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