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Dynamic Causal Modelling for event-related responses. J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK. Overview. DCM: introduction Neural ensembles dynamics Bayesian inference Example Conclusion. Overview. - PowerPoint PPT Presentation
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J. Daunizeau
Motivation, Brain and Behaviour group, ICM, Paris, FranceWellcome Trust Centre for Neuroimaging, London, UK
Dynamic Causal Modellingfor event-related responses
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. 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
u1 u1 X u2
A B
u2u1
A B
u2u1
localizing brain activity:functional segregation
effective connectivity analysis:functional integration
Introductionfrom functional segregation to functional integration
« Where, in the brain, didmy experimental manipulation
have an effect? »
« How did my experimental manipulationpropagate through the network? »
1 2
31 2
31 2
3
time
0 ?txt
0t
t t
t t
t
Introductionwhy do we use dynamical system theory?
1 2
3
32
21
13
u
13u 3
u
u x y
( , , )x f x u neural states dynamics
Electromagneticobservation model:spatial convolution
• simple neuronal model• slow time scale
• complicated neuronal model• fast time scale
fMRI EEG/MEG
inputs
IntroductionDCM: evolution and observation mappings
Hemodynamicobservation model:temporal convolution
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
rSTGrA1
lSTGlA1
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
Daunizeau, Kiebel et al., Neuroimage, 2009
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Neural ensembles dynamicsmulti-scale perspective
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 dynamics from micro- to meso-scale
''
11
Nj
j j
H x t H x t p x t dxN
S
S(x) H(x)
: 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
(Hz)
membrane depolarization (mV)
ense
mbl
e de
nsity
p(x
)
S(x)
Neural ensembles dynamics synaptic kinematics
1iK
time (ms)m
embr
ane
depo
lariz
atio
n (m
V)
1 2
2 22 / / 2 / 1( ) 2i e i e i eS
post-synaptic potential
1eK
IPSP
EPSP
Neural ensembles dynamics intrinsic 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
Sx
S
Neural ensembles dynamics from meso- to macro-scale
22 2 2 ( ) ( )
2
( ) ( ')'
'
32 , ,2
i i
i iii
i
c t c tt t
S
r r
late
ral (
hom
ogen
eous
)de
nsity
of c
onne
xion
s
local wave propagation equation (neural field):
r1 r2
( ) ( ) ,i it t r
0th-order approximation: standing wave
Neural ensembles dynamics extrinsic 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 Sx
S
Observation mappingsthe electromagnetic forward model
( ) ( ) ( )0
i i ijj
i j
t t t y L w 0, yt N Q
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. 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” :
mod
el e
vide
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
mean-field: approximate marginal posterior distributions: 1 2,q q
Bayesian inferencethe variational Bayesian approach
Evoked responsesSpecify generative forward model
(with prior distributions of parameters)
Expectation-Maximization algorithm
Iterative procedure: 1. Compute model response using current set of parameters
2. Compare model response with data3. Improve parameters, if possible
1. Posterior distributions of parameters
2. Model evidence )|( myp
),|( myp
Bayesian inferenceEM in a nutshell
1 2
3
32
21
13
u
13u 3
u
Bayesian inferenceDCM: key 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
diffe
renc
es in
log-
mod
el e
vide
nces
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
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Examplemodels for deviant response generation
Garrido et al., (2007), NeuroImage
Bayesian Model Comparison
Forward (F)
Backward (B)
Forward and Backward (FB)
subjects
log-evidence
Group level
Garrido et al., (2007), NeuroImage
Examplegroup-level model comparison
Peristimulus time 1
Peristimulus time 2
Do forward and backward connections operate as a function of time?
Models for Deviant Response Generation
ExampleMMN: temporal hypotheses
Garrido et al., PNAS, 2008
Garrido et al., PNAS, 2008
time (ms) time (ms)
ExampleMMN: (best) model fit
Garrido et al., PNAS, 2008
ExampleMMN: group-level model comparison across time
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Conclusionback to the auditory mismatch negativity
S-D: reorganisationof the connectivity structure
rIFG
rSTGrA1
lSTGlA1
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 (FIL, London, UK)Klaas E. Stephan (UZH, Zurich, Switzerland)
Stefan Kiebel (MPI, Leipzig, Germany)