Basics of Dynamic Causal Modelling

Report
Basics of Dynamic Causal Modelling (DCM)
Sudhir Shankar Raman
Translational Neuromodeling Unit,
UZH & ETH
* Thanks to Klaas for the slides
Methods & Models 2013
Overview
• Brain connectivity: types & definitions
– anatomical connectivity
– functional connectivity
– effective connectivity
• Psycho-physiological interactions (PPI)
• 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)
Figures 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, http://www.scholarpedia.org/article/Brain_connectivity
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
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
•
synaptic plasticity
=
change in the structure and
transmission properties of a
chemical synapse
NMDA
receptor
 critical for learning
 can occur both rapidly and slowly
 NMDA receptors play a critical role
 NMDA receptors are regulated by
modulatory neurotransmitters like
dopamine, serotonin, acetylcholine
Gu 2002, Neuroscience
Different approaches to analysing functional
connectivity
Seed voxel correlation analysis
Eigen-decomposition (PCA, SVD),
Independent component analysis (ICA)
Clustering Approaches – fuzzy, hierarchical
clustering
any other technique describing statistical
dependencies amongst regional time series
Seed-voxel based correlation analysis
SVCA
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 firstepisode schizophrenic
patients:
• voxels that showed
changes in functional
connectivity (p<0.005)
with the left ant.
cerebellum after
medication with
olanzapine
Stephan et al. 2001, Psychological Medicine.
Does functional connectivity not simply
correspond to co-activation in SPMs?
No, it does not - see the
fictitious example on the
right:
regional response A1
task T
regional response A2
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) = 0.71
but
r(A1,A2) = 0 !
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,
hallucinations, resting state
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 interest
models 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
Overview
• Brain connectivity: types & definitions
– anatomical connectivity
– functional connectivity
– effective connectivity
• Psycho-physiological interactions (PPI)
• 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
Psycho-physiological interaction (PPI)
Task factor
factor
Stim 2
Stimulus
Stim 1
Task A
TA/S1
TA/S2
Task B
TB/S1
TB/S2
We can replace one main effect in the
GLM by the time series of an area that
shows this main effect.
E.g. let's replace the main effect of
stimulus type by the time series of area
V1:
Friston et al. 1997, NeuroImage
GLM of a 2x2 factorial design:
y  (T A  T B )  1
main effect
of task
 ( S1  S 2 ) β 2
main effect
of stim. type
 (T A  T B ) ( S 1  S 2 ) β 3
interaction
e
y  (T A  T B )  1
 V 1β 2
 (T A  T B ) V 1 β 3
e
main effect
of task
V1 time series
 main effect
of stim. type
psychophysiological
interaction
Attentional modulation of V1→V5
Attention
V1
V5
V5 activity
SPM{Z}
time
V1 x Att.
Friston et al. 1997, NeuroImage
Büchel & Friston 1997, Cereb. Cortex
V5
V5 activity
=
attention
no attention
V1 activity
PPI: interpretation
y  (T A  T B )  1
 V 1β 2
 (T A  T B ) V 1 β 3
Two possible
interpretations of
the PPI term:
e
attention
V1
attention
V5
Modulation of V1V5 by
attention
V1
V5
Modulation of the impact of attention on V5
by V1
Pros & Cons of PPIs
Pros:
• given a single source region, we can
test for its context-dependent
connectivity across the entire brain
• easy to implement
Cons:
• very simplistic model:
only allows to model contributions
from a single area
• ignores time-series properties of data
• application to event-related data
relies on deconvolution procedure
(Gitelman et al. 2003, NeuroImage)
• operates at the level of BOLD time
series
sometimes very useful, but limited causal interpretability;
in most cases, we need more powerful models
Gitelman et al. 2003, NeuroImage
Overview
• Brain connectivity: types & definitions
– anatomical connectivity
– functional connectivity
– effective connectivity
• Psycho-physiological interactions (PPI)
• 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 approx. 200 published papers on DCM
Dynamic Causal Modeling (DCM)
Hemodynamic
forward model:
neural activityBOLD
Neural state equation:
dx
 f ( x, u , )
dt
fMRI
simple neuronal model
complicated forward model
Stephan & Friston 2007,
Handbook of Brain Connectivity
Electromagnetic
forward model:
neural activityEEG
MEG
LFP
EEG/MEG
complicated neuronal model
simple forward model
inputs
Bayesian inversion
Neural dynamics
u (t )
Design experimental inputs
dx dt  f ( x , u ,  )
Define likelihood model
Observer function
y  g ( x, )  
p ( y |  , m )  N ( g ( ),  ( ))
p ( , m )  N (   ,   )
Inference on model
structure
Inference on parameters
p( y | m) 
 p ( y |  , m ) p ( ) d 
p ( | y , m ) 
Specify priors
Invert model
p ( y |  , m ) p ( , m )
p( y | m)
Make inferences
Example:
a linear system
of dynamics in
visual cortex
x3
x1
FG
left
LG
left
FG
right
LG
right
x4
LG = lingual gyrus
FG = fusiform gyrus
x2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
RVF
LVF
u2
u1
x1  a 11 x1  a 12 x 2  a 13 x 3  c12 u 2
x 2  a 21 x1  a 22 x 2  a 24 x 4  c 21u1
x 3  a 31 x1  a 33 x 3  a 34 x 4
x 4  a 42 x 2  a 43 x 3  a 44 x 4
Example:
a linear system
of dynamics in
visual cortex
x3
x1
state
changes
effective
connectivity
 x1   a 11
  
a 21
x2
 
 x 3   a 31
  
 x4   0
a 12
a 13
a 22
0
0
a 33
a 42
a 43
FG
left
FG
right
LG
left
LG
right
x4
LG = lingual gyrus
FG = fusiform gyrus
x2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
RVF
LVF
u2
u1
system
state
0 

a 24

a 34 

a 44 
 x1 
 
x2
 
 x3 
 
 x4 
input
parameters
 0

c 21

 0

 0
external
inputs
c12 

0  u1 

 
0  u2 

0 
x  Ax  Cu
  { A, C }
Extension:
bilinear dynamic
system
x3
FG
left
FG
right
x4
m
x  (A 

u jB
( j)
)x  Cu
j 1
x1
RVF
u2
LG
left
LG
right
x2
LVF
CONTEXT
u1
u3
Modulatory
connections
 x1    a 11
  
a
x
 2     21
 x 3    a 31
  
 x4    0
a 12
a 13
a 22
0
0
a 33
a 42
a 43
0 
0


a 24
0
u 
3
0
a 34 


a 44 
0
b12
( 3)
0
0
0
0
0
0
0
0 

0 
( 3)
b34 

0 







 x1 
 
x
 2 
 x3 
 
 x4 
 0

c
 21
 0

 0
c12
0
0
0
0
  u1 
0  
 u
2
0  
  u 3 
0
Bilinear DCM
driving
input
Neural state equation:
dx
 f ( x, u , )
dt
Two-dimensional Taylor series (around x0=0, u0=0):
dx
dt
 f ( x, u )  f ( x0 , u 0 ) 
f
x
x
f
u
2
u
 f
xu
modulation
A
Bilinear state equation:

 A
dt

dx
m
uB
i
i 1
(i)

 x  Cu

B
( j)

C 
 x
x

 x
u j x
 x
u
ux  .........
DCM- So far….



activity
x2(t)
x
activity
x3(t)
activity
x1(t)
integration
neuronal
states
modulatory
input u2(t)
driving
input u1(t)
Neural state equation
t
Stephan & Friston (2007), Handbook of Brain Connectivity
A
endogenous
connectivity
modulation of
connectivity
t
( j)
x  ( A   u j B ) x  Cu
direct inputs
B
( j)

C 
 x
x

 x
u j x
 x
u
DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
dx
x ( t )  x 0 exp( at )
 ax
dt
Coupling parameter a is inversely
proportional to the half life  of z(t):
x ( )  0.5 x 0
0.5 x 0
ln 2

x
The coupling parameter
thus describes the speed of
the exponential change in x(t)
 x 0 exp( a )
a  
_
 
ln 2
a
Example:
context-dependent decay
stimuli
u1
_
+
_
context
u2
u1
u1
u2
u2
Z1
x
Z2 1
x1
+
x2
+
x2
_
x = A
x + u22B2 x+
C u1
(2)
_
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)
 x1   
  
 x 2   a 21
2

a 12 
b 11
x

u
2 
 
 0
0 
 c1
x
2
b 22   0
0   u1 
 
0  u2 
Importance of Hemodynamic forward models for connectivity
analyses of fMRI data
Goebel et al. 2003, Magn. Res. Med.
Granger causality
David et al. 2008, PLoS Biol.
DCM
BOLD
y
y
y


y
λ

activity
x2(t)
x
activity
x3(t)
activity
x1(t)
integration
neuronal
states
modulatory
input u2(t)
driving
input u1(t)
hemodynamic
model
Neural state equation
t
Stephan & Friston (2007), Handbook of Brain Connectivity
A
endogenous
connectivity
modulation of
connectivity
t
( j)
x  ( A   u j B ) x  Cu
direct inputs
B
( j)

C 
 x
x

 x
u j x
 x
u
The hemodynamic
model in DCM
u
stimulus functions
t

A

dt

dx
• 6 hemodynamic parameters:
m
u
j
B
j 1
( j)

 x  Cu


neural state equation
vasodilato ry signal

h
 { ,  ,  ,  ,  ,  }
s  x   s  γ ( f  1)
f
s
s
flow induc tion (rCBF)
important for model fitting,
but of no interest for
statistical inference
f  s

h
 { ,  ,  ,  ,  ,  }
τ v  f  v
1 /α
v
 ( q, v ) 
v
changes in dHb
τ q  f E ( f,E 0 ) qE 0  v
1 /α
q/v
q
S


 q
 V0 k1 1  q   k2 1    k3 1  v 
S0
 v


k1  4.30 E0TE
Friston et al. 2000, NeuroImage
Stephan et al. 2007, NeuroImage
hemodynamic
state equations
Balloon model
changes in volume
• Computed separately for each
area (like the neural
parameters)
 region-specific HRFs!
f
k2  r0 E0TE
k3  1  
BOLD signal
change equation
u
The hemodynamic
model in DCM
stimulus functions
t

A

dt

dx
m
u
j
B
j 1
( j)

 x  Cu


0.4
0.2
0
vasodilato ry signal
s  x   s  γ ( f  1)
f
0
2
4
6
8
10
12
14
RBM N,  = 0.5
CBM ,  = 0.5
s
s
N
RBM N,  = 1
CBM ,  = 1
N
RBM ,  = 2
1
flow induc tion (rCBF)
0.5
f  s
N
CBM N,  = 2
0
f
Balloon model
changes in volume
τ v  f  v
1 /α
v
 ( q, v ) 
v
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
0.2
0
changes in dHb
-0.2
τ q  f E ( f,E 0 ) qE 0  v
1 /α
q/v
q
-0.4
-0.6
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 equation
Stephan et al. 2007, NeuroImage
How interdependent are neural and hemodynamic
parameter estimates?
1
A
0.8
5
0.6
10
B
0.4
15
C
0.2
20
0
25
-0.2
h
ε
30
-0.4
35
-0.6
-0.8
40
5
10
15
20

25
h
30
35
40
 { ,  ,  ,  ,  ,  }
-1
Stephan et al. 2007, NeuroImage
Bayesian statistics
new data
prior knowledge
p( y | )
p ( )
p ( | y )  p ( y |  ) p ( )
posterior
 likelihood
∙ prior
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.
stimulus function u
Overview:
parameter estimation
•
•
•
•

A

dt

dx
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 Bayes
under Laplace approximation
Result:
Gaussian a posteriori
parameter distributions,
characterised by
mean ηθ|y and
covariance Cθ|y.
m

u jB
( j)
j 1
neural state
equation

 x  Cu


activity - dependent vasodilato ry signal
s  z   s  γ ( f  1)
s
s
f
parameters
flow - induction (rCBF)
hidden states
f  s
z  { x , s , f , v , q}
f
state equation
h
 { ,  ,  ,  ,  }

n
 { A , B ... B , C }
1
m
  { ,  }
h
z  F ( x, u, )
changes in volume
τv  f  v1 /α
changes
v
n
in dHb
τ q  f E ( f,  ) q  v
1 /α
q/v
q
v
ηθ|y

y   ( x)
y  h(u ,  )  X  e
modelled
BOLD response
observation model
Inference about DCM parameters:
Bayesian single-subject analysis
• 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:
 T
 c  y  
p  N 
 c T C y c






• By default, γ is chosen as zero ("does the effect exist?").
Bayesian single subject inference
LD|LVF
0.13
 0.19
FG
left
LD
p(cT>0|y)
= 98.7%
0.34
 0.14
FG
right
0.44
 0.14
0.29
 0.14
LG
left
0.01
 0.17
RVF
stim.
Stephan et al. 2005,
Ann. N.Y. Acad. Sci.
LD
LG
right
-0.08
 0.16
LD|RVF
LVF
stim.
Contrast:
Modulation LG right  LG links by LD|LVF
vs.
modulation LG left  LG right by LD|RVF
Inference about DCM parameters:
Bayesian fixed-effects group analysis
Likelihood distributions from different
subjects are independent
Under Gaussian assumptions this is
easy to compute:
 one can use the posterior from one
subject as the prior for the next
group
posterior
covariance
p  | y1
y N   p  y1
 p 
yN 
N

 p  
p  yi 

N
i 1
N
 p  y1   p  y i 
1
C  | y ,..., y
1
N

 p  y1 , y 2   p  y i 
  | y1 ,..., y N

i3
 p  y1


1
C |y
i
i 1
i2
N
individual
posterior
covariances
y N 1  p  y N 




N

i 1
1
C |y  |y
i
i

C
  | y1 ,..., y N


“Today’s posterior is tomorrow’s prior”
group
posterior
mean
individual posterior
covariances and means
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
Selection of 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
Overview
• Brain connectivity: types & definitions
– anatomical connectivity
– functional connectivity
– effective connectivity
• Psycho-physiological interactions (PPI)
• 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
What type of design is good for DCM?
Any design that is good for a GLM of fMRI data.
GLM vs. DCM
DCM 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.
Stephan 2004, J. Anat.
Multifactorial design:
explaining interactions with DCM
factor
Task B
Stim 1
Stimulus
Task A
TA/S1
TB/S1
Stim 2
Task factor
TA/S2
TB/S2
Let’s assume that an SPM analysis
shows a main effect of stimulus in X1
and a stimulus  task interaction in X2.
Stim1
How do we model this using DCM?
Stim2
Stim1/
Task A
Stim2/
Task A
X1
X2
Stim 1/
Task B
Stim 2/
Task B
X1
X2
Task A
Task B
GLM
DCM
Simulated data
X1
Stimulus 1
+++
–
+
X1
Stimulus 2
+
–
+++
+++
Task A
X2
Stim 1
Task A
+
Task B
X2
Stephan et al. 2007, J. Biosci.
Stim 2
Task A
Stim 1
Task B
Stim 2
Task B
X1
Stim 1
Task A
Stim 2
Task A
Stim 1
Task B
Stim 2
Task B
X2
plus added noise (SNR=1)
Example studies
of DCM for fMRI
• DCM now an
established tool for
fMRI & M/EEG
analysis
• approx. 200 studies
published
• combinations of
DCM with
computational
models
Task-driven
lateralisation
Does the word
contain the letter A
or not?
letter decisions > spatial decisions
•
•
•
group analysis (random effects),
n=16, p<0.05 corrected
analysis with SPM2
Is the red letter left
or right from the
midline of the
word?
spatial decisions > letter decisions
Stephan et al. 2003, Science
Theories on inter-hemispheric integration
during lateralised tasks
Information transfer
Hemispheric recruitment
T|RVF
T
(for left-lateralised task)

+
+
T|LVF
LVF
Inhibition/Competition
T
+
T
−
−
T
RVF
Predictions:
Predictions:
Predictions:
modulation by task
conditional on visual field
asymmetric connection
strengths
modulation by task only
modulation by task only
positive & symmetric
connection strengths
negative & symmetric
connection strengths
A
FG
left
LG
left
LG
right
RVF
stim.
B
C
FG
right
FG
left
VF
RVF
LD
LD
VF
LD
Bind
Bcond
LD
Bind
LD,RVF
LD,LVF
Bcond
LD|RVF
LD|LVF
LG
left
VF
D
VF LD Bind Bcond
inter
LVF
LVF
stim.
intra
FG
right
LVF
LD
LD
LG
right
Bind
Bcond
LVF LD
LD|LVF
FG
left
FG
right
LG
left
LG
right
16 models
RVF
RVF
LD|RVF
Ventral stream & letter decisions
LD|LVF
Left fusiform gyrus
(FG)
-44,-52,-18
0.12
 0.02
FG
left
LD>SD, p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
LD
Right fusiform gyrus
(FG)
38,-52,-20
0.25
 0.04
FG
right
0.36
 0.06
0.16
 0.05
LG
left
Left lingual gyrus
(LG)
-12,-64,-4
0.02
 0.02
RVF
stim.
LD
LG
right
Right lingual gyrus
(LG)
14,-68,-2
0.03
 0.03
LD|RVF
p<0.01 uncorrected
LVF
stim.
mean parameter estimates  SE (n=12)
LD>SD masked incl. with RVF>LVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Stephan et al. 2007, J. Neurosci.
significant modulation (p<0.05, Bonferroni-corrected)
significant modulation (p<0.05, uncorrected)
non-significant modulation
LD>SD masked incl. with LVF>RVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Ventral stream & letter decisions
Left MOG
-38,-90,-4
Left FG
-44,-52,-18
mean parameter estimates  SE (n=12)
significant modulation (p<0.05, corrected)
significant modulation (p<0.05, uncorrected)
non-significant
Right FG
38,-52,-20
LD|LVF
0.20
 0.04
0.00
 0.01
0.07
 0.02
LD>SD, p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
p<0.01 uncorrected
MOG
left
LD
Left LG
-12,-70,-6
FG
left
FG
right
MOG
right
0.27
 0.06
0.11
 0.03
0.00
 0.04
0.01
 0.03
LG
left
0.01
 0.01
LD>SD masked incl. with RVF>LVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Right MOG
-38,-94,0
RVF
stim.
0.01
 0.01
LG
right
LD
Left LG
-14,-68,-2
0.06
 0.02
LD|RVF
LVF
stim.
LD>SD masked incl. with LVF>RVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Stephan et al. 2007, J. Neurosci.
Asymmetric modulation of LG callosal
connections is consistent across subjects
0.5
left to right
right to left
0.4
MAP estimate
0.3
0.2
0.1
0.0
-0.1
1
2
3
4
5
6
7
8
9
10
11
12
-0.2
Subjects
Stephan et al. 2007, J. Neurosci.
y
y


Summary
BOLD
y
y

activity
x2(t)
λ
activity
x3(t)
activity
x1(t)
x
integration
neuronal
states
modulatory
input u2(t)
driving
input u1(t)
Neural state equation
t
Stephan & Friston (2007), Handbook of Brain Connectivity
( j)
x  ( A   u j B ) x  Cu
A
endogenous
connectivity
modulation of
connectivity
t
hemodynamic
model
direct inputs
B
( j)

C 
 x
x

 x
u j x
 x
u
References
•
The first DCM paper: Dynamic Causal Modelling (2003). Friston et al. NeuroImage 19:1273-1302.
•
On the role of general system theory for functional neuroimaging (2004). Klaas Enno Stephan. J. Anat 205
, 443–470
•
Statistical Parametric Mapping: The Analysis of Functional Brain Images. K.J. Friston, J. Ashburner, S.J.
Kiebel, T.E. Nichols, and W.D. Penny, editors. Academic Press, 2007.
•
Hemodynamic model: Comparing hemodynamic models with DCM (2007). Stephan et al. NeuroImage
38:387-401
•
Interhemispheric Integration of Visual Processing during Task-Driven Lateralization (2007). Stephan et al.
The Journal of Neuroscience
•
Physiological validation of DCM for fMRI: Identifying neural drivers with functional MRI: an
electrophysiological validation (2008). David et al. PLoS Biol. 6 2683–2697
•
Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008). Stephan et al. NeuroImage 42:649662
•
10 Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.
Thank you

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