Probabilistic Modelling of Brain Imaging Data

Probabilistic Modelling of Probabilistic Modelling of Brain Imaging DataBrain Imaging Data

Probabilistic Modelling of Probabilistic Modelling of Brain Imaging DataBrain Imaging Data

Will PennyWill Penny

The Wellcome Department of Imaging Neuroscience, The Wellcome Department of Imaging Neuroscience, UCLUCL

http//:www.fil.ion.ucl.ac.uk/~wpennyhttp//:www.fil.ion.ucl.ac.uk/~wpenny

OverviewOverviewOverviewOverview

1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference

2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise

3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal

1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference

2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise

3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal

First level of Bayesian InferenceFirst level of Bayesian InferenceFirst level of Bayesian InferenceFirst level of Bayesian Inference

)(

)()|()|(

yp

pypyp

First level of Inference: What are the best parameters ?

We have data, y, and some parameters,

Parameters are of model, M, ….

First and Second LevelsFirst and Second LevelsFirst and Second LevelsFirst and Second Levels

)|(

)|(),|(),|(

Myp

MpMypMyp

The first level again, writing in dependence on M:

)(

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yp

MpMypyMp

Second level of Inference: What’s the best model ?

Model SelectionModel SelectionModel SelectionModel Selection

We need to compute the Bayesian Evidence:

dpypMyp )()|()|(

We can’t always compute it exactly, but we can approximate it: Log p(y|M) ~ F(M)

Evidence = Accuracy - Complexity

Model AveragingModel AveragingModel AveragingModel Averaging

Revisiting the first level:

)|(),|()|( yMpMypypM

Model-dependent posteriors are weighted accordingto the posterior probability of each model

Multiple Levels

12112 )|()|()|( dwwwpwYpwYp

23223 )|()|()|( dwwwpwYpwYp

22211 )|(),|()|( dwYwpwYwpYwp

33322 )|(),|()|( dwYwpwYwpYwp

)(

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333

Yp

wpwYpYwp

w3

Y

w1

w2

333 )()|()( dwwpwYpYp

Evidence Up Posteriors Down

w3

Y

w1

w2

OverviewOverviewOverviewOverview

1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference

2. 2. A model of fMRI time series: The NoiseA model of fMRI time series: The Noise

3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal

1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference

2. 2. A model of fMRI time series: The NoiseA model of fMRI time series: The Noise

3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal

Noise sources in fMRINoise sources in fMRINoise sources in fMRINoise sources in fMRI

1. Slow drifts due to instrumentation instabilities1. Slow drifts due to instrumentation instabilities

2. Subject movement2. Subject movement

3. 3. Vasomotor oscillation ~ 0.1 HzVasomotor oscillation ~ 0.1 Hz

4. Respiratory activity ~ 0.25 Hz4. Respiratory activity ~ 0.25 Hz

5. Cardiac activity ~ 1 Hz5. Cardiac activity ~ 1 Hz

1. Slow drifts due to instrumentation instabilities1. Slow drifts due to instrumentation instabilities

2. Subject movement2. Subject movement

3. 3. Vasomotor oscillation ~ 0.1 HzVasomotor oscillation ~ 0.1 Hz

4. Respiratory activity ~ 0.25 Hz4. Respiratory activity ~ 0.25 Hz

5. Cardiac activity ~ 1 Hz5. Cardiac activity ~ 1 Hz

Remove with ICA/PCA – but non-automatic

fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model

• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:

y = X y = X + e + e

where X contains task-related regressors.where X contains task-related regressors.

• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:

y = X y = X + e + e

where X contains task-related regressors.where X contains task-related regressors.

fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model

= +

y X e

Time-seriesat one spatial

location

Putative effects ofexperimentalmanipulation

Sizeof effects

Residuals

fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model

• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:

y = X y = X + e + e

where X contains task-related regressors.where X contains task-related regressors.

• The errors are modelled as an AR(p) process.The errors are modelled as an AR(p) process. ((Parametric spectral estimationParametric spectral estimation))

• The order can be selected using Bayesian evidenceThe order can be selected using Bayesian evidence

• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:

y = X y = X + e + e

where X contains task-related regressors.where X contains task-related regressors.

• The errors are modelled as an AR(p) process.The errors are modelled as an AR(p) process. ((Parametric spectral estimationParametric spectral estimation))

• The order can be selected using Bayesian evidenceThe order can be selected using Bayesian evidence

Synthetic GLM-AR(3) DataSynthetic GLM-AR(3) DataSynthetic GLM-AR(3) DataSynthetic GLM-AR(3) Data

Map of AR model order, pMap of AR model order, pMap of AR model order, pMap of AR model order, p

p=0,1,2,3FaceData

AngiogramsAngiogramsAngiogramsAngiograms

Other subjects, aOther subjects, a11Other subjects, aOther subjects, a11

Ring ofvoxels with

highly correlatederror

Other subjects, aOther subjects, a11Other subjects, aOther subjects, a11

Unmodelledsignal

orincreasedcardiac

artifact due to increasedblood flow?

OverviewOverviewOverviewOverview

1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference

2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise

3. 3. A model of fMRI time series: The SignalA model of fMRI time series: The Signal

1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference

2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise

3. 3. A model of fMRI time series: The SignalA model of fMRI time series: The Signal

fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model

• Use a General Linear Model Use a General Linear Model

y = X y = X + e + e

• Priors factorise into groups:Priors factorise into groups:

p(p() = p() = p(11) p() p(22) p() p(33))

• Priors in each group may be smoothness Priors in each group may be smoothness priors or Gaussianspriors or Gaussians

• Use a General Linear Model Use a General Linear Model

y = X y = X + e + e

• Priors factorise into groups:Priors factorise into groups:

p(p() = p() = p(11) p() p(22) p() p(33))

• Priors in each group may be smoothness Priors in each group may be smoothness priors or Gaussianspriors or Gaussians

Rik’s dataRik’s dataRik’s dataRik’s 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 repetition suppresion

Every face presented twice

Modelling the SignalModelling the SignalModelling the SignalModelling the Signal

Assumption: Neuronal Event Stream is Identical to the Experimental Event Stream

Convolve event-stream with basis functions to account for the HRF

FIR modelsFIR modelsFIR modelsFIR models

Timeafterevent

Sizeof

signal

FIR modelFIR modelFIR modelFIR model

Separate smoothness priors for each event type

Design matrixfor FIR model with

8 time bins in a 20-second window

Q. Is this a good prior ?

FIR basis setFIR basis setFIR basis setFIR basis set

Left occipital cortex (x=-33, y=-81, z=-24)

FIR model average responses

FIR basis setFIR basis setFIR basis setFIR basis set

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

FIR model average responses

RFX-Event modelRFX-Event modelRFX-Event modelRFX-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

Non-stationary modelsNon-stationary modelsNon-stationary modelsNon-stationary models

As RFX-event but smoothness priors

Testing for smooth temporal variations statistically …

Comparing Types of ModelsComparing Types of ModelsComparing Types of ModelsComparing Types of Models

Left Occipital Right FusiformEvidence

Model averaging to get peak post-stimulus response

RFX-Event

FIR

NonStat

RFX-Event

FIR

NonStat

SummarySummarySummarySummary

• Bayesian inference provides a framework for Bayesian inference provides a framework for

model comparison and synthesismodel comparison and synthesis

• Appropriate for fMRI as we have some prior Appropriate for fMRI as we have some prior knowledgeknowledge

• We have focussed on temporal modelsWe have focussed on temporal models

• Bayesian inference provides a framework for Bayesian inference provides a framework for

model comparison and synthesismodel comparison and synthesis

• Appropriate for fMRI as we have some prior Appropriate for fMRI as we have some prior knowledgeknowledge

• We have focussed on temporal modelsWe have focussed on temporal models