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
03 December 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
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
Sporns 2007, Scholarpedia
• functional connectivity
= statistical dependencies between regional time series
• effective connectivity
= causal (directed) influences between neurons or
neuronal populations
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
Diffusion-weighted
imaging of the corticospinal tract
Parker, Stephan et al. 2002, NeuroImage
However,
knowing anatomical connectivity is not enough...
1. Connections are recruited in a context-dependent
fashion
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Local functions are context-sensitive:
They depend on network activity.
2. 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, serotonine, acetylcholine
Gu 2002, Neuroscience
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
olanzapine
Stephan 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:
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, 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
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)
Stim 1
Stim 2
Stimulus factor
Task factor
Task A
Task B
TA/S1
TB/S1
TA/S2
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  (TA  TB )  1
main effect
of task
 ( S1  S 2 ) β 2
main effect
of stim. type
 (TA  TB ) ( S1  S 2 ) β 3
interaction
e
y  (TA  TB )  1
 V 1β 2
 (TA  TB ) V 1β 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  (TA  TB )  1
 V 1β 2
 (TA  TB ) V 1β 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
Two PPI variants
• "Classical" PPI:
– Friston et al. 1997, NeuroImage
– depends on factorial design
– in the GLM, physiological time series replaces one experimental factor
– physio-physiological interactions:
two experimental factors are replaced by physiological time series
• Alternative PPI:
– Macaluso et al. 2000, Science
– interaction term is added to an existing GLM
– can be used with any design
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
Bilateral ACC activation in both tasks –
but asymmetric connectivity !
group analysis
random effects (n=15)
p<0.05, corrected (SVC)
IFG
left ACC (-6, 16, 42)
letter vs spatial
decisions
Left ACC  left inf. frontal gyrus (IFG):
increase during letter decisions.
IPS
right ACC (8, 16, 48)
Stephan et al. 2003, Science
spatial vs letter
decisions
Right ACC  right IPS:
increase during spatial decisions.
PPI single-subject example
bVS= -0.16
spatial
decisions
bL=0.63
Signal in right ant. IPS
Signal in left IFG
letter
decisions
spatial
decisions
bL= -0.19
letter
decisions
bVS=0.50
Signal in left ACC
Signal in right ACC
Left ACC signal plotted against left IFG
Right ACC signal plotted against right IPS
Stephan et al. 2003, Science
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 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
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
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  a11 x1  a12 x2  a13 x3  c12u2
x2  a21 x1  a22 x2  a24 x4  c21u1
x3  a31 x1  a33 x3  a34 x4
x4  a42 x2  a43 x3  a44 x4
Example:
a linear system
of dynamics in
visual cortex
x3
x1
FG
left
LG
left
  { A, C}
LG
right
x4
LG = lingual gyrus
FG = fusiform gyrus
x2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
RVF
LVF
u2
u1
state
changes
x  Az  Cu
FG
right
effective
connectivity
system
state
input
parameters
external
inputs
 x1   a11 a12 a13 0   x1   0 c12 
  x  c
 x  a a
 u
0
a
0
 1
24   2 
21
 2    21 22



 x3   a31 0 a33 a34   x3   0 0  u2 
   
  

0
a
a
a
x
x
0
0
42
43
44   4 


 4 
Extension:
bilinear
dynamic
system
x3
FG
left
FG
right
x4
m
x  ( A   u j B ( j ) ) x  Cu
j 1
x1
LG
left
LG
right
x2
RVF
CONTEXT
LVF
u2
u3
u1
0 b12(3)
 x1    a11 a12 a13 0 


 x   a a
0
a
0 0
24 

 2     21 22
 u3
0 0
 x3    a31 0 a33 a34 


  
0
a
a
a
x
42
43
44 
 4  
0 0
0 

0 0 
0 b34(3) 

0 0 
0







 x1   0 c12
 x  c
0
 2    21
 x3   0 0
  
 x4   0 0
0
 u1 

0  
 u2
0  
  u3 
0
Dynamic Causal Modelling (DCM)
Hemodynamic
forward model:
neural activityBOLD
Electromagnetic
forward model:
neural activityEEG
MEG
LFP
Neural state equation:
dx
 F ( x , u,  )
dt
fMRI
simple neuronal model
complicated forward model
Stephan & Friston 2007,
Handbook of Brain Connectivity
EEG/MEG
complicated neuronal model
simple forward model
inputs
Basic idea of DCM for fMRI
(Friston et al. 2003, NeuroImage)
• 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 (x) is
transformed into area-specific BOLD signals (y)
by a hemodynamic forward model (λ).
The aim of DCM is to estimate parameters at
the neuronal level such that the modelled
BOLD signals are maximally similar to the
experimentally measured BOLD signals.
x
λ
y
y


y
BOLD
y

activity
x2(t)
neuronal
states
t
Neural state equation
intrinsic connectivity
modulation of
connectivity
direct inputs
Stephan & Friston (2007),
Handbook of Brain Connectivity
hemodynamic
model
x
integration
modulatory
input u2(t)
t
λ
activity
x3(t)
activity
x1(t)
driving
input u1(t)
y
x  ( A   u j B( j ) ) x  Cu
x
x
 x

u j x
A
B( j)
x
C
u
Bilinear DCM
driving
input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
dx
f
f
2 f
 f ( x, u)  f ( x0 ,0) 
x u
ux  ...
dt
x
u
xu
Bilinear state equation:
m
dx 

  A   ui B ( i )  x  Cu
dt 
i 1

Example:
context-dependent decay
stimuli
u1
context
u2
+
-
x1
+
u1
u1
u2
u2
Z1
x
Z2 1
x2
+
x2
-
x  Ax  u2 B (2) x  Cu1
-
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)
 x1   
 x    a 21
 2 
2

a12 
b11
x

u
2
 
 0
0 
 c1 0  u1 
x
 u 
2
0
0
b 22     2 
DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
dx
 ax
dt
x(t )  x0 exp(at )
Coupling parameter a is inversely
proportional to the half life  of z(t):
x( )  0.5 x0
The coupling parameter a
thus describes the speed of
the exponential change in x(t)
0.5x0
 x0 exp( a )
a  ln 2 / 
  ln 2 / a
The hemodynamic
model in DCM
u
t
m

dx 
  A   u j B ( j )  x  Cu
dt 
j 1

• 6 hemodynamic parameters:
  { ,  , ,  ,  ,  }
stimulus functions
neural state equation
vasodilatory signal
s  x  s  γ( f  1)
h
f
s
s
flow induction (rCBF)
f  s
important for model fitting,
but of no interest for
statistical inference
Balloon model
changes in volume
• Computed separately for each
area (like the neural
parameters)
 region-specific HRFs!
τv  f  v1 /α
v
 ( q, v ) 
v
changes in dHb
τq  f E ( f,E0 ) qE0  v1 /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
f
k2  r0 E0TE
k3  1  
BOLD signal
change equation
u
The hemodynamic
model in DCM
stimulus functions
t
m

dx 
  A   u j B ( j )  x  Cu
dt 
j 1

neural state
equation
0.4
0.2
vasodilatory signal
0
s  x  s  γ( f  1)
f
0
2
4
6
8
10
12
s
s
N
RBM N,  = 1
CBM ,  = 1
N
RBM ,  = 2
1
flow induction (rCBF)
0.5
f  s
hemodynamic
state
equations
N
CBM N,  = 2
0
f
Balloon model
changes in volume
τv  f  v1 /α
v
 ( q, v ) 
14
RBM N,  = 0.5
CBM ,  = 0.5
v
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
0.2
changes in dHb
τq  f E ( f,E0 ) qE0  v1/α q/v
q
0
-0.2
-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 our 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
r,A
10
15
r,B
20
r,C
25
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.
Shrinkage priors
Small & variable effect
Large & variable effect
Small but clear effect
Large & clear effect
stimulus function u
Overview:
parameter estimation
•
•
•
•
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.
neural state
equation
x  ( A   u j B j ) x  Cu
activity - dependent vasodilatory signal
s  z  s  γ( f  1)
s
s
f
flow - induction (rCBF)
hidden states
z  {x, s, f , v, q}
state equation
 h  { ,  , ,  , }
f
 n  { A, B1...B m , C}
  { h , n }
z  F ( x , u,  )
changes in volume
τv  f  v1/α
v
ηθ|y
parameters
f  s
v
changes in dHb
τq  f E ( f, ) q  v1/α q/v
q
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:
 cT  
 y

p  N 
 cT C y c






ηθ|y
• 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
Because the likelihood distributions
from different subjects are independent,
one can combine their posterior
densities by using the posterior of one
subject as the prior for the next:
p ( | y1 )
 p ( y1 |  ) p ( )
p ( | y1 , y2 )  p ( y2 |  ) p ( y1 |  ) p( )
Under Gaussian assumptions this is
easy to compute:
group
posterior
covariance
N
C|1y1 ,..., y N   C|1yi
i 1
 p ( y2 |  ) p ( | y1 )
...
p ( | y1 ,..., y N )  p ( y N |  ) p( | y N 1 )... p( | y1 )
individual
posterior
covariances
 | y ,..., y
1
group
posterior
mean
N
 N 1
 1
   C | yi | yi C | y1 ,..., y N
 i 1

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 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
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 as a GLM, just in a different way:
It is a model, based on connectivity and its modulation, for explaining
experimentally controlled variance in local responses.
No activation detected by a GLM → inclusion of this region in a DCM is
useless!
Stephan 2004, J. Anat.
Multifactorial design:
explaining interactions with DCM
Stim 1
Stim 2
Stimulus factor
Task factor
Stim1/
Task A
Stim2/
Task A
Task A
Task B
TA/S1
TB/S1
X1
X2
TA/S2
TB/S2
Stim 1/
Task B
Stim 2/
Task B
X1
X2
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
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)
Recent fMRI
studies with DCM
• DCM now an
established tool for
fMRI analysis
• >50 studies
published, incl. highprofile journals
• combinations of
DCM with
computational
models
Thank you