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SPM2: Modelling and Inference

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1 SPM2: Modelling and Inference
Will Penny K. Friston, J. Ashburner, J.-B. Poline, R. Henson, S. Kiebel, D. Glaser Wellcome Department of Imaging Neuroscience, University College London, UK

2 What’s new in SPM2 ? Spatial transformation of images Batch Mode
Modelling and Inference

3 SPM99 fMRI time-series Kernel Design matrix
Inference with Gaussian field theory Statistical parametric map (SPM) Realignment Smoothing General linear model Normalisation Adjusted regional data spatial modes and effective connectivity Template Parameter estimates

4 What’s new in SPM2 ? Spatial transformation of images Batch Mode
Modelling and Inference Expectation-Maximisation (EM) Restricted Maximum Likelihood (ReML) Parametric Empirical Bayes (PEB)

5 Hierarchical models Parametric Hierarchical Empirical model
Bayes (PEB) Hierarchical model Restricted Maximimum Likelihood (ReML) Single-level model

6 Bayes Rule

7 Example 2:Univariate model
Likelihood and Prior Posterior Relative Precision Weighting

8 Example 2:Multivariate two-level model
Likelihood and Prior Data-determined parameters Assume diagonal precisions Posterior Precisions Assume Shrinkage Prior

9 Covariance constraints
General Case: Arbitrary error covariances Covariance constraints

10 General Case EM algorithm Friston, K. et al. (2002), Neuroimage ( ) å
E-Step ( ) y C X T 1 - = e q h M-Step r for i and j { } { Q tr J g i j ij k å + l Friston, K. et al. (2002), Neuroimage

11 Pooling assumption Decompose error covariance at each voxel, i, into
a voxel specific term, r(i), and voxel-wide terms.

12 What’s new in SPM2 ? Corrections for Non-Sphericity
Posterior Probability Maps (PPMs) Haemodynamic modelling Dynamic Causal Modelling (DCM)

13 Non-sphericity Relax assumption that errors are Independent and Identically Distributed (IID) Non-independent errors eg. repeated measures within subject Non-identical errors eg. unequal condition/subject error variances Correlation in fMRI time series Allows multiple parameters at 2nd level ie. RFX

14 Single-subject contrasts from Group FFX
PET Verbal Fluency SPMs,p<0.001 uncorrected Single-subject contrasts from Group FFX Non-identical error variances Sphericity Non-sphericity

15 Correlation in fMRI time series
Model errors for each subject as AR(1) + white noise.

16 The Interface PEB OLS Parameters Parameters, and REML Hyperparameters
No Priors Shrinkage priors

17 Bayesian estimation: Two-level model
1st level = within-voxel Likelihood Shrinkage Prior 2nd level = between-voxels

18 Bayesian Inference: Posterior Probability Maps
PPMs Posterior Likelihood Prior SPMs

19 SPMs and PPMs

20 Sensitivity

21 Spatio-temporal modelling of PET data
Simulated data Spatio-temporal modelling of PET data Real data (Word fluency)

22 The hemodynamic model

23 Hemodynamics

24 Inference with MISO models
fMRI study of attention to visual motion

25 Dynamical Causal Models
Functional integration and the modulation of specific pathways V1 V4 BA37 STG BA39 Cognitive set - u2(t) {e.g. semantic processing} Stimuli - u1(t) {e.g. visual words}

26 Extension to a MIMO system
The bilinear model neuronal changes intrinsic connectivity induced response Input u(t) activity x1(t) x3(t) x2(t) hemodynamics response y(t)=(X) Hemodynamic model

27 Dynamical systems theory
Connections   {A,B,C} Inputs u(t) Kernels () Outputs y(t) EM algorithm E-Step M-Step Connectivity constraints C Inference p() >  Bayesian Inference

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