Dynamic Causal Modelling (DCM ) for fMRI
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Dynamic Causal Modelling (DCM) for fMRI
Klaas Enno Stephan Laboratory for Social & Neural Systems Research (SNS) University of Zurich
Wellcome Trust Centre for NeuroimagingUniversity College London
SPM Course, FIL13 May 2011
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Structural, functional & effective connectivity
• anatomical/structural connectivity= presence of axonal connections
• functional connectivity = statistical dependencies between regional time series
• effective connectivity = directed influences between neurons or neuronal populations
Sporns 2007, Scholarpedia
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Some models of effective connectivity for fMRI data
• Structural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000
• regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997
• Volterra kernels Friston & Büchel 2000
• Time series models (e.g. MAR/VAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003
• Dynamic Causal Modelling (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008
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Dynamic causal modelling (DCM)
• DCM framework was introduced in 2003 for fMRI by Karl Friston, Lee Harrison and Will Penny (NeuroImage 19:1273-1302)
• part of the SPM software package
• currently more than 160 published papers on DCM
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),,( uxFdtdx
Neural state equation:
Electromagneticforward model:
neural activityEEGMEGLFP
Dynamic Causal Modeling (DCM)
simple neuronal modelcomplicated forward model
complicated neuronal modelsimple forward model
fMRI EEG/MEG
inputs
Hemodynamicforward model:neural activityBOLD
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LGleft
LGright
RVF LVF
FGright
FGleft
LG = lingual gyrusFG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF)visual field.x1 x2
x4x3
u2 u1
1 11 1 12 2 13 3 12 2
2 21 1 22 2 24 4 21 1
3 31 1 33 3 34 4
4 42 2 43 3 44 4
x a x a x a x c ux a x a x a x c ux a x a x a xx a x a x a x
Example: a linear model of interacting visual regions
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Example: a linear model of interacting visual regions
LG = lingual gyrusFG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF)visual field.
state changes
effectiveconnectivity
externalinputs
systemstate
inputparameters
11 12 131 1 12
21 22 242 2 121
31 33 343 3 2
42 43 444 4
0 00 0
0 0 0
0 0 0
a a ax x ca a ax x uca a ax x u
a a ax x
x Ax Cu
},{ CA
LGleft
LGright
RVF LVF
FGright
FGleft
x1 x2
x4x3
u2 u1
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Extension: bilinear model
LGleft
LGright
RVF LVF
FGright
FGleft
x1 x2
x4x3
u2 u1
CONTEXTu3
( )
1
( )m
jj
j
x A u B x Cu
(3)11 12 131 1 1212
121 22 242 2 21
3 2(3)31 33 343 334
342 43 444 4
0 0 00 0 00 0 00 0 0 0
0 0 0 00 0 0
0 0 0 00 0 0 0
a a ax x cbu
a a ax x cu u
a a ax xbu
a a ax x
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endogenous connectivity
direct inputs
modulation ofconnectivity
Neural state equation CuxBuAx jj )( )(
uxC
xx
uB
xxA
j
j
)(
hemodynamicmodelλ
x
y
integration
BOLDyyy
activityx1(t)
activityx2(t) activity
x3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
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Bilinear DCM
CuxBuAdtdx m
i
ii
1
)(
Bilinear state equation:
driving input
modulation
...)0,(),(2
0
uxuxfu
ufx
xfxfuxf
dtdxTwo-dimensional Taylor series (around x0=0, u0=0):
0
2
0
u
x
fAx
fBx ufCu
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DCM parameters = rate constants
dx axdt
0( ) exp( )x t x at
The coupling parameter a thus describes the speed ofthe exponential change in x(t)
0
0
( ) 0.5exp( )
x xx a
Integration of a first-order linear differential equation gives anexponential function:
/2lna
00.5x
a/2ln
Coupling parameter a is inverselyproportional to the half life of z(t):
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-
x2
stimuliu1
contextu2
x1
+
+
-
-
-+
u1
Z1
u2
Z2
2 1
(2)
2121 1111
2 2212 222
0 00 00
x Ax u B x Cu
x ua cx u x
x uab
b
Example: context-dependent decay u1
u2
x2
x1
Penny et al. 2004, NeuroImage
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The problem of hemodynamic convolution
Goebel et al. 2003, Magn. Res. Med.
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Hemodynamic forward models are important for connectivity analyses of fMRI data
David et al. 2008, PLoS Biol.
Granger causality
DCM
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0 2 4 6 8 10 12 14
0
0.2
0.4
0 2 4 6 8 10 12 14
0
0.5
1
0 2 4 6 8 10 12 14-0.6
-0.4
-0.2
0
0.2
RBMN, = 0.5CBMN, = 0.5RBMN, = 1CBMN, = 1RBMN, = 2CBMN, = 2sf
tionflow induc
(rCBF)
s
v
stimulus functions
vq q/vvEf,EEfqτ /α
dHbchanges in1
00 )( /αvfvτ
volumechanges in1
f
q
)1(
fγsxssignalryvasodilato
u
s
CuxBuAdtdx m
j
jj
1
)(
t
neural state equation
1
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111),(
3
002
001
32100
kTEErk
TEEk
vkvqkqkV
SSvq
hemodynamic state equations
f
Balloon model
BOLD signal change equation
The hemodynamic model in DCM
Stephan et al. 2007, NeuroImage
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5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
A
B
C
h
ε
How interdependent are neural and hemodynamic parameter estimates?
Stephan et al. 2007, NeuroImage
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DCM is a Bayesian approach
)()|()|( pypyp posterior likelihood ∙ prior
)|( yp )(p
Bayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities.
In DCM: empirical, principled & shrinkage priors.
The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.
new data prior knowledge
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sf (rCBF)induction -flow
s
v
f
stimulus function u
modelled BOLD response
vq q/vvf,Efqτ /α1)(
dHbin changes
/αvfvτ 1
in volume changes
f
q
)1(signalry vasodilatodependent -activity
fγszs
s
)(xy eXuhy ),(
observation model
hidden states{ , , , , }z x s f v q
state equation( , , )z F x u
parameters
},{
},...,{
},,,,{1
nh
mn
h
CBBA
• Combining the neural and hemodynamic states gives the complete forward model.
• An observation model includes measurement error e and confounds X (e.g. drift).
• Bayesian inversion: parameter estimation by means of variational EM under Laplace approximation
• Result:Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.
Overview:parameter estimation
ηθ|y
neural stateequation( )j
jx A u B x Cu
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• Gaussian assumptions about the posterior distributions of the parameters
• posterior probability that a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ:
• By default, γ is chosen as zero ("does the effect exist?").
Inference about DCM parameters:Bayesian single-subject analysis
cCc
cp
yT
yT
N
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Bayesian single subject inference
LGleft
LGright
RVFstim.
LVFstim.
FGright
FGleft
LD|RVF
LD|LVF
LD LD
0.34 0.14
-0.08 0.16
0.13 0.19
0.01 0.17
0.44 0.14
0.29 0.14
Contrast:Modulation LG right LG links by LD|LVFvs.modulation LG left LG right by LD|RVF
p(cT>0|y) = 98.7%
Stephan et al. 2005, Ann. N.Y. Acad. Sci.
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Likelihood distributions from different subjects are independent one can use the posterior from one subject as the prior for the next
NiiN
iN
yy
N
iyyyy
N
iyyy
CC
CC
,...,|1
|1|,...,|
1
1|
1,...,|
11
1
Under Gaussian assumptions this is easy to compute:
groupposterior covariance
individualposterior covariances
groupposterior mean
individual posterior covariances and means“Today’s posterior is tomorrow’s prior”
Inference about DCM parameters: Bayesian parameter averaging (FFX group analysis)
1 1
1
12
1 23
1 1
|
,
N N
N
ii
N
ii
N
ii
N N
p y y p y y p
p p y
p y p y
p y y p y
p y y p y
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Inference about DCM parameters:RFX group analysis (frequentist)• In analogy to “random effects” analyses in SPM, 2nd level
analyses can be applied to DCM parameters:
Separate fitting of identical models for each subject
Selection of (bilinear) parameters of interest
one-sample t-test:
parameter > 0 ?
paired t-test: parameter 1 > parameter 2 ?
rmANOVA: e.g. in case of
multiple sessions per subject
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inference on model structure or inference on model parameters?
inference on individual models or model space partition?
comparison of model families using
FFX or RFX BMS
optimal model structure assumed to be identical across subjects?
FFX BMS RFX BMS
yes no
inference on parameters of an optimal model or parameters of all models?
BMA
definition of model space
FFX analysis of parameter estimates
(e.g. BPA)
RFX analysis of parameter estimates(e.g. t-test, ANOVA)
optimal model structure assumed to be identical across subjects?
FFX BMS
yes no
RFX BMS
Stephan et al. 2010, NeuroImage
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Any design that is good for a GLM of fMRI data.
What type of design is good for DCM?
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GLM vs. DCMDCM tries to model the same phenomena (i.e. local BOLD responses) as a GLM, just in a different way (via connectivity and its modulation).
No activation detected by a GLM → no motivation to include this region in a deterministic DCM.However, a stochastic DCM could be applied despite the absence of a local activation.
Stephan 2004, J. Anat.
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Multifactorial design: explaining interactions with DCM
Task factorTask A Task B
Stim
1St
im 2
Stim
ulus
fact
or
TA/S1 TB/S1
TA/S2 TB/S2
X1 X2
Stim2/Task A
Stim1/Task A
Stim 1/Task B
Stim 2/Task B
GLM
X1 X2
Stim2
Stim1
Task A Task B
DCMLet’s assume that an SPM analysis shows a main effect of stimulus in X1 and a stimulus task interaction in X2. How do we model this using DCM?
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Stim 1Task A
Stim 2Task A
Stim 1Task B
Stim 2Task B
Simulated data
X1
X2
+++X1 X2
Stimulus 2
Stimulus 1
Task A Task B+++++
++++
– –
Stephan et al. 2007, J. Biosci.
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Stim 1Task A
Stim 2Task A
Stim 1Task B
Stim 2Task B
plus added noise (SNR=1)
X1
X2
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DCM10 in SPM8• DCM10 was released as part of SPM8 in July 2010 (version 4010).
• Introduced many new features, incl. two-state DCMs and stochastic DCMs
• This led to various changes in model defaults, e.g.– inputs mean-centred– changes in coupling priors– self-connections: separately estimated for each area
• For details, see: www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf
• Further changes in version 4290 (released April 2011) to accommodate new developments and give users more choice (e.g. whether or not to mean-centre inputs).
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The evolution of DCM in SPM• DCM is not one specific model, but a framework for Bayesian inversion of
dynamic system models
• The default implementation in SPM is evolving over time– better numerical routines for inversion– change in priors to cover new variants (e.g., stochastic DCMs,
endogenous DCMs etc.)
To enable replication of your results, you should ideally state which SPM version you are using when publishing papers.
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Factorial structure of model specification in DCM10• Three dimensions of model specification:
– bilinear vs. nonlinear
– single-state vs. two-state (per region)
– deterministic vs. stochastic
• Specification via GUI.
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bilinear DCM
CuxDxBuAdtdx m
i
n
j
jj
ii
1 1
)()(CuxBuAdtdx m
i
ii
1
)(
Bilinear state equation:
driving input
modulation
driving input
modulationnon-linear DCM
...)0,(),(2
0
uxuxfu
ufx
xfxfuxf
dtdx
Two-dimensional Taylor series (around x0=0, u0=0):
Nonlinear state equation:
...2
)0,(),(2
2
22
0
x
xfux
uxfu
ufx
xfxfuxf
dtdx
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0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
Neural population activity
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100-1
0
1
2
3
4
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
fMRI signal change (%)x1 x2
x3
CuxDxBuAdtdx n
j
jj
m
i
ii
1
)(
1
)(
Nonlinear dynamic causal model (DCM)
Stephan et al. 2008, NeuroImage
u1
u2
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V1 V5stim
PPC
attention
motion
-2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
%1.99)|0( 1,5 yDp PPCVV
1.25
0.13
0.46
0.390.26
0.50
0.26
0.10MAP = 1.25
Stephan et al. 2008, NeuroImage
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V1V5PPC
observedfitted
motion &attention
motion &no attention
static dots
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uinput
Single-state DCM
1x
Intrinsic (within-region)
coupling
Extrinsic (between-region)
coupling
NNNN
N
ijijij
x
xx
uBACuxx
1
1
111
Two-state DCM
Ex1
IN
EN
I
E
IINN
IENN
EENN
EENN
EEN
IIIE
EEN
EIEE
ijijijij
xx
xx
x
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Cuxx
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Ix1
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E
x
x
1
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Two-state DCM
Marreiros et al. 2008, NeuroImage
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0 200 400 600 800 1000 1200-1
-0.5
0
0.5
1inputs or causes - V2
0 200 400 600 800 1000 1200-0.1
-0.05
0
0.05
0.1hidden states - neuronal
0 200 400 600 800 1000 12000.8
0.9
1
1.1
1.2
1.3hidden states - hemodynamic
0 200 400 600 800 1000 1200-3
-2
-1
0
1
2predicted BOLD signal
time (seconds)
excitatorysignal
flowvolumedHb
observedpredicted
Stochastic DCM
, ,dx f x udt
Friston et al. (2008, 2011) NeuroImageDaunizeau et al. (2009) Physica D
• accounts for stochastic neural fluctuations
• can be fitted to resting state data
• has unknown precision and smoothness additional hyperparameters
Li et al. (2011) NeuroImage
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Thank you