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DCM: Advanced Topics
Klaas Enno Stephan
Translational Neuromodeling Unit (TNU)Institute for Biomedical EngineeringUniversity of Zurich & Swiss Federal Institute of Technology (ETH) Zurich
Wellcome Trust Centre for NeuroimagingInstitute of NeurologyUniversity College London
SPM Course 2012 @ FIL London18 May 2012
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fMRI signal change (%)
x1 x2
x3
x1 x2
x3
CuxDxBuAdtdx n
j
jj
m
i
ii
1
)(
1
)(
u1
u2
Overview
• Bayesian model selection (BMS) • Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Integrating tractography and DCM• Applications of DCM to clinical questions
),,( uxFdtdx
Neural state equation:
Electromagneticforward model:
neural activityEEGMEGLFP
Dynamic Causal Modeling (DCM)
simple neuronal modelcomplicated forward model
complicated neuronal modelsimple forward model
fMRI EEG/MEG
inputs
Hemodynamicforward model:neural activityBOLD
Generative models & model selection
• any DCM = a particular generative model of how the data (may) have been caused
• modelling = comparing competing hypotheses about the mechanisms underlying observed data a priori definition of hypothesis set (model space) is crucial determine the most plausible hypothesis (model), given the
data
• model selection model validation! model validation requires external criteria (external to the
measured data)
Model comparison and selectionGiven 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
( | ) ( | , ) ( | ) p y m p y m p m d
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. 2004a, NeuroImage
pmypAIC ),|(log
Logarithm is a monotonic function
Maximizing log model evidence= Maximizing model evidence
)(),|(log )()( )|(log
mcomplexitymypmcomplexitymaccuracymyp
SPM2 & SPM5 offered 2 approximations:
NpmypBIC log2
),|(log
Akaike Information Criterion:
Bayesian Information Criterion:
Log model evidence = balance between fit and complexity
Penny et al. 2004a, NeuroImagePenny 2012, NeuroImage
Approximations to the model evidence in DCM
No. of parameters
No. ofdata points
The (negative) free energy approximation• Under Gaussian assumptions about the posterior (Laplace
approximation):
log ( | )
log ( | , ) , | , | ,
p y m
p y m KL q p m KL q p y m
log ( | ) , | ,
log ( | , ) , |accuracy complexity
F p y m KL q p y m
p y m KL q p m
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: Since SPM8, only F is used for model selection !
y
Tyy CCC
mpqKL
|1
|| 21ln
21ln
21
)|(),(
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) Evidence1 to 3 50-75% weak
3 to 20 75-95% positive20 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 heterogeneous groups
Dirichlet parameters = “occurrences” of models in the population
Dirichlet distribution of model probabilities r
Multinomial distribution of model labels m
Measured data y
Model inversion by Variational Bayes (VB) or MCMC
Stephan et al. 2009a, NeuroImagePenny et al. 2010, PLoS Comp. Biol.
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Subj
ects
Log model evidence differences
MOG
LG LG
RVFstim.
LVFstim.
FGFG
LD|RVF
LD|LVF
LD LD
MOGMOG
LG LG
RVFstim.
LVFstim.
FGFG
LD
LD
LD|RVF LD|LVF
MOG
m2 m1
m1m2
Data: Stephan et al. 2003, ScienceModels: Stephan et al. 2007, J. Neurosci.
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1
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3.5
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4.5
5
r1
p(r 1|y
)
p(r1>0.5 | y) = 0.997
m1m2
1
1
11.884.3%r
2
2
2.215.7%r
%7.9921 rrp
Stephan et al. 2009a, NeuroImage
inference on model structure or inference on model parameters?
inference on individual models or model space partition?
comparison of model families using
FFX or RFX BMS
optimal model structure assumed to be identical across subjects?
FFX BMS RFX BMS
yes no
inference on parameters of an optimal model or parameters of all models?
BMA
definition of model space
FFX analysis of parameter estimates
(e.g. BPA)
RFX analysis of parameter estimates(e.g. t-test, ANOVA)
optimal model structure assumed to be identical across subjects?
FFX BMS
yes no
RFX BMS
Stephan et al. 2010, NeuroImage
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5
r1
p(r 1|y
)
p(r1>0.5 | y) = 0.986
Model space partitioning:
comparing model families
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Sum
med
log
evid
ence
(rel
. to
RB
ML)
CBMN CBMN(ε) CBML CBML(ε)RBMN RBMN(ε) RBML RBML(ε)
CBMN CBMN(ε) CBML CBML(ε)RBMN RBMN(ε) RBML RBML(ε)
nonlinear models linear models
FFX
RFX
4
1
*1
kk
8
5
*2
kk
nonlinear linear
log GBF
Model space partitioning
1 73.5%r 2 26.5%r
1 2 98.6%p r r
m1 m2
m1m2
Stephan et al. 2009, NeuroImage
Bayesian Model Averaging (BMA)
• abandons dependence of parameter inference on a single model
• uses the entire model space considered (or an optimal family of models)
• computes average of each parameter, weighted by posterior model probabilities
• represents a particularly useful alternative– when none of the models (or model
subspaces) considered clearly outperforms all others
– when comparing groups for which the optimal model differs
1..
1..
|
| , |n N
n n Nm
p y
p y m p m y
NB: p(m|y1..N) can be obtained by either FFX or RFX BMS
Penny et al. 2010, PLoS Comput. Biol.
Overview
• Bayesian model selection (BMS) • Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Integrating tractography and DCM• Applications of DCM to clinical questions
DCM10 in SPM8• DCM10 was released as part of SPM8 in July 2010 (version 4010).
• Introduced many new features, incl. two-state DCMs and stochastic DCMs
• This led to various changes in model defaults, e.g.– inputs mean-centred– changes in coupling priors– self-connections estimated separately for each area
• For details, see: www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf
• Further changes in version 4290 (released April 2011) to accommodate new developments and give users more choice (e.g., whether or not to mean-centre inputs).
The evolution of DCM in SPM• DCM is not one specific model, but a framework for Bayesian inversion of
dynamic system models
• The default implementation in SPM is evolving over time– improvements of numerical routines (e.g., for inversion)– change in priors to cover new variants (e.g., stochastic DCMs,
endogenous DCMs etc.)
To enable replication of your results, you should ideally state which SPM version (release number) you are using when publishing papers.
In the next SPM version, the release number will be stored in the DCM.mat.
endogenous connectivity
direct inputs
modulation ofconnectivity
Neural state equation CuxBuAx jj )( )(
uxC
xx
uB
xxA
j
j
)(
hemodynamicmodelλ
x
y
integration
BOLDyyy
activityx1(t)
activityx2(t) activity
x3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
The classical DCM:a deterministic, one-state, bilinear model
Factorial structure of model specification in DCM10• Three dimensions of model specification:
– bilinear vs. nonlinear
– single-state vs. two-state (per region)
– deterministic vs. stochastic
• Specification via GUI.
bilinear DCM
CuxDxBuAdtdx m
i
n
j
jj
ii
1 1
)()(CuxBuAdtdx m
i
ii
1
)(
Bilinear state equation:
driving input
modulation
driving input
modulationnon-linear DCM
...)0,(),(2
0
uxuxfu
ufx
xfxfuxf
dtdx
Two-dimensional Taylor series (around x0=0, u0=0):
Nonlinear state equation:
...2
)0,(),(2
2
22
0
x
xfux
uxfu
ufx
xfxfuxf
dtdx
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fMRI signal change (%)x1 x2
x3
CuxDxBuAdtdx n
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m
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1
)(
1
)(
Nonlinear dynamic causal model (DCM)
Stephan et al. 2008, NeuroImage
u1
u2
V1 V5stim
PPC
attention
motion
-2 -1 0 1 2 3 4 50
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%1.99)|0( 1,5 yDp PPCVV
1.25
0.13
0.46
0.390.26
0.50
0.26
0.10MAP = 1.25
Stephan et al. 2008, NeuroImage
uinput
Single-state DCM
1x
Intrinsic (within-region)
coupling
Extrinsic (between-region)
coupling
NNNN
N
ijijij
x
xx
uBACuxx
1
1
111
Two-state DCM
Ex1
IN
EN
I
E
IINN
IENN
EENN
EENN
EEN
IIIE
EEN
EIEE
ijijijij
xx
xx
x
uBA
Cuxx
1
1
1
1111
11111
000
000
)exp(
Ix1
I
E
x
x
1
1
Two-state DCM
Marreiros et al. 2008, NeuroImage
Estimates of hidden causes and states(Generalised filtering)
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1inputs or causes - V2
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time (seconds)
excitatorysignal
flowvolumedHb
observedpredicted
Stochastic DCM
( ) ( )
( )
( )j xjj
v
dx A u B x Cvdt
v u
Li et al. 2011, NeuroImage
• all states are represented in generalised coordinates of motion
• random state fluctuations w(x) account for endogenous fluctuations,have unknown precision and smoothness two hyperparameters
• fluctuations w(v) induce uncertainty about how inputs influence neuronal activity
• can be fitted to resting state data
Overview
• Bayesian model selection (BMS) • Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Integrating tractography and DCM• Applications of DCM to clinical questions
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(fa
ce)
trial
CS1
CS2
den Ouden et al. 2010, J. Neurosci.
Hierarchical 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
Behrens et al. 2007, Nat. Neurosci.
prior on volatility
Explaining RTs by different learning models
400 440 480 520 560 6000
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1
Trial
p(F)
TrueBayes VolHMM fixedHMM learnRW
0
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Categoricalmodel
Bayesianlearner
HMM (fixed) HMM (learn) Rescorla-Wagner
Exce
edan
ce p
rob.
Bayesian model selection: hierarchical Bayesian model performs best
5 alternative learning models: • categorical probabilities• hierarchical Bayesian learner• Rescorla-Wagner• Hidden Markov models
(2 variants)
0.1 0.3 0.5 0.7 0.9390
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RT
(ms)
p(outcome)
Reaction times
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
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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 incidentalsensory learning
O'Doherty et al. 2004, Science
den Ouden et al. 2009, Cerebral Cortex
Could the putamen be regulating trial-by-trial changes of task-relevant connections?
PE = “teaching signal” for synaptic plasticity during
learning
p < 0.05 (SVC)
PE during activesensory learning
Prediction errors control plasticity during adaptive cognition
• Modulation of visuo-motor connections by striatal prediction error activity
• Influence of visual areas on premotor cortex:– stronger for
surprising stimuli – weaker for expected
stimuli
den Ouden et al. 2010, J. Neurosci .
PPA FFA
PMd
Hierarchical Bayesian learning model
PUT
p = 0.010 p = 0.017
events in the world
association
volatility
Hierarchical variational Bayesian learning
sensory stimuli
11kx
1kx
12kx
2kx
3kx1
3kx
1ku ku
1 13 3 3| , ~ ,exp( )k k kp x x N x
1p
1 12 2 3 2 3| , ~ ,exp( )k k k k kp x x x N x x
1 2 2| ~k k kp x x Bernoulli S x
1 1 1| ~ 0, 1k k k kp u x MoG x x
Mean-field decomposition
1.. 11 2 3, , |k k k k k kp u x u q x q x q x q
Mathys et al. (2011), Front. Hum. Neurosci.
Overview
• Bayesian model selection (BMS) • Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Integrating tractography and DCM• Applications of DCM to clinical questions
Diffusion-weighted imaging
Parker & Alexander, 2005, Phil. Trans. B
R2R1
R2R1
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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 LG
FGFG
DCM
LGleft
LGright
FGright
FGleft
13 15.7%
34 6.5%
24 43.6%
12 34.2%
anatomical connectivity
probabilistictractography
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15.7%0.1070v
34.2%0.5268v
43.6%0.7746v
Proof of concept study
connection-specific priors for coupling parameters
Stephan, Tittgemeyer et al. 2009, NeuroImage
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1m6: a=-12,b=-24
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1m13: a=-8,b=-12
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1m14: a=-4,b=-32
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1m15: a=-4,b=-28
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1m16: a=-4,b=-24
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1m17: a=-4,b=-20
0 0.5 10
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1m18: a=-4,b=-16
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1m19: a=-4,b=-12
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1m20: a=-4,b=-8
0 0.5 10
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1m21: a=-4,b=-4
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1m22: a=-4,b=0
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1m23: a=-4,b=4
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1m24: a=0,b=-32
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1m25: a=0,b=-28
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1m26: a=0,b=-24
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1m27: a=0,b=-20
0 0.5 10
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1m28: a=0,b=-16
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1m29: a=0,b=-12
0 0.5 10
0.5
1m30: a=0,b=-8
0 0.5 10
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1m31: a=0,b=-4
0 0.5 10
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1m32: a=0,b=0
0 0.5 10
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1m33: a=0,b=4
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1m34: a=0,b=8
0 0.5 10
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1m35: a=0,b=12
0 0.5 10
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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
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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
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1m43: a=4,b=4
0 0.5 10
0.5
1m44: a=4,b=8
0 0.5 10
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1m45: a=4,b=12
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1m46: a=4,b=16
0 0.5 10
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1m47: a=4,b=20
0 0.5 10
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1m48: a=4,b=24
0 0.5 10
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1m49: a=4,b=28
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1m50: a=4,b=32
0 0.5 10
0.5
1m51: a=8,b=12
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1m52: a=8,b=16
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1m53: a=8,b=20
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1m54: a=8,b=24
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1m55: a=8,b=28
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1m56: a=8,b=32
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1m57: a=12,b=20
0 0.5 10
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1m58: a=12,b=24
0 0.5 10
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1m59: a=12,b=28
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1m60: a=12,b=32
0 0.5 10
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1m61: a=16,b=28
0 0.5 10
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1m62: a=16,b=32
0 0.5 10
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1m63 & m64
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
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Models with anatomically informed priors (of an intuitive form)
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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
Models with anatomically informed priors (of an intuitive form) were clearly superior to anatomically uninformed ones: Bayes Factor >109
Overview
• Bayesian model selection (BMS) • Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Integrating tractography and DCM• Applications of DCM to clinical questions
model structure
Model-based predictions for single patients
set of parameter estimates
BMS
model-based decoding
BMS: Parkison‘s disease and treatment
Rowe et al. 2010,NeuroImage
Age-matched controls
PD patientson medication
PD patientsoff medication
DA-dependent functional disconnection of the SMA
Selection of action modulates connections between PFC and SMA
Model-based decoding by generative embedding
Brodersen et al. 2011, PLoS Comput. Biol.
step 2 —kernel construction
step 1 —model inversion
measurements from an individual subject
subject-specificinverted generative model
subject representation in the generative score space
A → BA → CB → BB → C
A
CB
step 3 —support vector classification
separating hyperplane fitted to discriminate between groups
A
CB
jointly discriminativemodel parameters
step 4 —interpretation
Discovering remote or “hidden” brain lesions
Discovering remote or “hidden” brain lesions
detect “down-stream” network changes altered synaptic coupling among healthy regions
Model-based decoding of disease status: mildly aphasic patients (N=11) vs. controls (N=26)Connectional fingerprints from a 6-region DCM of auditory areas during speech perception
Brodersen et al. 2011, PLoS Comput. Biol.
MGB
PT
HG (A1)
S
MGB
PT
HG (A1)
S
Model-based decoding of disease status: mildly aphasic patients (N=11) vs. controls (N=26)
Classification accuracy
Sensitivity: 100 %Specificity: 96.2%
0
0.2
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0.6
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1
bala
nced
acc
urac
y
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1
FPR (1 - specificity)
TPR
(sen
sitiv
ity)
0 0.5 10
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0.6
0.8
1
TPR (recall)
PPV
(pre
cisi
on)
anatomical FS
search-lightFS
generative embedding
contrast FS
Brodersen et al. 2011, PLoS Comput. Biol.
L.MGB
L.PT
L.HG(A1)
R.MGB
R.PT
R.HG(A1)
auditory stimuli
16218917734729133230781 2433893600.4
0.5
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1
bala
nced
acc
urac
y
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FPR (1 - specificity)
TP
R (
sens
itivi
ty)
0 0.5 10
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1
TPR (recall)
PP
V (
prec
isio
n)
anatomicalcontrastsearchlightPCA
A B C
means correlationseigenvariates correlationseigenvariates z-correlationsgen.embed., original modelgen.embed., feedforwardgen.embed., left hemispheregen.embed., right hemisphere
Legend (cont’d)
Legend
activation-based
correlation-based
model-based
a c s p m e z o f l r
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10
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0.5-0.4
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generative em
bedding
L.H
G
L.H
G
Voxe
l (64
,-24,
4) m
m
L.MGB L.MGBVoxel (-42,-26,10) mmVoxel (-56,-20,10) mm R.HG L.HG
controlspatients
Voxel-based feature space Generative score space
Multivariate searchlightclassification analysis
Generative embedding using DCM
F
D
EC
B
A
Definition of ROIsAre regions of interest definedanatomically or functionally?
anatomically functionally
Functional contrastsAre the functional contrasts defined
across all subjects or between groups?
1 ROI definitionand nmodel inversionsunbiased estimate
Repeat n times:1 ROI definition and nmodel inversionsunbiased estimate
1 ROI definition and nmodel inversionsslightly optimistic estimate:voxel selection for training set and test set based on test data
Repeat n times:1 ROI definition and 1 model inversionslightly optimistic estimate:voxel selection for training set based on test data and test labels
Repeat n times:1 ROI definition and nmodel inversionsunbiased estimate
1 ROI definition and nmodel inversionshighly optimistic estimate:voxel selection for training set and test set based on test data and test labels
across subjects
between groups
Brodersen et al. 2011, PLoS Comput. Biol.
Key methods papers: DCM for fMRI and BMS – part 1• Brodersen KH, Schofield TM, Leff AP, Ong CS, Lomakina EI, Buhmann JM, Stephan KE (2011) Generative
embedding for model-based classification of fMRI data. PLoS Computational Biology 7: e1002079.• Daunizeau J, David, O, Stephan KE (2011) Dynamic Causal Modelling: A critical review of the biophysical and
statistical foundations. NeuroImage 58: 312-322.• Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. NeuroImage 19:1273-1302.• Friston K, Stephan KE, Li B, Daunizeau J (2010) Generalised filtering. Mathematical Problems in Engineering
2010: 621670.• Friston KJ, Li B, Daunizeau J, Stephan KE (2011) Network discovery with DCM. NeuroImage 56: 1202–1221.• Friston K, Penny W (2011) Post hoc Bayesian model selection. Neuroimage 56: 2089-2099.• 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.• Li B, Daunizeau J, Stephan KE, Penny WD, Friston KJ (2011). Stochastic DCM and generalised filtering.
NeuroImage 58: 442-457• 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.
Key methods papers: DCM for fMRI and BMS – part 2• Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing
Families of Dynamic Causal Models. PLoS Computational Biology 6: e1000709. • Penny WD (2012) Comparing dynamic causal models using AIC, BIC and free energy. Neuroimage 59:
319-330.• 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 (2009a) Bayesian model selection for
group studies. NeuroImage 46:1004-1017.• Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009b) 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.
Thank you