1. No category

DCM: Advanced topics
Klaas Enno Stephan
Zurich SPM Course 2014
14 February 2014
Overview
• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models into DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions
Generative models
p( y |  , m) p( | m)
p( | y, m)
Advantages:
1. force us to think mechanistically: how were the data caused?
2. allow one to generate synthetic data.
3. Bayesian perspective → inversion & model evidence
Dynamic Causal Modeling (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
EEG/MEG
complicated neuronal model
simple forward model
inputs
Bayesian system
identification
Neural dynamics
u (t )
Design experimental inputs
dx dt  f ( x, u,  )
Define likelihood model
Observer function
Inference on model
structure
Inference on parameters
y  g ( x,  )  
p( y |  , m)  N ( g ( ), ( ))
p ( , m)  N (  ,  )
Specify priors
p ( y | m)   p ( y |  , m) p ( )d
Invert model
p ( | y, m) 
p ( y |  , m) p ( , m)
p ( y | m)
Make inferences
VB in a nutshell (mean-field approximation)
 Neg. free-energy
approx. to model
evidence.
 Mean field approx.
 Maximise neg. free
energy wrt. q =
minimise divergence,
by maximising
variational energies
ln p  y | m   F  KL  q  ,   , p  ,  | y  
F  ln p  y,  ,   q  KL  q  ,   , p  ,  | m  
p  ,  | y   q  ,    q   q   
q    exp  I   exp  ln p  y,  ,  

q     exp  I    exp  ln p  y,  ,  

 Iterative updating of sufficient statistics of approx. posteriors by
gradient ascent.



q ( ) 
q ( )
Generative models
p  | y, m   p  y |  , m  p  | m 
• any DCM = a particular generative model of how the data (may) have been
caused
• modelling = comparing competing hypotheses about the mechanisms
underlying observed data
 model space: a priori definition of hypothesis set is crucial
 model selection: determine the most plausible hypothesis (model),
given the data
 inference on parameters: e.g., evaluate consistency of how model
mechanisms are implemented across subjects
• model selection  model validation!
 model validation requires external criteria (external to the measured data)
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)  log p( y |  , m)
 KL  q   , p  | m  
 KL  q   , p  | y, m  
accounts for both accuracy and
complexity of the model
p(y|m)
p( y | m)   p( y |  , m) p( | m) d
y
all possible datasets
Various approximations, e.g.:
- negative free energy, AIC, BIC
a measure of generalizability
McKay 1992, Neural Comput.
Penny et al. 2004a, 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
No. of
data points
Penny et al. 2004a, NeuroImage
The (negative) free energy approximation
• Under Gaussian assumptions about the posterior (Laplace
approximation):
log p( y | m)
 log p( y |  , m)  KL  q   , p  | m    KL  q   , p  | y, m  
F  log p( y | m)  KL  q   , p  | y, m  
 log p( y |  , m)  KL  q   , p  | m  
accuracy
complexity
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: Since SPM8, only F is used for model selection !
definition of model space
inference on model structure or inference on model parameters?
inference on
individual models or model space partition?
optimal model structure assumed
to be identical across subjects?
yes
FFX BMS
comparison of model
families using
FFX or RFX BMS
inference on
parameters of an optimal model or parameters of all models?
optimal model structure assumed
to be identical across subjects?
yes
no
FFX BMS
RFX BMS
no
RFX BMS
Stephan et al. 2010, NeuroImage
FFX analysis of
parameter estimates
(e.g. BPA)
RFX analysis of
parameter estimates
(e.g. t-test, ANOVA)
BMA
Random effects BMS for heterogeneous groups

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 r
Multinomial distribution of model labels m
Model inversion
by Variational
Bayes or MCMC
Measured data y
Stephan et al. 2009a, NeuroImage
Penny et al. 2010, PLoS Comput. Biol.
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





• By default, γ is chosen as zero ("does the effect exist?").
Inference about DCM parameters:
Random effects 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 model 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
Bayesian Model Averaging (BMA)
• uses the entire model space
considered (or an optimal family of
models)
• averages parameter estimates,
weighted by posterior model
probabilities
• particularly useful alternative when
p  n | y1.. N 
  p  n | yn , m  p  m | y1.. N 
m
NB: p(m|y1..N) can be obtained by
either FFX or RFX BMS
– none of the models (subspaces)
considered clearly outperforms all
others
– when comparing groups for which
the optimal model differs
Penny et al. 2010, PLoS Comput. Biol.
definition of model space
inference on model structure or inference on model parameters?
inference on
individual models or model space partition?
optimal model structure assumed
to be identical across subjects?
yes
FFX BMS
comparison of model
families using
FFX or RFX BMS
inference on
parameters of an optimal model or parameters of all models?
optimal model structure assumed
to be identical across subjects?
yes
no
FFX BMS
RFX BMS
no
RFX BMS
Stephan et al. 2010, NeuroImage
FFX analysis of
parameter estimates
(e.g. BPA)
RFX analysis of
parameter estimates
(e.g. t-test, ANOVA)
BMA
Overview
• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models into DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions
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 parameterization (e.g., self-connections, hemodynamic states
in log space)
– change in priors to accommodate 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.
The release number is stored in the DCM.mat file.
y


y
BOLD
y

activity
x2(t)
neuronal
states
t
Neural state equation
endogenous
connectivity
The classical DCM:
a deterministic, one-state,
bilinear model
hemodynamic
model
x
integration
modulatory
input u2(t)
t
λ
activity
x3(t)
activity
x1(t)
driving
input u1(t)
y
modulation of
connectivity
direct inputs
x  ( A   u j B( j ) ) x  Cu
x
x
 x

u j x
A
B( j)
x
C
u
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
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
0.1
0
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
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)
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
4
3
m
n

dx 
(i )
( j)
  A   ui B   x j D  x  Cu
dt 
i 1
j 1

2
1
0
-1
3
2
1
Stephan et al. 2008, NeuroImage
0
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
Two-state DCM
Single-state DCM
Two-state DCM
input
u
x1E
x1E
x1
x1I
x1I
x  x  Cu
ij  ij exp( Aij  uBij )
x  x  Cu
ij  Aij  uBij
 11  1N 
    
 
 N 1   NN 
Marreiros et al. 2008, NeuroImage
 x1 
x    
 x N 
EE
11
 IE
 11
 
 EE
 N 1
 0

EI
11
 1EEN
II
11
0

0
EE
NN
0
 IE
NN
Extrinsic
(between-region)
coupling
0 

0 
 

EE
NN 
IINN 
Intrinsic
(within-region)
coupling
 x1E 
 I
 x1 
x  
 E
 xN 
xI 
 N
Estimates of hidden causes and states
(Generalised filtering)
Stochastic DCM
inputs or causes - V2
1
dx
 ( A   j u j B ( j ) ) x  Cv   ( x )
dt
v  u   (v)
0.5
0
-0.5
-1
0
200
400
600
800
1000
hidden states - neuronal
0.1
excitatory
signal
0.05
• 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
1200
0
-0.05
-0.1
0
200
400
600
800
1000
1200
hidden states - hemodynamic
1.3
flow
volume
dHb
1.2
1.1
1
0.9
0.8
0
200
400
600
800
1000
1200
predicted BOLD signal
2
observed
predicted
1
0
-1
-2
Li et al. 2011, NeuroImage
-3
0
200
400
600
time (seconds)
800
1000
1200
Overview
• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions
Prediction errors drive synaptic plasticity
PE(t)

x3
R
x1
a0
x2
McLaren 1989
t
wt  w0    PEk
k 1
t
wt  a0    PE ( )d
0
synaptic plasticity during learning = f (prediction error)
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
prior on volatility
volatility
pk   1
k
vt-1
pvt 1 | vt , k  ~ N vt , exp( k ) 
vt
probabilistic association
rt
rt+1
observed events
ut
ut+1
Behrens et al. 2007, Nat. Neurosci.
prt 1 | rt , vt  ~ Dir rt , exp( vt ) 
Explaining RTs by different learning models
Reaction times
1
True
Bayes Vol
HMM fixed
HMM learn
RW
450
0.8
430
p(F)
RT (ms)
440
420
0.6
0.4
410
400
390
0.2
0.1
0.3
0.5
0.7
0.9
p(outcome)
0
400
0.7
• Rescorla-Wagner
• Hidden Markov models
(2 variants)
520
560
600
Bayesian model selection:
0.6
Exceedance prob.
• hierarchical Bayesian learner
480
Trial
5 alternative learning models:
• categorical probabilities
440
hierarchical Bayesian model
performs best
0.5
0.4
0.3
0.2
0.1
0
Categorical
model
den Ouden et al. 2010, J. Neurosci.
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 active
sensory learning
PE during incidental
sensory learning
p < 0.05
(SVC)
PE during
reinforcement learning
O'Doherty et al. 2004,
Science
den Ouden et al. 2009,
Cerebral Cortex
PE = “teaching signal” for
synaptic plasticity during
learning
Could the putamen be regulating trial-by-trial changes of
task-relevant connections?
Hierarchical
Bayesian
learning model
Prediction errors control plasticity
during adaptive cognition
• Influence of visual
areas on premotor
cortex:
PUT
– stronger for
surprising stimuli
– weaker for expected
stimuli
PMd
PPA
den Ouden et al. 2010, J. Neurosci .
p = 0.017
p = 0.010
FFA
Overview
• Generative models & analysis options
• 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
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
Proof of
concept
study
probabilistic
tractography
FG
34  6.5%
FG
left
FG
FG
right
24  43.6%
13  15.7%
LG
left
LG
LG
12  34.2%
LG
right
 anatomical
connectivity 
 DCM
 connectionspecific priors
for coupling
parameters
  6.5%
v  0.0384
2
1.8
  15.7%
1.6
v  0.1070
1.4
1.2
1
0.8
0.6
  34.2%
  43.6%
v  0.5268
v  0.7746
0.4
Stephan, Tittgemeyer et al.
2009, NeuroImage
0.2
0
-3
-2
-1
0
1
2
3
Connection-specific prior variance  as a function of
anatomical connection probability 
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
ij 
1  0 exp(  ij )
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
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 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
1
1
1
1
1
1
1
1
1
• 64 different mappings
by systematic search
across hyperparameters  and 
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 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
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 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
1
1
1
1
1
1
1
1
1
0.5
• yields anatomically
informed (intuitive and
counterintuitive) and
uninformed priors
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 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
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
0
0
0
0
0
0
0
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
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 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
1
1
1
1
1
1
1
1
1
0
0
0.5
1
0
0
0.5
1
0
0
0.5
1
0
0
0.5
1
0
0
0.5
1
0
0
0.5
1
0
0
0.5
1
0
0.5
0
0.5
1
0
0
0.5
1
log group Bayes factor
600
400
200
log group Bayes factor
0
0
10
20
30
model
40
50
60
0
10
20
30
model
40
50
60
40
50
60
700
695
690
685
680
post. model prob.
0.6
0.5
0.4
0.3
Models with anatomically informed
priors (of an intuitive form)
0.2
0.1
0
0
10
20
30
model
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
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 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
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 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
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 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
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 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
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 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
1
1
1
1
1
1
1
1
1
0.5
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0.5
0
0.5
1
0
0
0.5
Models with anatomically informed priors (of an intuitive form) were
clearly superior to anatomically uninformed ones: Bayes Factor >109
1
Overview
• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions
Model-based predictions for single patients
model structure
BMS
set of
parameter estimates
model-based decoding
BMS: Parkison‘s disease and treatment
Age-matched
controls
Rowe et al. 2010,
NeuroImage
PD patients
on medication
Selection of action modulates
connections between PFC and SMA
PD patients
off medication
DA-dependent functional disconnection
of the SMA
Model-based decoding by generative embedding
step 1 —
model inversion
step 2 —
kernel construction
A
B
C
measurements from
an individual subject
A
subject-specific
inverted generative model
B
C
Brodersen et al. 2011, PLoS Comput. Biol.
subject representation in the
generative score space
step 3 —
support vector classification
step 4 —
interpretation
jointly discriminative
model parameters
A→B
A→C
B→B
B→C
separating hyperplane fitted to
discriminate between groups
Discovering remote or “hidden” brain lesions
Discovering remote or “hidden” brain lesions
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
PT
PT
HG
(A1)
HG
(A1)
MGB
MGB
S
S
Brodersen et al. 2011, PLoS Comput. Biol.
Model-based decoding of disease status:
aphasic patients (N=11) vs. controls (N=26)
Classification accuracy
PT
PT
HG
(A1)
HG
(A1)
MGB
auditory stimuli
Brodersen et al. 2011, PLoS Comput. Biol.
MGB
Generative embedding
using DCM
Multivariate searchlight
classification analysis
Generative score space
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
-0.3
0
0
patients
-0.35
controls
-0.1
-0.5
-0.1
-0.5
0
Voxel (-42,-26,10) mm
-0.15
-0.2
-0.25
-10
0
0.5
10
-0.15
-10
-0.4
-0.4
0
0
Voxel (-56,-20,10) mm
0.5 10
-0.2
L.HG  L.HG
generative
embedding
Voxel (64,-24,4) mm
Voxel-based feature space
-0.25
-0.3
-0.35
0.5
-0.4
-0.4
-0.2
-0.2
0 -0.5
L.MGB  L.MGB
0
0
-0.5
0.5
0
R.HG  L.HG
Definition of ROIs
Are regions of interest defined
anatomically or functionally?
anatomically
A
1 ROI definition
and n model inversions
unbiased estimate
functionally
Functional contrasts
Are the functional contrasts defined
across all subjects or between groups?
across
subjects

between
groups
B
1 ROI definition and n model inversions
slightly optimistic estimate:
voxel selection for training set and test set
based on test data
D

1 ROI definition and n model inversions
highly optimistic estimate:
voxel selection for training set and test set
based on test data and test labels
C
Repeat n times:
1 ROI definition and n model inversions
unbiased estimate
E

Repeat n times:
1 ROI definition and 1 model inversion
slightly optimistic estimate:
voxel selection for training set based on test
data and test labels
F
Repeat n times:
1 ROI definition and n model inversions
unbiased estimate
Brodersen et al. 2011, PLoS Comput. Biol.
Generative embedding for detecting patient subgroups
Brodersen et al. 2014, NeuroImage: Clinical
Generative embedding of variational
Gaussian Mixture Models
Supervised:
SVM classification
Unsupervised:
GMM clustering
71%
number of clusters
number of clusters
• 42 controls vs. 41 schizophrenic patients
• fMRI data from working memory task (Deserno et al. 2012, J. Neurosci)
Brodersen et al. 2014, NeuroImage: Clinical
Detecting subgroups of patients in
schizophrenia
•
three distinct subgroups (total N=41)
•
subgroups differ (p < 0.05) wrt. negative symptoms
on the positive and negative symptom scale (PANSS)
Brodersen et al. 2014, NeuroImage: Clinical
Optimal
cluster
solution
TAPAS
www.translationalneuromodeling.org/tapas
• PhysIO: physiological
noise correction
Kasper et al., in prep.
• Hierarchical Gaussian
Filter (HGF)
• Variational Bayesian
linear regression
Brodersen et al. 2011, PLoS CB
Mathys et al. 2011,
Front. Hum. Neurosci.
• Mixed effects inference
for classification
studies
Brodersen et al. 2012, J Mach Learning Res
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.
•
Brodersen KH, Deserno L, Schlagenhauf F, Lin Z, Penny WD, Buhmann JM, Stephan KE (2014) Dissecting
psychiatric spectrum disorders by generative embedding. NeuroImage: Clinical 4: 98-111
•
Daunizeau J, David, O, Stephan KE (2011) Dynamic Causal Modelling: A critical review of the biophysical and
statistical foundations. NeuroImage 58: 312-322.
•
Daunizeau J, Stephan KE, Friston KJ (2012) Stochastic Dynamic Causal Modelling of fMRI data: Should we care about
neural noise? NeuroImage 62: 464-481.
•
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.
•
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 22:11571172.
•
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.
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