Models of effective connectivity & Dynamic Causal Modelling (DCM) Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London Methods & Models for fMRI data analysis in neuroeconomics April 2010
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Models of effective connectivity & Dynamic Causal Modelling (DCM) Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical.
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Models of effective connectivity &Dynamic Causal Modelling (DCM)
Klaas Enno Stephan
Laboratory for Social & Neural Systems Research Institute for Empirical Research in EconomicsUniversity of Zurich
Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London
Methods & Models for fMRI data analysis in neuroeconomics April 2010
• Dynamic causal models (DCMs)– DCM for fMRI: Neural and hemodynamic levels– Parameter estimation & inference
• Applications of DCM to fMRI data– Design of experiments and models– Some empirical examples and simulations
Connectivity
A central property of any system
Communication systems Social networks(internet) (Canberra, Australia)
FIgs. by Stephen Eick and A. Klovdahl;see http://www.nd.edu/~networks/gallery.htm
Structural, functional & effective connectivity
• anatomical/structural connectivity= presence of axonal connections
• functional connectivity = statistical dependencies between regional time series
• effective connectivity = causal (directed) influences between neurons or neuronal populations
Sporns 2007, Scholarpedia
Anatomical connectivity
• neuronal communication via synaptic contacts
• visualisation by tracing techniques
• long-range axons “association fibres”
Diffusion-weighted imaging
Parker & Alexander, 2005, Phil. Trans. B
Parker, Stephan et al. 2002, NeuroImage
Diffusion-weighted imaging of the
cortico-spinal tract
Why would complete knowledge of anatomical connectivity not be enough
to understand how the brain works?
Connections are recruited in a context-dependent fashion
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Synaptic strengths are context-sensitive: They depend on spatio-temporal patterns
of network activity.
Connections show plasticity
• critical for learning
• can occur both rapidly and slowly
• NMDA receptors play a critical role
• NMDA receptors are regulated by modulatory neurotransmitters like dopamine, serotonine, acetylcholine
• synaptic plasticity = change in the structure and transmission properties of a chemical synapse
Gu 2002, Neuroscience
NMDAreceptor
Short-term SP & neuromodulation
• NMDAR-independent– synaptic
depression/facilitation– effects due to dendritic
backpropagation & voltage-sensitive ion channels
• NMDAR-dependent– phosphorylation of AMPARs– modulation of EPSPs at
NMDARs by DA, ACh, 5HT (gating) through• phosphorylation• NMDAR trafficking• changes in membrane
potential
Reynolds et al. 2001, Nature
Tsodyks & Markram 1997, PNAS
pea
k P
SP
(m
V)
Different approaches to analysing functional connectivity
• Seed voxel correlation analysis
• Eigen-decomposition (PCA, SVD)
• Independent component analysis (ICA)
• any other technique describing statistical dependencies amongst regional time series
Seed-voxel correlation analyses
• Very simple idea:– hypothesis-driven choice of
a seed voxel → extract reference
time series
– voxel-wise correlation with time series from all other voxels in the brain
seed voxel
Drug-induced changes in functional connectivity
Finger-tapping task in first-episode schizophrenic patients:
voxels that showed changes in functional connectivity (p<0.005) with the left ant. cerebellum after medication with olanzapineStephan et al. 2001, Psychol. Med.
Does functional connectivity not simply correspond to co-activation in
SPMs?No, it does not - see the fictitious example on the right:
Here both areas A1 and A2 are correlated identically to task T, yet they have zero correlation among themselves:
r(A1,T) = r(A2,T) = 0.71butr(A1,A2) = 0 !
task T regional response A2regional response A1
Stephan 2004, J. Anat.
Pros & Cons of functional connectivity analyses
• Pros:– useful when we have no experimental control over
the system of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.)
• Cons:– interpretation of resulting patterns is difficult /
arbitrary – no mechanistic insight into the neural system of
interest– usually suboptimal for situations where we have a
priori knowledge and experimental control about the system of interestmodels of effective connectivity necessary
For understanding brain function mechanistically, we need models of effective connectivity, i.e.
models of causal interactions among neuronal populations.
Some models for computing effective connectivity from 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, Granger causality)Harrison et al. 2003, Goebel et al. 2003
• Dynamic Causal Modelling (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008
• Dynamic causal models (DCMs)– DCM for fMRI: Neural and hemodynamic levels– Parameter estimation & inference
• Applications of DCM to fMRI data– Design of experiments and models– Some empirical examples and simulations
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 100 published papers on DCM
),,( uxFdt
dx
Neural state equation:
Electromagneticforward model:
neural activityEEGMEGLFP
Dynamic Causal Modeling (DCM)
simple neuronal modelcomplicated forward model
complicated neuronal modelsimple forward model
fMRIfMRI EEG/MEGEEG/MEG
inputs
Hemodynamicforward model:neural activityBOLD
Stephan & Friston 2007, Handbook of Brain Connectivity
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 u
x a x a x a x c u
x a x a x a x
x a x a x a x
Example: a linear system of dynamics in visual cortex
Example: a linear system of dynamics in visual cortex
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 0
0 0
0 0 0
0 0 0
a a ax x c
a a ax x uc
a a ax x u
a a ax x
x Ax Cu
},{ CA
LGleft
LGright
RVF LVF
FGright
FGleft
x1 x2
x4x3
u2 u1
Extension: bilinear dynamic system
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 0
0 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
intrinsic connectivity
direct inputs
modulation ofconnectivity
Neural state equation CuxBuAx jj )( )(
u
xC
x
x
uB
x
xA
j
j
)(
hemodynamicmodelλ
x
y
integration
BOLDyyy
activityx1(t)
activityx2(t) activity
x3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
Stephan & Friston (2007),Handbook of Brain Connectivity
Bilinear DCM
CuxBuAdt
dx m
i
ii
1
)(
Bilinear state equation:
driving input
modulation
...)0,(),(2
0
uxux
fu
u
fx
x
fxfuxf
dt
dxTwo-dimensional Taylor series (around x0=0, u0=0):
The coupling parameter a thus describes the speed ofthe exponential change in x(t)
0
0
( ) 0.5
exp( )
x x
x 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):
The problem of hemodynamic convolution
Goebel et al. 2003, Magn. Res. Med.
Hemodynamic forward models are important for connectivity analyses of fMRI data
David et al. 2008, PLoS Biol.
Granger causality
DCM
sf
tionflow induc
(rCBF)
s
v
stimulus functions
v
q q/vvEf,EEfqτ /α
dHbchanges in
100 )( /αvfvτ
volumechanges in
1
f
q
)1(
fγsxs
signalryvasodilato
u
s
CuxBuAdt
dx m
j
jj
1
)(
t
neural state equation
1
3.4
111),(
3
002
001
32100
k
TEErk
TEEk
vkv
qkqkV
S
Svq
hemodynamic state equationsf
Balloon model
BOLD signal change equation
},,,,,{ h
important for model fitting, but of no interest for statistical inference
• 6 hemodynamic parameters:
• Computed separately for each area (like the neural parameters) region-specific HRFs!
The hemodynamic model in DCM
Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage
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
, = 0.5
CBMN
, = 0.5
RBMN
, = 1
CBMN
, = 1
RBMN
, = 2
CBMN
, = 2sf
tionflow induc
(rCBF)
s
v
stimulus functions
v
q q/vvEf,EEfqτ /α
dHbchanges in
100 )( /αvfvτ
volumechanges in
1
f
q
)1(
fγsxs
signalryvasodilato
u
s
CuxBuAdt
dx m
j
jj
1
)(
t
neural state equation
1
3.4
111),(
3
002
001
32100
k
TEErk
TEEk
vkv
qkqkV
S
Svq
hemodynamic state equations
f
Balloon model
BOLD signal change equation
The hemodynamic model in DCM
Stephan et al. 2007, NeuroImage
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
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1
A
B
C
h
ε
How interdependent are neural and hemodynamic parameter estimates?
Stephan et al. 2007, NeuroImage
Bayesian statistics
)()|()|( 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
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 parameter estimation by means of a Levenberg-Marquardt gradient ascent, embedded into an EM algorithm.
• 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
• Gaussian assumptions about the posterior distributions of the parameters
• Use of the cumulative normal distribution to test the 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
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.
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 fixed-effects 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
Inference about DCM parameters:group analysis (classical)
• 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
Separate fitting of identical models for each subject
Selection of bilinear parameters of interestSelection of bilinear