Bayesian Modelling of Functional Imaging Data Will Penny The Wellcome Department of Imaging...

Bayesian Modelling of Bayesian Modelling of Functional Imaging DataFunctional Imaging DataBayesian Modelling of Bayesian Modelling of

Functional Imaging DataFunctional 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

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

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

4.4. The fMRI Inverse ProblemThe fMRI Inverse Problem





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

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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

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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 SelectionModel SelectionModel SelectionModel Selection

We need to compute the Bayesian Evidence:

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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:

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

Multiple Levels

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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:Use a General Linear Model:

y = X y = X + e + e

• The errors are modelled as an AR(p) processThe errors are modelled as an AR(p) process

• The order can be selected using Bayesian The order can be selected using Bayesian evidenceevidence

• Use a General Linear Model:Use a General Linear Model:

y = X y = X + e + e

• The errors are modelled as an AR(p) processThe errors are modelled as an AR(p) process

• The order can be selected using Bayesian The order can be selected using Bayesian evidenceevidence

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?




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




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 for the signal :Use a General Linear Model for the signal :

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 for the signal :Use a General Linear Model for the signal :

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 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 …

Simpler DesignsSimpler DesignsSimpler DesignsSimpler Designs

Canon. + Temp. Deriv Gammas

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

Left Occipital

Canon. + Temp. Deriv

Gammas

RFX-Event

FIR

Right Fusiform

Gammas

RFX-Event

FIR

Canon. + Temp. Deriv

Evidence

Model averaging to get peak post-stimulus response

NonStat NonStat

The fMRI Inverse ProblemThe fMRI Inverse ProblemThe fMRI Inverse ProblemThe fMRI Inverse Problem

• In EEG there is an ill-posed spatial inverse In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical problem. We wish to recover the electrical activity at a particular voxel from scalp activity at a particular voxel from scalp electrical activity.electrical activity.

• It is solved via modelling.It is solved via modelling.

• In fMRI there is an ill-posed temporal inverse In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at activity at a voxel from hemodynamic activity at that voxel.that voxel.

• In EEG there is an ill-posed spatial inverse In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical problem. We wish to recover the electrical activity at a particular voxel from scalp activity at a particular voxel from scalp electrical activity.electrical activity.

• It is solved via modelling.It is solved via modelling.

• In fMRI there is an ill-posed temporal inverse In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at activity at a voxel from hemodynamic activity at that voxel.that voxel.

HDM & DCM: Conceptual shiftHDM & DCM: Conceptual shiftHDM & DCM: Conceptual shiftHDM & DCM: Conceptual shift

• For a given subject and point in brain, the HRF For a given subject and point in brain, the HRF is fixed ! is fixed !

• Need two-stage models Need two-stage models

(i) How do experimental events affect neurodynamics ?(i) How do experimental events affect neurodynamics ?

A. Via a bilinear dynamical modelA. Via a bilinear dynamical model

(ii) How do neurodynamics affect hemodynamics ?(ii) How do neurodynamics affect hemodynamics ?

A. Via the balloon modelA. Via the balloon model

• For a given subject and point in brain, the HRF For a given subject and point in brain, the HRF is fixed ! is fixed !

• Need two-stage models Need two-stage models

(i) How do experimental events affect neurodynamics ?(i) How do experimental events affect neurodynamics ?

A. Via a bilinear dynamical modelA. Via a bilinear dynamical model

(ii) How do neurodynamics affect hemodynamics ?(ii) How do neurodynamics affect hemodynamics ?

A. Via the balloon modelA. Via the balloon model

Bilinear DynamicsBilinear DynamicsBilinear DynamicsBilinear Dynamics

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Neuronal Transients and BOLD: INeuronal Transients and BOLD: INeuronal Transients and BOLD: INeuronal Transients and BOLD: I

300ms 500ms

More enduring transients produce bigger BOLD signals

SecondsSeconds

Bigger transients produce bigger BOLD signals

The interaction changes the shape of the response

Neuronal Transients and BOLD: IINeuronal Transients and BOLD: IINeuronal Transients and BOLD: IINeuronal Transients and BOLD: II

BOLD is sensitive to frequencycontent of transients

Seconds

Seconds

Seconds

Relative timings of transients areamplified in BOLD

Inferences about Neuronal TransientsInferences about Neuronal TransientsInferences about Neuronal TransientsInferences about Neuronal Transients

CuuBzAzz U1,U2,F1,F2

F2

Z1 -

-+ Even for a single area we can ask eg.:

Does the second presentation of a familiar face

(a) increase the magnitude of the neuronal transient ?,

(b) increase its time constant ?

t

m(or fast v. slow responses)

ConclusionsConclusionsConclusionsConclusions

• Bayesian model selection and averaging can Bayesian model selection and averaging can help in the choice of signal and noise modelshelp in the choice of signal and noise models

• I have described some useful exploratary toolsI have described some useful exploratary tools

• Spatial ModelsSpatial Models

• Need to solve fMRI inverse problemNeed to solve fMRI inverse problem

• Bayesian model selection and averaging can Bayesian model selection and averaging can help in the choice of signal and noise modelshelp in the choice of signal and noise models

• I have described some useful exploratary toolsI have described some useful exploratary tools

• Spatial ModelsSpatial Models

• Need to solve fMRI inverse problemNeed to solve fMRI inverse problem

Gaussian-smoothed contrast imagesGaussian-smoothed contrast imagesGaussian-smoothed contrast imagesGaussian-smoothed contrast images

Wavelet-smoothed contrast imagesWavelet-smoothed contrast imagesWavelet-smoothed contrast imagesWavelet-smoothed contrast images

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Analogy: Processing in sensory cortex


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“Reagan”

Bayesian Modelling of Functional Imaging Data Will Penny The Wellcome Department of Imaging...

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