Bayesian Methods in Brain Imaging

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Bayesian 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@

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Bayesian Methods in Brain Imaging. Will Penny. Thanks to Karl Friston & collaborators@. Wellcome Department of Imaging Neuroscience, University College London, UK. Institute for Adaptive and Neural Computation, University of Edinburgh, 27 January 2004. Overview. - PowerPoint PPT Presentation

Transcript of Bayesian Methods in Brain Imaging

Page 1: 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@

Page 2: Bayesian Methods in Brain Imaging

Overview

1. Bayesian Inference

2. Brain Imaging: Functional Segregation

3. Brain Imaging: Functional Integration

Page 3: Bayesian Methods in Brain Imaging

Overview

1. Bayesian Inference

2. Brain Imaging: Functional Segregation

3. Brain Imaging: Functional Integration

Page 4: Bayesian Methods in Brain Imaging

First level of Bayesian Inference

)(

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yp

pypyp

We have data, y, and some parameters,

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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

Page 6: Bayesian Methods in Brain Imaging

Two parameters

Page 7: Bayesian Methods in Brain Imaging

First level of Bayesian Inference

)(

)()|()|(

yp

pypyp

We have data, y, and some parameters,

Parameters are of model, M, ….

Page 8: Bayesian Methods in Brain Imaging

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 ?

Page 9: Bayesian Methods in Brain Imaging

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

Page 10: Bayesian Methods in Brain Imaging

Overview

1. Bayesian Inference

2. Brain Imaging: Functional Segregation

3. Brain Imaging: Functional Integration

Page 11: Bayesian Methods in Brain Imaging

realignment &motion

correction

smoothing

normalisation

General Linear Modelmodel fittingstatistic image

image data

Parameterestimates

designmatrix

anatomicalreference

kernel

Probabilistic map of activations

fMRI data

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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)

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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

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FIR models

Timeafterevent

Sizeof

signal

5s

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FIR model

Separate smoothness priors for each event type

Design matrixfor FIR model with

8 time bins in a 20-second window

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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

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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

Page 18: Bayesian Methods in Brain Imaging

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

wpenny
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Activation MapMain Effect of Faces

p(cT > 0.5%)

Structural MRI

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FIR basis set

Right fusiform cortex (x=45, y=-60, z=-18)

FIR model average responses

Larger response to first presentation – priming effect

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FIR basis setLeft occipital cortex

(x=-33, y=-81, z=-24)FIR model average responses

Priming effect only for unfamiliar faces

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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

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Synthetic GLM-AR(3) Data

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Map of AR model order, p

p=0,1,2,3

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Angiograms

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Current work – with Nelson Trujillo-Bareto

Spatial Priors – Laplace priors

u1 u2 q1 q2

z W a

Y

[TxN] = [TxK][KxN] + [TxN]

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Maps of regression coefficients – Image (wk)

TRUE ESTIMATE – w/o Spatial Prior

ESTIMATE

1024 regression coefficientsbut only 280 `effective’ coefficients.

Page 28: Bayesian Methods in Brain Imaging

Overview

1. Bayesian Inference

2. Brain Imaging: Functional Segregation

3. Brain Imaging: Functional Integration

Page 29: Bayesian Methods in Brain Imaging

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

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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.

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Network Analysis

IFG

SPC

V5

V1

Photic

Motion

Attention

Inputs andOutputs

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DCM: A network model for fMRI

CuzuBAz )(

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Input

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Friston et al. 2003CuzBuAzz i

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PRIORS ….

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Bilinear Dynamics: Positive transients

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Hemodynamics

Impulseresponse

BOLD is sluggish

),,(iiii

qvzgy

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Bayes factors: ( / )

( / )ij

p D iB

p D j

12 3.5B

1914 10B 13 2.8B

…with Andrea Mechelliand Klaas Stephan

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Current work

Stochastic Neurodynamics – with Zoubin G.

DCMs for EEG/ERPs – with Olivier David/Lee H.

EEG-fMRI sensor fusion …………..