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					Methods for Dummies 2009

   Dynamic Causal Modelling
        Part I: Theory




                      18th February 2009
                       Stephanie Burnett
                       Christian Lambert
Last time, in MfD…
   Psychophysiological
    interactions (PPI) and
    structural equation modelling
    (SEM)
   Functional vs. effective
    connectivity
        Functional connectivity:
         temporal correlation between
         spatially remote                 Standard fMRI analysis
         neurophysiological events
        Effective connectivity: the
         influence that the elements of   PPIs, SEM, DCM
         a neuronal system exert over
         each other
Introduction: DCM and its place in
the methods family tree
                                          Task                   BOLD signal
   Standard fMRI analysis:
       The BOLD signal (related to
        brain activity in some implicit
        way) in some set of brain is
        correlated, and is also
        correlated with your task

                                                 “This is a fronto-parietal
                                                 network collection of brain
                                                 regions involved in activated
                                                 while processing coffee”
Introduction: DCM and its place in
the methods family tree
                                                        attention

   PPIs
       Represent how the
        (experimental) context
        modulates connectivity
        between a brain region of               V1                    V5
        interest, and anywhere else     DCM models how neuronal activity
                                        causes the BOLD signal (forward model)
       E.g. (Whatever gives rise
        to the) signal in one brain                      attention
                                        That is, your conclusions are about
        region (V1) will lead to a      neural events
        signal in V5, and the
        strength of this signal in V5   DCM models bidirectional and modulatory
        depends on attention            interactions, between multiple brain regions
                                                 V1                    V5
Introduction: DCM and its place in
the methods family tree
   DCM
       Your experimental task
        causes neuronal activity in
        an input brain region, and
        this generates a BOLD
        signal.
       The neuronal activity in this
        input region, due to your
        task, then causes or
        modulates neuronal activity
        in other brain regions (with
        resultant patterns of BOLD      “This sounds more
        signals across the brain)       like something I’d
                                        enjoy writing up!”
DCM basics
    DCM models
     interactions between
     neuronal populations
        fMRI, MEG, EEG
    The aim is to estimate,
     and make inferences
     about:
    1.   The coupling among
         brain areas
    2.   How that coupling is
         influenced by changes in
         experimental context
DCM basics
   DCM starts with a realistic
    model of how brain regions
    interact and where the inputs     Neural and hemodynamic models
    can come in
                                      (more on this in a few minutes)
   Adds a forward model of how
    neuronal activity causes the
    signals you observe (e.g.
    BOLD)
   …and estimates the
    parameters in your model
    (effective connectivity), given
    your observed data
DCM basics
   Inputs
   State variables
   Outputs
DCM basics
   Inputs
       In functional connectivity
        models (e.g. standard fMRI
        analysis), conceptually
        your input could have
        entered anywhere
       In effective connectivity
        models (e.g. DCM), input
        only enters at certain
        places
DCM basics
   Inputs can exert their
    influence in two ways:
       1. Direct influence
           e.g. visual input to V1

       2. Vicarious (indirect)
        influence
           e.g. attentional
            modulation of the
            coupling between V1
            and V5
DCM basics
   State variables
     Neuronal   activities,
       and other neuro- or
       bio-physical variables
       needed to form the
       outputs
            Neuronal priors
            Haemodynamic priors


What you’re modelling is how the
inputs modulate the coupling among
these state variables
DCM basics
   Output
       The BOLD signal (for
        example) that you’ve
        measured in the brain
        regions specified in your
        model
            Dynamic Modelling (i)

   Generate equations to model the dynamics of physical
    systems.

   These will be LINEAR or NON-LINEAR

   Linear models provide good approximation

   However neuronal dynamics are non-linear in nature
                     Linear Dynamic Model
                   INPUT U1                          INPUT U2
                            C11                              C22



                                     A21
       A11
                       X1            A12
                                                        X2               A22




X1= A11X1 + A21X2 + C11U1                                  X2= A22X2 + A12X1 + C22U2


                             The Linear Approximation
                                  fL(x,u)=Ax + Cu

                      
                     x1   A11 A12   x1  C11 0  U 1 
                                    x 2   0 C 22  U 2 
                     
                     x 2   A21 A22                  
       Intrinsic Connectivity                             Extrinsic (input) Connectivity
                 Dynamic Modelling (ii)
   In DCM we are modelling the brain as a:

               “Deterministic non-linear dynamic system”

   Effective connectivity is parameterised in terms of coupling
    between unobserved brain states


   Bilinear approximation is useful:
       Reduces the parameters of the model to three sets
            1) Direct/extrinsic
            2) Intrinsic/Latent
            3) Changes in intrinsic coupling induced by inputs

   The idea behind DCM is not limited to bilinear forms
                     AIM:
Estimate the parameters by perturbing the system
           and observing the response.

   Important in experimental design:

     1)   One factor controls sensory perturbation

     2)One factor manipulates the context of sensory
      evoked responses
      INPUT U1         INPUT U2
          C11              C22



                 A21
A11
       X1        A12
                        X2        A22
      INPUT U1                INPUT U2
          C11                     C22
                       B221


                 A21
A11
       X1        A12
                               X2        A22
                   Bi-Linear Dynamic Model (DCM)
                        INPUT U1                         INPUT U2
                             C11                              C22
                                              B221


                                        A21
           A11
                         X1             A12
                                                          X2             A22



                    2
X1= A11X1 + (A21+ B 12U1(t))X + C11U1                      X2= A22X2 + A12X1 + C22U2

                              The Bilinear Approximation
                                                     j
                               fB(x,u)=(A+jUjB )x + Cu

                
              x1    A11 A12  0 B 2 12    x1  C11 0  U 1 
                                    U 2     
              
              x2    A21 A22  0 0    x 2  0 C 22  U 2 
                                                              
       Intrinsic                                                       Extrinsic (input)
                            INDUCED CONNECTIVITY
    Connectivity                                                         Connectivity
     Bilinear state equation in DCM

 state      intrinsic           modulation of system   direct   m external
changes   connectivity          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
    Bilinear state equation in DCM

 state      intrinsic      modulation of   system   direct   m external
changes   connectivity     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
   ( A   u j B j ) z  Cu
 z                                      {A, B ...B , C}
                                           n          1      m
            j 1
         FG            FG
    z3   left         right z4
                                              m
                                   z  ( A   u j B ) z  Cu
                                                     j

                                              j 1
         LG            LG
    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  
        DCM for fMRI: the basic idea
   Using a bilinear state equation, a cognitive
    system is modelled at its underlying neuronal
    level (which is not directly accessible for
    fMRI).                                                   z

   The modelled neuronal dynamics (z) is                    λ
    transformed into area-specific BOLD signals
    (y) 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.
Priors on biophysical parameters
          The hemodynamic “Balloon” model

• 5 hemodynamic                          Neural activity
  parameters:                                       z (t )



   { ,  , ,  , }
   h                                       vasodilatory signal
                                           Vasodilatory signal
                                          s  z  s  γ( f  1)
                                          

                                 f                   s


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


• Computed separately
  for each area (like the
                                         BOLD signal
  neural parameters).
                                         y (t )   v, q 
Conceptual                   Neural state equation           z  F ( z, u, n )
                                                             
 overview                    The bilinear model            z  ( A  u j B j ) z  Cu
                                                           
                                                               F z  
                                  effective connectivity    A    
                                                               z z
                                                                2F      z
                                       modulation of       B 
                                                            j
                                                                     
                                       connectivity            zu j u j z
            Input
                                                                  F z
                                                                      
             u(t)                      direct inputs        C      
                                                                  u u


c1                                                               integration
                                          neuronal           z
                 activity   b23
           a12
                  z2(t)
                                          states
                            activity
activity                                                             hemodynamic
 z1(t)
                             z3(t)
                                                             λ       model
                     y
     y                            y      BOLD                y          Friston et al. 2003,
                                                                        NeuroImage
          Estimating model parameters

                                           Bayes Theorem
   DCMs are biologically
    plausible (i.e. complicated) -   posterior    likelihood   ∙ prior

                                     p( | y )  p( y |  )  p( )
    they have lots of free
    parameters

   A Bayesian framework is a
    good way to embody the
    constraints on these
    parameters
    Use Bayes’ theorem to estimate
    model parameters
   Priors – empirical                       Bayes Theorem
    (haemodynamic
    parameters) and non-               posterior     likelihood     ∙ prior

                                      p( | y )  p( y |  )  p( )
    empirical (eg. shrinkage
    priors, temporal scaling)

   Likelihood derived from
    error and confounds (eg.
    drift)

   Calculate the Posterior
    probability for each effect,
    and the probability that it
    exceeds a set threshold      Inferences about the strength (= speed) of
                                   connections between the brain regions in
                                   your model
      Interpretation of parameters
                                     Single subject analysis
-   EM algorithm – works out             Use the cumulative normal
                                          distribution to test the probability
    the parameters in a model             with which a certain parameter is
                                          above a chosen threshold γ:




-   Bayesian model selection to                         ηθ|y
    test between alternative
    models
Model comparison and selection
   A good model of your data
    will balance model fit with
    complexity (overfitting
    models noise)

   You find this by taking
    evidence ratios (the “Bayes
    factor”)

   The “Bayes factor” is a
    summary of the evidence in
    favour of one model as
    opposed to another
       Bayesian Model Selection
Bayes’   theorem:                  p( y |  , m) p( | m)
                      p( | y, m) 
                                           p ( y | m)


Model    evidence:   p( y | m)   p( y |  , m)  p( | m) d


The log model        log p ( y | m)  accuracy(m) 
evidence can be                        complexity(m)
represented as:
                             p( y | m  i)
Bayes    factor:      Bij 
                             p( y | m  j )
                                                  Penny et al. 2004, NeuroImage
        Interpretation of parameters
    - Group analysis:
• Like “random effects” analysis in SPM, 2nd level analysis
  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:                     rm ANOVA:
    Parameter > 0?         Parameter 1 > parameter 2?         For multiple sessions
                                                              per subject
               New stuff in DCM
   1. DCM now accounts for the slice timing problem
             Extension I: Slice timing model




                                           slice acquisition
 Potential timing problem in DCM:
Temporal shift between regional time                                              2
    series because of multi-slice
                                                                                  1
             acquisition
                                                                                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)
              New stuff in DCM
1. DCM now accounts for the slice timing problem (SPM5)

2. Biological plausibility: each brain area can have two states
   (SPM8) (exc./inh.)
              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 A11
                                                           EE
                                                                  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


                                              Extrinsic (between-            Intrinsic (within-
  z  Az   u B z  Cu
              j    j
                                               region) coupling              region) coupling
               New stuff in DCM
1. DCM now accounts for the slice timing problem (SPM5)



2. Biological plausibility: each brain area can have two states
   (SPM8) (exc./inh.)



3. Biological plausibility: more complex balloon model (SPM5)



4. Non-linear version of DCM as well as bilinear (SPM8)
              Dynamic Causal Modelling of fMRI



   Network                                                 Model
   dynamics                                              comparison



                             Model inversion
Priors    State space
                                   using
            Model
                         Expectation-maximization



Haemodynamic                                        Posterior distribution
   response                       fMRI                  of parameters
                                 data (y)
Practical steps of a DCM study - I
1. Definition of the hypothesis & the model (on paper)
   •   Structure: which areas, connections and inputs?
   •   Which parameters in the model concern my hypothesis?
   •   How can I demonstrate the specificity of my results?
   •   What are the alternative models to test?

2. Defining criteria for inference:
   •   single-subject analysis: stat. threshold? contrast?
   •   group analysis:          which 2nd-level model?

3. Conventional SPM analysis (subject-specific)
   •   DCMs are fitted separately for each session (subject)
       →     for multi-session experiments, consider concatenation
             of sessions or adequate 2nd level analysis
Practical steps of a DCM study - II
4. Extraction of time series, e.g. via VOI tool in SPM
   •   caveat: anatomical & functional standardisation important
               for group analyses

5. Possibly definition of a new design matrix, if the “normal”
   design matrix does not represent the inputs appropriately.
   •   NB: DCM only reads timing information of each input from the
       design matrix, no parameter estimation necessary.

6. Definition of model
   •   via DCM-GUI or directly
       in MATLAB
Practical steps of a DCM study - III
7.       DCM parameter estimation
     •      caveat: models with many regions & scans can crash
            MATLAB!

8.       Model comparison and selection:
     •      Which of all models considered is the optimal one?
             Bayesian model selection

9.       Testing the hypothesis
         Statistical test on
         the relevant parameters
         of the optimal model
DCM button



‘specify’
             NB: in order!
                            Summary
   DCM is NOT EXPLORATORY

   Used to test the hypothesis that motivated the experimental
    design
      BUILD A MODEL TO EXPRESS HYPOTHESIS IN TERMS
       OF NEURAL CONNECTIVITY

   The GLM used in typical fMRI data analysis uses the same
    architecture as DCM but embodies more assumptions

   Note: In DCM a “Strong Connection” means an influence that is
    expressed quickly or with a small time constant.

   When constructing experiments, consider whether you want to
    use DCM early

   When in doubt, ask the experts………
                   REFERENCES

   Karl J. Friston. Dynamic Causal Modelling. Human brain
    function. Chapter 22. Second Edition.
     http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/

   K.J Friston, L. Harrison and W. Penny. Dynamic Causal
    Modelling. Neuroimage 2003; 19:1273-1302.

   SPM Manual

   Last year’s presentation
    ANY
QUESTIONS???