Dynamic causal modelling
of brain-behaviour relationships
J. Daunizeau
Brain and Spine Institute, Paris, France
Wellcome Trust Centre for Neuroimaging, London, UK
Overview
 DCM: introduction
 Augmenting DCM with behavioural outputs
 Proof of concept: inhibitory control
Overview
 DCM: introduction
 Augmenting DCM with behavioural outputs
 Proof of concept: inhibitory control
Brain connectivities
structural connectivity
functional connectivity
effective connectivity
O. Sporns 2007, Scholarpedia
•
structural connectivity
= presence of axonal connections
•
functional connectivity
= statistical dependencies between regional time series
•
effective connectivity
= causal (directed) influences between neuronal populations
! connections are recruited in a context-dependent fashion
Functional segregation / integration
localizing brain activity:
functional segregation
A
effective connectivity analysis:
functional integration
A
B
A
u1
u2
u1
B
B
u1
u2
u1 X u2
« Where, in the brain, did
my experimental manipulation
have an effect? »
« How did my experimental manipulation
propagate through the network? »
DCM for fMRI: example
auditory cue
visual outcome
or
P(outcome|cue)
or
Put
response
0
200
400
600
800
2000
time (ms)
PMd
PPA
FFA
PPA
cue-dependent
surprise
Put
FFA
PMd
cue-independent
surprise
Den Ouden et al., J. Neurosci., 2010
Dynamical systems theory

1

u 
 x 
y
1
2
2
3
 21
1
1
2
3
2
32
13
3
13u
time
3
u
t  t
t  0
 3u
t
u
t  t
x t 0

x
t
Evolution and observation mappings
Hemodynamic
observation model:
temporal convolution
Electromagnetic
observation model:
spatial convolution
neural states dynamics
x  f ( x, u , )
fMRI
EEG/MEG
• simple neuronal model
• realistic observation model
• realistic neuronal model
• simple observation model
inputs
System identification: agnostic neural dynamics
a24
b12
d24
gating effect
2
4
u2
modulatory effect
1
3
c1
u1
driving input
f
f
2 f
 2 f x2
x  f ( x, u )  f  x0 ,0   x  u 
ux  2
 ...
x
u
xu
x 2
0
nonlinear state equation:
m
n

(i )
( j) 
x   A   ui B   x j D  x  Cu
i 1
j 1


Stephan et al., 2008
The neuro-vascular coupling
u
t
m
n


x   A   ui B (i )   x j D ( j )  x  Cu
i 1
j 1


experimentally controlled
stimulus
neural states dynamics
vasodilatory signal
s  x   s   ( f  1)
f
s
s
flow induction (rCBF)
f s
 h  { ,  , ,  , E0 ,  }
hemodynamic
states dynamics
f
 n  {A, B(i ) , C, D( j ) }
Balloon model
changes in volume
v  f v
1/ 
v
 ( q, v ) 
v
changes in dHb
1/ 
 q  f E ( f,E0 ) E
q0 v q / v
q
S


 q
 V0  k1 1  q   k2 1    k3 1  v  
S0
 v


k1  4.30 E0TE
k2   r0 E0TE
k3  1  
BOLD signal change
observation
Friston et al., 2003
Parametric statistical approach
• DCM: model structure
 y  g  x,    

 x  f  x, u, 
24
2
likelihood
 p  y  ,, m
4
3
1
u
• DCM: Bayesian inference
parameter estimates:
model evidence:


priors on parameters
ˆ    p  y  ,  , m  p  m  p  m  d d

p  y m    p  y  ,  , m  p  m  p  m  d d

The variational Bayesian approach

ln p  y m   ln p  , y m   S  q   DKL q   ; p  y, m 
q

free energy : functional of q
mean-field: approximate marginal posterior distributions:
q   , q  
1
2
p 1 ,2 y, m 
2
p 1 or 2 y, m 
1
q 1 or 2 
Overview
 DCM: introduction
 Augmenting DCM with behavioural outputs
 Proof of concept: inhibitory control
Identifying the brain-behaviour mapping
dk2
ak2
x2
x4
u2
x1
sensory
u
input 1
bk3
ok
behavioural
output
x3
u3
 modelling the brain input-output transform (through the network)
 decomposing the relative contribution of brain regions and their
interactions to the behavioural response
Identifying the brain-behaviour mapping
dk2
ak2
x2
x4
u2
x1
bk3
ok
behavioural
output
x3
sensory
u
input 1
u3
p  o x   s  r  1  s  r  
o
1 o
t

r  t    h  x   , u    e  d  r  h  x, u    r


h
h
2h
2h x2
h  x, u   h  0,0   x  u 
ux  2
 ...
x
u
xu
x 2
bDCM: face validity
B
u2
0
1
u1
0
r
0
time (s)
F
u1
0
35
0
time (s)
35
time (s)
35
H
3
4
0
1
2
0
0
-2
r
0
beh
u2
0
BOLD
r
2
-2
y
inverted
1
G
Estimated value
E
BOLD signal
Inputs
u2
u1
4
1
Response
simulated
D
C
y
A
-2
Parameter
0
g(x)BOLD
4
0
g(x) beh
1
bDCM: face validity
u2
Functional connectivity
u2
u2
u1
u1
u1
r
Response encoding
u1
u1
r
A
B
C
D
explained variance
bDCM: behavioural susceptibility analysis
u1
u2
u1
u2
u1
u2
u1
u2
?
r
r
r
r
bDCM: predicting the effect of lesions
to u1
on
off
B
to u2
u1
permutation
0
u2
0 1 2 3 4 5 6
time (s)
u2
B
1
u1
C
0.8
0.6
0.4
0.2
0

A
u
contribution
B
C
A
connection
B
C
C
2
A
susceptibility
1
r
response rate
Volterra kernel
A
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
u2 = 0
u2 = 1
condition
normal
lesioned
u2 = 0
u2 = 1
condition
Overview
 DCM: introduction
 Augmenting DCM with behavioural outputs
 Proof of concept: inhibitory control
Go/noGo: paradigm and fMRI results
A
B
Execute (NOGO)
t-value
8.5
Prepare
C
Go
cue
0.0
z=28
No-Go
z=62
Execute (GO)
prep
exec
stop
z=62
Go/noGo: model comparison set
H2
H3
H4
stop
direct
1
drive
cue
2
exec
3
potent
cue
cue
cue
4
serial
5
6
exec
exec
stop
H1
partial
prep
prep
prep
withold
prep
parallel
cue
cue
cue
cue
stop
full
1. left dPFC
2. right dPFC
3. left pre-motor 4. rigth pre-motor
5. left M1
6. right M1
modulation
Go/noGo: Bayesian model selection
5
H4 partial
H3 modulate
H1 drive
H2 direct
stop
- 4.85
cue
cue
exec
- 4.90
H0: - 6.73 x 103
- 4.95
+1
−1
1


prepare left


execute (Go)


stop (NoGo)
0
nnormalized eural activity
models
response predictor
log model evidence
x103
- 4.80
Go/noGo: behavioural fit
100
accuracy (%)
accuracy (%)
100
90
80
70
60
90
bDCM
70
fMRI decoding
60
50
50
0
2
train
test
train
test
evaluation set
4 6 8 10
time (s)
Left response
Right response
12
Volterra kernel
12
Volterra kernel
80
8
4
0
-4
-8
8
cue left
4
cue right
0
exec
-4
stop
-8
0
1
time (s)
2
0
1
time (s)
2
Go/noGo: behavioural susceptibility analysis
stop
cue
cue
exec
right response
left response
preparation
execution (Go)
stop (NoGo)
Go/noGo: lesion-induced behavioural deficits
stop
cue
cue
exec
1
response rate
1
0
1
L
R
1
0
0
L
R
L
R
L
R
1
L
R
R
response side
condition:
Go
No Go
0
L
R
1
0
1
node state:
normal
lesioned
0
L
R
0
Overview
 DCM: introduction
 Augmenting DCM with behavioural outputs
 Proof of concept: inhibitory control
Many thanks to Lionel Rigoux