DCM_fMRI by liwenting

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									Dynamic Causal Modelling (DCM)
           for fMRI


            Andre Marreiros




      Wellcome Trust Centre for Neuroimaging
      University College London
Thanks to...


    Stefan Kiebel
    Lee Harrison
    Klaas Stephan
     Karl Friston
Overview

Dynamic Causal Modelling of fMRI



    Definitions & motivation


The neuronal model(bilinear dynamics)
    The Haemodynamic model



   Estimation: Bayesian framework


       DCM latest Extensions
 Principles of organisation

Functional specialization   Functional integration
     Neurodynamics: 2 nodes with input

                                            u1

                  a11                       u2

                                            z1
            a21

                                            z2
                        a22




 z1   a11 0   z1  c 
                                          activity in z 2 is coupled to z1 via
 z   a               u
  2   21 a22   z2  0 1
                                 a21  0
                                     coefficient a21
     Neurodynamics: positive modulation



                                                   u1
                    a11


                                                   u2

              a21
                                                   z1
                          a22
                                                   z2


 z1   a11 0   z1 
                              0 0  z1  c 
                                                          modulatory input u2 activity
z  a                  u2  2            u1   b 0
                                                    2

 2   21 a22   z2 
                           b21 0  z2  0
                                       
                                                    21
                                                            through the coupling a21
 Neurodynamics: reciprocal connections


                                                    u1
               a11

                                                    u2


        a12
                                                    z1
              a21


                     a22                            z2


                                                                                   reciprocal
 z1   a11 a12   z1 
                               0 0  z1  c 
z  a                   u2  2            u1    a21  0   a12  0   b21  0
                                                                          2
                                                                                   connection
 2   21 a22   z2 
                            b21 0  z2  0
                                        
                                                                                   disclosed by u2
Haemodynamics: reciprocal connections


                                                           Simulated response
              a11

                                            Bold
                                          Response


       a12

              a21
                                           Bold
                    a22                   Response




                                                                                   green: neuronal activity
 z1   a11 a12   z1 
                               0 0  z1  c                                   red:   bold response
z  a                   u2  2            u1    a21  0   a12  0   b21  0
                                                                          2

 2   21 a22   z2 
                            b21 0  z2  0
                                        
Haemodynamics: reciprocal connections


              a11

                                           Bold
                                           with
                                        Noise added

       a12

              a21
                                            Bold
                    a22
                                           with
                                        Noise added




                                                                                    green: neuronal activity
 z1   a11 a12   z1 
                               0 0  z1  c                                    red:   bold response
z  a                   u2  2            u1     a21  0   a12  0   b21  0
                                                                           2

 2   21 a22   z2 
                            b21 0  z2  0
                                        
                 Example: modelled BOLD signal
  Underlying model
  (modulatory inputs not shown)                                                   left LG
           FG               FG
          left             right




           LG               LG
          left             right
                                                                                 right LG


 RVF                               LVF
LG = lingual gyrus    Visual input in the
FG = fusiform gyrus   - left (LVF)
                      - right (RVF)
                      visual field.
                                            blue:   observed BOLD signal
                                            red:    modelled BOLD signal (DCM)
         Use differential equations to describe
           mechanistic model of a system

                                                      z1 (t )  overall
• System dynamics =                         z (t )     system state
                                                               represented
  change of state vector in time                                 by state variables
                                                      zn (t )
                                                              
• Causal effects in the system:           z1  z1   f1 ( z1...zn , u,1 ) 
                                            
   – interactions between elements
                                               change of
                                     z      in time                      
                                                                                
                                                               state vector
   – external inputs u
                                          z n  zn   f n ( z1...zn , u, n )
                                                                         
• System parameters  :
  specify exact form of system                z  F ( z, u, )
                                              
Example:              FG
                 z3 left
                                 FG
                                        z4
                                              LG = lingual gyrus
                                right         FG = fusiform gyrus
linear                                        Visual input in the
dynamic                LG        LG
                                              - left (LVF)
                                              - right (RVF)
                 z1                     z2
system                left      right         visual field.


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


                  z1   a11 a12 a13 0   z1   0 c12 
                   
 z  Az  Cu
                 z  a a
                   2              0 a24   z2  c21 0   u1 
                     21 22                      
                  z3  a31 0 a33 a34   z3   0 0  u2 
                                                           
   { A, C}       
                   
                                           
                  z4   0 a42 a43 a44   z4   0 0 
                                                         
Extension:               FG          FG
                   z3   left        right   z4
bilinear                                                        m

dynamic                                               z  ( A   u j B j ) z  Cu
                                                      
                                                                j 1
                         LG          LG
system             z1   left        right   z2



                 RVF       CONTEXT          LVF
                  u2           u3           u1



  z1   a11 a12 a13 0 
                              0 b12
                                   3
                                            0 0    z1   0 c12     0
  z  a a
   2  21 22 0 a24                                                u1 
                      u 0 0         0 0    z2  c21 0      0  
                                               3  
                                                                         u2 
  z3  a31 0 a33 a34  3 0 0
                                           0 b34   z3   0 0       0
                                                              u3 
   
  z4   0 a42 a43 a44      0 0
                                           0 0    z4   0 0
                                                                      0  
 Bilinear state equation in DCM/fMRI

 state                       modulation of system   direct   m external
          connectivity
changes                      connectivity   state   inputs     inputs


  z1   a11  a1n  m b11  b1n    z1  c11  c1m   u1 
                               j      j
   
           u                  
                   j                               
  zn  an1  ann 
                 
                       j 1
                            bnj1  bnn    zn  cn1  cnm  um 
                            
                                      j
                                                          
         
                               m
                    z  ( A   u j B ) z  Cu
                                      j

                               j 1
Conceptual                                                       z  F ( z, u, n )
                                         Neuronal state equation 

overview                                   The bilinear model        z  ( A  u j B j ) z  Cu
                                                                     
                                                                                 F z  
                                            effective connectivity            A    
                                                                                 z z
                                            modulation of                         2F      z
                                                                             B 
                                                                              j
                                                                                       
                   Input                    connectivity                         zu j u j z
                    u(t)                                                           F z
                                                                                       
                                            direct inputs                    C      
                                                                                   u u
      c1                       b23
                                                                         integration

                                              neuronal               z
             a12    activity                  states
                     z2(t)      activity
  activity                       z3(t)                                        hemodynamic
   z1(t)                                                             λ        model
                       y
                                     y
     y                                        BOLD                   y                 Friston et al. 2003,
                                                                                       NeuroImage
      The hemodynamic “Balloon” model
• 5 hemodynamic                                             activity
  parameters:                                                  z(t )


   h  { , g ,t , a , }                                vasodilatory signal
                                                     s  z  s  γ( f  1)
                                                     
                                           f                    s
                                                            flow induction
  important for model fitting, but                              
                                                                f s
  of no interest for statistical                                    f

  inference
                                     changes in volume                          changes in dHb
• Empirically determined              τv  f  v
                                                  1 /α
                                                                v
                                                                                         
                                                                        τq  f E ( f, ) q  v1/α q/v
                                                                         
  a priori distributions.                      v                                     q


• Computed separately for
  each area
                                                           BOLD signal
                                                           y (t )   v, q 
                         Diagram

                         Dynamic Causal Modelling of fMRI


Network             Haemodynamic
dynamics               response

                                         Priors                    Model
                                                                 comparison

       State space
         Model
                                   Model inversion
                                         using
                               Expectation-maximization     Posterior distribution
           fMRI                                                 of parameters
           data y
      Estimation: Bayesian framework
                  Models of                             Constraints on
        •Hemodynamics in a single region                 •Connections
            •Neuronal interactions                  •Hemodynamic parameters

likelihood term    p( y |  )                                 p( )           prior

         posterior              p( | y)  p( y |  ) p( )

                                 Bayesian estimation
                                                                    stimulus function u
Overview:
parameter estimation                                            z  ( A   u j B j ) z  Cu
                                                                
                                                                                                                       neuronal state
                                                                                                                       equation
•   Specify model (neuronal and
    hemodynamic level)                                              activity - dependent vasodilatory signal

•   Make it an observation model                                            s  z  s  γ( f  1)
                                                                            
    by adding measurement error e                               f                    s                            s
    and confounds X (e.g. drift).                                          flow - induction (rCBF)
                                                                                                                parameters
                                        hidden states                                
                                                                                     f s
•   Bayesian parameter estimation                                                                               h  { , g ,t , a , }
                                        x  {z, s, f , v, q}
    using Bayesian version of an                                                     f                          n  { A, B1... B m , C}
    expectation-maximization            state equation                                                           { h , n }
                                        x  F ( x, u, )
                                        
    algorithm.
                                                        changes in volume           v           changes in dHb
•   Result:                                                    τv  f  v1/α
                                                                                         τq  f E ( f, ) q  v1/α q/v
                                                                                           
    (Normal) posterior parameter                                      v                               q
    distributions, given by mean ηθ|y
    and Covariance Cθ|y.

                                                                               y   (x )                                    y  h(u, )  X  e
                                        ηθ|y
                                                                            modelled                                        observation model
                                                                          BOLD response
    Haemodynamics: 2 nodes with input

                               Dashed Line: Real BOLD response
           a11




                  a21
            a22




Activity in z1 is coupled to
                                   p( | y)  p( y |  ) p( )
  z2 via coefficient a21
       Inference about DCM parameters:
             single-subject analysis
• Bayesian parameter estimation in DCM: Gaussian assumptions about the
  posterior distributions of the parameters
• Use of the cumulative normal distribution to test the probability by which
  a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen
  threshold γ:


                          g
                                      ηθ|y
         Model comparison and selection

Given competing hypotheses,
which model is the best?




log p( y | m)  accuracy (m) 
                complexity(m)

           p( y | m  i )
     Bij 
           p( y | m  j )          Pitt & Miyung (2002), TICS
         Comparison of three simple models
Model 1:                                     Model 2:                               Model 3:
attentional modulation                       attentional modulation                 attentional modulation
of V1→V5                                     of SPC→V5                              of V1→V5 and SPC→V5

                                                          Attention                               Attention
Photic                               SPC      Photic                         SPC     Photic                         SPC
         0.85                                                     0.55                                     0.03
                            0.70                       0.86                                   0.85
                                      0.84                            0.75                                     0.70
                     1.36
                                                               1.42        0.89                        1.36         0.85
                V1                                        V1                                      V1


         0.57           -0.02                      0.56         -0.02                      0.57           -0.02
                                V5                                      V5                                        V5
Motion        0.23                              Motion                                  Motion         0.23

      Attention                                                                               Attention


Bayesian model selection:                                      Model 1 better than model 2,
                                                               model 1 and model 3 equal
                                                               log p( y | m1 )  log p( y | m3 )  log p( y | m2 )
            → Decision for model 1:                            in this experiment, attention
                                                               primarily modulates V1→V5
             Extension I: Slice timing model
•   potential timing problem in DCM:




                                                slice acquisition
      temporal shift between regional
                                                                      2
          time series because of multi-
          slice acquisition
                                                                      1

                                                                    visual
                                                                    input

•   Solution:
      – Modelling of (known) slice timing of each area.

                 Slice timing extension now allows for any slice
                 timing differences.
                 Long TRs (> 2 sec) no longer a limitation.
                 (Kiebel et al., 2007)
                 Extension II: Two-state model

                       Single-state DCM                           Two-state DCM



    input
                                                                                                              x1E
    u (t )
                                             x1                                          x1E , I                    exp( A11  uB11 )
                                                                                                                           IE     IE


                                                                                                              x1I

                            Aij  uBij                              exp( Aij  uBij )


                                                                                     x
                                                                                         ( ABu ) x  Cu
       x                                                                            t
           ( A  uB) x  Cu
       t
                                                           e A11E
                                                                 E
                                                                           e A11
                                                                                   EI
                                                                                                  e A1 N       0                 x1E 
   A11          A1N               x1                  A11                                                                   I
                                                                           e A11
                                                               IE               II

                                                           e                                        0          0                 x1 
A 
                       x(t )    
                                                      A                                                           x(t )    
   AN 1         ANN               xN                  A                                                     EI              E
                                   
                                                                                                        EE
                                                           e N1               0             e ANN           e ANN              xN 
                                                           0                                   IE
                                                                                           e ANN             e ANN 
                                                                                                                  II
                                                                                                                                  xI 
                                                                              0                                                  N


  z  Az   u B z  Cu
                  j    j
                                                  Extrinsic (between-region)            Intrinsic (within-region)
                                                            coupling                            coupling
           Extension III: Nonlinear DCM for fMRI
Here DCM can model activity-
                                     dz          m            n            
dependent changes in connectivity;       A   ui B (i )   z j D ( j )  z  Cu
how connections are enabled or gated dt                                    
                                                 i 1         j 1          
by activity in one or more areas.    The D matrices encode which of the n neural
                      attention             units gate which connections in the system.
                        0.19
                     (100%)

                                          Can V5 activity during attention to
                         SPC              motion be explained by allowing
              0.03             0.01       activity in SPC to modulate the V1-
           (100%)              (97.4%)
                                          to-V5 connection?
   1.65                                                                    ( SPC )
 (100%)                                   The posterior density of D
          V1                         V5                                    V 5,V 1
                                          indicates that this gating existed with
                  0.04                    97.4% confidence.
               (100%)

                 motion
  Conclusions

Dynamic Causal Modelling (DCM) of fMRI is mechanistic model that is
        informed by anatomical and physiological principles.


DCM uses a deterministic differential equation to model neuro-dynamics
                (represented by matrices A,B and C)

          DCM uses a Bayesian framework to estimate these

 DCM combines state-equations for dynamics with observation model
         (fMRI: BOLD response, M/EEG: lead field).


  DCM is not model or modality specific (Models will change and the
          method extended to other modalities e.g. ERPs)

								
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