Transcript DCM - University College London

DCM Advanced, Part II
Will Penny (Klaas Stephan)
Wellcome Trust Centre for Neuroimaging
Institute of Neurology
University College London
SPM Course 2014 @ FIL
Overview
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications
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)
C
x
u
Factorial structure of model specification in DCM
• 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
f
f
2 f
2 f x2
 f ( x, u)  f ( x0 ,0) 
x u
ux  ... 2
 ...
dt
x
u
xu
x 2
Bilinear state equation:
m
dx 
(i ) 
  A   ui B  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
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u2
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u1
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x3
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0
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0.1
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x1
x2
3
fMRI signal change (%)
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0
Nonlinear dynamic causal model (DCM)
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m
n

dx 
(i )
( j)
  A   ui B   x j D  x  Cu
dt 
i 1
j 1

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0
-1
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2
1
Stephan et al. 2008, NeuroImage
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attention
MAP = 1.25
0.10
0.8
0.7
PPC
0.6
0.26
0.5
0.39
1.25
stim
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V1
0.13
0.46
0.50
V5
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0.1
0
-2
motion
Stephan et al. 2008, NeuroImage
-1
0
1
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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    
 xN 
11EE
 IE
 11
  
 EE
 N 1
 0

11EI
11II
 1EE
N
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
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hidden states - neuronal
0.1
excitatory
signal
0.05
0
-0.05
• random state fluctuations w(x) account for
endogenous fluctuations,
• fluctuations w(v) induce uncertainty about
how inputs influence neuronal activity
-0.1
0
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hidden states - hemodynamic
1.3
flow
volume
dHb
1.2
1.1
1
0.9
• can be fitted to resting state data
0.8
0
200
400
600
800
1000
1200
predicted BOLD signal
2
observed
predicted
1
0
-1
Li et al. 2011, NeuroImage
-2
-3
0
200
400
600
time (seconds)
800
1000
1200
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
1200
hidden states - neuronal
0.1
excitatory
signal
0.05
0
• Good working knowledge of dDCM
• sDCMs (esp. for nonlinear models) can
have richer dynamics than dDCM
• Model selection may be easier than with
dDCM
-0.05
-0.1
200
400
600
800
1000
1200
hidden states - hemodynamic
1.3
flow
volume
dHb
1.2
1.1
1
0.9
0.8
• See Daunizeau et al. ‘sDCM: Should we
care about neuronal noise ?’,
Neuroimage, 2012
0
0
200
400
600
800
1000
1200
predicted BOLD signal
2
observed
predicted
1
0
-1
-2
-3
0
200
400
600
time (seconds)
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1000
1200
Overview
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications
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?
Prediction errors control
plasticity during adaptive
cognition
• Modulation of visuomotor connections by
striatal prediction
error activity
Hierarchical
Bayesian
learning model
PUT
PMd
• Influence of visual
areas on premotor
cortex:
– stronger for
surprising stimuli
– weaker for expected
stimuli
den Ouden et al. 2010, J. Neurosci .
p = 0.017
p = 0.010
PPA
FFA
Overview
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications
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
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
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0.2
0.2
0.1
0.1
-0.3
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patients
-0.35
controls
-0.1
-0.5
-0.1
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Voxel (-42,-26,10) mm
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-0.2
-0.25
-10
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10
-0.15
-10
-0.4
-0.4
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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
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-0.4
-0.4
-0.2
-0.2
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L.MGB  L.MGB
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R.HG  L.HG
Summary
• Model Selection
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications