j. daunizeau wellcome trust centre for neuroimaging, london, uk uzh – foundations of human social...
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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
Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
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
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
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
Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
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
Neural ensembles dynamicsfrom micro- to meso-scale: mean-field treatment
( ) ( ) ( ) ( )'
'
( )
1
1Ni i i i
jj j
i
H x t H x t p x t dxN
S
( )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)
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
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
Neural ensembles dynamicsfrom meso- to macro-scale: neural fields
22 2 2 ( ) ( )
2
( ) ( ')'
'
32 , ,
2i i
i iii
i
c t c tt t
S
r r
loca
l (ho
mog
ene
ous)
den
sity
of
conn
exi
ons
local wave propagation equation:
r1 r2
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
(( ) ( )) 2
(( ) ( ) ) 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
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
, ,
,
x f x u
y g x
likelihood function?
Neural ensembles dynamicsthe observation function
, ,p y m
Overview
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
Bayesian inferenceforward and inverse problems
,p y m
forward problem
likelihood
,p y m
inverse problem
posterior distribution
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
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
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
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
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
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
1 DCM: introduction
2 Neural ensembles dynamics
3 Bayesian inference
4 Conclusion
Overview
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
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
Many thanks to:
Karl J. Friston (London, UK)Rosalyn Moran (London, UK)
Stefan J. Kiebel (Leipzig, Germany)
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