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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
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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Myp
MpMypMyp
The first level again, writing in dependence on M:
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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
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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
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
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?
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 SignalA model of fMRI time series: The Signal
4.4. The fMRI Inverse ProblemThe fMRI Inverse Problem
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 SignalA model of fMRI time series: The Signal
4.4. The fMRI Inverse ProblemThe fMRI Inverse Problem
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
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
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
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
CuuBzAzz -
Z2
Stimuliu1
Setu2
Z1
+
+
-
-
-+
u1
Z1
u2
Z2
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”