Connectivity_DCM_den_Ouden

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					         Models of Effective Connectivity &
           Dynamic Causal Modelling

Hanneke den Ouden
Wellcome Trust Centre for Neuroimaging,
  University College London, UK
Donders Institute for Brain, Cognition and
  Behaviour, Nijmegen, the Netherlands




                                                             SPM course
Thanks to Klaas Stephan and Meike Grol for slides   Zurich, February 2009
     Systems analysis in functional neuroimaging

Functional specialisation:             Functional integration:
What regions respond to a particular   How do regions influence each other?
experimental input?                     Brain Connectivity




                                                                 ?
                                                          ?
                               Overview

• Brain connectivity: types & definitions
   – anatomical connectivity
   – functional connectivity
   – effective connectivity

• Functional connectivity
• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)

• Applications of DCM to fMRI data
    Structural, functional & effective connectivity




                                                               Sporns 2007, Scholarpedia
• anatomical/structural connectivity
  = presence of axonal connections

• functional connectivity
  =    statistical dependencies between regional time series

• effective connectivity
  =    causal (directed) influences between neurons or neuronal populations
                      Anatomical connectivity
• presence of axonal
  connections

• neuronal communication
  via synaptic contacts

• visualisation by

   – tracing techniques




   – diffusion tensor imaging
                  However,
 knowing anatomical connectivity is not enough...

• Connections are recruited in a context-dependent fashion:
   – Local functions depend on network activity

                                    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
                  However,
 knowing anatomical connectivity is not enough...

• Connections are recruited in a context-dependent fashion:
   – Local functions depend on network activity

• Connections show plasticity
   – Synaptic plasticity =    change in the structure and transmission
                                     properties of a synapse
   – Critical for learning
   – Can occur both rapidly and slowly




          Need to look at functional and effective connectivity
                         Overview

• Brain connectivity: types & definitions

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)

• Applications of DCM to fMRI data
    Different approaches to analysing functional
                    connectivity

Definition: statistical dependencies between regional time series
• 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
                    SVCA example:
     Task-induced changes in functional connectivity
2 bimanual finger-tapping tasks:


During task that required more
bimanual coordination, SMA,
PPC, M1 and PM showed
increased functional connectivity
(p<0.001) with left M1


 No difference in SPMs!




Sun et al. 2003, Neuroimage
           Does functional connectivity not simply
           correspond to co-activation in SPMs?

                                regional      task T   regional response A2
                                response A1
No, it does not - see the
fictitious example 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
For understanding brain function mechanistically,
  we need models of effective connectivity, i.e.

 models of causal interactions among neuronal
                  populations

to explain regional effects in terms of interregional
                    connectivity
 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

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)

• Applications of DCM to fMRI data
     Psycho-physiological interaction (PPI)

• bilinear model of how the influence of area A on
  area B changes by the psychological context C:

                       AxCB

• a PPI corresponds to differences in regression
  slopes for different contexts.
                                Psycho-physiological interaction (PPI)
                                      Task factor                  GLM of a 2x2 factorial design:
                                 Task A         Task B
                                                                   y  (TA  TB )  1                  main effect
                                                                                                       of task
  Stimulus factor
                    Stim 1




                                 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
                             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
      Example PPI: 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
                       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
   – operates at the level of BOLD time series

            sometimes very useful, but limited causal interpretability;
            in most cases, we need more powerful models

            DCM!
                           Overview
• Brain connectivity: types & definitions

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – Basic idea
   – Neural level
   – Hemodynamic level
   – Priors & Parameter estimation

• Applications of DCM to fMRI data
                    Basic idea of DCM for fMRI
                         (Friston et al. 2003, NeuroImage)


• Investigate functional integration & modulation of specific cortical
  pathways

• Using a bilinear state equation, a cognitive system is modelled at its   x
  underlying neuronal level (which is not directly accessible for fMRI).
                                                                           λ
• The modelled neuronal dynamics (x) is 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 and measured
    BOLD signals are maximally similar.
                           Overview
• Brain connectivity: types & definitions

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – Basic idea
   – Neural level
   – Hemodynamic level
   – Priors & Parameter estimation

• Applications of DCM to fMRI data
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
                                                  u1
                      u2



                  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 
      Az 
  x  Ax Cu        x  a a        0 a24   x2   c21 0   u1 
                    2    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        0 a24                                 x  c               u1 
 2     21 22                                                        0      0  
                           u3 0 0         0 0 
                                                             2    21          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  
                 y                        y               y         BOLD                  y

                                                                                                    hemodynamic
                                       activity                                           λ         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)
                                                                  endogenous                                   x
                                                                                                                
                                                                  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
                           Overview
• Brain connectivity: types & definitions

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – Basic idea
   – Neural level
   – Hemodynamic level
   – Priors & Parameter estimation

• Applications of DCM to fMRI data
                      The hemodynamic model in DCM
• 6 hemodynamic                                                u                                  stimulus functions
  parameters:                                              t

                                                           activity
                                                                                                  neural state equation
  h  { ,  , ,  ,  ,  }                                 x (t )

                                                         vasodilatory signal

   important for model fitting, but                 s  x  s  γ ( f  1)
                                                    
                                                                                         s
   of no interest for statistical
                                            f                      s

   inference
                                                     flow induction (rCBF)                        hemodynamic state
                                                               
                                                               f s                               equations
                                                                f

• Computed separately for                           Balloon model
  each area (like the neural          changes in volume        v               changes in dHb

  parameters)                           τv  f  v1 /α
                                                                      τq  f E ( f,E 0 ) q 0  v1 /α q/v
                                                                        
                                                                                     q
                                                                                           E

   region-specific HRFs!
                                                v




                                                    BOLD signal                                   Estimated BOLD
                                                     y (t )   v, q 
 Friston et al. 2000, NeuroImage
                                                                                                  response
 Stephan et al. 2007, NeuroImage
Example: modelled BOLD signal
             FG         FG
             left      right




             LG         LG
             left      right




       RVF                     LVF




     black: observed BOLD signal
     red:   modelled BOLD signal
                           Overview
• Brain connectivity: types & definitions

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)
   – Basic idea
   – Neural level
   – Hemodynamic level
   – Priors & Parameter estimation

• Applications of DCM to fMRI data
                          Bayesian statistics

            new data     prior knowledge

            p( y |  )        p( )

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



Bayes theorem allows us to express         The posterior probability of the
our prior knowledge or “belief” about      parameters given the data is an
parameters of the model                    optimal combination of prior knowledge
                                           and new data, weighted by their
                                           relative precision.
                                 Priors in DCM
• embody constraints on parameter estimation
   – hemodynamic parameters: empirical priors
   – coupling parameters of self-connections: principled priors
   – coupling parameters other connections: shrinkage priors

      Small & variable effect                      Large & variable effect




      Small but clear effect                       Large & clear effect
            DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
  dx
      ax                       x (t )  x0 exp( at )
  dt
Coupling parameter a is inversely                     The coupling parameter a
proportional to the half life  of x(t):              thus describes the speed of
 x( )  0.5x0                                        the exponential change in x(t)
                                       0.5x0
         x0 exp(a )

            a  ln 2 / 
                                                 ln 2 / a
 If AB is 0.10 s-1 this means that, per unit time, the increase in activity in B
 corresponds to 10% of the activity in A
   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
                                                                             0              0 0 u2 
                                                                                     b 22     
                                                                                       2
                                               2                 
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)                                                                        
                                 DCM Summary
Select areas you want to model
•   Extract timeseries of these areas
                                                       Modulatory input             Driving input
    (x(t))                                      (e.g. context/learning/drugs)    (e.g. sensory stim)
•   Specify at neuronal level
                                                                     b12
     – what drives areas (c)                                                     c1
     – how areas interact (a)                                                                 c2
                                         neuronal
     – what modulates interactions (b)   states           activity
•   State-space model with 2 levels:                       x1(t)           a12   activity
                                                                                      x2(t)
     – Hidden neural dynamics
     – Predicted BOLD response                                y
                                         BOLD                                           y
•   Estimate model parameters:
    Gaussian a posteriori parameter          ηθ|y
    distributions, characterised by
    mean ηθ|y and
    covariance Cθ|y.
            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?").
            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

• Functional connectivity

• Psycho-physiological interactions (PPI)

• Dynamic causal models (DCMs)

• Applications of DCM to fMRI data
   – Design of experiments and models
   – Some empirical examples and simulations
              Planning a DCM-compatible study

• Suitable experimental design:
   – any design that is suitable for a GLM
   – preferably multi-factorial (e.g. 2 x 2)
       • e.g. one factor that varies the driving (sensory) input
       • and one factor that varies the contextual input


• Hypothesis and model:
   – Define specific a priori hypothesis
   – Which parameters are relevant to test this hypothesis?
   – If you want to verify that intended model is suitable to test this hypothesis,
     then use simulations
   – Define criteria for inference
   – What are the alternative models to test?
                                   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
                              A1


    +++
Stim1           +
                                   Stim 1   Stim 2   Stim 1   Stim 2
          A1            A2         Task A   Task A   Task B   Task B
                +++

Stim2          +++     +
    +
          Task A     Task B
                              A2
X1




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




X2




                                plus added noise (SNR=1)
               Final point: 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.

 If there is no evidence for an experimental effect (no activation
  detected by a GLM) → inclusion of this region in a DCM is not
                            meaningful.
                  Thank you


           Stay tuned to find out how to

 … select the best model comparing various DCMs
… test whether one region influences the connection
               between other regions
      … do DCM on your M/EEG & LFP data
                 … and lots more!

				
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