DCM: Advanced topics
Klaas Enno Stephan
Neural population activity
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Laboratory for Social & Neural Systems
Research
Institute for Empirical Research in Economics
University of Zurich
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u2
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u1
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x3
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x1
x2
3
fMRI signal change (%)
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Wellcome Trust Centre for Neuroimaging
Institute of Neurology
University College London
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m
n

dx 
  A   ui B (i )   x j D ( j )  x  Cu
dt 
i 1
j 1

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Methods & models for fMRI data analysis in Neuroeconomics,
April 2010
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Integrating tractography and DCM
Model comparison and selection
Given competing hypotheses
on structure & functional
mechanisms of a system, which
model is the best?
Which model represents the
best balance between model
fit and model complexity?
For which model m does p(y|m)
become maximal?
Pitt & Miyung (2002) TICS
Bayesian model selection (BMS)
Model evidence:
Gharamani, 2004
 log p( y |  , m)
 KLq , p | m 
 KLq , p | y, m 
p(y|m)
p( y | m)   p( y |  , m) p( | m) d
y
all possible datasets
accounts for both accuracy and
complexity of the model
allows for inference about
structure (generalisability) of the
model
Various approximations, e.g.:
- negative free energy, AIC, BIC
Model comparison via Bayes factor:
p( y | m1 )
BF 
p ( y | m2 )
Penny et al. 2004, NeuroImage Stephan et al. 2007, NeuroImage
Approximations to the model evidence in DCM
Logarithm is a
monotonic function
Maximizing log model evidence
= Maximizing model evidence
Log model evidence = balance between fit and complexity
log p( y | m )  accuracy ( m )  complexity( m )
 log p( y |  , m)  complexity( m )
No. of
parameters
In SPM2 & SPM5, interface offers 2 approximations:
Akaike Information Criterion:
Bayesian Information Criterion:
AIC  log p( y |  , m)  p
p
BIC  log p ( y |  , m )  log N
2
AIC favours more complex models,
BIC favours simpler models.
No. of
data points
Penny et al. 2004, NeuroImage
Bayes factors
To compare two models, we can just compare their log evidences.
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is made
possible by Bayes factors:
positive value, [0;[
p( y | m1 )
B12 
p( y | m2 )
Kass & Raftery classification:
Kass & Raftery 1995, J. Am. Stat. Assoc.
B12
p(m1|y)
Evidence
1 to 3
50-75%
weak
3 to 20
75-95%
positive
20 to 150
95-99%
strong
 150
 99%
Very strong
The negative free energy approximation
• Under Gaussian assumptions about the posterior (Laplace
approximation), the negative free energy F is a lower bound on
the log model evidence:
log p( y | m)
 log p( y |  , m)  KLq , p | m  KLq , p | y, m
 F  KLq , p | y, m
 F  log p( y | m)  KLq , p | y, m
The complexity term in F
• In contrast to AIC & BIC, the complexity term of the negative
free energy F accounts for parameter interdependencies.
KLq( ), p( | m)
1
1
1
T
 ln C  ln C | y   | y    C1  | y   
2
2
2
• The complexity term of F is higher
– the more independent the prior parameters ( effective DFs)
– the more dependent the posterior parameters
– the more the posterior mean deviates from the prior mean
• NB: SPM8 only uses F for model selection !
BMS in SPM8: an example
attention
M1
stim
M1
M2
M3
M4
M3
stim
PPC
V1
attention
V1
V5
PPC
M2
M2 better than M1
BF 2966
F = 7.995
PPC
attention
stim
V1
V5
M3 better than M2
BF  12
F = 2.450
V5
M4
attention
PPC
M4 better than M3
BF  23
F = 3.144
stim
V1
V5
Fixed effects BMS at group level
Group Bayes factor (GBF) for 1...K subjects:
GBFij   BF
(k )
ij
k
Average Bayes factor (ABF):
ABFij  K  BF
(k )
ij
k
Problems:
- blind with regard to group heterogeneity
- sensitive to outliers
Random effects BMS for group studies

r ~ Dir (r ;  )
mk ~ p(mk | p)
mk ~ p(mk | p)
mk ~ p(mk | p)
m1 ~ Mult (m;1, r )
y1 ~ p( y1 | m1 )
y1 ~ p( y1 | m1 )
y2 ~ p( y2 | m2 )
y1 ~ p( y1 | m1 )
Dirichlet parameters
= “occurrences” of models in the population
Dirichlet distribution of model probabilities
Multinomial distribution of model labels
Model inversion
by Variational
Bayes (VB)
Measured data y
Stephan et al. 2009, NeuroImage
LD
m2
MOG
FG
LD|LVF
MOG
FG
LD|RVF
MOG
LD|LVF
RVF
stim.
LD
Subjects
-30
-25
-20
LD
LG
LG
LVF
stim.
RVF LD|RVF
stim.
m2
-35
-15
MOG
FG
LD
LG
LG
FG
m1
LVF
stim.
m1
-10
-5
Log model evidence differences
0
5
Stephan et al. 2009, NeuroImage
p(r >0.5 | y) = 0.997
1
5
4.5
4
pr1  r2   99.7%
m2
3.5
m1
p(r 1|y)
3
2.5
2
1.5
r1  84.3%
r2  15.7%
0.5
0
0
1  11.8
2  2.2
1
0.1
0.2
0.3
0.4
0.5
r
0.6
1
0.7
0.8
0.9
1
Validation of VB estimates by sampling
30
1  11.8
Sampling Approach
Variational Bayes
25
F
20
15
10
5
0
2
4
6

1
8
10
12
14
Simulation study: sampling
subjects from a heterogenous
population
• Population where 70% of all
subjects' data are generated
by model m1 and 30% by
model m2
LD|LVF
m1
MOG
FG
LD
LD
LG
LG
RVF LD|RVF
stim.
• Random sampling of
subjects from this population
and generating synthetic
data with observation noise
Stephan et al. 2009, NeuroImage
LVF
stim.
LD
m2
MOG
• Fitting both m1 and m2 to all
data sets and performing
BMS
MOG
FG
FG
MOG
FG
LD|RVF
LD|LVF
LG
LG
RVF
stim.
LD
LVF
stim.
18
true values:
1=220.7=15.4
2=220.3=6.6
16
14
12

mean estimates:
1=15.4, 2=6.6
10
8
1
true values:
r1 = 0.7, r2=0.3
0.9
0.8
mean estimates:
r1 = 0.7, r2=0.3
0.7
0.6
<r> 0.5
0.4
6
0.3
4
0.2
2
0.1
0
m1
m2
1
0
m1
m2
700
0.9
true values:
1 = 1, 2=0
0.8
0.7
mean estimates:
1 = 0.89, 2=0.11
0.6

0.5
600
500
400
300
0.4
0.3
200
0.2
100
0.1
0
0
m1
m2
log GBF12
nonlinear
Model space partitioning:
Nonlinear hemodynamic
models vs. linear ones
log
GBF
80
Summed log evidence (rel. to RBML)
FFX
linear
60
40
20
p(r >0.5 | y) = 0.986
1
0
5
CBMN CBMN(ε) RBMN RBMN(ε) CBML CBML(ε) RBML RBML(ε)
RFX

12
4.5
10
4
m2
m1
alpha
8
3.5
p  r1  r2   98.6%
6
3
p(r 1|y)
4
2
2.5
2
0
CBMN CBMN(ε) RBMN RBMN(ε) CBML CBML(ε) RBML RBML(ε)
1.5

1
16
14
12
m1
m2
4
8
1*    k
r2  26.5%
r1  73.5%
0.5
0
0
10
alpha
Model
space
partitioning
0.1
0.2
0.3
0.4
0.5
r
k 1
0.6
0.7
0.8
0.9
1
1
6
4
8
 2*    k
2
k 5
0
nonlinear models
linear models
Stephan et al. 2009, NeuroImage
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Integrating tractography and DCM
Dynamic causal modelling (DCM)
Hemodynamic
forward model:
neural activityBOLD
(nonlinear)
Electric/magnetic
forward model:
neural activityEEG
MEG
LFP
(linear)
Neural state equation:
x  F ( x, u,  )
fMRI
Neural model:
1 state variable per region
bilinear state equation
no propagation delays
ERPs
Neural model:
8 state variables per region
nonlinear state equation
propagation delays
inputs
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
non-linear DCM
modulation
driving
input
driving
input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
dx
2 f x2
f
f
2 f
 f ( x, u )  f ( x0 ,0) 
 ...
x u
ux  ... 2
dt
x 2
x
u
xu
Bilinear state equation:
m
dx 

  A   ui B ( i )  x  Cu
dt 
i 1

Nonlinear state equation:
m
n
dx 
(i )
( j) 

  A   ui B   x j D  x  Cu
dt 
i 1
j 1

Neural population activity
0.4
0.3
0.2
u2
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90
100
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80
90
100
0
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20
30
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50
60
70
80
90
100
0.6
u1
0.4
x3
0.2
0
0.3
0.2
0.1
0
x1
x2
3
fMRI signal change (%)
2
1
0
Nonlinear dynamic causal model (DCM):
4
m
n

dx 
(i )
( j)
  A   ui B   x j D  x  Cu
dt 
i 1
j 1

2
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
3
1
0
-1
3
2
1
Stephan et al. 2008, NeuroImage
0
Nonlinear DCM: Attention to motion
Stimuli + Task
Previous bilinear DCM
Attention
Photic
.52 (98%)
.37
(90%)
.42
(100%)
Büchel & Friston
(1997)
250 radially moving dots
(4.7 °/s)
Conditions:
F – fixation only
A – motion + attention
(“detect changes”)
N – motion without attention
S – stationary dots
V1
Motion
SPC
.56
(99%)
.69 (100%)
.47
(100%)
.82
(100%)
.65 (100%)
IFG
V5
Friston et al. (2003)
Friston et al. (2003):
attention modulates backward connections
IFG→SPC and SPC→V5.
Q: Is a nonlinear mechanism (gain control) a
better explanation of the data?
attention
 modulation of back-
M1
PPC
ward or forward
connection?
stim
 additional driving
effect of attention
on PPC?
M3
stim
M2
M2 better than M1
BF = 2966
V1
attention
PPC
Stephan et al. 2008, NeuroImage
V1
V5
BF = 12
M3 better than M2
V1
V5
M4
 bilinear or nonlinear
modulation of
forward connection?
attention
stim
V5
PPC
attention
PPC
BF = 23
M4 better than M3
stim
V1
V5
attention
MAP = 1.25
0.10
0.8
0.7
PPC
0.6
0.26
0.5
0.39
1.25
stim
0.26
V1
0.13
0.46
0.50
V5
0.4
0.3
0.2
0.1
0
-2
motion
Stephan et al. 2008, NeuroImage
-1
0
1
2
3
4
p( DVPPC
5,V 1  0 | y )  99.1%
5
motion &
attention
static
motion &
no attention dots
V1
V5
PPC
observed
fitted
Learning of dynamic audio-visual associations
1
Conditioning Stimulus
CS1
Target Stimulus
CS2
0.8
or
p(face)
or
CS
0
Response
TS
200
400
600
Time (ms)
800
0.6
0.4
0.2
2000
±
650
0
0
200
400
600
trial
den Ouden et al. 2010, J. Neurosci .
800
1000
Hierarchical Bayesian learning model
pk   1
k
volatility
vt-1
pvt 1 | vt , k  ~ N vt , exp( k ) 
vt
prt 1 | rt , vt  ~ Dir rt , exp( vt ) 
probabilistic association
rt
rt+1
observed events
ut
ut+1
1


prediction: p rt , vt , K u1:t 1 
t


rt 1 , vt 1  p  vt vt 1 , K  p rt 1 , vt 1 , K u1:t 1 drt 1dvt 1


update: p rt , vt , K u1:t 


0.2
 p  r , v , K u  p u r  dr dv dK
t
1:t 1
Behrens et al. 2007, Nat. Neurosci.
0.6
0.4
p rt , vt , K u1:t 1 p  ut rt 
t
p(F)
 p  r
0.8
t
t
t
t
0
400
440
480
520
Trial
560
600
Comparison of different learning models
1
Reaction times
450
True
Bayes Vol
HMM fixed
HMM learn
RW
0.8
p(F)
RT (ms)
440
430
420
0.6
0.4
410
0.2
400
390
0.1
0.3
0.5
0.7
0.9
0
400
440
480
True probabilities
Rescorla-Wagner
Hidden Markov models
(2 variants)
den Ouden et al. 2010, J. Neurosci .
0.7
560
600
Bayesian model selection:
0.6
Exceedance prob.
Alternative learning
models:
520
Trial
p(outcome)
hierarchical Bayesian
learner performs best
0.5
0.4
0.3
0.2
0.1
0
Categorical
model
Bayesian
learner
HMM (fixed) HMM (learn)
RescorlaWagner
Stimulus-independent prediction error
Putamen
Premotor cortex
p < 0.05
(cluster-level wholebrain corrected)
0
-0.5
0
-0.5
-1
-1.5
-2
BOLD resp. (a.u.)
BOLD resp. (a.u.)
p < 0.05
(SVC)
-1
-1.5
p(F)
p(H)
den Ouden et al. 2010, J. Neurosci .
-2
p(F)
p(H)
Prediction error (PE) activity in the putamen
PE during
reinforcement learning
O'Doherty et al. 2004,
Science
PE during
incidental
sensory learning
den Ouden et al. 2009,
Cerebral Cortex
According to the FEP (and other learning theories):
synaptic plasticity during learning = PE dependent changes in connectivity
Prediction error gates visuo-motor connections
• Modulation of
visuo-motor
connections by
striatal PE activity
• Influence of visual
areas on premotor
cortex:
– stronger for
surprising stimuli
– weaker for
expected stimuli
den Ouden et al. 2010, J. Neurosci .
p(H)
p(F)
PUT
d = 0.011 0.004
d = 0.010 0.003
p = 0.010
PPA
PMd
p = 0.017
FFA
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Integrating tractography and DCM
Diffusion-weighted imaging
Parker & Alexander, 2005,
Phil. Trans. B
Probabilistic tractography: Kaden et al. 2007, NeuroImage
• computes local fibre orientation
density by spherical deconvolution of
the diffusion-weighted signal
• estimates the spatial probability
distribution of connectivity from given
seed regions
• anatomical connectivity = proportion
of fibre pathways originating in a
specific source region that intersect
a target region
• If the area or volume of the source
region approaches a point, this
measure reduces to method by
Behrens et al. (2003)
1.6
Integration of
tractography
and DCM
1.4
1.2
1
R1
R2
0.8
0.6
0.4
0.2
0
-2
-1
0
1
2
low probability of anatomical connection
 small prior variance of effective connectivity parameter
1.6
1.4
1.2
1
R1
R2
0.8
0.6
0.4
0.2
0
Stephan, Tittgemeyer et al.
2009, NeuroImage
-2
-1
0
1
high probability of anatomical connection
 large prior variance of effective connectivity parameter
2
LD|LVF
FG
(x3)
probabilistic
tractography
FG
(x4)
LD
 DCM
structure
FG
left
*
 24
 1.50  102
 24  43.6%
LG
(x2)
LG
right
LG
left
12*  1.17  102
12  34.2%
LD|RVF
RVF
stim.
FG
right
13*  5.37  103
13  15.7%
LD
LG
(x1)
*
34
 2.23  103
34  6.5%
BVF
stim.
LVF
stim.
  6.5%
v  0.0384
2
1.8
  15.7%
1.6
 connectionspecific priors
for coupling
parameters
v  0.1070
1.4
1.2
1
0.8
0.6
  34.2%
  43.6%
v  0.5268
v  0.7746
0.4
0.2
0
-3
-2
-1
0
1
2
3
 anatomical
connectivity
Connection-specific prior variance  as a function of
anatomical connection probability 
ij 
0
1  0 exp(  ij )
m 1: a=-32,b=-32 m 2: a=-16,b=-32 m 3: a=-16,b=-28 m 4: a=-12,b=-32 m 5: a=-12,b=-28 m 6: a=-12,b=-24 m 7: a=-12,b=-20 m 8: a=-8,b=-32 m 9: a=-8,b=-28
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 10: a=-8,b=-24 m 11: a=-8,b=-20 m 12: a=-8,b=-16 m 13: a=-8,b=-12 m 14: a=-4,b=-32 m 15: a=-4,b=-28 m 16: a=-4,b=-24 m 17: a=-4,b=-20 m 18: a=-4,b=-16
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• 64 different mappings
by systematic search
across hyperparameters  and 
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m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0
m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
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m 28: a=0,b=-16 m 29: a=0,b=-12 m 30: a=0,b=-8
m 31: a=0,b=-4
m 32: a=0,b=0
m 33: a=0,b=4
m 34: a=0,b=8
m 35: a=0,b=12 m 36: a=0,b=16
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m 37: a=0,b=20 m 38: a=0,b=24 m 39: a=0,b=28 m 40: a=0,b=32 m 41: a=4,b=-32
m 42: a=4,b=0
m 43: a=4,b=4
m 44: a=4,b=8
m 45: a=4,b=12
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• yields anatomically
informed (intuitive and
counterintuitive) and
uninformed priors
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m 46: a=4,b=16 m 47: a=4,b=20 m 48: a=4,b=24 m 49: a=4,b=28 m 50: a=4,b=32 m 51: a=8,b=12 m 52: a=8,b=16 m 53: a=8,b=20 m 54: a=8,b=24
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m 55: a=8,b=28 m 56: a=8,b=32 m 57: a=12,b=20 m 58: a=12,b=24 m 59: a=12,b=28 m 60: a=12,b=32 m 61: a=16,b=28 m 62: a=16,b=32
m 63 & m 64
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log group Bayes factor
600
400
200
log group Bayes factor
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30
model
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model
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model
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700
695
690
685
680
post. model prob.
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m 1: a=-32,b=-32 m 2: a=-16,b=-32 m 3: a=-16,b=-28 m 4: a=-12,b=-32 m 5: a=-12,b=-28 m 6: a=-12,b=-24 m 7: a=-12,b=-20 m 8: a=-8,b=-32 m 9: a=-8,b=-28
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m 10: a=-8,b=-24 m 11: a=-8,b=-20 m 12: a=-8,b=-16 m 13: a=-8,b=-12 m 14: a=-4,b=-32 m 15: a=-4,b=-28 m 16: a=-4,b=-24 m 17: a=-4,b=-20 m 18: a=-4,b=-16
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m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0
m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
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m 28: a=0,b=-16 m 29: a=0,b=-12 m 30: a=0,b=-8
m 31: a=0,b=-4
m 32: a=0,b=0
m 33: a=0,b=4
m 34: a=0,b=8
m 35: a=0,b=12 m 36: a=0,b=16
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m 37: a=0,b=20 m 38: a=0,b=24 m 39: a=0,b=28 m 40: a=0,b=32 m 41: a=4,b=-32
m 42: a=4,b=0
m 43: a=4,b=4
m 44: a=4,b=8
m 45: a=4,b=12
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m 46: a=4,b=16 m 47: a=4,b=20 m 48: a=4,b=24 m 49: a=4,b=28 m 50: a=4,b=32 m 51: a=8,b=12 m 52: a=8,b=16 m 53: a=8,b=20 m 54: a=8,b=24
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m 55: a=8,b=28 m 56: a=8,b=32 m 57: a=12,b=20 m 58: a=12,b=24 m 59: a=12,b=28 m 60: a=12,b=32 m 61: a=16,b=28 m 62: a=16,b=32
m 63 & m 64
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Stephan, Tittgemeyer et al. 2009, NeuroImage
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1
Further reading: Methods papers on DCM for fMRI – part 1
•
Chumbley JR, Friston KJ, Fearn T, Kiebel SJ (2007) A Metropolis-Hastings algorithm
for dynamic causal models. Neuroimage 38:478-487.
•
Daunizeau J, David, O, Stephan KE (2010) Dynamic Causal Modelling: A critical
review of the biophysical and statistical foundations. NeuroImage, in press.
•
Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuroimage
19:1273-1302.
•
Kasess CH, Stephan KE, Weissenbacher A, Pezawas L, Moser E, Windischberger C
(2010) Multi-Subject Analyses with Dynamic Causal Modeling. NeuroImage 49:30653074.
•
Kiebel SJ, Kloppel S, Weiskopf N, Friston KJ (2007) Dynamic causal modeling: a
generative model of slice timing in fMRI. Neuroimage 34:1487-1496.
•
Marreiros AC, Kiebel SJ, Friston KJ (2008) Dynamic causal modelling for fMRI: a twostate model. Neuroimage 39:269-278.
•
Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Comparing dynamic causal
models. Neuroimage 22:1157-1172.
•
Penny WD, Stephan KE, Mechelli A, Friston KJ (2004b) Modelling functional
integration: a comparison of structural equation and dynamic causal models.
Neuroimage 23 Suppl 1:S264-274.
Further reading: Methods papers on DCM for fMRI – part 2
•
Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI
responses. Curr Opin Neurobiol 14:629-635.
•
Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing
hemodynamic models with DCM. Neuroimage 38:387-401.
•
Stephan KE, Harrison LM, Kiebel SJ, David O, Penny WD, Friston KJ (2007) Dynamic
causal models of neural system dynamics: current state and future extensions. J
Biosci 32:129-144.
•
Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing
hemodynamic models with DCM. Neuroimage 38:387-401.
•
Stephan KE, Kasper L, Harrison LM, Daunizeau J, den Ouden HE, Breakspear M,
Friston KJ (2008) Nonlinear dynamic causal models for fMRI. Neuroimage 42:649662.
•
Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009) Bayesian model
selection for group studies. Neuroimage 46:1004-1017.
•
Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009) Tractographybased priors for dynamic causal models. Neuroimage 47: 1628-1638.
•
Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, Friston KJ (2010)
Ten simple rules for Dynamic Causal Modelling. NeuroImage 49: 3099-3109.
Dynamic causal modelling (DCM)
Hemodynamic
forward model:
neural activityBOLD
(nonlinear)
Electric/magnetic
forward model:
neural activityEEG
MEG
LFP
(linear)
Neural state equation:
x  F ( x, u, )
fMRI
Neural model:
1 state variable per region
bilinear state equation
no propagation delays
ERPs
Neural model:
8 state variables per region
nonlinear state equation
propagation delays
inputs
Take-home messages
• Bayesian model selection (BMS):
generic approach to selecting an optimal model from a set of competing
models
• random effects BMS for group studies:
posterior model probabilities and exceedance probabilities
• nonlinear DCM:
enables one to investigate synaptic gating processes via activity-dependent
changes in connection strengths
• DCM & tractography:
probabilities of anatomical connections can be used to inform the prior
variance of DCM coupling parameters
• DCM implementations do not only exist for fMRI data, but also for
electrophysiological data
Thank you