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  • Applications  of  Variational  Bayes  &  DAGs   in  Neuroimaging  

    ECE  6504:     Advanced  Topics  in  Machine  Learning  

         

    Rosalyn  Moran   [email protected]  

  • ì   Overview  

    1.  Dynamics  in  Dynamic  Causal  Modeling  

    2.  Graphical  Model   -­‐  VariaFonal  Inversion  

    -­‐   StaFsFcal  Inference  from  VB  

    3.  Examples     -­‐  ANenFon  in  the  Human  Brain  

    -­‐  Synesthesia  

  • Dynamic  Causal  Modelling  

    Friston  et  al  2003;  Stephan  et  al  2008  

    Kiebel  et  al,  2006;  Garrido  et  al,  2007  

    David  et  al,  2006;  Moran  et  al,  2007    

    dx dt

    Tim e  Series  

    DCM  is  not  intended  for  ‘modelling’       DCM  is  an  analysis  framework  for  empirical  data     DCM  uses  a  Fmes  series  to  test  mechanisFc  hypotheses     Hypotheses  are  constrained  by  the  underlying  dynamic   generaFve  (biological)  model  

  • ),,( θuxF dt dx

    =

    Neural state equation:

    Electromagnetic forward model:


    neural activity→EEG
 MEG LFP

    simple neuronal model complicated forward model

    complicated neuronal model simple forward model

    fMRI EEG/MEG

    Hemodynamic
 forward model:
 neural activity→BOLD

    Dynamic  Causal  Modelling  (DCM)  

  • DCM  for  fMRI  

    u1   A(1,1)        

    A(2,1)        

    A(1,2)        

    A(2,2)        

    x1  

    x = (A+uB)x +Cu y = g(x,H )+ε ε ~ N(0,σ )

    u2   B(1,2)        

    H{1}  

    y  

    H{2}  

    y  

    x2  

    C(1)  

  • ),,( θuxF dt dx

    =

    x1   x2   x3  

    System  states  xt  

    ConnecFvity  parameters  θ  

    Inputs  ut  

    Aim:  model  temporal  evoluFon  of  a  set  of  neuronal  states  xt  

    Neuronal  model  

    State  changes  are  dependent   on:  

    –  the  current  state  x   –  external  inputs  u   –  its  connecFvity  θ  

  • Example:  a  linear  model  of  interacFng  visual  regions  

    Visual  input  in  the    visual  field    -­‐  le\  (LVF)    -­‐  right  (RVF)     LG  =  lingual  gyrus   FG  =  fusiform  gyrus  

    LG   le\  

    LG   right  

    RVF   LVF  

    FG   right  

    FG   le\  

    x1   x2  

    x4  x3  

    u2   u1  

    x3 = a31x1 + a33x3 + a34x4

    x1 = a11x1 + a12x2 + a13x3 + c12u2

    x4 = a42x2 + a43x3 + a44x4

    x2 = a21x1 + a22x2 + a24x4 + c21u1

  • Example:  a  linear  model  of  interacFng  visual  regions  

    x1 = a11x1 + a12x2 + a13x3 + c12u2 x2 = a21x1 + a22x2 + a24x4 + c21u1 x3 = a31x1 + a33x3 + a34x4 x4 = a42x2 + a43x3 + a44x4

    Visual  input  in  the    visual  field    -­‐  le\  (LVF)    -­‐  right  (RVF)     LG  =  lingual  gyrus   FG  =  fusiform  gyrus  

    LG   le\  

    LG   right  

    RVF   LVF  

    FG   right  

    FG   le\  

    x1   x2  

    x4  x3  

    u2   u1  

  • Visual  input  in  the     visual  field    -­‐  le\  (LVF)    -­‐  right  (RVF)     LG  =  lingual  gyrus   FG  =  fusiform  gyrus  

    Example:  a  linear  model  of  interacFng  visual  regions  

    state 
 changes

    effective connectivity

    external
 inputs

    system
 state

    input parameters

    x1 x2 x3 x4

    !

    "

    # # # # # #

    $

    %

    & & & & & &

    =

    a11 a12 a13 0

    a21 a22 0 a24 a31 0 a33 a34 0 a42 a43 a44

    !

    "

    # # # # # #

    $

    %

    & & & & & &

    x1 x2 x3 x4

    !

    "

    # # # # # #

    $

    %

    & & & & & &

    +

    0 c21 0 0

    c12 0 0 0

    !

    "

    # # # # #

    $

    %

    & & & & &

    u1 u2

    !

    "

    # #

    $

    %

    & &

    x = Ax +Cu

    },{ CA=θ

    LG   le\  

    LG   right  

    RVF   LVF  

    FG   right  

    FG   le\  

    x1   x2  

    x4  x3  

    u2   u1  

  • Example:  a  linear  model  of  interacFng  visual  regions  

    LG   le\  

    LG   right  

    RVF   LVF  

    FG   right  

    FG   le\  

    x1   x2  

    x4  x3  

    u2   u1  

    ATTENTION   u3  

    x = (A+ u jB ( j )

    j=1

    m

    ∑ )x +Cu

    x1 x2 x3 x4

    !

    "

    # # # # # #

    $

    %

    & & & & & &

    =

    a11 a12 a13 0

    a21 a22 0 a24 a31 0 a33 a34 0 a42 a43 a44

    !

    "

    # # # # # #

    $

    %

    & & & & & &

    +u3

    0 b12 (3) 0 0

    0 0 0 0 0 0 0 b34

    (3)

    0 0 0 0

    !

    "

    # # # # #

    $

    %

    & & & & &

    '

    (

    ) ))

    *

    ) ) )

    +

    ,

    ) ))

    -

    ) ) )

    x1 x2 x3 x4

    !

    "

    # # # # # #

    $

    %

    & & & & & &

    +

    0 c21 0 0

    c12 0 0 0

    0 0 0 0

    !

    "

    # # # # #

    $

    %

    & & & & &

    u1 u2 u3

    !

    "

    # # # #

    $

    %

    & & & &

  • DeterminisFc  Bilinear  DCM  

    CuxBuA dt dx m

    i

    i i +⎟

    ⎠

    ⎞ ⎜ ⎝

    ⎛ += ∑

    =1

    )(

    Bilinear state equation:

    driving input

    modulation

    ...)0,(),( 2

    0 +∂∂ ∂

    + ∂ ∂

    + ∂ ∂

    +≈= ux ux fu

    u fx

    x fxfuxf

    dt dx

    Simply a two-dimensional taylor expansion (around x0=0, u0=0):

    A= ∂f ∂x u=0

    C = ∂f ∂u x=0

    B = ∂ 2 f

    ∂x∂u

  • u2

    u1

    x1

    x2

    stimulus u1

    context u2

    x1

    x 2

    21a

    Context-­‐dependent  enhancement  

    ( )

    ( ) ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ +⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ +⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ =⎥

    ⎦

    ⎤ ⎢ ⎣

    ⎡

    ++=

    2

    111

    2

    1 2 21

    2 2

    1

    2221

    11

    2

    1

    2 2

    00 0

    0 000

    u uc

    x x

    b u

    x x

    aa a

    x x

    CuxBuAxx

     

  • endogenous   connecFvity  

    direct  inputs  

    modulaFon  of   connecFvity  

    Neural  state  equaFon   CuxBuAx jj ++= ∑ )( )(

    u x

    C

    x x

    u B

    x x

    A

    j

    j

    ∂ =

    ∂ =

    ∂ =

    )(

    hemodynamic   model  H  

    x  

    y  

    integraFon  

    Stephan & Friston (2007), Handbook of Brain Connectivity

    BOLD  y  y  y  

    ac#vity   x1(t)  

    ac#vity   x2(t)   ac#vity  

    x3(t)  

    Neuronal   states  

    t  

    driving   input  u1(t)  

    modulatory   input  u2(t)  

    t  

    DCM  for  fMRI:  the  full  picture  

  • ì  Cognitive system is modelled at its underlying neuronal level (not directly accessible for fMRI).

    ì  The modelled neuronal dynamics (x) are transformed into area-specific BOLD signals (y) by a hemodynamic model (λ).

    ì  Overcomes regional variability at the hemodynamic level

    ì  DCM not based on temporal precedence at measurement level

    DCM:  Neuronal  and  hemodynamic  level  

    hemodynamic   model  

    H