbiased statistics
Unbiased Statistics
  • P-P plots for FILM on 6 null datasets

Boxcar

Single Event

session effects investigation
Session Effects Investigation
  • 3 paradigms x 33 “identical” sessions (McGonigle 2000)
  • Variety of 1st-level analyses - use group-level mixed-effects-Z to judge efficiency of first-level analysis
autocorrelation conclusions
Autocorrelation Conclusions
  • Precolouring is nearly as sensitive as prewhitening for boxcar designs
  • Single-event designs require prewhitening for increased sensitivity
  • Local autocorrelation estimation using a Tukey taper with nonlinear spatial smoothing produces close to zero bias when prewhitening
  • More advanced: need spatiotemporal noise model:
    • Model-based (Woolrich)
    • Model-“free” (Beckmann)
signal modelling

Signal Modelling

  • Start with stimulation timings
    • Several conditions (original Evs)?
  • Convolve with HRF to blur and delay
  • What choice of HRF?
    • Does it vary across subjects?
    • Does it vary across the brain?
  • “Advanced” issues:
    • Allow signal height to change over time (dynamic)?
    • Use nonlinear convolution (events interact)?
    • Spatiotemporal modelling
linear time invariant system
Linear (time invariant) System

Experimental Stimulus, e.g. boxcar

Parameterised HRF, e.g. Gamma function

Assumed response

hrf parameterisation
HRF Parameterisation

Half-cosine parameterisation

Prior samples

?

hrf parameterisation1
HRF Parameterisation

Half-cosine parameterisation

Model Selection

Model 1(no undershoot): c2 = 0

Model 2(undershoot): c2 ≠ 0

?

automatic relevance determination ard prior mackay 1995
Automatic Relevance Determination (ARD) Prior(Mackay 1995)
  • Relevance of a parameter is automatically determined by the parameter

then

with high precision : Model 1

then

is non-zero : Model 2

MCMC

ard of undershoot
ARD of Undershoot

Simulated data with no undershoot

No ARD prior

ARD prior

True value

True value

hrf results

Prior samples

HRF Results

Boxcar

Jittered single-event

Randomised single-event

Boxcar

Jittered Single-event

Randomised Single-event

Response fits

Marginal posterior samples

Posterior samples

Posterior samples

Posterior samples

basis sets for hrf modelling
Basis Sets for HRF Modelling
  • Basis functions in the GLM: instead of one fixed HRF we can have several
  • Here an F across all 3 betas finds the best linear combination of the 3 HRFs
  • 3 Original EVs now become 6 (2 submodels each with 3 HRFs)
slide30

4 HRF basis functions

partial

model fits

full

model fits

basis sets for hrf modelling1
Basis Sets for HRF Modelling

MCMC

  • Instead of a parameterised HRF
  • We can use a linear basis set to span the space of expected HRF shapes

Variational

Bayes

WHY? : We can then use the basis set in an easier to infer GLM

generating hrf basis sets
Generating HRF Basis Sets

(1) Take samples of the HRF

generating hrf basis sets1
Generating HRF Basis Sets

(1) Take samples of the HRF

(2) Perform SVD

generating hrf basis sets2
Generating HRF Basis Sets

(1) Take samples of the HRF

(2) Perform SVD

(3) Select the top eigenvectors as the optimal basis set

unconstrained basis set
Unconstrained Basis Set

BUT: The basis set spans a wider range of HRF shapes than we want to allow:

HRF samples from prior

Unconstrained basis set

constrained basis set
Constrained Basis Set

We can regress the HRF samples back on to the basis set

Basis set

HRF samples from prior

constrained basis set1
Constrained Basis Set

and fit a multivariate normal to the basis set parameter space

Basis set

HRF samples from prior

constrained basis set2
Constrained Basis Set

This contrains the basis set to give only sensible looking HRF shapes

HRF samples from prior

Samples from basis set

Unconstrained

Constrained

using the constrained basis set
Using the Constrained Basis Set
  • The multivariate normal on the basis set parameters can then be used as a prior on those parameters in the GLM

Constrained basis set

Variational Bayes

acknowledgements

Acknowledgements

FMRIB Analysis Group

UK EPSRC, MRC, MIAS-IRC