The General Linear Model and Statistical Parametric Mapping

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MEG Analysis in SPM Rik Henson (MRC CBU,
Cambridge) Jeremie Mattout, Christophe
Phillips, Stefan Kiebel, Olivier David, Vladimir
Litvak, ... Karl Friston (UCL, London)
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Overview

• 0. Standard preprocessing (filtering,
  epoching...)
• Random Field Theory for Space-Time images
• Empirical Bayesian approach to the Inverse
  Problem
• A Canonical Cortical mesh and Group Analyses
• Dynamic Causal Modelling (DCM)

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Overview

• Random Field Theory for Space-Time images
• Empirical Bayesian approach to the Inverse
  Problem
• A Canonical Cortical mesh and Group Analyses
• Dynamic Causal Modelling (DCM)

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(No Transcript)
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1. Localising in Space/Freq/Time

• Random Field Theory is a method for correcting
  for multiple statistical comparisons with
  N-dimensional spaces (for parametric statistics,
  eg Z-, T-, F- statistics)
• When is there an effect in time, eg GFP (1D)?
• Where is there an effect in time-frequency space
  (2D)?
• Where is there an effect in time-sensor space
  (3D)?
• Where is there an effect in time-source space
  (4D)?

Worsley Et Al (1996). Human Brain Mapping, 458-73
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Where is an effect in time-sensor space (3D)?

• F-statistic for ANOVA across EEG subjects (Henson
  et al, 2008, Neuroimage)
• MEG data first requires sensor-level realignment,
  using e.g, SSS...

Taylor Henson (2008) Biomag
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1. Localising in Space/Freq/Time

• Extended to 2D cortical mesh surface
• RFT generally requires Gaussian smoothing, but
  exerts exact FWE control for sufficient smoothing
  
• Nonparametric (permutation) methods of FWE
  control make fewer distributional assumptions (do
  not require smoothing), but do require
  exchangeability

Pantazis Et Al (2005) NeuroImage, 25383-394
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Overview

• Random Field Theory for Space-Time images
• Empirical Bayesian approach to the Inverse
  Problem
• A Canonical Cortical mesh and Group Analyses
• Dynamic Causal Modelling (DCM)

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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Weighted Minimum Norm, Regularisation
Linear system to be inverted
Y Data, n sensors x t1 time-samples J
Sources, p sources x t time-samples L Forward
model, n sensors x p sources E Multivariate
Gaussian noise, n x t Ce error covariance
over sensors
Since nltp, need to regularise, eg weighted
minimum (L2) norm (WMN)
W Weighting matrix W I
minimum norm W DDT coherent W
diag(LTL)-1 depth-weighted Wp
(LpTCy-1Lp)-1 SAM W
.
Phillips Et Al (2002) Neuroimage, 17, 287301
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Equivalent Bayesian Formulation
Equivalent Parametric Empirical Bayes
formulation
Y Data, n sensors x t1 time-samples J
Sources, p sources x t time-samples L Forward
model, n sensors x p sources C(e) covariance
over sensors C(j) covariance over sources
Posterior is product of likelihood and prior
W Weighting matrix W I
minimum norm W DDT coherent W
diag(LTL)-1 depth-weighted Wp
(LpTCy-1Lp)-1 SAM W
.
Maximal A Posteriori (MAP) estimate is


(Contrasting with Tikhonov)
Phillips Et Al (2005) Neuroimage, 997-1011
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Covariance Constraints (Priors)
How parameterise C(e) and C(j)?
Q (co)variance components (Priors) ?
estimated hyperparameters
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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Expectation-Maximisation (EM)
How estimate ?? . Use EM algorithm
to maximise the (negative) free energy (F)
(Note estimation in nxn sensor space)
Once estimated hyperparameters (iterated
M-steps), get MAP for parameters (single E-step)
(Can also estimate conditional covariance of
parameters, allowing inference)
Phillips et al (2005) Neuroimage
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Multiple Constraints (Priors)
Multiple constraints Smooth sources (Qs), plus
valid (Qv) or invalid (Qi) focal prior
Mattout Et Al (2006) Neuroimage, 753-767
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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Model Evidence
A (generative) model, M, is defined by the set of
Q(e), Q(j), L
The model log-evidence is bounded by the free
energy
Friston Et Al (2007) Neuroimage, 34, 220-34
(F can also be viewed the difference of an
accuracy term and a complexity term)
Two models can be compared using the Bayes
factor
Also useful when comparing different forward
models, ie Ls, Henson et al (submitted-b)
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Model Comparison (Bayes Factors)
Multiple constraints Smooth sources (Qs), plus
valid (Qv) or invalid (Qi) focal prior
Mattout Et Al (2006) Neuroimage, 753-767
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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Temporal Correlations
To handle temporally-extended solutions, first
assume temporal-spatial factorisation
In general, temporal correlation of signal
(sources) and noise (sensors) will differ, but
can project onto a temporal subspace (via S) such
that
Friston Et Al (2006) Human Brain Mapping,
27722735
V typically Gaussian autocorrelations
S typically an SVD into Nr temporal modes
Then turns out that EM can simply operate on
prewhitened data (covariance), where Y size n x
t


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Localising Power (eg induced)
Friston Et Al (2006) Human Brain Mapping,
27722735
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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Automatic Relevance Detection (ARD)
When have many constraints (Qs), pairwise model
comparison becomes arduous
Moreover, when Qs are correlated, F-maximisation
can be difficult (eg local maxima), and
hyperparameters can become negative (improper for
covariances)
Note Even though Qs may be uncorrelated in
source space, they can become correlated when
projected through L to sensor space (where F is
optimised)
Henson Et Al (2007) Neuroimage, 38, 422-38
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Automatic Relevance Detection (ARD)
When have many constraints (Qs), pairwise model
comparison becomes arduous
Moreover, when Qs are correlated, F-maximisation
can be difficult (eg local maxima), and
hyperparameters can become negative (improper for
covariances)
To overcome this, one can
1) impose positivity constraint on
hyperparameters
2) impose (sparse) hyperpriors on the
(log-normal) hyperparameters
Complexity
(where ? and S? are the posterior mean and
covariance of hyperparameters)
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Automatic Relevance Detection (ARD)
When have many constraints (Qs), pairwise model
comparison becomes arduous
Moreover, when Qs are correlated, F-maximisation
can be difficult (eg local maxima), and
hyperparameters can become negative (improper for
covariances)
Henson Et Al (2007) Neuroimage, 38, 422-38
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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Multiple Sparse Priors (MSP)
So why not use ARD to select from a large number
of sparse source priors.!?


Friston Et Al (2008) Neuroimage
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Multiple Sparse Priors (MSP)
So why not use ARD to select from a large number
of sparse source priors.!
Friston Et Al (2008) Neuroimage
No depth bias!
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2. Parametric Empirical Bayes (PEB)

• Weighted Minimum Norm Bayesian equivalent
• EM estimation of hyperparameters
  (regularisation)
• Model evidence and Model Comparison
• Spatiotemporal factorisation and Induced Power
• Automatic Relevance Detection (hyperpriors)
• Multiple Sparse Priors
• MEG and EEG fusion (simultaneous inversion)

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Fusion of MEG/EEG
Separate Error Covariance components for each of
i1..M modalities (Ci(e))
Data and leadfields scaled (with mi spatial
modes)
Remember, EM returns conditional precisions (S)
of sources (J), which can be used to compare
separate vs fused inversions
Henson Et Al (submitted-a)
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Fusion of MEG/EEG
Magnetometers (MEG)
Gradiometers (MEG)
Electrodes (EEG)
Fused
Henson Et Al (submitted-a)
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Overview

• Random Field Theory for Space-Time images
• Empirical Bayesian approach to the Inverse
  Problem
• A Canonical Cortical mesh and Group Analyses
• Dynamic Causal Modelling (DCM)

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3. Canonical Mesh Group Analyses

• A canonical (Inverse-normalised) cortical mesh
• Group analyses in 3D
• Use of fMRI spatial priors (in MNI space)
• Group-based inversions

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A Canonical Cortical Mesh
Given the difficulty in (automatically)
creating accurate cortical meshes from MRIs, how
about inverse-normalising a (quality) template
mesh in MNI space?
Ashburner Friston (2005) Neuroimage
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A Canonical Cortical Mesh
N1
Apply inverse of warps from spatial
normalisation of whole MRI to a template cortical
mesh
Mattout Et Al (2007) Comp. Intelligence
Neuroscience
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A Canonical Cortical Mesh
But warps from cortex not appropriate to
skull/scalp, so use individually (and easily)
defined skull/scalp meshes
N9

• Statistical tests of model evidence over N9 MEG
  subjects show
• MSP gt MMN
• BEMs gt Spheres (for CanInd)
• (7000 gt 3000 dipoles)
• (Normal gt Free for MSP)

Henson Et Al (submitted-b)
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3. Canonical Mesh Group Analyses

• A canonical (Inverse-normalised) cortical mesh
• Group analyses in 3D
• Use of fMRI spatial priors (in MNI space)
• Group-based inversions

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Group Analyses in 3D
Once have a 1-to-1 mapping from M/EEG source to
MNI space, can create 3D normalised images (like
fMRI) and use SPM machinery to perform
group-level classical inference
N19, MNI space, PseudowordsgtWords 300-400ms with
gt95 probability
Smoothed, Interpolated J
Taylor Henson (2008), Biomag
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3. Canonical Mesh Group Analyses

• A canonical (Inverse-normalised) cortical mesh
• Group analyses in 3D
• Use of fMRI spatial priors (in MNI space)
• Group-based inversions

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fMRI spatial priors
Group fMRI results in MNI space can be used as
spatial priors on individual source
space... ...importantly each fMRI cluster is
separate prior, so is weighted independently
Flandin Et Al (in prep)
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3. Canonical Mesh Group Analyses

• A canonical (Inverse-normalised) cortical mesh
• Group analyses in 3D
• Use of fMRI spatial priors (in MNI space)
• Group-based inversions

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Group-based source priors
Concatenate data and leadfields over i1..N
subjects
projecting data and leadfields to a reference
subject (0)


Common source-level priors
Subject-specific sensor-level priors
Litvak Friston (2008), Neuroimage
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Group-based source priors
N19, MNI space, PseudowordsgtWords, 300-400ms
with gt95 probability
Group Inversion
Individual Inversions
Taylor Henson (in prep)
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Summary

• SPM also implements Random Field Theory for
  principled correction of multiple comparisons
  over space/time/freq
• SPM implements a variant of the L2-distributed
  norm that
• effectively automatically regularises in
  principled fashion
• allows for multiple constraints (priors), valid
  invalid
• allows model comparison, or automatic relevance
  detection
• to the extent that multiple (100s) of sparse
  priors possible
• also offers a framework for MEGEEG fusion
• SPM can also inverse-normalise a template
  cortical mesh that
• obviates manual cortex meshing
• allows use of fMRI priors in MNI space
• allows using group constraints on individual
  inversions

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Overview

• Random Field Theory for Space-Time images
• Empirical Bayesian approach to the Inverse
  Problem
• A Canonical Cortical mesh and Group Analyses
• Dynamic Causal Modelling (DCM)

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Functional vs Effective Connectivity
Correlations A B C 1 0.49 1 0.30 0.12 1
B and C correlated because of common input from
A, eg A time-series B 0.5 A e1 C 0.3
A e2
B
A
C

• Effective connectivity is model-dependent
• Real interest is changes in effective
  connectivity induced by (experimental) inputs

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Basic DCM Approach
Design experimental inputs
Define likelihood model
Specify priors
Invert model
Make inferences
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1. Neural Dynamics
David et al. (2006) NeuroImage
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2. Observer Model

• Measurements assumed to reflect currents in
  (large) pyramidal cells (x0)

B
A

• One option is a small number of equivalent
  current dipoles (ECDs)

C

• Fix their locations, but allow orientations to
  be estimated as 3 parameters (q)

y1
y2
Kiebel et al. (2006) NeuroImage
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EEG example MisMatch Negativity (MMN)
Standards (1kHz)
Deviants (2kHz)
HEOG
VEOG
MMN deviants standards
gtSeed 5 ECDs
Garrido et al. (2007) NeuroImage
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EEG example MisMatch Negativity (MMN)
IFG
IFG
IFG
-
STG
STG
STG
STG
STG
STG
A1
A1
A1
A1
A1
A1
input
input
input
Forward
Forward
Forward
Backward
Backward
Backward
Lateral
Lateral
Lateral
Garrido et al. (2007) NeuroImage
53
EEG example MisMatch Negativity (MMN)
MisMatch reflects changes in forward and backward
connections Invalid top-down predictions fail
to suppress bottom-up prediction error?
Group-based posterior densities of connections in
FB model
IFG
0.93 (55)
1.41 (99)
STG
STG
1.74 (96)
4.50 (100)
2.41 (100)
5.40 (100)
A1
A1
input
Garrido et al. (2007) NeuroImage
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The End (Really)
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If want to try

• http//www.fil.ion.ucl.ac.uk/spm
• SPM5 Manual (/spm5/man/manual.pdf)
• http//www.fil.ion.ucl.ac.uk/spm/data/mmfaces.html
  

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Future Directions
Variational Bayes (VB), relaxing Gaussian
assumptions e.g, VB for ECD (Kiebel et al,
Neuroimage, 2007) Dynamic Causal Modelling
(DCM), for ECD or MSP Multi-level heirarchical
models (e.g, across subjects) Nonstationary
hyperparameters Proper Data Fusion (single
forward model from neural activity to both M/EEG
and fMRI)
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Optimal Regularisation
Single hyper-parameter for a coherent
(smoothness) constraint on sources (cf. LORETA)
Mattout Et Al (2006) Neuroimage, 753-767
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Where is an effect in time-frequency (2D)?
Kilner Et Al (2005) Neuroscience Letters 374,
174178
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Expectation Maximisation
How estimate J and ? simultaneously? Maximise
the free energy (F)
Y Data, n sensors x t time-samples ?
hyperparameter(s) q(J) any distribution over J
using EM algorithm, where E-step
Friston Et Al (2006) Neuroimage, 20, 220-234
and M-step is

• For Gaussian distributions, equivalent to ReML
  objective function

In practice, this gives ReML estimates of ?,
which can then be used to give MAP estimates of J
(via Cj and Ce)
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Temporal Correlations
To handle temporally-extended solutions, first
assume temporal-spatial factorisation
In general, temporal correlation of signal
(sources) and noise (sensors) will differ, but
can project onto a temporal subspace (via S) such
that
Friston Et Al (2006) Human Brain Mapping,
27722735
V typically Gaussian autocorrelations
S typically an SVD into Nr temporal modes
Then turns out that EM can simply operate on
prewhitened data (covariance), where Y size n x
t


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Localising Power (eg induced)
Similarly, can extend over trials as well as
samples, such that given i1..N trials, Yi
Can be shown that expected energy for one trial
in a time-frequency window defined by W
and total energy (induced and evoked) across
i1..N trials becomes
Friston Et Al (2006) Human Brain Mapping,
27722735
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Localising Power (eg induced)
Friston Et Al (2006) Human Brain Mapping,
27722735
63
Where is an effect in time-sensor space (3D)?

• SPM of F-statistic for EEG condition effect
  across subjects
• (NB MEG data requires sensor-level realignment,
  e.g SSS)

Henson Et Al (2008) Neuroimage