Post on 27-Dec-2015
transcript
DCM Advanced, Part II
Will Penny (Klaas Stephan)
Wellcome Trust Centre for NeuroimagingInstitute of NeurologyUniversity College London
SPM Course 2014 @ FIL
Overview
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications
endogenous connectivity
direct inputs
modulation ofconnectivity
Neural state equation CuxBuAx jj )( )(
u
xC
x
x
uB
x
xA
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 DCM• Three dimensions of model specification:
– bilinear vs. nonlinear
– single-state vs. two-state (per region)
– deterministic vs. stochastic
• Specification via GUI.
bilinear DCM
CuxDxBuAdt
dx m
i
n
j
jj
ii
1 1
)()(CuxBuA
dt
dx m
i
ii
1
)(
Bilinear state equation:
driving input
modulation
driving input
modulation
non-linear DCM
...)0,(),(2
0
uxux
fu
u
fx
x
fxfuxf
dt
dx
Two-dimensional Taylor series (around x0=0, u0=0):
Nonlinear state equation:
...2
)0,(),(2
2
22
0
x
x
fux
ux
fu
u
fx
x
fxfuxf
dt
dx
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
Neural population 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
fMRI signal change (%)
x1 x2
x3
CuxDxBuAdt
dx n
j
jj
m
i
ii
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
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
uinput
Single-state DCM
1x
Intrinsic (within-region)
coupling
Extrinsic (between-region)
coupling
NNNN
N
ijijij
x
x
x
uBA
Cuxx
1
1
111
Two-state DCM
Ex1
IN
EN
I
E
IINN
IENN
EENN
EENN
EEN
IIIE
EEN
EIEE
ijijijij
x
x
x
x
x
uBA
Cuxx
1
1
1
1111
11111
00
0
00
0
)exp(
Ix1
I
E
x
x
1
1
Two-state DCM
Marreiros et al. 2008, NeuroImage
Estimates of hidden causes and states(Generalised filtering)
0 200 400 600 800 1000 1200-1
-0.5
0
0.5
1inputs or causes - V2
0 200 400 600 800 1000 1200-0.1
-0.05
0
0.05
0.1hidden states - neuronal
0 200 400 600 800 1000 12000.8
0.9
1
1.1
1.2
1.3hidden states - hemodynamic
0 200 400 600 800 1000 1200-3
-2
-1
0
1
2predicted BOLD signal
time (seconds)
excitatorysignal
flowvolumedHb
observedpredicted
Stochastic DCM
( ) ( )
( )
( )j xjj
v
dxA u B x Cv
dt
v u
Li et al. 2011, NeuroImage
• random state fluctuations w(x) account for endogenous fluctuations,
• fluctuations w(v) induce uncertainty about how inputs influence neuronal activity
• can be fitted to resting state data
Estimates of hidden causes and states(Generalised filtering)
0 200 400 600 800 1000 1200-1
-0.5
0
0.5
1inputs or causes - V2
0 200 400 600 800 1000 1200-0.1
-0.05
0
0.05
0.1hidden states - neuronal
0 200 400 600 800 1000 12000.8
0.9
1
1.1
1.2
1.3hidden states - hemodynamic
0 200 400 600 800 1000 1200-3
-2
-1
0
1
2predicted BOLD signal
time (seconds)
excitatorysignal
flowvolumedHb
observedpredicted
Stochastic DCM
( ) ( )
( )
( )j xjj
v
dxA u B x Cv
dt
v u
• Good working knowledge of dDCM
• sDCMs (esp. for nonlinear models) can have richer dynamics than dDCM
• Model selection may be easier than with dDCM
• See Daunizeau et al. ‘sDCM: Should we care about neuronal noise ?’, Neuroimage, 2012
Overview
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications
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.
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
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.
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
400
410
420
430
440
450
RT
(m
s)
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
-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 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
Overview
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications
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 → B
A → C
B → B
B → C
A
CB
step 3 —support vector classification
separating hyperplane fitted to discriminate between groups
A
CB
jointly discriminativemodel parameters
step 4 —interpretation
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: aphasic patients (N=11) vs. controls (N=26)
Classification accuracy
Brodersen et al. 2011, PLoS Comput. Biol.
MGB
PT
HG(A1)
MGB
PT
HG(A1)
auditory stimuli
-10
0
10
-0.5
0
0.5
-0.1
0
0.1
0.2
0.3
0.4
-0.4-0.2
0 -0.5
0
0.5-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-10
0
10
-0.5
0
0.5
-0.1
0
0.1
0.2
0.3
0.4
-0.4-0.2
0 -0.5
0
0.5-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
gen
erative em
bedd
ing
L.H
G
L.H
G
Vo
xel (
64,-
24,4
) mm
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
Summary
• Model Selection
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Clinical Applications