Bayesian Methods in Brain Imaging
-
Upload
austin-barry -
Category
Documents
-
view
41 -
download
2
description
Transcript of Bayesian Methods in Brain Imaging
Bayesian Methods in Brain ImagingBayesian Methods in Brain Imaging
Will Penny
Wellcome Department of Imaging Neuroscience, University College London, UK
Institute for Adaptive and Neural Computation, University of Edinburgh, 27 January 2004.
Thanks to Karl Friston & collaborators@
Overview
1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration
Overview
1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration
First level of Bayesian Inference
)(
)()|()|(
yp
pypyp
We have data, y, and some parameters,
One parameterLikelihood and Prior
(1) (1) (1)
(1) (2) (2)
( | ) ( , )
( ) ( , )
p y N
p N
(1)
(1) (2)
(1) (2)(1) (2)
( | ) ( , )p y N m P
P
mP P
Posterior
)2( m )1(
Relative Precision Weighting
Prior
LikelihoodPosterior
Two parameters
First level of Bayesian Inference
)(
)()|()|(
yp
pypyp
We have data, y, and some parameters,
Parameters are of model, M, ….
First and Second Levels
)|(
)|(),|(),|(
Myp
MpMypMyp
The first level again, writing in dependence on M:
)(
)()|()|(
yp
MpMypyMp
Second level of Inference: What’s the best model ?
Model Selection
dpypMyp )()|()|(We need to compute the Bayesian Evidence:
We can’t always compute it exactly, but we can approximate it: Log p(y|M) ~ F(M)
Laplace Approximations
VariationalBayes
Overview
1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration
realignment &motion
correction
smoothing
normalisation
General Linear Modelmodel fittingstatistic image
image data
Parameterestimates
designmatrix
anatomicalreference
kernel
Probabilistic map of activations
fMRI data
Face processing data
24 Transverse Slices acquired with TR=2s
Press left key if famous, right key if not
Time series of 351 images
Part of larger study lookingat factors influencing
priming
Every face presented twice
R. Henson et al. (2001)
Modelling the Signal
Assumption: Neuronal Event Stream is Identical to the Experimental Event Stream
Convolve event-stream with basis functions to account for the hemodynamic response function
FIR models
Timeafterevent
Sizeof
signal
5s
FIR model
Separate smoothness priors for each event type
Design matrixfor FIR model with
8 time bins in a 20-second window
fMRI time series modelUse a General Linear Model
y = X + e
Priors factorise into groups:
p() = p(1) p(2) p(3)
Priors in each group may be smoothness priors or Gaussians
Noise sources in fMRI
1. Slow drifts due to instrumentation instabilities
2. Subject movement
3. Vasomotor oscillation ~ 0.1 Hz
4. Respiratory activity ~ 0.25 Hz
5. Cardiac activity ~ 1 HzRemove with ICA/PCA – but non-automatic
fMRI time series model – with Stefan Kiebel
Use a General Linear Model
y = X + e;
Priors factorise into groups:
p() = p(1) p(2) p(3)
Priors in each group may be smoothness priors or Gaussians
1
p
t i t i ti
e a e z
Activation MapMain Effect of Faces
p(cT > 0.5%)
Structural MRI
FIR basis set
Right fusiform cortex (x=45, y=-60, z=-18)
FIR model average responses
Larger response to first presentation – priming effect
FIR basis setLeft occipital cortex
(x=-33, y=-81, z=-24)FIR model average responses
Priming effect only for unfamiliar faces
RFX-Event model
Design Matrix
97 parameters ! But only 24 effective parameters
Responses to each event of type A are randomly distributed about some typical “type A” response
Synthetic GLM-AR(3) Data
Map of AR model order, p
p=0,1,2,3
Angiograms
Current work – with Nelson Trujillo-Bareto
Spatial Priors – Laplace priors
u1 u2 q1 q2
z W a
Y
[TxN] = [TxK][KxN] + [TxN]
Maps of regression coefficients – Image (wk)
TRUE ESTIMATE – w/o Spatial Prior
ESTIMATE
1024 regression coefficientsbut only 280 `effective’ coefficients.
Overview
1. Bayesian Inference
2. Brain Imaging: Functional Segregation
3. Brain Imaging: Functional Integration
Attention to Visual MotionAttention to Visual Motion
StimuliStimuli
250 radially moving dots at 4.7 degrees/s250 radially moving dots at 4.7 degrees/s
Pre-ScanningPre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)5 x 30s trials with 5 speed changes (reducing to 1%)
Task - detect change in radial velocityTask - detect change in radial velocity
ScanningScanning (no speed changes) (no speed changes)
6 normal subjects, 4 100 scan sessions;6 normal subjects, 4 100 scan sessions;
each session comprising 10 scans of 4 different conditioneach session comprising 10 scans of 4 different condition
e.g. F A F N F A F N S .................e.g. F A F N F A F N S .................
F – fixationF – fixation
S – stationary dots S – stationary dots
N – moving dotsN – moving dots
A – attended moving dotsA – attended moving dots
1. Photic Stimulation, S,N,A2. Motion, N,A3. Attention, A
Experimental Factors
Buchel et al. 1997
Motion Sensitive Areas
y = Xw + e
Mass Univariate Analyses
Where is effect, w, eg. of motion, significantlynon-zero.
Analysis of 360 images each containing 100,000 voxels, ie. 100,000 time series. New imageevery 3 seconds.
Network Analysis
IFG
SPC
V5
V1
Photic
Motion
Attention
Inputs andOutputs
DCM: A network model for fMRI
CuzuBAz )(
( , , )i i i iy g z v q e
Setu2
Stimuliu1
1111111 uczaz
5353333 zazaz
454353
5555
zaza
xaz
11c
21a
23a
54a242b
),( 11 qvg
1z
1y
),( 22 qvg
2z
2y),( 33 qvg
3z
3y
),( 44 qvg
4z
4y
),( 55 qvg
5z
5y
2242242545
4444
)( zbuaza
zaz
3223223121
2222
)( zbuaza
zaz
Input
State
Output
Friston et al. 2003CuzBuAzz i
ii
PRIORS ….
Bilinear Dynamics: Positive transients
-
Z2
Stimuliu1
Setu2
Z1
+
+
-
-
-+
u1
Z1
u2
Z2
CuzBuAzz i
ii
Hemodynamics
Impulseresponse
BOLD is sluggish
),,(iiii
qvzgy
Bayes factors: ( / )
( / )ij
p D iB
p D j
12 3.5B
1914 10B 13 2.8B
…with Andrea Mechelliand Klaas Stephan
Current work
Stochastic Neurodynamics – with Zoubin G.
DCMs for EEG/ERPs – with Olivier David/Lee H.
EEG-fMRI sensor fusion …………..