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

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One parameterLikelihood and Prior

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(1) (2) (2)

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Relative Precision Weighting

Prior

LikelihoodPosterior

Two parameters

First level of Bayesian Inference

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We have data, y, and some parameters,

Parameters are of model, M, ….

First and Second Levels

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The first level again, writing in dependence on M:

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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

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Bilinear Dynamics: Positive transients

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Hemodynamics

Impulseresponse

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Bayes factors: ( / )

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12 3.5B

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…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 …………..