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Bayesian fMRI models with
Spatial Priors
Will Penny (1), Nelson Trujillo-Barreto (2)
Guillaume Flandin (1)
Stefan Kiebel(1), Karl Friston (1)
(1) Wellcome Department of Imaging Neuroscience, UCL
http://www.fil.ion.ucl.ac.uk/~wpenny
(2) Cuban Neuroscience Center, Havana, Cuba.
Motivation
Even without applied spatial smoothing, activation maps
(and maps of eg. AR coefficients) have spatial structure
Contrast
AR(1)
We can increase the sensitivity of our inferences by smoothing
data with Gaussian kernels (SPM2). This is worthwhile, but crude.
Can we do better with a spatial model (SPM5) ?
Aim: For SPM5 to remove the need for spatial smoothing just as
SPM2 removed the need for temporal smoothing
The Model
u1
u2
q1
q2
a
r2
r1
b
Spatial
Priors
l
W
Y
Y=XW+E
[TxN] [TxK] [KxN] [TxN]
Spatial
Priors
A
Spatio-Temporal Model for fMRI
Laplacian Prior on
regression coefficients W
Spatial domain
General Linear Model
Time domain
Spatial domain, voxels g=1..G
Time domain, at voxel g
W
Laplacian Prior:
Spatial operator D
Spatial precision a
General Linear Model:
Data yg
Design matrix X
Temporal precision lg
Yg=Xwg+eg
Spatial domain, voxels g=1..G
Time domain, at voxel g
rg
Spatial operator D
Spatial precision a
g
1
Data yg
Design matrix X
Temporal precision lg
 l g X X  diag (a ) D gg
T
w g   g ( l g X y g  diag (a ) r g )
T
SPATIO-TEMPORAL DECONVOLUTION
Synthetic Data: blobs
True
Smoothing
Spatial prior
Sensitivity
1-Specificity
Face data
Convolve event-stream with basis functions to account for
the hemodynamic response function
Event-related fMRI: Faces versus chequerboard
Smoothing
Spatial Prior
Event-related fMRI: Familiar faces versus unfamiliar faces
Smoothing
Spatial Prior
PAPER - 1
W. Penny, S. Kiebel and K. Friston (2003)
Variational Bayesian Inference for fMRI
time series. NeuroImage 19, pp 727-741.
• GLMs with voxel-wise AR(p) models
• Model order selection shows p=0,1,2 or 3 is sufficient
• Voxel-wise AR(p) modelling can improve effect size
estimation accuracy by 15%
PAPER - 2
W.Penny, N. Trujillo-Barreto and K. Friston
(2005). Bayesian fMRI time series
analysis with spatial priors. NeuroImage
24(2), pp 350-362.
• Spatial prior for regression coefficients
• Shown to be more sensitive than ‘smoothing the data’
PAPER - 3
W.Penny and G. Flandin (2005). Bayesian
analysis of single-subject fMRI: SPM
implementation. Technical Report. WDIN,
UCL.
• Describes more efficient implementation of algorithm in papers
1 and 2.
• Describes how contrasts are evaluated when specified post-hoc
• Roadmap to code (? Erm, OK, not done yet)
PAPER - 4
W.Penny, G. Flandin and N.Trujillo-Barreto.
Bayesian Comparison of Spatially
Regularised General Linear Models.
Human Brain Mapping, Accepted for
publication.
• ROI-based Bayesian model comparison for selecting optimal
hemodynamic basis set.
• Above Bayesian approach can be twice as sensitive as the
classical F-test method
• Defined spatial priors for AR coefficients
• Bayesian model comparison shows these to be better than
(i) global AR value, (ii) tissue-specific AR values
Using model evidence to select hemodynamic basis sets
Optimality of non-nested model comparison
Bayesian
Cluster of
Interest
Analysis
Non-nested
model
comparison
Nested Model
Comparison
FIR basis (truth) versus Inf-3 basis in low SNR environment
Using model evidence to select spatial noise model
Tissue
Specific
Priors
Spatial
Smoothness
Priors
PAPER - 5
W. Penny. Bayesian Analysis of fMRI data with spatial
Priors. To appear in Proceedings of the Joint Statistical
Meeting (JSM), 2005.
• Describes Bayesian inference for multivariate contrasts based
on chi-squared statistics
• Describes `default thresholds’ for generating PPMs: effect size
threshold is 0, probability threshold is 1-1/S where S is the number
of voxels in the search volume
• This gives approximately 0, 1 or 2 False Positives per PPM
• Describes a recent bug-fix !!! ?
PAPER - 6
G. Flandin and W. Penny. Bayesian
Analysis of fMRI data with spatial
basis set priors. Proceedings of Human
Brain Mapping Conference, 2005.
• Uses spatial prior based on spatial basis functions eg.
wavelets
• This is much faster than previous approach
• And provides yet more sensitivity …… ?!!
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