Transcript mbi2008 7826

Dynamic Causal Modelling (DCM)
Marta I. Garrido
[email protected]
Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros,
Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner
Motivation
Functional specialisation
Functional integration
Varela et al. 2001, Nature Rev Neuroscience
Interactions between
distant regions
Analysis of regionally
specific effects
Effective Connectivity
Functional Connectivity
•
Correlations between activity
spatially remote regions
•
independent
of
how
dependencies are caused
MODEL-FREE
in
•
The influence one neuronal system
exerts over another
the
•
Requires a mechanism or a
generative model of measured
brain responses
MODEL-DRIVEN
Outline
I.
DCM: the neuronal and the hemodynamic models
II. Estimation and Bayesian inference
III. Application: Attention to motion in the visual system
IV. Extensions for fMRI and EEG data
I. DCM: the basic idea
The aim of DCM is to estimate and make inferences
about the coupling among brain areas, and how that
coupling is influences by changes in the experimental
contex. (Friston et al. 2003, Neuroimage)
• Using a bilinear state equation, a cognitive system is
modelled at its underlying neuronal level (which is not
directly accessible for fMRI).
z
λ
• The modelled neuronal dynamics (z) is transformed
into area-specific BOLD signals (y) by a hemodynamic
forward model (λ).
y
direct inputs
I. Conceptual overview
y
y
y
a12
BOLD
λ
b23
activity
z2(t)
activity
z3(t)
activity
z1(t)
y
hemodynamic
model
z
integration
c1
modulatory
input u2(t)
driving
input u1(t)
Neuronal state equation
t
intrinsic connectivity
modulation of
connectivity
t
Stephan & Friston 2007, Handbook of Connectivity
direct inputs
z  ( A   u j B( j ) ) z  Cu
z
z
 z

u j z
A
B( j)
C
z
u
I. The hemodynamic “Balloon” model
5 hemodynamic
parameters:
neuronal input
z (t )
 h  { ,  , ,  , }
vasodilatory signal
s  z  s  γ( f  1)
s
f
flow induction
f  s
f
changes in volume
τv  f  v
1 /α
v
changes in dHb
τq  f E ( f, ) q  v1/α q/v
q
v
BOLD signal
y (t )   v, q 
Buxton et al. 1998
Mandeville et al. 1999
Friston et al. 2000, NeuroImage
I. Elements of a dynamic neuronal system
• State vector
– Changes with time
• Rate of change of state vector
– Interactions between
elements
– External inputs, u
• System parameters 
 z1 (t ) 
z (t )    
 zn (t )
 z1   f1 ( z1...zn , u,1 ) 



  

 zn   f n ( z1...zn , u, n )
z  f ( z, u,  )
I. Connectivity parameters = rate constants
Generic solution to the ODEs in DCM:
dz1
  sz1
dt
z1 (t )  z1 (0)exp(st ),
Decay function
Half-life:
z1 ( )  0.5 z1 (0)
Coupling parameter
describes the speed of
the exponential decay
0.5 z1 (0)
 z1 (0) exp( s )
s  ln 2 / 
z1 (0)  1

I. Linear dynamics: 2 nodes
s
z1
s  4; a21  1
a21
z2
s  4; a21  2
s
z1   sz1
z2  s  a21 z1  z2 
z1 (0)  1
s  8; a21  1
z2 (0)  0
z1 (t )  exp( st )
z2 (t )  sa21t exp( st )
a21  0
z1
sa21t
z2
I. Neurodynamics: 2 nodes with input
u1
z1
u1
u2
Stimulus function
z1
a21
z2
z2
 z1    1 0   z1  c 
 z   s a  1  z   0u1
 2   21
 2   
a21  0
activity in z2 is coupled to z1 via coefficient a21
I. Neurodynamics: modulatory effect
u1
u2
z1
u1
u2
z1
a21
z2
z2
index, not squared
 z1    1 0   z1 
 0 0  z1  c 
 z   s a  1  z   u2 b 2 0  z   0u1
 2   21
 2 
 21   2   
modulatory input u2 activity through the coupling a21
b212  0
I. Neurodynamics: reciprocal connections
u1
u1
u2
u2
z1
z1
a21
z2
a12
z2
 z1    1 a12   z1 
 0 0  z1  c 

s

u
 z  a  1  z  2 b 2 0  z   0u1
 2   21
 2 
 21   2   
a12 , a21, b212  0
reciprocal
connection
disclosed by u2
I. Hemodynamics
blue: neuronal activity
red: bold response
u1
u2
z1
h1
BOLD
(no noise)
4
2
0
0
20
40
60
0
20
40
seconds
60
4
z2
h2
2
0
h(u,θ) represents the BOLD response (balloon model) to input
I. Hemodynamics (with noise)
blue: neuronal activity
red: bold response
u1
u2
z1
y1
4
BOLD
2
noise added
0
0
20
40
60
0
20
40
seconds
60
4
z2
y2
2
0
y represents simulated observation of BOLD response, i.e. includes noise
y  h(u, )  e
I. Bilinear state equation in DCM for fMRI
context-dependent
state
latent
changes connectivity
induced
connectivity
state
vector
driving
inputs
j
j


   z1  c11  c1m   u1 

z
a

a
b

b
 1   11
1n 
11
1
n
m
          u                  
j
   
  
 



 zn  an1  ann  j 1 bnj1  bnnj    zn  cn1  cnm  um 


n regions
m modulatory inputs
m
z  ( A   u j B ) z  Cu
j
j 1
m drv inputs
II. Estimation: Bayesian framework
Models of
Constraints on
•Hemodynamics in a single region
•Neuronal interactions
•Hemodynamic parameters
•Connections
p( y |  )
likelihood term
posterior
p ( )
p( | y )  p( y |  ) p( )
Bayesian estimation

post-1
p-1
d-1
Mp Mpost Md

prior
probability that a
parameter (or
contrast of
parameters cT ηθ|y) is
above a chosen
threshold γ
ηθ|y
II. Parameter estimation
stimulus function u
neuronal state
equation
z  ( A   u j B j ) z  Cu
• Specify model
(neuronal and
hemodynamic level)
• Make it an observation
model by adding
measurement error e
and confounds X (e.g.
drift).
• Bayesian parameter
estimation using
expectationmaximization.
• Result:
(Normal) posterior
parameter
distributions, given by
mean ηθ|y and
Covariance Cθ|y.
vasodilatory signal
s  z  s  γ( f  1)
s
f
parameters
flow induction
hidden states
x  {z, s, f , v, q}
f  s
state equation
f
 h  { ,  , ,  , }
 n  { A, B1...B m , C}
  { h , n }
x  F ( x, u ,  )
changes in volume
τv  f  v1/α
v
changes in dHb
τq  f E ( f, )   v1/α q/v
v
q
ηθ|y
h( x, u, )
modeled
BOLD response
y  h( x, u, )  X   e
observation model
II. Bayesian model comparison
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
III. Application: Attention to motion in the
visual system
Model 1:
attentional modulation
of V1→V5
Photic
SPC
0.70
0.
1.36
V1
84
0.85
0.57
-0.02
V5
0.23
Motion
Attention
Model 2:
attentional modulation
of SPC→V5
Photic
Attention
SPC
0.55
0.86
0.75
1.42
0.89
V1
0.56
Motion
-0.02
V5
log p( y | m1 )  log p( y | m2 )
Büchel & Friston
•
•
potential timing problem in DCM:
temporal shift between regional
time series because of multi-slice
acquisition
slice acquisition
IV. Extensions: Slice timing model
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, Neuroimage
2
1
visual
input
IV. Extensions: Two-state model
Single-state DCM
Two-state DCM
input
x1E
u (t )
x1E , I
x1
exp( A11IE  uB11IE )
x1I
Aij  uBij
 A11  A1N 
A      
 AN 1  ANN 
 x1 
x(t )    
 xN 
Marreiros et al. 2008, Neuroimage
exp( Aij  uBij )
  eA11EE
 AIE
 e 11
A  
 A
 e N1

 0
Extrinsic (betweenregion) coupling
 eA11
II
 eA11
EI

A
e 1N
0
0
0

 eANN
 eANN
II
 eANN

0
0
EE

IE
eANN
EI
Intrinsic (withinregion) coupling

x1E

 I

 x1
 x (t)   

 E

xN

xI
 N









IV. Extensions: Nonlinear DCM
bilinear DCM
non-linear DCM
modulation
driving
input
driving
input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
dx
f
f
2 f
2 f x2
 f ( x, u)  f ( x0 ,0)  x  u 
ux  2
 ...
dt
x
u
xu
x 2
Bilinear state equation:
m
dx 

  A   ui B ( i )  x  Cu
dt 
i 1

Nonlinear state equation:
m
n
dx 
(i )
( j) 
  A   ui B   x j D  x  Cu
dt 
i 1
j 1

Here DCM can model activity-dependent changes in connectivity; how
connections are enabled or gated by activity in one or more areas.
Stephan et al. 2008, Neuroimage
IV. Extensions: DCM for ERPs
.
x  f (x,u, )

Jansen and Rit 1995
David et al. 2006, Kiebel et al. 2006, Neuroimage
IV. Extensions: DCM for ERPs
a
b
with backward connections
c
and without
IFG
IFG
FB
F
STG
STG
STG
STG
rIFG
lA1
A1
A1
A1
A1
lSTG
rA1
rSTG
input
input
deviants
standards
Forward
Backward
Lateral
Garrido et al. 2007, PNAS
IV. Extensions: DCM for ERPs
a
Grand mean ERPs
b
128 EEG electrodes
c
ERP oddball
Garrido et al. 2007, PNAS
model inversion from 0 to t where
t = 120:10:400 ms for F and FB
IV. Extensions: DCM for ERPs
Garrido et al. 2008, Neuroimage
The DCM cycle
Hypotheses about
a neural system
Statistical test
on parameters
of optimal
model
DCMs specification
models the system
Design a study to
investigate
that system
Bayesian model
selection of
optimal DCM
Parameter estimation
for all DCMs considered
Data acquisition
Extraction of
time series
from SPMs
DCM roadmap
Neuronal
dynamics
Hemodynamics
State space
Model
Posterior densities
of parameters
Priors
Model inversion using
Expectation-Maximization
fMRI data
Model
comparison
Dynamic Causal Modelling (DCM)
Marta I. Garrido
[email protected]
Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros,
Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner