EffectiveConnectivity_DCM

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					             Models of effective connectivity &
             Dynamic Causal Modelling (DCM)

Klaas Enno Stephan
Laboratory for Social & Neural Systems
Research
Institute for Empirical Research in Economics
University of Zurich

Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London




                    Methods & Models for fMRI data analysis
                             03 December 2008
                           Overview
• Brain connectivity: types & definitions
   – anatomical connectivity
   – functional connectivity
   – effective connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – DCM for fMRI: Neural and hemodynamic levels
   – Parameter estimation & inference

• Applications of DCM to fMRI data
   – Design of experiments and models
   – Some empirical examples and simulations
                     Connectivity
A central property of any system




   Communication systems            Social networks
   (internet)                       (Canberra, Australia)

                                      FIgs. by Stephen Eick and A. Klovdahl;
                                      see http://www.nd.edu/~networks/gallery.htm
   Structural, functional & effective connectivity




• anatomical/structural connectivity              Sporns 2007, Scholarpedia

  = presence of axonal connections

• functional connectivity
  = statistical dependencies between regional time series

• effective connectivity
  = causal (directed) influences between neurons or
      neuronal populations
                    Anatomical connectivity
• neuronal communication
  via synaptic contacts

• visualisation by tracing
  techniques

• long-range axons 
  “association fibres”
Diffusion-weighted imaging




             Parker & Alexander, 2005,
             Phil. Trans. B
  Diffusion-weighted
imaging of the cortico-
      spinal tract




        Parker, Stephan et al. 2002, NeuroImage
                 However,
knowing anatomical connectivity is not enough...
1. Connections are recruited in a context-dependent
                      fashion

                            0.4
                            0.3
                            0.2
                            0.1
                             0
                                  0   10   20   30   40   50   60   70   80   90   100

                            0.6

                            0.4

                            0.2

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


                            0.3

                            0.2

                            0.1

                             0

                                  0   10   20   30   40   50   60   70   80   90   100




                        Local functions are context-sensitive:
                          They depend on network activity.
                 2. Connections show plasticity
•   synaptic plasticity
    =   change in the structure and
        transmission properties of a
        chemical synapse


                                                                  NMDA
                                                                  receptor


                                               • critical for learning
                                               • can occur both rapidly and slowly
                                               • NMDA receptors play a critical role
                                               • NMDA receptors are regulated by
                                                 modulatory neurotransmitters like
                                                 dopamine, serotonine, acetylcholine
                       Gu 2002, Neuroscience
    Different approaches to analysing functional
                    connectivity

• Seed voxel correlation analysis

• Eigen-decomposition (PCA, SVD)

• Independent component analysis (ICA)

• any other technique describing statistical dependencies
  amongst regional time series
            Seed-voxel correlation analyses

• Very simple idea:
                                              seed voxel
  – hypothesis-driven choice of
    a seed voxel
    → extract reference
       time series

  – voxel-wise correlation with
    time series from all other
    voxels in the brain
  Drug-induced changes in functional connectivity

Finger-tapping task in
first-episode
schizophrenic patients:
voxels that showed
changes in functional
connectivity (p<0.005)
with the left ant.
cerebellum after
medication with
olanzapine


Stephan et al. 2001, Psychol. Med.
         Does functional connectivity not simply
         correspond to co-activation in SPMs?
No, it does not - see        regional      task T   regional response A2
the fictitious example       response A1
on the right:

Here both areas A1 and
A2 are correlated
identically to task T, yet
they have zero
correlation among
themselves:

r(A1,T) = r(A2,T) = 0.71
but
r(A1,A2) = 0 !
                                                       Stephan 2004, J. Anat.
Pros & Cons of functional connectivity analyses
• Pros:
  – useful when we have no experimental control over the
    system of interest and no model of what caused the data
    (e.g. sleep, hallucinatons, etc.)

• Cons:
  – interpretation of resulting patterns is difficult / arbitrary
  – no mechanistic insight into the neural system of interest
  – usually suboptimal for situations where we have a priori
    knowledge and experimental control about the system of
    interest

          models of effective connectivity necessary
For understanding brain function mechanistically,
 we need models of effective connectivity, i.e.

models of causal interactions among neuronal
                populations.
 Some models for computing effective connectivity
                from fMRI data
• Structural Equation Modelling (SEM)
  McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000

• regression models
  (e.g. psycho-physiological interactions, PPIs)
  Friston et al. 1997

• Volterra kernels
  Friston & Büchel 2000

• Time series models (e.g. MAR, Granger causality)
  Harrison et al. 2003, Goebel et al. 2003

• Dynamic Causal Modelling (DCM)
  bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008
                           Overview
• Brain connectivity: types & definitions
   – anatomical connectivity
   – functional connectivity
   – effective connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – DCM for fMRI: Neural and hemodynamic levels
   – Parameter estimation & inference

• Applications of DCM to fMRI data
   – Design of experiments and models
   – Some empirical examples and simulations
                                Psycho-physiological interaction (PPI)
                                      Task factor                    GLM of a 2x2 factorial design:
                                 Task A          Task B
                                                                     y  (TA  TB )  1                  main effect
                                                                                                         of task
                    Stim 1
  Stimulus factor




                                 TA/S1           TB/S1
                                                                         ( S1  S 2 ) β 2               main effect
                                                                                                         of stim. type

                                                                         (TA  TB ) ( S1  S 2 ) β 3    interaction
                    Stim 2




                                 TA/S2           TB/S2                  e

                             We can replace one main effect in
                             the GLM by the time series of an
                                                                       y  (TA  TB )  1             main effect
                                                                                                      of task
                                                                                                      V1 time series
                             area that shows this main effect.              V 1β 2                    main effect
                                                                                                      of stim. type
                             E.g. let's replace the main effect of
                             stimulus type by the time series of            (TA  TB ) V 1β 3        psycho-
                                                                                                      physiological
                             area V1:                                                                 interaction
                                                                           e
Friston et al. 1997, NeuroImage
                       Attentional modulation of V1→V5

                                                                                     SPM{Z}
                     Attention




                                                    V5 activity
             V1                        V5
                                                                  time

                            =
                                                                         attention




                                            V5 activity
        V1 x Att.                      V5
                                                                                 no attention


Friston et al. 1997, NeuroImage
Büchel & Friston 1997, Cereb. Cortex                               V1 activity
                        PPI: interpretation
y  (TA  TB )  1
   V 1β 2
   (TA  TB ) V 1β 3
  e                           Two possible
                             interpretations of
                               the PPI term:
          attention                                    attention




   V1                   V5                        V1                V5

 Modulation of V1V5 by               Modulation of the impact of attention on V5
 attention                            by V1
                         Two PPI variants

• "Classical" PPI:
   – Friston et al. 1997, NeuroImage
   – depends on factorial design
   – in the GLM, physiological time series replaces one experimental factor
   – physio-physiological interactions:
     two experimental factors are replaced by physiological time series


• Alternative PPI:
   – Macaluso et al. 2000, Science
   – interaction term is added to an existing GLM
   – can be used with any design
Task-driven
lateralisation
    Does the word
    contain the letter
    A or not?
                                                           letter decisions > spatial decisions

                                 •
                             •                            group analysis (random effects),
                         •                                    n=16, p<0.05 corrected
                                                                analysis with SPM2




                                     Is the red letter
                                     left or right from
                                     the midline of the    spatial decisions > letter decisions
                                     word?
                                                                            Stephan et al. 2003, Science
                           Bilateral ACC activation in both tasks –
                                but asymmetric connectivity !
                                          group analysis
                                       random effects (n=15)
                                      p<0.05, corrected (SVC)

                                                                      IFG




       left ACC (-6, 16, 42)          letter vs spatial         Left ACC  left inf. frontal gyrus (IFG):
                                          decisions             increase during letter decisions.




                                                                     IPS



                                      spatial vs letter
       right ACC (8, 16, 48)            decisions               Right ACC  right IPS:
Stephan et al. 2003, Science
                                                                increase during spatial decisions.
                                         PPI single-subject example


                                                       letter
                                                       decisions                                                                    spatial




                                                                   Signal in right ant. IPS
                                                                                                                                    decisions
Signal in left IFG




                     bVS= -0.16                                                               bL= -0.19
                                                       spatial                                                                       letter
                                                       decisions                                                                     decisions

                                                                                              bVS=0.50
                     bL=0.63




                                  Signal in left ACC                                                      Signal in right ACC


          Left ACC signal plotted against left IFG                 Right ACC signal plotted against right IPS



                                                                                                                  Stephan et al. 2003, Science
                       Pros & Cons of PPIs
• Pros:
   – given a single source region, we can test for its context-dependent
     connectivity across the entire brain
   – easy to implement

• Cons:
   – very simplistic model:
     only allows to model contributions from a single area
   – ignores time-series properties of data
   – application to event-related data relies deconvolution procedure (Gitelman
     et al. 2003, NeuroImage)
   – operates at the level of BOLD time series

            sometimes very useful, but limited causal interpretability;
            in most cases, we need more powerful models
                           Overview
• Brain connectivity: types & definitions
   – anatomical connectivity
   – functional connectivity
   – effective connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – DCM for fMRI: Neural and hemodynamic levels
   – Parameter estimation & inference

• Applications of DCM to fMRI data
   – Design of experiments and models
   – Some empirical examples and simulations
Example:                     FG          FG            LG = lingual gyrus
                        x3                      x4
a linear system              left       right          FG = fusiform gyrus

of dynamics in                                         Visual input in the
visual cortex                                           - left (LVF)
                                                        - right (RVF)
                             LG          LG
                        x1   left       right   x2     visual field.


                     RVF                        LVF
                      u2                          u1




                  x1  a11 x1  a12 x2  a13 x3  c12u2
                  
                  x2  a21 x1  a22 x2  a24 x4  c21u1
                  
                  x3  a31 x1  a33 x3  a34 x4
                  
                  x4  a42 x2  a43 x3  a44 x4
                  
Example:                 FG        FG               LG = lingual gyrus
                    x3                    x4
a linear system          left     right             FG = fusiform gyrus

of dynamics in                                      Visual input in the
visual cortex                                        - left (LVF)
                                                     - right (RVF)
                         LG        LG
                    x1   left     right   x2        visual field.


                  RVF                     LVF
                  u2                           u1
                    state        effective           system       input     external
                   changes      connectivity          state    parameters    inputs


                    x1   a11 a12 a13 0   x1   0 c12 
                     
  x  Az  Cu
                   x  a a
                     2              0 a24   x2   c21 0   u1 
                       21 22                       
                    x3   a31 0 a33 a34   x3   0 0  u2 
                                                              
   { A, C}         
                   
                                              
                     x4   0 a42 a43 a44   x4   0 0 
                                                            
Extension:                FG           FG
                     x3                       x4
bilinear                  left        right

dynamic
                                                                     m
                                                         x  ( A   u j B ( j ) ) x  Cu
                                                         
system                                                              j 1
                          LG           LG
                     x1   left        right   x2



                   RVF       CONTEXT          LVF
                    u2           u3           u1



 x1    a11 a12 a13 0 
                               0 b12
                                    (3)
                                              0  0          x1   0 c12      0
 x   a a
  2  21 22 0 a24                                        x  c               u1 
                      u3 0 0          0 0           2    21
                                                                          0      0  
                                                                                   u2
                                                         
 x3    a31 0 a33 a34 
                               0 0          0 b34 
                                                 (3)
                                                             x3   0 0        0  
                                                                         u3 
 x4    0 a42 a43 a44       0 0          0 0       
                                                             x4   0 0        0  
                       Dynamic Causal Modelling (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



Stephan & Friston 2007,          inputs
Handbook of Brain Connectivity
              Basic idea of DCM for fMRI
                  (Friston et al. 2003, NeuroImage)

• Using a bilinear state equation, a cognitive
  system is modelled at its underlying neuronal
  level (which is not directly accessible for fMRI).   x
• The modelled neuronal dynamics (x) is
  transformed into area-specific BOLD signals (y)      λ
  by a hemodynamic forward model (λ).
                                                       y
   The aim of DCM is to estimate parameters at
   the neuronal level such that the modelled
   BOLD signals are maximally similar to the
   experimentally measured BOLD signals.
                  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
                                                                                                                   
                                                                    intrinsic connectivity                 A
                                                                                                                  x
                                                                    modulation of                                   x 
                                 t
                                                                                                        B( j)   
                                                                    connectivity                                  u j x
                                                                                                             x
                                                                                                              
                                                                    direct inputs                         C
Stephan & Friston (2007),                                                                                    u
Handbook of Brain Connectivity
                    Bilinear DCM

          driving
          input




                    modulation


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

                                    Bilinear state equation:

                                    dx        m
                                                          
                                        A   ui B( i )  x  Cu
                                    dt       i 1        
   Example:
                                                         u1
   context-dependent decay                                 u1
                                                         u2
               stimuli              context
                  u1                  u2                   u2
                               -                         Z1
                 +
                               -
                                                           x
                  x1                                     Z2 1
                   +
                                                           x2
             +
                  x2
                   -                          x  Ax  u2 B (2) x  Cu1
                                              
                           -                   x1   
                                                               a12          b 2    0      c1 0  u1 
                                               x    a 21          x  u 2  11        x
                                               2                         0              0 0 u2 
                                                                                     b 22     
                                                                                       2
                                                                    
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)                                                                        
             DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:

  dx                            x (t )  x0 exp( at )
      ax
  dt
Coupling parameter a is inversely                  The coupling parameter a
proportional to the half life  of z(t):           thus describes the speed of
                                                   the exponential change in x(t)
                                     0.5x0
  x( )  0.5x0
          x0 exp(a )
             a  ln 2 /                        ln 2 / a
The hemodynamic                                                        u          stimulus functions

model in DCM                                                   t


                                                   dx        m
                                                                          
                                                       A   u j B( j )  x  Cu
                                                   dt                                          neural state equation
• 6 hemodynamic parameters:                                 j 1         


                                                           vasodilatory signal


 h  { ,  , ,  ,  ,  }                  f
                                                         s  x  s  γ ( f  1)
                                                         
                                                                       s
                                                                                             s

                                                         flow induction (rCBF)
                                                                   
                                                                   f s
  important for model fitting,                                                                                  hemodynamic
                                                                   f
  but of no interest for                                                                                        state equations
  statistical inference                                 Balloon model
                                       changes in volume           v               changes in dHb
                                          τv  f  v1 /α
                                                                          τq  f E ( f,E 0 ) q 0  v1 /α q/v
                                                                                              E
• Computed separately for each                      v                                    q

  area (like the neural
  parameters)
   region-specific HRFs!                          S                        q                
                                     ( q, v )        V0 k1 1  q   k2 1    k3 1  v 
                                                   S0                        v                
                                          k1  4.30 E0TE
  Friston et al. 2000, NeuroImage         k2  r0 E0TE                    BOLD signal
  Stephan et al. 2007, NeuroImage         k3  1                         change equation
                                          u          stimulus functions
                                                                                                      The hemodynamic
                                                                                                          model in DCM
                                  t


                      dx        m
                                             
neural state              A   u j B( j )  x  Cu
equation              dt      j 1
                                             
                                                                                  0.4


                                                                                   0.2


                              vasodilatory signal                                    0

                            s  x  s  γ ( f  1)
                                                                                         0   2   4    6    8       10      12        14
                                                                s                                                                RBM N,  = 0.5
                  f                       s                                                                                      CBM ,  = 0.5
                                                                                                                                     N
                                                                                     1                                           RBM N,  = 1
                            flow induction (rCBF)                                                                                CBM ,  = 1
                                                                                                                                     N
                                                                                   0.5                                           RBM ,  = 2
hemodynamic                           
                                      f s
                                                                                                                                     N
                                                                                                                                 CBM N,  = 2

state                                                                                0
                                      f                                                   0   2   4    6    8       10      12        14
equations
                           Balloon model                                           0.2
                                                                                     0
          changes in volume           v               changes in dHb
                                                                                   -0.2
             τv  f  v1 /α
                                             τq  f E ( f,E 0 ) q 0  v1 /α q/v
                                                                 E                -0.4

                       v                                    q                      -0.6
                                                                                          0   2   4    6    8       10      12        14




                      S                        q                
        ( q, v )        V0 k1 1  q   k2 1    k3 1  v 
                      S0                        v                
             k1  4.30 E0TE
             k2  r0 E0TE                    BOLD signal
             k3  1                         change equation
                                                                                                           Stephan et al. 2007, NeuroImage
How interdependent are our neural and hemodynamic
              parameter estimates?
                                                                             1


     A     5                                                                 0.8


                                                                             0.6
          10

     B                                                                       0.4
          15

     C                                                                       0.2
          20
                                                                             0

          25
                                                                             -0.2


     h   30                                                                 -0.4


          35                                                                 -0.6


                                                                             -0.8
          40
     ε
                                                                             -1
                      5   10          15   20     25   30   35   40


               r,A            r,B        r,C                       Stephan et al. 2007, NeuroImage
                          Bayesian statistics
            new data     prior knowledge

            p( y |  )        p( )

p( | y)  p( y |  ) p( )
posterior       likelihood   ∙ prior


Bayes theorem allows one to formally
incorporate prior knowledge into           The “posterior” probability of the
computing statistical probabilities.       parameters given the data is an
                                           optimal combination of prior knowledge
In DCM: empirical, principled &            and new data, weighted by their
shrinkage priors.                          relative precision.
                          Shrinkage priors
Small & variable effect             Large & variable effect




Small but clear effect              Large & clear effect
                                                                 stimulus function u
    Overview:
    parameter estimation                                     x  ( A   u j B j ) x  Cu
                                                             
                                                                                                                   neural state
                                                                                                                   equation
•   Combining the neural and
    hemodynamic states gives
    the complete forward model.                                  activity- dependent vasodilatory signal
                                                                         s  z  s  γ( f  1)
                                                                         
•   An observation model                                                          s                           s
    includes measurement
                                                             f

    error e and confounds X                                             flow - induction (rCBF)
                                                                                                            parameters
    (e.g. drift).                    hidden states                                
                                                                                  f s                      h  { ,  , ,  , }
                                     z  {x, s, f , v, q}
                                                                                  f                         n  { A, B1... B m , C}
•   Bayesian parameter               state equation                                                          { h , n }
    estimation by means of a         z  F ( x , u,  )
                                     
    Levenberg-Marquardt
    gradient ascent, embedded                         changes in volume         v            changes in dHb

    into an EM algorithm.                                   τv  f  v1/α
                                                                                      τq  f E ( f, ) q  v1/α q/v
                                                                                        
                                                                   v                               q
•   Result:
    Gaussian a posteriori
    parameter distributions,
    characterised by              ηθ|y
    mean ηθ|y and                                                           y   (x )                                   y  h(u, )  X  e
    covariance Cθ|y.                                                                                                    observation model
                                                                      modelled
                                                                    BOLD response
            Inference about DCM parameters:
             Bayesian single-subject analysis
• Gaussian assumptions about the posterior distributions of the
  parameters
• Use of the cumulative normal distribution to test the probability that
  a certain parameter (or contrast of parameters cT ηθ|y) is above a
  chosen threshold γ:

                 cT                       ηθ|y
                     y        
         p  N                
                 cT C y c     
                               

• By default, γ is chosen as zero ("does the effect exist?").
                              Bayesian single subject inference

                                   LD|LVF

                                0.13    0.34                                                    p(cT>0|y)
                               0.19     0.14
                                                                                                = 98.7%
                       FG                         FG
                       left                      right


     LD                                                              LD
             0.44                                           0.29
            0.14                                            0.14

                       LG                         LG
                       left                      right
                            0.01       -0.08
                           0.17        0.16

             RVF           LD|RVF                        LVF
             stim.                                       stim.            Contrast:
                                                                          Modulation LG right  LG links by LD|LVF
                                                                          vs.
Stephan et al. 2005,
                                                                          modulation LG left  LG right by LD|RVF
Ann. N.Y. Acad. Sci.
                   Inference about DCM parameters:
                  Bayesian fixed-effects group analysis

Because the likelihood distributions                               Under Gaussian assumptions this is
from different subjects are independent,                           easy to compute:
one can combine their posterior
densities by using the posterior of one                               group                     individual
subject as the prior for the next:                                    posterior                 posterior
                                                                      covariance                covariances



p( | y1 )        p( y1 |  ) p( )                                                           N

p( | y1 , y2 )  p( y2 |  ) p( y1 |  ) p( )                      C|1y1 ,..., y N   C|1yi
                                                                                              i 1
                  p( y2 |  ) p( | y1 )
                                                                                             N 1              1
...                                                                   | y ,..., y          C | yi | yi C | y1 ,..., y N
 p( | y1 ,..., y N )  p( y N |  ) p( | y N 1 )...p( | y1 )                             i 1             
                                                                            1         N




                                                                      group                  individual posterior
                                                                      posterior              covariances and means
                                                                      mean
            Inference about DCM parameters:
                 group analysis (classical)
• In analogy to “random effects” analyses in SPM, 2nd level analyses
  can be applied to DCM parameters:

                  Separate fitting of identical models
                          for each subject


                  Selection of bilinear parameters of
                                interest


 one-sample t-test:          paired t-test:                 rmANOVA:
  parameter > 0 ?           parameter 1 >            e.g. in case of multiple
                            parameter 2 ?             sessions per subject
                           Overview
• Brain connectivity: types & definitions
   – anatomical connectivity
   – functional connectivity
   – effective connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – DCM for fMRI: Neural and hemodynamic levels
   – Parameter estimation & inference

• Applications of DCM to fMRI data
   – Design of experiments and models
   – Some empirical examples and simulations
    What type of design is good for DCM?


Any design that is good for a GLM of fMRI data.
                             GLM vs. DCM
DCM tries to model the same phenomena as a GLM, just in a different way:
It is a model, based on connectivity and its modulation, for explaining
experimentally controlled variance in local responses.
No activation detected by a GLM → inclusion of this region in a DCM is
useless!




                                                                    Stephan 2004, J. Anat.
                                   Multifactorial design:
                             explaining interactions with DCM
                                Task factor                Stim1/      Stim2/
                                                           Task A      Task A
                           Task A        Task B
                  Stim 1
Stimulus factor




                           TA/S1          TB/S1              X1          X2      GLM
                  Stim 2




                                                           Stim 1/     Stim 2/
                           TA/S2          TB/S2            Task B      Task B




Let’s assume that an SPM analysis                  Stim1
shows a main effect of stimulus in X1
and a stimulus  task interaction in X2.                     X1          X2      DCM
How do we model this using DCM?                    Stim2

                                                            Task A   Task B
    Simulated data

                                                       X1


                        +++        –               –
  Stimulus 1                             +

                                  X1     +++    X2          Stim 1
                                                            Task A
                                                                     Stim 2
                                                                     Task A
                                                                              Stim 1
                                                                              Task B
                                                                                       Stim 2
                                                                                       Task B

  Stimulus 2            +
                                       +++     +

                              Task A         Task B
                                                       X2




Stephan et al. 2007, J. Biosci.
X1




     Stim 1   Stim 2   Stim 1   Stim 2
     Task A   Task A   Task B   Task B




X2




                                plus added noise (SNR=1)
Recent fMRI
studies with DCM

• DCM now an
  established tool for
  fMRI analysis

• >50 studies
  published, incl. high-
  profile journals

• combinations of
  DCM with
  computational
  models
Thank you

				
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