Convolution Models

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Convolution Models for fMRI Rik Henson With
thanks to Karl Friston, Oliver Josephs
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Overview
1. BOLD impulse response 2. Convolution and
GLM 3. Temporal Basis Functions 4.
Filtering/Autocorrelation 5. Timing Issues 6.
Nonlinear Models
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BOLD Impulse Response

• Function of blood oxygenation, flow, volume
  (Buxton et al, 1998)
• Peak (max. oxygenation) 4-6s poststimulus
  baseline after 20-30s
• Initial undershoot can be observed (Malonek
  Grinvald, 1996)
• Similar across V1, A1, S1
• but differences across other regions
  (Schacter et al 1997) individuals (Aguirre et
  al, 1998)

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BOLD Impulse Response

• Early event-related fMRI studies used a long
  Stimulus Onset Asynchrony (SOA) to allow BOLD
  response to return to baseline
• However, if the BOLD response is explicitly
  modelled, overlap between successive responses at
  short SOAs can be accommodated
• particularly if responses are assumed to
  superpose linearly
• Short SOAs are more sensitive

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Overview
1. BOLD impulse response 2. Convolution and
GLM 3. Temporal Basis Functions 4.
Filtering/Autocorrelation 5. Timing Issues 6.
Nonlinear Models
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General Linear (Convolution) Model
GLM for a single voxel y(t) u(t) ??
h(t) ?(t) u(t) neural causes (stimulus
train) u(t) ? ? (t - nT) h(t)
hemodynamic (BOLD) response h(t) ? ßi
fi (t) fi(t) temporal basis functions
y(t) ? ? ßi fi (t - nT) ?(t) y
X ß e
sampled each scan
Design Matrix
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General Linear Model (in SPM)

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A word about down-sampling
x2
x3
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Overview
1. BOLD impulse response 2. Convolution and
GLM 3. Temporal Basis Functions 4.
Filtering/Autocorrelation 5. Timing Issues 6.
Nonlinear Models
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Temporal Basis Functions

• Fourier Set
• Windowed sines cosines
• Any shape (up to frequency limit)
• Inference via F-test

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Temporal Basis Functions

• Finite Impulse Response
• Mini timebins (selective averaging)
• Any shape (up to bin-width)
• Inference via F-test

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Temporal Basis Functions

• Fourier Set
• Windowed sines cosines
• Any shape (up to frequency limit)
• Inference via F-test
• Gamma Functions
• Bounded, asymmetrical (like BOLD)
• Set of different lags
• Inference via F-test

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Temporal Basis Functions

• Fourier Set
• Windowed sines cosines
• Any shape (up to frequency limit)
• Inference via F-test
• Gamma Functions
• Bounded, asymmetrical (like BOLD)
• Set of different lags
• Inference via F-test
• Informed Basis Set
• Best guess of canonical BOLD response Variabilit
  y captured by Taylor expansion Magnitude
  inferences via t-test?

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Temporal Basis Functions
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Temporal Basis Functions

• Informed Basis Set
• (Friston et al. 1998)
• Canonical HRF (2 gamma functions)

Canonical
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Temporal Basis Functions

• Informed Basis Set
• (Friston et al. 1998)
• Canonical HRF (2 gamma functions)
• plus Multivariate Taylor expansion in
• time (Temporal Derivative)

Canonical
Temporal
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Temporal Basis Functions

• Informed Basis Set
• (Friston et al. 1998)
• Canonical HRF (2 gamma functions)
• plus Multivariate Taylor expansion in
• time (Temporal Derivative)
• width (Dispersion Derivative)

Canonical
Temporal
Dispersion
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Temporal Basis Functions

• Informed Basis Set
• (Friston et al. 1998)
• Canonical HRF (2 gamma functions)
• plus Multivariate Taylor expansion in
• time (Temporal Derivative)
• width (Dispersion Derivative)
• F-tests allow for canonical-like responses

Canonical
Temporal
Dispersion
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Temporal Basis Functions

• Informed Basis Set
• (Friston et al. 1998)
• Canonical HRF (2 gamma functions)
• plus Multivariate Taylor expansion in
• time (Temporal Derivative)
• width (Dispersion Derivative)
• F-tests allow for any canonical-like responses
• T-tests on canonical HRF alone (at 1st level) can
  be improved by derivatives reducing residual
  error, and can be interpreted as amplitude
  differences, assuming canonical HRF is good fit

Canonical
Temporal
Dispersion
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(Other Approaches)

• Long Stimulus Onset Asychrony (SOA)
• Can ignore overlap between responses (Cohen et
  al 1997)
• but long SOAs are less sensitive
• Fully counterbalanced designs
• Assume response overlap cancels (Saykin et al
  1999)
• Include fixation trials to selectively average
  response even at short SOA (Dale Buckner,
  1997)
• but unbalanced when events defined by subject
• Define HRF from pilot scan on each subject
• May capture intersubject variability (Zarahn et
  al, 1997)
• but not interregional variability
• Numerical fitting of highly parametrised
  response functions
• Separate estimate of magnitude, latency,
  duration (Kruggel et al 1999)
• but computationally expensive for every voxel

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Temporal Basis Sets Which One?
In this example (rapid motor response to faces,
Henson et al, 2001)
FIR
Dispersion
Temporal
Canonical
canonical temporal dispersion derivatives
appear sufficient may not be for more complex
trials (eg stimulus-delay-response) but then
such trials better modelled with separate neural
components (ie activity no longer delta
function) constrained HRF (Zarahn, 1999)
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Overview
1. BOLD impulse response 2. Convolution and
GLM 3. Temporal Basis Functions 4.
Filtering/Autocorrelation 5. Timing Issues 6.
Nonlinear Models
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Low frequency noise

• Low frequency noise
• Physical (scanner drifts)
• Physiological (aliased)
• cardiac (1 Hz)
• respiratory (0.25 Hz)

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Temporal autocorrelation

• Because the data are typically correlated from
  one scan to the next, one cannot assume the
  degrees of freedom (dfs) are simply the number of
  scans minus the dfs used in the model need
  effective degrees of freedom
• In other words, the residual errors are not
  independent
• Y Xb e e N(0,s2V) V ? I, VAA'
• where A is the intrinsic autocorrelation
• Generalised least squares
• KY KXb Ke Ke N(0, s2V) V KAA'K'
• (autocorrelation is a special case of
  nonsphericity)

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Temporal autocorrelation (History)
KY KXb Ke Ke N(0, s2V) V KAA'K'

• One method is to estimate A, using, for example,
  an AR(p) model, then
• K A-1 V I (allows OLS)
• This pre-whitening is sensitive, but can be
  biased if K mis-estimated
• Another method (SPM99) is to smooth the data with
  a known autocorrelation that swamps any intrinsic
  autocorrelation
• K S V SAA'S SS' (use GLS)
• Effective degrees of freedom calculated with
  Satterthwaite approximation ( df
  trace(RV)2/trace(RVRV) )
• This is more robust (providing the temporal
  smoothing is sufficient, eg 4s FWHM Gaussian),
  but less sensitive
• Most recent method (SPM2/5) is to restrict K to
  highpass filter, and estimate residual
  autocorrelation A using voxel-wide, one-step ReML

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Nonsphericity and ReML (SPM2/SPM5)

• Nonsphericity means (kind of) that
• Ce cov(e) ? s2I
• Nonsphericity can be modelled by set of variance
  components
• Ce ?1Q1 ?2Q2 ?3Q3 ...
• (?i are hyper-parameters)
• - Non-identical (inhomogeneous) (e.g, two
  groups of subjects)

- Non-independent (autocorrelated) (e.g,
white noise AR(1))
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Nonsphericity and ReML (SPM2/SPM5)

• Joint estimation of parameters and
  hyper-parameters requires ReML
• ReML gives (Restricted) Maximum Likelihood (ML)
  estimates of (hyper)parameters, rather than
  Ordinary Least Square (OLS) estimates
• ML estimates are more efficient, entail exact dfs
  (no Satterthwaite approx)
• but computationally expensive ReML is iterative
  (unless only one hyper-parameter)
• To speed up
• Correlation of errors (V) estimated by pooling
  over voxels
• Covariance of errors (s2V) estimated by single,
  voxel-specific scaling hyperparameter

Ce ReML( yyT, X, Q )
V ReML( ? yjyjT, X, Q )
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Overview
1. BOLD impulse response 2. Convolution and
GLM 3. Temporal Basis Functions 4.
Filtering/Autocorrelation 5. Timing Issues 6.
Nonlinear Models
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Timing Issues Practical
TR4s
Scans

• Typical TR for 48 slice EPI at 3mm spacing is 4s

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Timing Issues Practical
TR4s
Scans

• Typical TR for 48 slice EPI at 3mm spacing is
  4s
• Sampling at 0,4,8,12 post- stimulus may miss
  peak signal

Stimulus (synchronous)
SOA8s
Sampling rate4s
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Timing Issues Practical
TR4s
Scans

• Typical TR for 48 slice EPI at 3mm spacing is
  4s
• Sampling at 0,4,8,12 post- stimulus may miss
  peak signal
• Higher effective sampling by 1. Asynchrony eg
  SOA1.5TR

Stimulus (asynchronous)
SOA6s
Sampling rate2s
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Timing Issues Practical
TR4s
Scans

• Typical TR for 48 slice EPI at 3mm spacing is
  4s
• Sampling at 0,4,8,12 post- stimulus may miss
  peak signal
• Higher effective sampling by 1. Asynchrony eg
  SOA1.5TR 2. Random Jitter eg
  SOA(20.5)TR

Stimulus (random jitter)
Sampling rate2s
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Timing Issues Practical
TR4s
Scans

• Typical TR for 48 slice EPI at 3mm spacing is
  4s
• Sampling at 0,4,8,12 post- stimulus may miss
  peak signal
• Higher effective sampling by 1. Asynchrony eg
  SOA1.5TR 2. Random Jitter eg
  SOA(20.5)TR
• Better response characterisation (Miezin et al,
  2000)

Stimulus (random jitter)
Sampling rate2s
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Timing Issues Practical

• but Slice-timing Problem
• (Henson et al, 1999)
• Slices acquired at different times, yet
  model is the same for all slices

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Timing Issues Practical
Bottom Slice
Top Slice

• but Slice-timing Problem
• (Henson et al, 1999)
• Slices acquired at different times, yet
  model is the same for all slices
• gt different results (using canonical HRF) for
  different reference slices

TR3s
SPMt
SPMt
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Timing Issues Practical
Bottom Slice
Top Slice

• but Slice-timing Problem
• (Henson et al, 1999)
• Slices acquired at different times, yet
  model is the same for all slices
• gt different results (using canonical HRF) for
  different reference slices
• Solutions
• 1. Temporal interpolation of data but less
  good for longer TRs

TR3s
SPMt
SPMt
Interpolated
SPMt
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Timing Issues Practical
Bottom Slice
Top Slice

• but Slice-timing Problem
• (Henson et al, 1999)
• Slices acquired at different times, yet
  model is the same for all slices
• gt different results (using canonical HRF) for
  different reference slices
• Solutions
• 1. Temporal interpolation of data but less
  good for longer TRs
• 2. More general basis set (e.g., with temporal
  derivatives) use a composite estimate,
  or use F-tests

TR3s
SPMt
SPMt
Interpolated
SPMt
Derivative
SPMF
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Timing Issues Latency

• Assume the real response, r(t), is a scaled (by
  ?) version of the canonical, f(t), but delayed
  by a small amount dt

r(t) ? f(tdt) ? f(t) ? f (t) dt
1st-order Taylor

• If the fitted response, R(t), is modelled by
  the canonical temporal derivative

R(t) ß1 f(t) ß2 f (t)
GLM fit

• If want to reduce estimate of BOLD impulse
  response to one composite value, with some
  robustness to latency issues (e.g, real, or
  induced by slice-timing)

• (similar logic applicable to other partial
  derivatives)

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Timing Issues Latency

• Assume the real response, r(t), is a scaled (by
  ?) version of the canonical, f(t), but delayed
  by a small amount dt

r(t) ? f(tdt) ? f(t) ? f (t) dt
1st-order Taylor

• If the fitted response, R(t), is modelled by
  the canonical temporal derivative

R(t) ß1 f(t) ß2 f (t)
GLM fit

• or if want to estimate latency directly
  (assuming 1st-order approx holds)

• ? ß1 dt ß2 / ß1

(Henson et al, 2002) (Liao et al, 2002)

• ie, Latency can be approximated by the ratio of
  derivative-to-canonical parameter estimates
  (within limits of first-order approximation,
  /-1s)

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Overview
1. BOLD impulse response 2. Convolution and
GLM 3. Temporal Basis Functions 4.
Filtering/Autocorrelation 5. Timing Issues 6.
Nonlinear Models
41
Nonlinear Model

• Volterra series - a general nonlinear
  input-output model
• y(t) ?1u(t) ?2u(t)
  .... ?nu(t) ....
• ?nu(t) ?.... ? hn(t1,..., tn)u(t -
  t1) .... u(t - tn)dt1 .... dtn

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Nonlinear Model

• Friston et al (1997)

kernel coefficients - h
SPMF p lt 0.001
SPMF testing H0 kernel coefficients, h 0
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Nonlinear Model

• Friston et al (1997)

kernel coefficients - h
SPMF p lt 0.001
SPMF testing H0 kernel coefficients, h
0 Significant nonlinearities at SOAs 0-10s
(e.g., underadditivity from 0-5s)
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Nonlinear Effects

Underadditivity at short SOAs
Linear Prediction
Volterra Prediction
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Nonlinear Effects

Underadditivity at short SOAs
Linear Prediction
Volterra Prediction
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Nonlinear Effects

Underadditivity at short SOAs
Linear Prediction
Implications for Efficiency
Volterra Prediction
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The End
This talk appears as Chapter 14 in the SPM
bookhttp//www.mrc-cbu.cam.ac.uk/rh01/Henson_Co
nv_SPMBook_2006_preprint.pdf
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Temporal Basis Sets Which One?
Group Analysis (N12)
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Timing Issues Latency
Face repetition reduces latency as well as
magnitude of fusiform response
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Timing Issues Latency
A. Decreased B. Advanced C. Shortened (same
integrated) D. Shortened (same maximum)
A. Smaller Peak B. Earlier Onset C. Earlier
Peak D. Smaller Peak and earlier Peak