DCM: Advanced topics Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Wellcome Trust Centre for Neuroimaging Institute of Neurology University College London SPM Course 2010 University of Zurich, 17-19 February 2010 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 90 100 0 0.2 0.4 0.6 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 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 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 90 100 0 0.2 0.4 0.6 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 N euralpopulation 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 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 fM R Isignalchange (% ) x 1 x 2 x 3 x 1 x 2 x 3 Cu x D x B u A dt dx n j j j m i i i 1 ) ( 1 ) ( u 1 u 2
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DCM: Advanced topics Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich.
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DCM: Advanced topics
Klaas Enno Stephan
Laboratory for Social & Neural Systems Research Institute for Empirical Research in EconomicsUniversity of Zurich
Wellcome Trust Centre for NeuroimagingInstitute of NeurologyUniversity College London
SPM Course 2010University of Zurich, 17-19 February 2010
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fMRI signal change (%)
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fMRI signal change (%)
x1 x2
x3
x1 x2
x3
CuxDxBuAdt
dx n
j
jj
m
i
ii
1
)(
1
)(
u1
u2
),,( 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
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Embedding computational models in DCMs
• Integrating tractography and DCM
Model comparison and selection
Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?
For which model m does p(y|m) become maximal?
Which model represents thebest balance between model fit and model complexity?
Pitt & Miyung (2002) TICS
mypqKL
mpqKL
myp
dmpmypmyp
,|,
|,
),|(log
)|(),|()|(
Model evidence:
Various approximations, e.g.:- negative free energy, AIC, BIC
Bayesian model selection (BMS)
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
all possible datasets
y
p(y
|m
)
Gharamani, 2004
McKay 1992, Neural Comput.Penny et al. 2004, NeuroImage Stephan et al. 2007, NeuroImage
pmypAIC ),|(log
Logarithm is a monotonic function
Maximizing log model evidence= Maximizing model evidence
)(),|(log
)()( )|(log
mcomplexitymyp
mcomplexitymaccuracymyp
In SPM2 & SPM5, interface offers 2 approximations:
Np
mypBIC log2
),|(log
Akaike Information Criterion:
Bayesian Information Criterion:
Log model evidence = balance between fit and complexity
Penny et al. 2004, NeuroImage
Approximations to the model evidence in DCM
No. of parameters
No. ofdata points
AIC favours more complex models,BIC favours simpler models.
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:
mypqKLF
mypqKLmpqKLmyp
myp
,|,
,|,|,),|(log
)|(log
mypqKLmypF ,|,)|(log
The complexity term in F
• In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies.
• 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 !
y
Tyy CCC
mpqKL
|1
|| 2
1ln
2
1ln
2
1
)|(),(
Bayes factors
)|(
)|(
2
112 myp
mypB
positive value, [0;[
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is made possible by Bayes factors:
To compare two models, we could just compare their log evidences.
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
Kass & Raftery classification:
Kass & Raftery 1995, J. Am. Stat. Assoc.
V1 V5stim
PPCM2
attention
V1 V5stim
PPCM1
attention
V1 V5stim
PPCM3attention
V1 V5stim
PPCM4attention
BF 2966F = 7.995
M2 better than M1
BF 12F = 2.450
M3 better than M2
BF 23F = 3.144
M4 better than M3
M1 M2 M3 M4
BMS in SPM8: an example
Fixed effects BMS at group level
Group Bayes factor (GBF) for 1...K subjects:
Average Bayes factor (ABF):
Problems:- blind with regard to group heterogeneity- sensitive to outliers
k
kijij BFGBF )(
( )kKij ij
k
ABF BF
)|(~ 111 mypy)|(~ 111 mypy
)|(~ 222 mypy)|(~ 111 mypy
)|(~ pmpm kk
);(~ rDirr
)|(~ pmpm kk )|(~ pmpm kk),1;(~1 rmMultm
Random effects BMS for group studies
Dirichlet parameters= “occurrences” of models in the population
Dirichlet distribution of model probabilities
Multinomial distribution of model labels
Measured data y
Model inversion by Variational Bayes (VB)
Model inversion by Variational Bayes (VB)
Stephan et al. 2009, NeuroImage
Is the red letter left or right from the midline of the word?
group analysis (random effects),n=16, p<0.05 corrected
analysis with SPM2
group analysis (random effects),n=16, p<0.05 corrected
analysis with SPM2
Task-driven lateralisation
letter decisions > spatial decisions
time
•••
Does the word contain the letter A or not?
spatial decisions > letter decisions
Stephan et al. 2003, Science
Theories on inter-hemispheric integration during lateralised
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?
Büchel & Friston (1997)
modulation of back-ward or forward connection?
additional drivingeffect of attentionon PPC?
bilinear or nonlinearmodulation offorward connection?
V1 V5stim
PPCM2
attention
V1 V5stim
PPCM1
attention
V1 V5stim
PPCM3attention
V1 V5stim
PPCM4attention
BF = 2966
M2 better than M1
M3 better than M2
BF = 12
M4 better than M3
BF = 23
Stephan et al. 2008, NeuroImage
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
0.39
0.26
0.50
0.26
0.10MAP = 1.25
Stephan et al. 2008, NeuroImage
V1
V5PPC
observedfitted
motion &attention
motion &no attention
static dots
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Embedding computational models in DCMs
• Integrating tractography and DCM
Learning of dynamic audio-visual associations
CS Response
Time (ms)
0 200 400 600 800 2000 ± 650
or
Target StimulusConditioning Stimulus
or
TS
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
p(f
ace)
trial
CS1
CS2
den Ouden et al. 2010, J. Neurosci .
Bayesian learning model
observed events
probabilistic association
volatility
k
vt-1 vt
rt rt+1
ut ut+1
)exp(,~,|1 ttttt vrDirvrrp
)exp(,~,|1 kvNkvvp ttt
1kp
1: 1 1 1 1 1 1 1: 1 1 1
1: 1
1:
1: 1
prediction: , , , , , ,
, ,update: , ,
, ,
t t t t t t t t t t t t t
t t t t t
t t t
t t t t t t t
p r v K u p r r v p v v K p r v K u dr dv
p r v K u p u rp r v K u
p r v K u p u r dr dv dK
Behrens et al. 2007, Nat. Neurosci.
Random effects BMS
0.1 0.3 0.5 0.7 0.9390
400
410
420
430
440
450
RT
(m
s)
Reaction times
0 5 10 15 20-5
0
5
10
15
20
25
30
35
40
log
mo
del
evi
den
ce
subject
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
r1
p(r 1
|y)
p(r1>0.5 | y) = 1.000
true probabilities
Bayesian learner
400 440 480 520 560 600trial
p(F
)
den Ouden et al. 2010, J. Neurosci .
Comparison with competing learning models
400 440 480 520 560 6000
0.2
0.4
0.6
0.8
1
Trial
p(F
)
TrueBayes VolHMM fixedHMM learnRW
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Categoricalmodel
Bayesianlearner
HMM (fixed) HMM (learn) Rescorla-Wagner
Exc
eed
ance
pro
b.BMS:
hierarchical Bayesian learner performs best
Alternative learning models:
Rescorla-Wagner
HMM (2 variants)
True probabilities
den Ouden et al. 2010, J. Neurosci .
Putamen Premotor cortex
Stimulus-independent prediction error
p < 0.05 (SVC)
p < 0.05 (cluster-level whole- brain corrected)
p(F) p(H)-2
-1.5
-1
-0.5
0
BO
LD
re
sp.
(a.u
.)
p(F) p(H)-2
-1.5
-1
-0.5
0
BO
LD
re
sp.
(a.u
.)
den Ouden et al. 2010, J. Neurosci .
Prediction error (PE) activity in the putamen
PE during reinforcement learning
PE during incidental sensory learning
O'Doherty et al. 2004, Science
den Ouden et al. 2009, Cerebral Cortex
According to the free energy principle (and other learning theories):
synaptic plasticity during learning = PE dependent changes in connectivity
PPA FFA
PMd
p(F)p(H)
PUT
d = 0.010 0.003
p = 0.010
Prediction error gates visuo-motor connections
d = 0.011 0.004
p = 0.017
• 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 .
Prediction error in PMd: cause or effect?
Model 1 Model 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-4
-3
-2
-1
0
1
2
3
4
5
log
mo
del
evi
den
ce
subject
Model 1 minus Model 2
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
r1p(
r 1|y
)
p(r1>0.5 | y) = 0.991
den Ouden et al. 2010, J. Neurosci .
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Embedding computational models in DCMs
• 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)
R2R1
R2R1
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
low probability of anatomical connection small prior variance of effective connectivity parameter
high probability of anatomical connection large prior variance of effective connectivity parameter
Integration of tractography and DCM
Stephan, Tittgemeyer et al. 2009, NeuroImage
LG(x1)
LG(x2)
RVFstim.
LVFstim.
FG(x4)
FG(x3)
LD|LVF
LD LD
BVFstim.
LD|RVF DCM structure
LGleft
LGright
FGright
FGleft
* 313
13
5.37 10
15.7%
* 334
34
2.23 10
6.5%
* 224
24
1.50 10
43.6%
* 212
12
1.17 10
34.2%
anatomical connectivity
probabilistictractography
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
6.5%
0.0384v
15.7%
0.1070v
34.2%
0.5268v
43.6%
0.7746v
connection-specific priors for coupling parameters
Stephan, Tittgemeyer et al. 2009, NeuroImage
0 0.5 10
0.5
1m1: a=-32,b=-32
0 0.5 10
0.5
1m2: a=-16,b=-32
0 0.5 10
0.5
1m3: a=-16,b=-28
0 0.5 10
0.5
1m4: a=-12,b=-32
0 0.5 10
0.5
1m5: a=-12,b=-28
0 0.5 10
0.5
1m6: a=-12,b=-24
0 0.5 10
0.5
1m7: a=-12,b=-20
0 0.5 10
0.5
1m8: a=-8,b=-32
0 0.5 10
0.5
1m9: a=-8,b=-28
0 0.5 10
0.5
1m10: a=-8,b=-24
0 0.5 10
0.5
1m11: a=-8,b=-20
0 0.5 10
0.5
1m12: a=-8,b=-16
0 0.5 10
0.5
1m13: a=-8,b=-12
0 0.5 10
0.5
1m14: a=-4,b=-32
0 0.5 10
0.5
1m15: a=-4,b=-28
0 0.5 10
0.5
1m16: a=-4,b=-24
0 0.5 10
0.5
1m17: a=-4,b=-20
0 0.5 10
0.5
1m18: a=-4,b=-16
0 0.5 10
0.5
1m19: a=-4,b=-12
0 0.5 10
0.5
1m20: a=-4,b=-8
0 0.5 10
0.5
1m21: a=-4,b=-4
0 0.5 10
0.5
1m22: a=-4,b=0
0 0.5 10
0.5
1m23: a=-4,b=4
0 0.5 10
0.5
1m24: a=0,b=-32
0 0.5 10
0.5
1m25: a=0,b=-28
0 0.5 10
0.5
1m26: a=0,b=-24
0 0.5 10
0.5
1m27: a=0,b=-20
0 0.5 10
0.5
1m28: a=0,b=-16
0 0.5 10
0.5
1m29: a=0,b=-12
0 0.5 10
0.5
1m30: a=0,b=-8
0 0.5 10
0.5
1m31: a=0,b=-4
0 0.5 10
0.5
1m32: a=0,b=0
0 0.5 10
0.5
1m33: a=0,b=4
0 0.5 10
0.5
1m34: a=0,b=8
0 0.5 10
0.5
1m35: a=0,b=12
0 0.5 10
0.5
1m36: a=0,b=16
0 0.5 10
0.5
1m37: a=0,b=20
0 0.5 10
0.5
1m38: a=0,b=24
0 0.5 10
0.5
1m39: a=0,b=28
0 0.5 10
0.5
1m40: a=0,b=32
0 0.5 10
0.5
1m41: a=4,b=-32
0 0.5 10
0.5
1m42: a=4,b=0
0 0.5 10
0.5
1m43: a=4,b=4
0 0.5 10
0.5
1m44: a=4,b=8
0 0.5 10
0.5
1m45: a=4,b=12
0 0.5 10
0.5
1m46: a=4,b=16
0 0.5 10
0.5
1m47: a=4,b=20
0 0.5 10
0.5
1m48: a=4,b=24
0 0.5 10
0.5
1m49: a=4,b=28
0 0.5 10
0.5
1m50: a=4,b=32
0 0.5 10
0.5
1m51: a=8,b=12
0 0.5 10
0.5
1m52: a=8,b=16
0 0.5 10
0.5
1m53: a=8,b=20
0 0.5 10
0.5
1m54: a=8,b=24
0 0.5 10
0.5
1m55: a=8,b=28
0 0.5 10
0.5
1m56: a=8,b=32
0 0.5 10
0.5
1m57: a=12,b=20
0 0.5 10
0.5
1m58: a=12,b=24
0 0.5 10
0.5
1m59: a=12,b=28
0 0.5 10
0.5
1m60: a=12,b=32
0 0.5 10
0.5
1m61: a=16,b=28
0 0.5 10
0.5
1m62: a=16,b=32
0 0.5 10
0.5
1m63 & m64
0
01 exp( )ijij
0
01 exp( )ijij
Connection-specific prior variance as a function of anatomical connection probability
• 64 different mappings by systematic search across hyper-parameters and
• yields anatomically informed (intuitive and counterintuitive) and uninformed priors
0 10 20 30 40 50 600
200
400
600
model
log
gro
up
Bay
es f
acto
r
0 10 20 30 40 50 60
680
685
690
695
700
model
log
gro
up
Bay
es f
acto
r
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
model
po
st.
mo
del
pro
b.
0 0.5 10
0.5
1m1: a=-32,b=-32
0 0.5 10
0.5
1m2: a=-16,b=-32
0 0.5 10
0.5
1m3: a=-16,b=-28
0 0.5 10
0.5
1m4: a=-12,b=-32
0 0.5 10
0.5
1m5: a=-12,b=-28
0 0.5 10
0.5
1m6: a=-12,b=-24
0 0.5 10
0.5
1m7: a=-12,b=-20
0 0.5 10
0.5
1m8: a=-8,b=-32
0 0.5 10
0.5
1m9: a=-8,b=-28
0 0.5 10
0.5
1m10: a=-8,b=-24
0 0.5 10
0.5
1m11: a=-8,b=-20
0 0.5 10
0.5
1m12: a=-8,b=-16
0 0.5 10
0.5
1m13: a=-8,b=-12
0 0.5 10
0.5
1m14: a=-4,b=-32
0 0.5 10
0.5
1m15: a=-4,b=-28
0 0.5 10
0.5
1m16: a=-4,b=-24
0 0.5 10
0.5
1m17: a=-4,b=-20
0 0.5 10
0.5
1m18: a=-4,b=-16
0 0.5 10
0.5
1m19: a=-4,b=-12
0 0.5 10
0.5
1m20: a=-4,b=-8
0 0.5 10
0.5
1m21: a=-4,b=-4
0 0.5 10
0.5
1m22: a=-4,b=0
0 0.5 10
0.5
1m23: a=-4,b=4
0 0.5 10
0.5
1m24: a=0,b=-32
0 0.5 10
0.5
1m25: a=0,b=-28
0 0.5 10
0.5
1m26: a=0,b=-24
0 0.5 10
0.5
1m27: a=0,b=-20
0 0.5 10
0.5
1m28: a=0,b=-16
0 0.5 10
0.5
1m29: a=0,b=-12
0 0.5 10
0.5
1m30: a=0,b=-8
0 0.5 10
0.5
1m31: a=0,b=-4
0 0.5 10
0.5
1m32: a=0,b=0
0 0.5 10
0.5
1m33: a=0,b=4
0 0.5 10
0.5
1m34: a=0,b=8
0 0.5 10
0.5
1m35: a=0,b=12
0 0.5 10
0.5
1m36: a=0,b=16
0 0.5 10
0.5
1m37: a=0,b=20
0 0.5 10
0.5
1m38: a=0,b=24
0 0.5 10
0.5
1m39: a=0,b=28
0 0.5 10
0.5
1m40: a=0,b=32
0 0.5 10
0.5
1m41: a=4,b=-32
0 0.5 10
0.5
1m42: a=4,b=0
0 0.5 10
0.5
1m43: a=4,b=4
0 0.5 10
0.5
1m44: a=4,b=8
0 0.5 10
0.5
1m45: a=4,b=12
0 0.5 10
0.5
1m46: a=4,b=16
0 0.5 10
0.5
1m47: a=4,b=20
0 0.5 10
0.5
1m48: a=4,b=24
0 0.5 10
0.5
1m49: a=4,b=28
0 0.5 10
0.5
1m50: a=4,b=32
0 0.5 10
0.5
1m51: a=8,b=12
0 0.5 10
0.5
1m52: a=8,b=16
0 0.5 10
0.5
1m53: a=8,b=20
0 0.5 10
0.5
1m54: a=8,b=24
0 0.5 10
0.5
1m55: a=8,b=28
0 0.5 10
0.5
1m56: a=8,b=32
0 0.5 10
0.5
1m57: a=12,b=20
0 0.5 10
0.5
1m58: a=12,b=24
0 0.5 10
0.5
1m59: a=12,b=28
0 0.5 10
0.5
1m60: a=12,b=32
0 0.5 10
0.5
1m61: a=16,b=28
0 0.5 10
0.5
1m62: a=16,b=32
0 0.5 10
0.5
1m63 & m64
Stephan, Tittgemeyer et al. 2009, NeuroImage
Methods papers on DCM for fMRI and BMS – 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: 3065-3074.
• 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 two-state 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.
• Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing Families of Dynamic Causal Models. PLoS Computational Biology, in press.
Methods papers on DCM for fMRI and BMS – 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 42:649-662.
• 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) Tractography-based 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.