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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