J. Daunizeau Wellcome Trust Centre for Neuroimaging, London, UK
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Transcript of J. Daunizeau Wellcome Trust Centre for Neuroimaging, London, UK
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J. Daunizeau
Wellcome Trust Centre for Neuroimaging, London, UK
UZH – Foundations of Human Social Behaviour, Zurich, Switzerland
Dynamic Causal Modelling:basics
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Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
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Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
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Introductionstructural, functional and effective connectivity
• structural connectivity= presence of axonal connections
• functional connectivity = statistical dependencies between regional time series
• effective connectivity = causal (directed) influences between neuronal populations
! connections are recruited in a context-dependent fashion
O. Sporns 2007, Scholarpedia
structural connectivity functional connectivity effective connectivity
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IntroductionDCM: a parametric statistical approach
,
, ,
y g x
x f x u
• DCM: model structure
1
2
4
3
24
u
, ,p y m likelihood
• DCM: Bayesian inference
ˆ , ,p y m p m p m d d
, ,p y m p y m p m p m d d
model evidence:
parameter estimate:
priors on parameters
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IntroductionDCM for EEG-MEG: auditory mismatch negativity
S-D: reorganisationof the connectivity structure
rIFG
rSTG
rA1
lSTG
lA1
rIFG
rSTG
rA1
lSTG
lA1
standard condition (S)
deviant condition (D)
t ~ 200 ms
……S S S D S S S S D S
sequence of auditory stimuli
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Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
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Neural ensembles dynamicssystems of neural populations
Golgi Nissl
internal granularlayer
internal pyramidallayer
external pyramidallayer
external granularlayer
mean-field firing rate synaptic dynamics
macro-scale meso-scale micro-scale
EP
EI
II
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Neural ensembles dynamicsfrom micro- to meso-scale: mean-field treatment
( ) ( ) ( ) ( )'
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1
1Ni i i i
jj j
i
H x t H x t p x t dxN
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( )iS(x) H(x)
( )i
: post-synaptic potential of j th neuron within its ensemble jx t
mean-field firing rate
mean membrane depolarization (mV)
mea
n fir
ing
rate
(H
z)
membrane depolarization (mV)
ense
mbl
e de
nsi
ty p(
x)
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Neural ensembles dynamicssynaptic dynamics
1iK
time (ms)m
emb
rane
dep
olar
izat
ion
(mV
)1 2
2 22 / / 2 / 1( ) 2i e i e i eS
post-synaptic potential
1eK
IPSP
EPSP
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Neural ensembles dynamicsintrinsic connections within the cortical column
Golgi Nissl
internal granularlayer
internal pyramidallayer
external pyramidallayer
external granularlayer
spiny stellate cells
inhibitory interneurons
pyramidal cells
intrinsic connections 1 2
3 4
7 8
2 28 3 0 8 7
1 4
2 24 1 0 4 1
0 5 6
2 5
2 25 2 1 5 2
3 6
2 26 4 7 6 3
( ) 2
( ) 2
( ) 2
( ) 2
e e e
e e e
e e e
i i i
S
S
S
x
S
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Neural ensembles dynamicsfrom meso- to macro-scale: neural fields
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den
sity
of
conn
exi
ons
local wave propagation equation:
r1 r2
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Neural ensembles dynamicsextrinsic connections between brain regions
extrinsicforward
connections
spiny stellate cells
inhibitory interneurons
pyramidal cells
0( )F S
0( )LS
0( )BS extrinsic backward connections
extrinsiclateral connections
1 2
3 4
7 8
2 28 3 0 8 7
1 4
2 24 1 0 4 1
0 5 6
2 5
2 25 0 2 1 5 2
3 6
2 26 4 7 6 3
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(( ) ( ) ) 2
(( ) ( ) ( )) 2
( ) 2
e B L e e
e F L u e e
e B L e e
i i i
I S
I S u
S S
x
S
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Neural ensembles dynamicssystems of neural populations
Golgi Nissl
internal granularlayer
internal pyramidallayer
external pyramidallayer
external granularlayer
macro-scale meso-scale micro-scale
EP
EI
II
mean-field firing rate synaptic dynamics
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, ,
,
x f x u
y g x
likelihood function?
Neural ensembles dynamicsthe observation function
, ,p y m
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Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
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Bayesian inferenceforward and inverse problems
,p y m
forward problem
likelihood
,p y m
inverse problem
posterior distribution
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generative model m
likelihood
prior
posterior
Bayesian inferencelikelihood and priors
,p y m
p m
,,
p y m p mp y m
p y m
u
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Principle of parsimony :« plurality should not be assumed without necessity »
“Occam’s razor” :
mo
de
l evi
de
nce
p(y
|m)
space of all data sets
y=f(
x)y
= f(
x)
x
Bayesian inferencemodel comparison
Model evidence:
,p y m p y m p m d
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ln ln , ; ,KLqp y m p y m S q D q p y m
free energy : functional of q
1 or 2q
1 or 2 ,p y m
1 2, ,p y m
1
2
approximate (marginal) posterior distributions: 1 2,q q
Bayesian inferencethe variational Bayesian approach
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1 2
31 2
31 2
3
time
, ,x f x u
0t
t t
t t
t
Bayesian inferencea note on causality
1 2
3
32
21
13
u
13u 3
u
u x y
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1 2
3
32
21
13
u
13u 3
u
Bayesian inferencekey model parameters
state-state coupling 21 32 13, ,
3u input-state coupling
13u input-dependent modulatory effect
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Bayesian inferencemodel comparison for group studies
m1
m2
diff
eren
ces
in lo
g- m
ode
l evi
denc
es
1 2ln lnp y m p y m
subjects
fixed effect
random effect
assume all subjects correspond to the same model
assume different subjects might correspond to different models
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1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
Overview
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Conclusionback to the auditory mismatch negativity
S-D: reorganisationof the connectivity structure
rIFG
rSTG
rA1
lSTG
lA1
rIFG
rSTG
rA1
lSTG
lA1
standard condition (S)
deviant condition (D)
t ~ 200 ms
……S S S D S S S S D S
sequence of auditory stimuli
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ConclusionDCM for EEG/MEG: variants
DCM for steady-state responses
second-order mean-field DCM
DCM for induced responses
DCM for phase coupling
0 100 200 3000
50
100
150
200
250
0 100 200 300-100
-80
-60
-40
-20
0
0 100 200 300-100
-80
-60
-40
-20
0
time (ms)
input depolarization
time (ms) time (ms)
auto-spectral densityLA
auto-spectral densityCA1
cross-spectral densityCA1-LA
frequency (Hz) frequency (Hz) frequency (Hz)
1st and 2d order moments
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Many thanks to:
Karl J. Friston (London, UK)Rosalyn Moran (London, UK)
Stefan J. Kiebel (Leipzig, Germany)