Dynamic Causal Modelling (DCM)
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Dynamic Causal Modelling (DCM)
Marta I. Garrido
Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros, Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner
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Motivation
Functional specialisationFunctional specialisation
Analysis of regionally specific effects
Analysis of regionally specific effects
Functional integrationFunctional integration
Interactions between distant regions
Interactions between distant regions
Varela et al. 2001, Nature Rev Neuroscience
Functional Connectivity
• Correlations between activity in spatially remote regions
• independent of how the dependencies are caused
MODEL-FREE MODEL-DRIVEN
Effective Connectivity
• The influence one neuronal system exerts over another
• Requires a mechanism or a generative model of measured brain responses
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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
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I. DCM: the basic idea
• Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI).
• The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ).
λ
z
y
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)
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intrinsic connectivity
direct inputs
modulation ofconnectivity
Neuronal state equation CuzBuAz jj ++= ∑ )( )(&
u
zC
z
z
uB
z
zA
j
j
∂∂
=
∂∂
∂∂
=
∂∂
=
&
&
&
)(
hemodynamicmodelλ
z
y
integration
t
drivinginput u1(t)
modulatoryinput u2(t)
t
BOLDy
y
yactivity
z2(t)
activityz1(t)
activityz3(t)
direct inputs
c1
b23a12
I. Conceptual overview
Stephan & Friston 2007, Handbook of Connectivity
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tionflow induc
=&
s
v
f
v
q q/vvf,Efqô /á
dHbchanges in
1)( −= ρρ&/ávfvô
volumechanges in
1−=&
f
q
)1( −−−= fãszs ry signalvasodilato
κ&
},,,,{ ρατγκθ =h},,,,{ ρατγκθ =h
( ) ,)(
signal BOLD
qvty λ=
I. The hemodynamic “Balloon” model
)(
input neuronal
tz
5 hemodynamic parameters:
Buxton et al. 1998Mandeville et al. 1999Friston et al. 2000, NeuroImage
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⎥⎥⎥
⎦
⎤
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⎣
⎡=
)(
)(
)(1
tz
tz
tz
n
M• State vector
– Changes with time
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),,...(
1
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nnn
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uzzf
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z
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θ
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&
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&• Rate of change of state vector
– Interactions between elements
– External inputs, u
( , , )z f z u θ=&• System parameters θ
I. Elements of a dynamic neuronal system
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11
dzsz
dt=−
Decay function
Half-life τ:
Generic solution to the ODEs in DCM:
ln 2 /s τ=
10.5 (0)z
τ
1 1
1
( ) 0.5 (0)
(0)exp( )
z z
z s
ττ
== −
1 1 1( ) (0)exp( ), (0) 1z t z st z= − =
I. Connectivity parameters = rate constants
Coupling parameter describes the speed ofthe exponential decay
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( )1 1
2 21 1 2
1
2
1
2 21
21
(0) 1
(0) 0
( ) exp( )
( ) exp( )
0
z sz
z s a z z
z
z
z t st
z t sa t st
a
= −
= −
=
=
= −
= −
>
&
&
1;4 21 == as
2;4 21 == as
1;8 21 == as
z2
21a
z1
s
s
z1 sa21t z2
I. Linear dynamics: 2 nodes
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u2
u1
z1
z2
activity in z2 is coupled to z1 via coefficient a21
u1
21a
001
01211
2
1
212
1 >⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
−=⎥
⎦
⎤⎢⎣
⎡au
czz
as
zz&
&
z1
z2
I. Neurodynamics: 2 nodes with input
Stimulus function
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000
00
1
01 2211
2
1221
22
1
212
1 >⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
−=⎥
⎦
⎤⎢⎣
⎡bu
czz
bu
zz
as
zz&
&
u2
u1
z1
z2
modulatory input u2 activity through the coupling a21
u1
u2
index, not squared
z1
z2
I. Neurodynamics: modulatory effect
21a
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0,,00
00
1
1 22121121
2
1221
22
1
21
12
2
1 >⎥⎦
⎤⎢⎣
⎡+⎥
⎦
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⎡+⎥
⎦
⎤⎢⎣
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⎤⎢⎣
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−=⎥
⎦
⎤⎢⎣
⎡baau
czz
bu
zz
aa
szz&
&
u2
u1
z1
z2
reciprocal connection
disclosed by u2
u1
u2 z1
z2
I. Neurodynamics: reciprocal connections
21a12a
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0 20 40 60
0
2
4
0 20 40 60
0
2
4
seconds
blue: neuronal activity
red: bold response
h1
h2
u1
u2 z1
z2
h(u,θ) represents the BOLD response (balloon model) to input
BOLD(no noise)
I. Hemodynamics
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0 20 40 60
0
2
4
0 20 40 60
0
2
4
seconds
BOLDnoise addedy1
y2
u1
u2 z1
z2
euhy += ),( θy represents simulated observation of BOLD response, i.e. includes noise
blue: neuronal activity
red: bold response
I. Hemodynamics (with noise)
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I. Bilinear state equation in DCM for fMRI
state changes
latentconnectivity
drivinginputs
state vector
CuzBuAzm
j
jj ++= ∑
=
)(1
&
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∑=
mnmn
m
n
m
j jnn
jn
jn
j
j
nnn
n
n u
u
cc
cc
z
z
bb
bb
u
aa
aa
z
z
M
L
MOM
L
M
L
MOM
L
L
MOM
L
&
M
& 1
1
1111
11
111
1
1111
induced connectivity
n regions m drv inputsm modulatory inputs
context-dependent
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Constraints on•Hemodynamic parameters•Connections
Models of•Hemodynamics in a single region•Neuronal interactions
Bayesian estimation
)(θp
)()|()|( θθθ pypyp ∝
)|( θyp
posterior
priorlikelihood term
II. Estimation: Bayesian framework
Mp
p-1
Mpost
post-1
d-1
Md θ
γηθ|y
probability that a parameter (or
contrast of parameters cT ηθ|y) is
above a chosen threshold γ
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sf
tionflow induc
=&
s
f
vq/vvf,Efqô /á
dHbchanges in
1)( −= ρρ&/ávfvô
volumechanges in
1−=&
f
)1( −−−= fãszs ry signalvasodilato
κ&
II. Parameter estimation
• 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 expectation-maximization.
• Result:(Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y.
ηθ|y
v
stimulus function u
modeled BOLD response
q
( , , )h x u θ ( , , )y h x u X eθ β= + +
observation model
hidden states},,,,{ qvfszx =
state equation( , , )x F x u θ=&
parameters
},{
},...,{
},,,,{1
nh
mn
h
CBBA
θθθ
θ
ρατγκθ
=
=
=
neuronal stateequation∑ ++= CuzBuAz j
j )(&
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Given competing hypotheses, which model is the best?
Pitt & Miyung 2002, TICS
)(
)()|(log
mcomplexity
maccuracymyp −=
)|(
)|(
jmyp
imypBij =
==
II. Bayesian model comparison
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V1
V5
SPC
Motion
Photic
Attention
0.85
0.57 -0.02
1.360.70
0.84
0.23
Model 1:attentional modulationof V1→V5
V1
V5
SPC
Motion
PhoticAttention
0.86
0.56 -0.02
1.42
0.550.75
0.89
Model 2:attentional modulationof SPC→V5
1 2log ( | ) log ( | )p y m p y m>>
III. Application: Attention to motion in the visual system
Büchel & Friston
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• potential timing problem in DCM:
temporal shift between regional time series because of multi-slice acquisition
• Solution:– Modelling of (known) slice timing of each area.
1
2
slic
e ac
quis
ition
visualinput
Slice timing extension now allows for any slice timing differences
Long TRs (> 2 sec) no longer a limitation.
Kiebel et al. 2007, Neuroimage
IV. Extensions: Slice timing model
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)(tu
ijij uBA +
input
Single-state DCM
1x
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
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⎡=
NNNN
N
x
x
tx
AA
AA
A M
K
MOM
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)(
Two-state DCM
Ex1
)exp( ijij uBA +
Ix1
11 11exp( )IE IEA uB+IEx ,1
IN
EN
I
E
AA
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IENN
EINN
EENNN
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M
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)(
000
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1
1111
11111
IV. Extensions: Two-state model
Marreiros et al. 2008, Neuroimage
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bilinear DCM
CuxDxBuAdt
dx m
i
n
j
jj
ii +⎟⎟
⎠
⎞⎜⎜⎝
⎛++= ∑ ∑
= =1 1
)()(CuxBuA
dt
dx m
i
ii +⎟
⎠
⎞⎜⎝
⎛ += ∑=1
)(
Bilinear state equation:
driving input
modulation
non-linear DCM
driving input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
Nonlinear state equation:
...2
)0,(),(2
2
22
0 +∂∂
+∂∂
∂+
∂∂
+∂∂
+≈=x
xf
uxuxf
uuf
xxf
xfuxfdtdx
Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas.
IV. Extensions: Nonlinear DCM
Stephan et al. 2008, Neuroimage
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Jansen and Rit 1995
David et al. 2006, Kiebel et al. 2006, Neuroimage
€
x.
= f (x,u,θ)
IV. Extensions: DCM for ERPs
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Forward
Backward
Lateral
with backward connections and without
a b c
A1 A1
STG
input
STG
IFG
FB
A1 A1
STG
input
STG
IFG
F
rIFG
rSTG
rA1lA1
lSTG
IV. Extensions: DCM for ERPs
Garrido et al. 2007, PNAS
standardsdeviants
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Grand mean ERPsa b
c
ERP oddball
model inversion from 0 to t where t = 120:10:400 ms for F and FB
128 EEG electrodes
Garrido et al. 2007, PNAS
IV. Extensions: DCM for ERPs
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IV. Extensions: DCM for ERPs
Garrido et al. 2008, Neuroimage
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The DCM cycle
Design a study to investigatethat system
Extraction of time seriesfrom SPMs
Parameter estimationfor all DCMs considered
Bayesian modelselection of optimal DCM
Statistical test on parameters
of optimal model
Hypotheses abouta neural system
DCMs specificationmodels the system
Data acquisition
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fMRI data
Posterior densities of parameters
Neuronal dynamics Hemodynamics
Model comparison
DCM roadmap
Model inversion usingExpectation-Maximization
State space Model
Priors
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Dynamic Causal Modelling (DCM)
Marta I. Garrido
Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros, Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner