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DCM: Advanced issues. Klaas Enno Stephan Centre for the Study of Social & Neural Systems Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London. SPM Course 2008 Zurich. - PowerPoint PPT Presentation
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DCM: Advanced issues
Klaas Enno Stephan
Centre for the Study of Social & Neural SystemsInstitute for Empirical Research in EconomicsUniversity of Zurich
Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London
SPM Course 2008Zurich
intrinsic connectivity
direct inputs
modulation ofconnectivity
Neural state equation CuzBuAz jj )( )(
u
zC
z
z
uB
z
zA
j
j
)(
hemodynamicmodelλ
z
y
integration
BOLDyyy
activityz1(t)
activityz2(t) activity
z3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
Stephan & Friston (2007),Handbook of Connectivity
Overview
• Bayesian model selection (BMS)
• Timing errors & sampling accuracy
• The hemodynamic model in DCM
• Advanced DCM formulations for fMRI
– two-state DCMs
– nonlinear DCMs
• An outlook to the future
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
dmpmypmyp )|(),|()|( Model evidence:
Bayesian model selection (BMS)
)|(
)|(),|(),|(
myp
mpmypmyp
Bayes’ rule:
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability)of the model
integral usually not analytically solvable, approximations necessary (e.g. AIC or BIC)
dmpmypmyp )|(),|()|(
Model evidence p(y|m)Gharamani, 2004
p(y
|m
)
all possible datasets y
a specific y
Balance between fit and complexity
Generalisability of the model
Model evidence: probability of generating data y from parameters that are randomly sampled from the prior p(m).
Maximum likelihood: probability of the data y for the specific parameter vector that maximises p(y|,m).
pmypAIC ),|(log
Logarithm is a monotonic function
Maximizing log model evidence= Maximizing model evidence
)(),|(log
)()( )|(log
mcomplexitymyp
mcomplexitymaccuracymyp
At the moment, two approximations available in SPM interface:
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.
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 can 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
Raftery classification:
AIC:
BF = 3.3
BIC:
BF = 3.3
BMS result:
BF = 3.3
Two models with identical numbers of parameters
AIC:
BF = 0.1
BIC:
BF = 0.7
BMS result:
BF = 0.7
Two models with different numbers of parameters
&
compatible AIC/BIC based decisions about models
AIC:
BF = 0.3
BIC:
BF = 2.2
BMS result:
“AIC and BIC disagree about which model is superior - no decision can be made.”
Two models with different numbers of parameters
&
incompatible AIC/BIC based decisions about models
Further reading on BMS of DCMs
• Theoretical papers:– Penny et al. (2004) Comparing dynamic causal models. NeuroImage 22:
1157-1172.
– Stephan et al. (2007) Comparing hemodynamic models with DCM. NeuroImage 38: 387-401.
• Applications of BMS & DCM (selection):– Grol et al. (2007) Parieto-frontal connectivity during visually-guided
grasping. J. Neurosci. 27: 11877-11887.
– Kumar et al. (2007) Hierarchical processing of auditory objects in humans. PLoS Computat. Biol. 3: e100.
– Smith et al. (2006) Task and content modulate amygdala-hippocampal connectivity in emotional retrieval. Neuron 49: 631-638.
– Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27: 3512-3522.
Overview
• Bayesian model selection (BMS)
• Timing errors & sampling accuracy
• The hemodynamic model in DCM
• Advanced DCM formulations for fMRI
– two-state DCMs
– nonlinear DCMs
• An outlook to the future
Timing problems at long TRs/TAs
• Two potential timing problems in DCM:
1. wrong timing of inputs2. temporal shift between
regional time series because of multi-slice acquisition
• DCM is robust against timing errors up to approx. ± 1 s – compensatory changes of σ and θh
• Possible corrections:– slice-timing (not for long TAs)– restriction of the model to neighbouring regions– in both cases: adjust temporal reference bin in SPM
defaults (defaults.stats.fmri.t0)
1
2
slic
e a
cquis
itio
n
visualinput
Slice timing in DCM: three-level model
),,( hhzzgv
),( Tvhx
),,( uzfz n
3rd level
2nd level
1st level
sampled BOLD response
BOLD response
neuronal response
z = neuronal states u = inputszh = hemodynamic states v = BOLD responsesn, h = neuronal and hemodynamic parameters T = sampling time points
Kiebel et al. 2007, NeuroImage
Slice timing in DCM: an example
t
1 TR 2 TR 3 TR 4 TR 5 TR
t
1 TR 2 TR 3 TR 4 TR 5 TR
OriginalDCM
PresentDCM
1T
2T1T
2T1T
2T1T
2T1T
2T
1T 1T 1T 1T 1T2T 2T 2T 2T 2T
Overview
• Bayesian model selection (BMS)
• Timing errors & sampling accuracy
• The hemodynamic model in DCM
• Advanced DCM formulations for fMRI
– two-state DCMs
– nonlinear DCMs
• An outlook to the future
LGleft
LGright
RVF LVF
FGright
FGleft
Example: BOLD signal modelled with DCM
black: measured BOLD signalred: predicted BOLD signal
sf
tionflow induc
(rCBF)
s
v
stimulus functions
v
q q/vvEf,EEfqτ /α
dHbchanges in
100 )( /αvfvτ
volumechanges in
1
f
q
)1(
fγsxs
signalryvasodilato
u
s
CuxBuAdt
dx m
j
jj
1
)(
t
neural state equation
1
3.4
111),(
3
002
001
32100
k
TEErk
TEEk
vkv
qkqkV
S
Svq
hemodynamic state equationsf
Balloon model
BOLD signal change equation
},,,,,{ h},,,,,{ h
important for model fitting, but of no interest for statistical inference
• 6 hemodynamic parameters:
• Empirically determineda priori distributions.
• Computed separately for each area (like the neural parameters) region-specific HRFs!
The hemodynamic model in DCM
Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage
Recent changes in the hemodynamic model
(Stephan et al. 2007, NeuroImage)
• new output non-linearity, based on new exp. data and mathematical derivations
less problematic to apply DCM to high-field fMRI data
• field-dependency of output coefficients is handled better, e.g. by estimating intra-/extravascular BOLD signal ratio
BMS indicates that new model performs better than original Buxton model
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
r,Br,A r,C
A
B
C
h
ε
How independent are our neural and hemodynamic parameter estimates?
Stephan et al. 2007, NeuroImage
Overview
• Bayesian model selection (BMS)
• Timing errors & sampling accuracy
• The hemodynamic model in DCM
• Advanced DCM formulations for fMRI
– two-state DCMs
– nonlinear DCMs
• An outlook to the future
)(tu
ijij uBA
input
Single-state DCM
1x
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
NNNN
N
x
x
tx
AA
AA
A
CuxuBAt
x
1
1
111
)(
)(
Two-state DCM
Ex1
IN
EN
I
E
AA
AAA
AA
AAA
u
x
x
x
x
tx
ee
eee
ee
eee
A
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x
IINN
IENN
EINN
EENNN
IIIE
NEIEE
1
1
)(
00
0
00
0
)(
1
1111
11111
)exp( ijij uBA
Ix1
IEx ,1
Marreiros et al. 2008, NeuroImage
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
non-linear DCM
driving input
modulation
...)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
BOLD signal change (%)
x1 x2u1
x3
u2
–
– –
++
++++
+++
+++
+
2
1
32
11
3
2
1)3(
213
3332
232221
1211
0
0
0
0
000
00
000
0
0
u
u
c
c
x
x
x
dx
aa
aaa
aa
dt
dx
Neuronal state equation:
Stephan et al., submitted
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., submitted
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
A B
Stephan et al., submitted
V1
V5PPC
observedfitted
motion &attention
motion &no attention
static dots
Stephan et al., submitted
Overview
• Bayesian model selection (BMS)
• Timing errors & sampling accuracy
• The hemodynamic model in DCM
• Advanced DCM formulations for fMRI
– two-state DCMs
– nonlinear DCMs
• An outlook to the future
),,( uxFx Neural state equation:
Electric/magneticforward model:
neural activityEEGMEGLFP
(linear)
DCM: generative model for fMRI and ERPs
Neural model:1 state variable per regionbilinear state equationno propagation delays
Neural model:8 state variables per region
nonlinear state equationpropagation delays
fMRIfMRI ERPsERPs
inputs
Hemodynamicforward model:neural activityBOLD(nonlinear)
Neural mass model of a cortical macrocolumn
ExcitatoryInterneurons
He, e
PyramidalCells
He, e
InhibitoryInterneurons
Hi, e
Extrinsic inputs
Excitatory connection
Inhibitory connection
e, i : synaptic time constant (excitatory and inhibitory) He, Hi: synaptic efficacy (excitatory and inhibitory) 1,…,: intrinsic connection strengths propagation delays
21
43
MEG/EEGsignal
MEG/EEGsignal
Parameters:
Parameters:
Jansen & Rit (1995) Biol. Cybern.David et al. (2006) NeuroImage
mean firing rate
mean postsynaptic
potential (PSP)
mean PSP
mean firing rate
236
746
63
225
1205
52
650
214
014
41
278
038
87
2)(
2))()()((
2))()((
2))()((
iii
i
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LB
e
e
ee
LF
e
e
ee
LB
e
e
xxxS
Hx
xx
xxxSxSAA
Hx
xx
xxx
xxCuxSIAA
Hx
xx
xxxSIAA
Hx
xx
spiny stellate
cells
inhibitory interneurons
pyramidal cells
4 3
1 2)( 0xSAF
)( 0xSAL
)( 0xSAB
Extrinsicforward
connections
Extrinsic backward connections
Intrinsic connections
neuronal (source) model
Extrinsic lateral connections
State equations ,,uxFx
DCM for ERPs: neural state equations
David et al. (2006) NeuroImage
MEG/EEGsignal
MEG/EEGsignal
mV
Inhibitory cells in supra/infragranular layers
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in supra/infragranular layers
activity
DCM for LFPs
• extended neural mass models that can be fitted to LFP data (both frequency spectra and ERPs)
• explicit model of spike-frequency adaptation (SFA)
• current validation work to establish the sensitivity of various parameters wrt. specific neurotransmitter effects
• validation of this model by LFP recordings in rats, combined with pharmacological manipulations
Moran et al. (2007, 2008) NeuroImage
standards deviants
A1
A2
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