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					                    DCM: Advanced Topics


Klaas Enno Stephan                                                                          0.4
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                                                                                                            Neural population activity
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                                                                        u1        x3
Translational Neuromodeling Unit (TNU)
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Institute for Biomedical Engineering
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University of Zurich & Swiss Federal                                    x1             x2
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Institute of Technology (ETH) Zurich
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                                                                                                                fMRI signal change (%)
                                                                                             1

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                                         dx        m             n            
Wellcome Trust Centre for Neuroimaging
                                                                                             1



                                             A   ui B (i )   x j D ( j )  x  Cu
                                                                                             0




                                         dt                                   
                                                                                             -1
                                                                                                  0   10   20    30   40   50   60   70   80   90   100



                                                  i 1          j 1                       3




Institute of Neurology
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University College London
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                       SPM Course 2012 @ FIL London
                               18 May 2012
                         Overview


• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions
Dynamic Causal Modeling (DCM)

            Hemodynamic                         Electromagnetic
            forward model:                       forward model:
            neural activityBOLD            neural activityEEG
                                                            MEG
                                                             LFP


                                 Neural state equation:
                                     dx
                                         F ( x , u,  )
 fMRI                                dt                               EEG/MEG

simple neuronal model                                      complicated neuronal model
complicated forward model                                        simple forward model



                            inputs
Generative models & model selection

• any DCM = a particular generative model of how the data (may)
  have been caused

• modelling = comparing competing hypotheses about the
  mechanisms underlying observed data
    a priori definition of hypothesis set (model space) is crucial
    determine the most plausible hypothesis (model), given the
     data

• model selection  model validation!
    model validation requires external criteria (external to the
     measured data)
Model comparison and selection
Given competing hypotheses
on structure & functional
mechanisms of a system, which
model is the best?




Which model represents the
best balance between model
fit and model complexity?




For which model m does p(y|m)
become maximal?
                                 Pitt & Miyung (2002) TICS
Bayesian model selection (BMS)

Model evidence:                                                             Gharamani, 2004


p( y | m)   p( y |  , m) p( | m) d




                                          p(y|m)
         accounts for both accuracy                                     y
         and complexity of the
         model
                                                          all possible datasets
         allows for inference about
         structure (generalisability)
         of the model
                                                   Various approximations, e.g.:
                                                   - negative free energy, AIC, BIC

                                                                      McKay 1992, Neural Comput.
                                                                     Penny et al. 2004a, NeuroImage
Approximations to the model evidence in DCM

Logarithm is a                        Maximizing log model evidence
monotonic function                    = Maximizing model evidence

Log model evidence = balance between fit and complexity
       log p( y | m)  accuracy ( m)  complexity ( m)
                      log p( y |  , m)  complexity ( m)
                                                                         No. of
                                                                         parameters
SPM2 & SPM5 offered 2 approximations:

Akaike Information Criterion:     AIC  log p( y |  , m)  p            No. of
                                                                         data points
                                                            p
Bayesian Information Criterion:   BIC  log p( y |  , m)  log N
                                                            2
                                                            Penny et al. 2004a, NeuroImage
                                                            Penny 2012, NeuroImage
The (negative) free energy approximation

• Under Gaussian assumptions about the posterior (Laplace
  approximation):
  log p( y | m)
   log p( y |  , m)  KL q   , p  | m   KL q   , p  | y, m 
                                                                          


   F  log p( y | m)  KL  q   , p  | y, m  
                                                  
        log p( y |  , m)  KL  q   , p  | m  
                                                     
                   accuracy                        complexity
The complexity term in F
• In contrast to AIC & BIC, the complexity term of the negative
  free energy F accounts for parameter interdependencies.

    KLq ( ), p ( | m)

     ln C  ln C | y   | y    C  | y   
     1            1       1             T 1

     2            2       2

• The complexity term of F is higher
   – the more independent the prior parameters ( effective DFs)
   – the more dependent the posterior parameters
   – the more the posterior mean deviates from the prior mean

• NB: Since SPM8, only F is used for model selection !
Bayes factors
To compare two models, we could just compare their log
evidences.

But: the log evidence is just some number – not very intuitive!

A more intuitive interpretation of model comparisons is made
possible by Bayes factors:
                                                positive value, [0;[
                              p ( y | m1 )
                        B12 
                              p ( y | m2 )
                                               B12       p(m1|y)    Evidence
                                              1 to 3     50-75%          weak
                                              3 to 20    75-95%      positive
Kass & Raftery classification:               20 to 150   95-99%      strong
                                               150        99%    Very strong
Kass & Raftery 1995, J. Am. Stat. Assoc.
                                    BMS in SPM8: an example
                                    attention
                    M1                                                           M2
                                         PPC                                                        PPC
                                                       M2 better than M1
                                                                                 attention
                                                           BF 2966
                    stim       V1         V5               F = 7.995            stim        V1     V5


M1   M2   M3   M4
                    M3     attention
                                         PPC
                                                             M3 better than M2
                                                                    BF  12
                    stim       V1         V5                        F = 2.450


                                                                                 M4     attention
                                                                                                    PPC
                                                M4 better than M3
                                                    BF  23
                                                    F = 3.144                   stim        V1     V5
Fixed effects BMS at group level

Group Bayes factor (GBF) for 1...K subjects:

       GBFij   BFij( k )
                  k


Average Bayes factor (ABF):

      ABFij  K  BF     (k )
                        ij
                  k

Problems:
- blind with regard to group heterogeneity
- sensitive to outliers
Random effects BMS for heterogeneous groups

                               Dirichlet parameters 
                                = “occurrences” of models in the population



       r ~ Dir(r; )            Dirichlet distribution of model probabilities r




 mk ~ p (mk | p )               Multinomial distribution of model labels m
    mk ~ p (mk | p )
       mk ~ p (mk | p )
         m1 ~ Mult (m;1, r )                                      Model inversion
                                                                  by Variational
 y1 ~ p( y1 | m1 )                                                Bayes (VB) or
    y1 ~ p( y1 | m1 )           Measured data y                   MCMC
       y2 ~ p( y2 | m2 )
            y1 ~ p( y1 | m1 )
                                                        Stephan et al. 2009a, NeuroImage
                                                        Penny et al. 2010, PLoS Comp. Biol.
                                           LD                                                          LD|LVF
    m2                                                                                                                              m1

                    MOG         FG              FG           MOG                         MOG     FG         FG           MOG

           LD|RVF                                                  LD|LVF           LD                                         LD


                                LG              LG                                               LG         LG



                          RVF        LD              LVF                                       RVF LD|RVF        LVF
                          stim.                      stim.                                     stim.             stim.




                                                m2                                                m1
Subjects




                                                                                     Data:   Stephan et al. 2003, Science
                                                                                     Models: Stephan et al. 2007, J. Neurosci.
           -35      -30   -25        -20        -15          -10     -5     0   5
                           Log model evidence differences
                                                    p(r >0.5 | y) = 0.997
                                                          1
                        5

                       4.5

                        4

                       3.5         m2         pr1  r2   99 .7%                         m1
                        3
            p(r 1|y)




                       2.5

                        2

                       1.5

                        1
                                   2  2.2                                   1  11.8
                       0.5
                                    r2  15.7%                                 r1  84.3%
                        0
                         0   0.1        0.2   0.3   0.4       0.5       0.6    0.7   0.8   0.9   1
                                                                r
                                                                    1

Stephan et al. 2009a, NeuroImage
                                                       definition of model space



                                      inference on model structure or inference on model parameters?




                             inference on                                                   inference on
             individual models or model space partition?           parameters of an optimal model or parameters of all models?




  optimal model structure assumed       comparison of model           optimal model structure assumed             BMA
   to be identical across subjects?        families using              to be identical across subjects?
                                         FFX or RFX BMS

                                                                       yes                      no
      yes                   no

                                                                    FFX BMS                RFX BMS




  FFX BMS                 RFX BMS                         FFX analysis of                    RFX analysis of
                                                        parameter estimates                parameter estimates
                                                            (e.g. BPA)                     (e.g. t-test, ANOVA)

Stephan et al. 2010, NeuroImage
                                                              nonlinear                     linear
FFX            log                                      80

                                                                                                                                     Model space partitioning:

                   Summed log evidence (rel. to RBML)
               GBF

                                                                                                                                     comparing model families
                                                        60




                                                        40




                                                        20




                                                                                                                                                      p(r >0.5 | y) = 0.986
                                                         0                                                                                                  1
                                                             CBMN CBMN(ε) RBMN RBMN(ε) CBML CBML(ε) RBML RBML(ε)               5


RFX                                                    12
                                                                                                                              4.5

                                                        10
                                                                                                                               4
                                                                                                                                    m2                                                        m1
                                                        8

                                                                                                                              3.5


                                                                                                                                           p  r1  r2   98.6%
                   alpha




                                                        6

                                                                                                                               3
                                                        4




                                                                                                                   p(r 1|y)
                                                                                                                              2.5
                                                        2



                                                        0                                                                      2
                                                             CBMN CBMN(ε) RBMN RBMN(ε) CBML CBML(ε) RBML RBML(ε)

                                                                                                                              1.5


Model                                                  16                                                                     1
                                                                                                                                     r2  26.5%                             r1  73.5%
space
                                                        14


                                                        12
                                                                    m1                        m2                              0.5

partitioning                                            10                                                                     0
                                                                          4
                                                                                                                                0   0.1   0.2   0.3   0.4       0.5       0.6   0.7   0.8   0.9   1
                   alpha




                                                        8           1*    k                                                                                   r
                                                                         k 1                                                                                         1
                                                        6


                                                        4
                                                                                                   8
                                                                                             2  k
                                                                                              *
                                                        2
                                                                                                  k 5

                                                        0
                                                                                                                                                                 Stephan et al. 2009, NeuroImage
                                                             nonlinear models           linear models
   Bayesian Model Averaging (BMA)

• abandons dependence of parameter
  inference on a single model               p  n | y1.. N 
• uses the entire model space
  considered (or an optimal family of         p  n | yn , m  p  m | y1.. N 
  models)                                      m

• computes average of each parameter,            NB: p(m|y1..N) can be obtained
  weighted by posterior model                    by either FFX or RFX BMS
  probabilities
• represents a particularly useful
  alternative
    – when none of the models (or model
      subspaces) considered clearly
      outperforms all others
    – when comparing groups for which the
      optimal model differs                              Penny et al. 2010, PLoS Comput. Biol.
                        Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions
DCM10 in SPM8
• DCM10 was released as part of SPM8 in July 2010 (version 4010).

• Introduced many new features, incl. two-state DCMs and stochastic DCMs

• This led to various changes in model defaults, e.g.
    – inputs mean-centred
    – changes in coupling priors
    – self-connections estimated separately for each area

• For details, see:
  www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf

• Further changes in version 4290 (released April 2011) to accommodate new
  developments and give users more choice (e.g., whether or not to mean-
  centre inputs).
The evolution of DCM in SPM
• DCM is not one specific model, but a framework for Bayesian inversion of
  dynamic system models

• The default implementation in SPM is evolving over time
   – improvements of numerical routines (e.g., for inversion)
   – change in priors to cover new variants (e.g., stochastic DCMs,
     endogenous DCMs etc.)


            To enable replication of your results, you should ideally state
            which SPM version (release number) you are using when
            publishing papers.

            In the next SPM version, the release number will be stored in
            the DCM.mat.
          y                  y               y           BOLD                y
                      
                                                                           λ
                                                                                       hemodynamic
                          activity                                                     model
                           x2(t)          activity
       activity                            x3(t)
        x1(t)
                                         neuronal                            x
                                         states
                  modulatory                                                     integration
                  input u2(t)


 driving                             t               Neural state equation   x  ( A  u j B( j ) ) x  Cu
                                                                             
 input u1(t)
                                                                                                  x
                                                                                                   
                                                     endogenous                            A
                                                     connectivity                                 x
                                                     modulation of                                  x 
                  t
                                                                                        B( j)   
                                                     connectivity                                 u j x
The classical DCM:
                                                                                                  x
                                                                                                   
a deterministic, one-state,                          direct inputs                        C
                                                                                                  u
bilinear model
Factorial structure of model specification in DCM10
• Three dimensions of model specification:
   – bilinear vs. nonlinear
   – single-state vs. two-state (per region)
   – deterministic vs. stochastic

• Specification via GUI.
            bilinear DCM                                          non-linear DCM
                                                     modulation

 driving                                         driving
 input                                           input




           modulation


Two-dimensional Taylor series (around x0=0, u0=0):
    dx                             f    f    2 f      2 f x2
        f ( x, u )  f ( x0 ,0)     x    u      ux  ... 2    ...
    dt                             x    u    xu       x 2

Bilinear state equation:                     Nonlinear state equation:

      dx             (i ) 
                                             dx       m          n
                                                                             
                                                 A   ui B   x j D ( j )  x  Cu
                 m
          A   ui B  x  Cu                           (i )

                                             dt                             
      dt       i 1                                 i 1       j 1        
                                                 0.4             Neural population activity
                                                 0.3
                                                 0.2


                                       u2        0.1
                                                  0
                                                       0   10   20    30   40   50   60   70   80   90   100

                                                 0.6

                                                 0.4


                                  u1   x3        0.2

                                                  0
                                                       0   10   20    30   40   50   60   70   80   90   100


                                                 0.3

                                                 0.2

                                                 0.1

                                                  0


                                  x1        x2
                                                       0   10   20    30   40   50   60   70   80   90   100


                                                  3

                                                  2                  fMRI signal change (%)
                                                  1

                                                  0

                                                       0   10   20    30   40   50   60   70   80   90   100

Nonlinear dynamic causal model (DCM)              4
                                                  3
                                                  2


dx        m          n                          1



    A   ui B   x j D  x  Cu
                (i )      ( j)                    0




dt                            
                                                  -1
                                                       0   10   20    30   40   50   60   70   80   90   100


         i 1       j 1                        3

                                                  2

                                                  1

                                                  0
Stephan et al. 2008, NeuroImage                        0   10   20    30   40   50   60   70   80   90   100
               attention
                                                                   MAP = 1.25
                 0.10
                                                0.8


                                                0.7

                   PPC                          0.6

                             0.26               0.5

                                         0.39
                           1.25                 0.4

       0.26
stim            V1         0.13                 0.3

                                    V5          0.2

                   0.46
                                                0.1
                          0.50
                                                 0
                                                  -2      -1   0      1    2    3   4   5




                 motion                                p( DV 5,V 1  0 | y)  99.1%
                                                           PPC




Stephan et al. 2008, NeuroImage
   Two-state DCM
                      Single-state DCM                           Two-state DCM


         input
           u                                                                                          x1E
                                          x1                                         x1E
                                                                                     x1I
                                                                                                      x1I




                                                                        x  x  Cu
                                                                        
                       x  x  Cu
                                                                 ij  exp( Aij   uBij  )
                                                                 ij
                                                                                          


                      ij  Aij  uBij
                                                       11
                                                          EE
                                                                  11  1N
                                                                   EI     EE
                                                                                            0         x1E 
                                                        IE                                           I
               11  1N                  x1        11      11
                                                                    II
                                                                          0                 0         x1 
              
                                     x  
                                                                                           x  
                                                        EE                                           E
               N 1   NN 
                                          xN 
                                                      N 1      0     EE
                                                                          NN               EE 
                                                                                             NN        xN 
                                                        0         0   NN
                                                                          IE                 II 
                                                                                            NN      xI 
                                                                                                      N
                                                         Extrinsic                Intrinsic
                                                     (between-region)          (within-region)
Marreiros et al. 2008, NeuroImage                        coupling                 coupling
                                                                 Estimates of hidden causes and states
   Stochastic DCM                                                        (Generalised filtering)
                                                                                  inputs or causes - V2
                                                        1


                                                       0.5

     dx
         ( A   j u j B ( j ) ) x  Cv   ( x )      0


                                                      -0.5

     dt                                                 -1
                                                             0       200    400              600          800   1000            1200



      v  u   (v)                                    0.1
                                                                             hidden states - neuronal
                                                                                                                       excitatory
                                                     0.05                                                              signal


                                                        0

    • all states are represented in generalised      -0.05

      coordinates of motion                           -0.1
                                                             0       200    400              600          800   1000            1200


    • random state fluctuations w(x) account for       1.3
                                                                           hidden states - hemodynamic

      endogenous fluctuations,                         1.2
                                                                                                                         flow
                                                                                                                         volume
                                                                                                                         dHb
      have unknown precision and smoothness            1.1


       two hyperparameters                             1

                                                       0.9



    • fluctuations w(v) induce uncertainty about
                                                       0.8
                                                             0       200    400              600          800   1000            1200



      how inputs influence neuronal activity            2
                                                                                  predicted BOLD signal
                                                                                                                       observed
                                                        1

    • can be fitted to resting state data
                                                                                                                       predicted

                                                        0

                                                        -1

                                                        -2

                                                        -3
Li et al. 2011, NeuroImage                                   0       200    400               600         800   1000            1200
                                                                                        time (seconds)
                        Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions
  Learning of dynamic audio-visual associations


                                                                     1
   Conditioning Stimulus         Target Stimulus                                                                 CS1
                                                                                                                 CS2
                                                                    0.8
              or                            or




                                                          p(face)
                                                                    0.6


                                                                    0.4

       CS                   TS         Response
                                                                    0.2
   0        200       400        600       800     2000
                                                   ±                 0
                            Time (ms)              650                0   200   400           600   800   1000
                                                                                      trial




den Ouden et al. 2010, J. Neurosci.
Hierarchical Bayesian learning model




 prior on volatility                         k                pk   1
 volatility                           vt-1        vt          pvt 1 | vt , k  ~ N vt , exp( k ) 

 probabilistic association                   rt        rt+1   prt 1 | rt , vt  ~ Dir rt , exp( vt ) 

 observed events                             ut        ut+1




Behrens et al. 2007, Nat. Neurosci.
   Explaining RTs by different learning models
                      Reaction times                        1
                                                                                                                                                True
                450                                                                                                                             Bayes Vol
                                                                                                                                                HMM fixed
                                                           0.8                                                                                  HMM learn
                440                                                                                                                             RW

                430                                        0.6
      RT (ms)




                                                    p(F)
                420
                                                           0.4
                410

                400                                        0.2

                390
                      0.1   0.3   0.5   0.7   0.9           0
                            p(outcome)                      400                            440           480            520          560               600
                                                                                                                Trial

                                                                               0.7
      5 alternative learning models:                                                                            Bayesian model selection:
                                                                               0.6
      • categorical probabilities                                                                               hierarchical Bayesian model
                                                            Exceedance prob.   0.5
      • hierarchical Bayesian learner                                                                           performs best
                                                                               0.4

      • Rescorla-Wagner                                                        0.3

      • Hidden Markov models                                                   0.2

        (2 variants)                                                           0.1

                                                                                0
                                                                                     Categorical   Bayesian    HMM (fixed) HMM (learn)     Rescorla-
                                                                                       model        learner                                 Wagner
den Ouden et al. 2010, J. Neurosci.
  Stimulus-independent prediction error


                                        Putamen                                       Premotor cortex




                                                                                                        p < 0.05
                                                      p < 0.05                                          (cluster-level whole-
                                                      (SVC)                                             brain corrected)
             BOLD resp. (a.u.)




                                                                 BOLD resp. (a.u.)
                                   0
                                                                                     0
                                 -0.5                                            -0.5
                                  -1                                                 -1
                                 -1.5                                            -1.5
                                  -2    p(F)   p(H)                                  -2   p(F)   p(H)




den Ouden et al. 2010, J. Neurosci .
Prediction error (PE) activity in the putamen

PE during active                            PE during incidental
sensory learning                               sensory learning


                                 p < 0.05
                                                 den Ouden et al. 2009,
                                   (SVC)               Cerebral Cortex



           PE during
reinforcement learning                        PE = “teaching signal” for
                                              synaptic plasticity during
        O'Doherty et al. 2004,                        learning
                       Science



       Could the putamen be regulating trial-by-trial changes of
                     task-relevant connections?
  Prediction errors control
  plasticity during adaptive
                                                          Hierarchical
                                                          Bayesian
  cognition                                               learning model



   • Modulation of visuo-
     motor connections by                          PUT
     striatal prediction
     error activity
                                       p = 0.010         p = 0.017
                                                   PMd
   • Influence of visual
     areas on premotor
     cortex:
          – stronger for
            surprising stimuli
          – weaker for expected              PPA         FFA
            stimuli


den Ouden et al. 2010, J. Neurosci .
Hierarchical variational Bayesian learning


                                                                p    1
volatility          x3k 1              x3k                      p  x3 | x3 1 ,  ~ N  x3 1 ,exp( ) 
                                                                      k    k                k



association                  x   k 1
                                 2               x   k
                                                     2
                                                                 p  x2 | x2 1 , x3  ~ N  x2 1 ,exp( x3  ) 
                                                                      k    k       k          k            k



events in the world                     x1k 1           x1k                                     
                                                                 p  x1k | x2  ~ Bernoulli S  x2 
                                                                            k                    k
                                                                                                               
sensory stimuli                         u k 1           u   k
                                                                 p  u k | x1k  ~ MoG  x1k  0, x1k  1

Mean-field decomposition

p  u k , x , | u1..k 1   q  x1k  q  x2  q  x3  q  
           k                                  k        k


                                                                                   Mathys et al. (2011), Front. Hum. Neurosci.
                        Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions
Diffusion-weighted imaging




                             Parker & Alexander, 2005,
                             Phil. Trans. B
                                                              1.6


 Integration of
                                                              1.4
                                                              1.2


 tractography
                                                               1
                                 R1              R2           0.8


 and DCM
                                                              0.6
                                                              0.4
                                                              0.2
                                                               0
                                                                    -2   -1    0      1    2


                              low probability of anatomical connection
                               small prior variance of effective connectivity parameter


                                                              1.6
                                                              1.4
                                                              1.2
                                                               1
                                 R1              R2           0.8
                                                              0.6
                                                              0.4
                                                              0.2
                                                               0
                                                                    -2   -1    0      1    2

                              high probability of anatomical connection
Stephan, Tittgemeyer et al.    large prior variance of effective connectivity parameter
2009, NeuroImage
                                                probabilistic
                                                tractography                                 34  6.5%
Proof of                      FG           FG
                                                                            FG                               FG
                                                                            left                            right
concept
study                                                           13  15.7%                                             24  43.6%



                                                                              LG                             LG
                              LG           LG
                                                                              left                          right
                                                                                             12  34.2%
                                    DCM                                                                            anatomical
                                                                                                                     connectivity 

                                            connection-                         6.5%
                                                                                v  0.0384

                                             specific priors
                                                    2

                                             for coupling
                                                   1.8


                                             parameters
                                                   1.6
                                                                                               15.7%
                                                                                              v  0.1070

                                                   1.4



                                                   1.2



                                                    1



                                                   0.8



                                                   0.6      34.2%                                          43.6%
                                                          v  0.5268                                       v  0.7746
                                                   0.4



Stephan, Tittgemeyer et al.                        0.2



2009, NeuroImage                                    0
                                                     -3    -2          -1            0                 1       2            3
 Connection-specific prior variance  as a function of
        anatomical connection probability 
                             m 1: a=-32,b=-32 m 2: a=-16,b=-32 m 3: a=-16,b=-28 m 4: a=-12,b=-32 m 5: a=-12,b=-28 m 6: a=-12,b=-24 m 7: a=-12,b=-20 m 8: a=-8,b=-32 m 9: a=-8,b=-28



                  0
                             1                1                1                1                1                1                1                1               1

                            0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5


  ij                       0
                               0    0.5     1
                                              0
                                                0    0.5    1
                                                               0
                                                                 0    0.5     1
                                                                                0
                                                                                  0    0.5    1
                                                                                                 0
                                                                                                   0    0.5     1
                                                                                                                  0
                                                                                                                    0    0.5     1
                                                                                                                                   0
                                                                                                                                     0    0.5    1
                                                                                                                                                    0
                                                                                                                                                      0    0.5     1
                                                                                                                                                                     0
                                                                                                                                                                       0    0.5    1


        1   0 exp(  ij )
                             m 10: a=-8,b=-24 m 11: a=-8,b=-20 m 12: a=-8,b=-16 m 13: a=-8,b=-12 m 14: a=-4,b=-32 m 15: a=-4,b=-28 m 16: a=-4,b=-24 m 17: a=-4,b=-20 m 18: a=-4,b=-16
                             1                1                1                1                1                1                1                1                1

                            0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

                             0                0               0               0              0               0               0               0               0
                               0    0.5     1   0    0.5    1   0    0.5    1   0   0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1
                             m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0  m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
                             1                1               1               1              1               1               1               1               1

                            0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

• 64 different mappings      0
                               0    0.5    1
                                             0
                                               0    0.5    1
                                                             0
                                                               0   0.5    1
                                                                            0
                                                                              0    0.5    1
                                                                                            0
                                                                                              0     0.5    1
                                                                                                             0
                                                                                                               0     0.5    1
                                                                                                                              0
                                                                                                                                0     0.5    1
                                                                                                                                               0
                                                                                                                                                 0     0.5   1
                                                                                                                                                               0
                                                                                                                                                                 0   0.5    1
                             m 28: a=0,b=-16 m 29: a=0,b=-12 m 30: a=0,b=-8  m 31: a=0,b=-4   m 32: a=0,b=0    m 33: a=0,b=4    m 34: a=0,b=8   m 35: a=0,b=12 m 36: a=0,b=16
  by systematic search       1               1               1              1               1                1                1                1               1

                            0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5
  across hyper-              0               0              0              0              0               0                0                0                0

  parameters  and 
                               0     0.5   1   0   0.5    1   0    0.5   1   0   0.5    1   0    0.5    1   0     0.5    1   0     0.5    1   0     0.5    1   0    0.5    1
                              m 37: a=0,b=20 m 38: a=0,b=24 m 39: a=0,b=28 m 40: a=0,b=32 m 41: a=4,b=-32   m 42: a=4,b=0    m 43: a=4,b=4    m 44: a=4,b=8   m 45: a=4,b=12
                             1               1              1              1              1               1                1                1                1

                            0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5


• yields anatomically
                             0               0              0              0              0              0              0              0              0
                               0     0.5   1   0   0.5    1   0    0.5   1   0   0.5    1   0    0.5   1   0    0.5   1   0   0.5    1   0    0.5   1   0   0.5    1
                              m 46: a=4,b=16 m 47: a=4,b=20 m 48: a=4,b=24 m 49: a=4,b=28 m 50: a=4,b=32 m 51: a=8,b=12 m 52: a=8,b=16 m 53: a=8,b=20 m 54: a=8,b=24
                             1               1              1              1              1              1              1              1              1
  informed (intuitive and   0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

  counterintuitive) and      0
                               0     0.5   1   0
                                                 0
                                                   0.5    1   0
                                                                     0
                                                                   0.5    1   0    0.5
                                                                                         0
                                                                                          1   0    0.5    1
                                                                                                             0
                                                                                                              0    0.5    1   0
                                                                                                                                 0
                                                                                                                                   0.5    1   0
                                                                                                                                                     0
                                                                                                                                                   0.5    1
                                                                                                                                                            0
                                                                                                                                                              0    0.5
                                                                                                                                                                         0
                                                                                                                                                                           1
                              m 55: a=8,b=28 m 56: a=8,b=32 m 57: a=12,b=20 m 58: a=12,b=24 m 59: a=12,b=28 m 60: a=12,b=32 m 61: a=16,b=28 m 62: a=16,b=32    m 63 & m 64
  uninformed priors          1               1              1               1               1               1               1               1               1

                            0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

                             0                   0                   0                   0                   0                   0                   0                   0                   0
                                  0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1
log group Bayes factor
                          600


                          400


                          200


                           0
                                0      10          20         30        40   50   60
                                                                model


                          700
log group Bayes factor




                          695

                          690

                          685

                          680
                                0      10          20         30        40   50   60
                                                                model


                          0.6
      post. model prob.




                          0.5
                          0.4       Models with anatomically informed
                          0.3       priors (of an intuitive form)
                          0.2
                          0.1
                           0
                                0      10          20         30        40   50   60
                                                                model
 m 1: a=-32,b=-32 m 2: a=-16,b=-32 m 3: a=-16,b=-28 m 4: a=-12,b=-32 m 5: a=-12,b=-28 m 6: a=-12,b=-24 m 7: a=-12,b=-20 m 8: a=-8,b=-32 m 9: a=-8,b=-28
 1                1                1                1                1                1                1                1               1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0                0                0                0                0                0                0                0                0
   0    0.5     1   0    0.5    1    0    0.5     1   0    0.5    1    0    0.5     1   0    0.5     1   0    0.5    1    0    0.5     1   0    0.5    1
 m 10: a=-8,b=-24 m 11: a=-8,b=-20 m 12: a=-8,b=-16 m 13: a=-8,b=-12 m 14: a=-4,b=-32 m 15: a=-4,b=-28 m 16: a=-4,b=-24 m 17: a=-4,b=-20 m 18: a=-4,b=-16
 1                1                1                1                1                1                1                1                1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0                0               0               0              0               0               0               0               0
   0    0.5     1   0    0.5    1   0    0.5    1   0   0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1
 m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0  m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
 1                1               1               1              1               1               1               1               1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0               0               0              0               0                0                0                0               0
   0    0.5    1   0    0.5    1   0   0.5    1   0    0.5    1   0     0.5    1   0     0.5    1   0     0.5    1   0     0.5   1   0   0.5    1
 m 28: a=0,b=-16 m 29: a=0,b=-12 m 30: a=0,b=-8  m 31: a=0,b=-4   m 32: a=0,b=0    m 33: a=0,b=4    m 34: a=0,b=8   m 35: a=0,b=12 m 36: a=0,b=16
 1               1               1              1               1                1                1                1               1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0               0              0              0              0               0                0                0                0
   0     0.5   1   0   0.5    1   0    0.5   1   0   0.5    1   0    0.5    1   0     0.5    1   0     0.5    1   0     0.5    1   0    0.5    1
  m 37: a=0,b=20 m 38: a=0,b=24 m 39: a=0,b=28 m 40: a=0,b=32 m 41: a=4,b=-32   m 42: a=4,b=0    m 43: a=4,b=4    m 44: a=4,b=8   m 45: a=4,b=12
 1               1              1              1              1               1                1                1                1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0               0              0              0              0              0              0              0              0
   0     0.5   1   0   0.5    1   0    0.5   1   0   0.5    1   0    0.5   1   0    0.5   1   0   0.5    1   0    0.5   1   0   0.5    1
  m 46: a=4,b=16 m 47: a=4,b=20 m 48: a=4,b=24 m 49: a=4,b=28 m 50: a=4,b=32 m 51: a=8,b=12 m 52: a=8,b=16 m 53: a=8,b=20 m 54: a=8,b=24
 1               1              1              1              1              1              1              1              1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0               0              0               0               0               0               0               0               0
   0     0.5   1   0   0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5    1   0    0.5     1
  m 55: a=8,b=28 m 56: a=8,b=32 m 57: a=12,b=20 m 58: a=12,b=24 m 59: a=12,b=28 m 60: a=12,b=32 m 61: a=16,b=28 m 62: a=16,b=32    m 63 & m 64
 1               1              1               1               1               1               1               1               1

0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5                 0.5

 0                   0                   0                   0                   0                   0                   0                   0                   0
      0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1         0   0.5   1




      Models with anatomically informed priors (of an intuitive form) were
      clearly superior to anatomically uninformed ones: Bayes Factor >109
                        Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions
Model-based predictions for single patients



       model structure

       BMS




       set of
       parameter estimates
       model-based decoding
         BMS: Parkison‘s disease and treatment




            Age-matched         PD patients                           PD patients
            controls            on medication                         off medication

                          Selection of action modulates     DA-dependent functional disconnection
Rowe et al. 2010,
                          connections between PFC and SMA   of the SMA
NeuroImage
 Model-based decoding by generative embedding
                         step 1 —                       A                   step 2 —                   A→B
                         model inversion                                    kernel construction
                                                                                                       A→C
                                                                                                       B→B
                                                                B                                      B→C
                                                    C



measurements from                                   subject-specific                              subject representation in the
an individual subject                         inverted generative model                             generative score space




      A
                             step 4 —                                            step 3 —
                             interpretation                                      support vector classification

               B
  C


jointly discriminative                        separating hyperplane fitted to
 model parameters                              discriminate between groups


Brodersen et al. 2011, PLoS Comput. Biol.
Discovering remote or “hidden” brain lesions
Discovering remote or “hidden” brain lesions




                 detect “down-stream” network changes
            altered synaptic coupling among healthy regions
Model-based decoding of disease status:
mildly aphasic patients (N=11) vs. controls (N=26)

Connectional fingerprints
from a 6-region DCM of
auditory areas during speech
perception

  PT                                        PT




              HG                   HG
             (A1)                 (A1)




                MGB          MGB




                    S         S




Brodersen et al. 2011, PLoS Comput. Biol.
  Model-based decoding of disease status:
  mildly aphasic patients (N=11) vs. controls (N=26)

                                                                                      Classification accuracy
                                                                                1
L.PT                                                R.PT
                                                                               0.8




                                                           balanced accuracy
                 L.HG                        R.HG
                 (A1)                        (A1)                              0.6


                                                                               0.4

                    L.MGB                R.MGB
                                                                               0.2


                                                                                0
                                                                                     anatomical contrast   search- generative
                          auditory stimuli
                                                                                         FS       FS        light  embedding
                                                                                                             FS

                                                                                                              Sensitivity: 100 %
 Brodersen et al. 2011, PLoS Comput. Biol.                                                                    Specificity: 96.2%
  A                  1                                            B                    1                                           C                  1

                    0.9
                                                                                      0.8                                                            0.8
balanced accuracy




                                                                  TPR (sensitivity)




                                                                                                                                   PPV (precision)
                    0.8
                                                                                      0.6                                                            0.6
                    0.7                                                                                                                                        Legend (cont’d)
                                                                                                                                                                     means correlations
                                                                                      0.4                                                            0.4
                                                                                                                                                                     eigenvariates correlations
                    0.6                                                                                     Legend
                                                                                                                                                                     eigenvariates z-correlations
                                                                                                                 anatomical                                          gen.embed., original model
                                                                                      0.2                        contrast                            0.2             gen.embed., feedforward
                    0.5                                                                                          searchlight                                         gen.embed., left hemisphere
                                                                                                                 PCA                                                 gen.embed., right hemisphere
                    0.4                                                                0                                                              0
                           a 189 s 347 m 332 z o f 389 r
                          162 c 177 p 291 e 30781 243 l 360                                 0           0.5                    1                           0             0.5                        1
                                                                                                FPR (1 - specificity)                                                 TPR (recall)
                            activation-   correlation-   model-
                              based         based        based
                             Multivariate searchlight                                                           Generative embedding
                              classification analysis                                                                using DCM

                               Voxel-based feature space                                                              Generative score space



                      0.4        0.4                                       -0.15                       -0.15




                                                                   embedding
                      0.3        0.3                                            -0.2                    -0.2




                                                                   generative
Voxel (64,-24,4) mm




                                                                                         L.HG  L.HG
                      0.2        0.2                                       -0.25                       -0.25

                      0.1         0.1                                           -0.3                    -0.3
                                                                     patients
                        0          0                                    -0.35
                                                                     controls                          -0.35

                                                             -10                   -10                                                     0.5          0.5
                      -0.1       -0.1                                           -0.4                     -0.4
                      -0.5       -0.5                  0             0          -0.4                     -0.4                      0               0
                                  0           0        Voxel (-56,-20,10) mm                            -0.2           -0.2                       R.HG  L.HG
           Voxel (-42,-26,10) mm            0.5   10    0.5 10                                                            0 -0.5       0   -0.5
                                                                                                               L.MGB  L.MGB
                                       Definition of ROIs
                                 Are regions of interest defined
                                  anatomically or functionally?

                                anatomically          functionally



A   1 ROI definition                                               Functional contrasts
    and n model inversions                                   Are the functional contrasts defined
    unbiased estimate                                      across all subjects or between groups?

                                                                 across             between
                                                                subjects            groups




                                                                            
                     B   1 ROI definition and n model inversions              D   1 ROI definition and n model inversions
                         slightly optimistic estimate:                            highly optimistic estimate:
                         voxel selection for training set and test set            voxel selection for training set and test set
                         based on test data                                       based on test data and test labels




                                                                            
                     C   Repeat n times:                                      E   Repeat n times:
                         1 ROI definition and n model inversions                  1 ROI definition and 1 model inversion
                         unbiased estimate                                        slightly optimistic estimate:
                                                                                  voxel selection for training set based on test
                                                                                  data and test labels

                                                                              F   Repeat n times:
                                                                                  1 ROI definition and n model inversions
                                                                                  unbiased estimate

Brodersen et al. 2011, PLoS Comput. Biol.
           Key methods papers: DCM for fMRI and BMS – part 1
•   Brodersen KH, Schofield TM, Leff AP, Ong CS, Lomakina EI, Buhmann JM, Stephan KE (2011) Generative
    embedding for model-based classification of fMRI data. PLoS Computational Biology 7: e1002079.
•   Daunizeau J, David, O, Stephan KE (2011) Dynamic Causal Modelling: A critical review of the biophysical and
    statistical foundations. NeuroImage 58: 312-322.
•   Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. NeuroImage 19:1273-1302.
•   Friston K, Stephan KE, Li B, Daunizeau J (2010) Generalised filtering. Mathematical Problems in Engineering
    2010: 621670.
•   Friston KJ, Li B, Daunizeau J, Stephan KE (2011) Network discovery with DCM. NeuroImage 56: 1202–1221.
•   Friston K, Penny W (2011) Post hoc Bayesian model selection. Neuroimage 56: 2089-2099.
•   Kasess CH, Stephan KE, Weissenbacher A, Pezawas L, Moser E, Windischberger C (2010) Multi-Subject
    Analyses with Dynamic Causal Modeling. NeuroImage 49: 3065-3074.
•   Kiebel SJ, Kloppel S, Weiskopf N, Friston KJ (2007) Dynamic causal modeling: a generative model of slice
    timing in fMRI. NeuroImage 34:1487-1496.
•   Li B, Daunizeau J, Stephan KE, Penny WD, Friston KJ (2011). Stochastic DCM and generalised filtering.
    NeuroImage 58: 442-457
•   Marreiros AC, Kiebel SJ, Friston KJ (2008) Dynamic causal modelling for fMRI: a two-state model. NeuroImage
    39:269-278.
•   Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Comparing dynamic causal models. NeuroImage
    22:1157-1172.
•   Penny WD, Stephan KE, Mechelli A, Friston KJ (2004b) Modelling functional integration: a comparison of
    structural equation and dynamic causal models. NeuroImage 23 Suppl 1:S264-274.
          Key methods papers: DCM for fMRI and BMS – part 2
•   Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing
    Families of Dynamic Causal Models. PLoS Computational Biology 6: e1000709.
•   Penny WD (2012) Comparing dynamic causal models using AIC, BIC and free energy. Neuroimage 59:
    319-330.
•   Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI responses. Curr
    Opin Neurobiol 14:629-635.
•   Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic
    models with DCM. NeuroImage 38:387-401.
•   Stephan KE, Harrison LM, Kiebel SJ, David O, Penny WD, Friston KJ (2007) Dynamic causal models of
    neural system dynamics: current state and future extensions. J Biosci 32:129-144.
•   Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic
    models with DCM. NeuroImage 38:387-401.
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posted:12/28/2012
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