Bayesian Modelling of Functional Imaging Data Will Penny The Wellcome Department of Imaging Neuroscience, UCL http//:

OverviewOverview 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem

First level of Bayesian Inference First level of Inference: What are the best parameters ? We have data, y, and some parameters,  Parameters are of model, M, ….

First and Second Levels The first level again, writing in dependence on M: Second level of Inference: What’s the best model ?

Model Selection 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) Evidence = Accuracy - Complexity

Model Averaging Revisiting the first level: Model-dependent posteriors are weighted according to the posterior probability of each model

Multiple Levels w3w3 Y w1w1 w2w2 Evidence Up Posteriors Down w3w3 Y w1w1 w2w2

OverviewOverview 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem

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 Hz 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 Hz Remove with ICA/PCA – but non-automatic

fMRI time series model Use a General Linear Model:Use a General Linear Model: y = X  + e y = X  + 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 evidenceThe order can be selected using Bayesian evidence Use a General Linear Model:Use a General Linear Model: y = X  + e y = X  + 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 evidenceThe order can be selected using Bayesian evidence

Synthetic GLM-AR(3) Data

Map of AR model order, p p=0,1,2,3 Face Data

AngiogramsAngiograms

Other subjects, a 1 Ring of voxels with highly correlated error

Other subjects, a 1 Unmodelled signal or increased cardiac artifact due to increased blood flow?

OverviewOverview 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem

fMRI time series model Use a General Linear Model for the signal :Use a General Linear Model for the signal : y = X  + e y = X  + e Priors factorise into groups:Priors factorise into groups: p(  ) = p(  1 ) p(  2 ) p(  3 ) p(  ) = p(  1 ) p(  2 ) p(  3 ) Priors in each group may be smoothnessPriors in each group may be smoothness priors or Gaussians priors or Gaussians Use a General Linear Model for the signal :Use a General Linear Model for the signal : y = X  + e y = X  + e Priors factorise into groups:Priors factorise into groups: p(  ) = p(  1 ) p(  2 ) p(  3 ) p(  ) = p(  1 ) p(  2 ) p(  3 ) Priors in each group may be smoothnessPriors in each group may be smoothness priors or Gaussians priors or Gaussians

Rik’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 looking at factors influencing repetition suppresion Every face presented twice

Modelling 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 model Separate smoothness priors for each event type Design matrix for FIR model with 8 time bins in a 20-second window Q. Is this a good prior ?

FIR basis set Left occipital cortex (x=-33, y=-81, z=-24) FIR model average responses

FIR basis set Right fusiform cortex (x=45, y=-60, z=-18) FIR model average responses

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

Non-stationary models As RFX-event but smoothness priors Testing for smooth temporal variations statistically …

Simpler Designs Canon. + Temp. Deriv Gammas

Comparing 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

OverviewOverview 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem 1.Multiple levels of Bayesian Inference 2.A model of fMRI time series: The Noise 3.A model of fMRI time series: The Signal 4.The fMRI Inverse Problem

The fMRI Inverse Problem In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical activity at a particular voxel from scalp electrical activity.In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical activity at a particular voxel from scalp electrical activity. It is solved via modelling.It is solved via modelling. In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at that voxel.In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at that voxel. In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical activity at a particular voxel from scalp electrical activity.In EEG there is an ill-posed spatial inverse problem. We wish to recover the electrical activity at a particular voxel from scalp electrical activity. It is solved via modelling.It is solved via modelling. In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at that voxel.In fMRI there is an ill-posed temporal inverse problem. We wish to recover the electrical activity at a voxel from hemodynamic activity at that voxel.

HDM & DCM: Conceptual shift For a given subject and point in brain, the HRF is fixed !For a given subject and point in brain, the HRF is fixed ! Need two-stage modelsNeed two-stage models (i) How do experimental events affect neurodynamics ? (i) How do experimental events affect neurodynamics ? A. Via a bilinear dynamical model (ii) How do neurodynamics affect hemodynamics ? A. Via the balloon model For a given subject and point in brain, the HRF is fixed !For a given subject and point in brain, the HRF is fixed ! Need two-stage modelsNeed two-stage models (i) How do experimental events affect neurodynamics ? (i) How do experimental events affect neurodynamics ? A. Via a bilinear dynamical model (ii) How do neurodynamics affect hemodynamics ? A. Via the balloon model

Bilinear Dynamics - Z2Z2 Stimuli u 1 Set u 2 Z1Z u 1 Z 1 u 2 Z 2

Neuronal Transients and BOLD: I 300ms500ms More enduring transients produce bigger BOLD signals Seconds Bigger transients produce bigger BOLD signals The interaction changes the shape of the response

Neuronal Transients and BOLD: II BOLD is sensitive to frequency content of transients Seconds Relative timings of transients are amplified in BOLD

Inferences about Neuronal Transients U1,U2,F1,F2 F2 Z1Z 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t mm (or fast v. slow responses)

ConclusionsConclusions Bayesian model selection and averaging can help in the choice of signal and noise modelsBayesian model selection and averaging can help 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 help in the choice of signal and noise modelsBayesian model selection and averaging can help 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 images

Wavelet-smoothed contrast images

w3w3 Y w1w1 w2w2 Analogy: Processing in sensory cortex Evidence Up Posteriors Down w3w3 Y w1w1 w2w2 “Reagan”