j. daunizeau motivation, brain and behaviour group, icm, paris, france wellcome trust centre for...
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![Page 1: J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK Dynamic Causal Modelling for](https://reader033.vdocuments.mx/reader033/viewer/2022052603/56649dbc5503460f94aae7c0/html5/thumbnails/1.jpg)
J. Daunizeau
Motivation, Brain and Behaviour group, ICM, Paris, France
Wellcome Trust Centre for Neuroimaging, London, UK
Dynamic Causal Modelling for EEG/MEG:principles
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Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
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Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 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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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? »
?
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( , , )x f x u neural states dynamics
Electromagneticobservation model:spatial convolution
• simple neuronal model• realistic observation model
• realistic neuronal model• simple observation model
fMRIfMRI EEG/MEGEEG/MEG
inputs
IntroductionDCM: evolution and observation mappings
Hemodynamicobservation model:temporal convolution
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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 ,E y m
, ,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
Daunizeau, Kiebel et al., Neuroimage, 2009
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response
or or
time (ms)0 200 400 600 800 2000
Put
PPA FFA
PMd
P(outcome|cue)
PMdPutPPAFFA
auditory cue visual outcome
cue-independent surprise
cue-dependent surprise
Den Ouden, Daunizeau et al., J. Neurosci., 2010
IntroductionDCM for fMRI: audio-visual associative learning
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Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
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1 2
31 2
31 2
3
time
0 ?tx
t
0t
t t
t t
t
Dynamical systems theorymotivation
1 2
3
32
21
13
u
13u 3
u
u x y
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Dynamical systems theoryexponentials
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Dynamical systems theoryinitial values and fixed points
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Dynamical systems theorytime constants
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Dynamical systems theorymatrix exponential
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Dynamical systems theoryeigendecomposition of the Jacobian
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Dynamical systems theorydynamical modes
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Dynamical systems theoryspirals
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Dynamical systems theoryspirals
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Dynamical systems theoryspiral state-space
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Dynamical systems theoryembedding
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Dynamical systems theorykernels and convolution
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Dynamical systems theorysummary
• Motivation: modelling reciprocal influences
• System stability and dynamical modes can be derived from the system’s Jacobian:o D>0: fixed pointso D>1: spiralso D>1: limit cycles (e.g., action potentials)o D>2: metastability (e.g., winnerless competition)
• Link between the integral (convolution) and differential (ODE) forms
limit cycle (Vand Der Pol) strange attractor (Lorenz)
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Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
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Neural ensembles dynamicsDCM for M/EEG: systems 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 dynamics DCM for M/EEG: from micro- to meso-scale
''
1
1N
jj 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
(H
z)
membrane depolarization (mV)
ense
mbl
e de
nsi
ty p(
x)
S(x)
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Neural ensembles dynamics DCM for M/EEG: synaptic 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 dynamics DCM for M/EEG: 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
S
x
S
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Neural ensembles dynamics DCM for M/EEG: from meso- to macro-scale
22 2 2 ( ) ( )
2
( ) ( ')'
'
32 , ,
2i i
i iii
i
c t c tt t
S
r r
late
ral (
hom
ogen
eous
)de
nsi
ty o
f co
nne
xio
ns
local wave propagation equation (neural field):
r1 r2
( ) ( ) ,i it t r
0th-order approximation: standing wave
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Neural ensembles dynamics DCM for M/EEG: 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 S
x
S
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Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 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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( ) ( ) ( )0
i i ijj
i j
t t t y L w
Bayesian inferencethe electromagnetic forward problem
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Bayesian paradigmderiving the likelihood function
- Model of data with unknown parameters:
y f e.g., GLM: f X
- But data is noisy: y f
- Assume noise/residuals is ‘small’:
22
1exp
2p
4 0.05P
→ Distribution of data, given fixed parameters:
2
2
1exp
2p y y f
f
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Likelihood:
Prior:
Bayes rule:
Bayesian paradigmlikelihood, priors and the model evidence
generative model m
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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
mean-field: approximate marginal posterior distributions: 1 2,q q
Bayesian inferencethe variational Bayesian approach
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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
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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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Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
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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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• Suitable experimental design:
– any design that is suitable for a GLM
– preferably multi-factorial (e.g. 2 x 2)
• e.g. one factor that varies the driving (sensory) input
• and one factor that varies the modulatory input
• Hypothesis and model:
– define specific a priori hypothesis
– which models are relevant to test this hypothesis?
– check existence of effect on data features of interest
– there exists formal methods for optimizing the experimental design
for the ensuing bayesian model comparison
[Daunizeau et al., PLoS Comp. Biol., 2011]
Conclusionplanning a compatible DCM study
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Many thanks to:
Karl J. Friston (UCL, London, UK)Will D. Penny (UCL, London, UK)
Klaas E. Stephan (UZH, Zurich, Switzerland)Stefan Kiebel (MPI, Leipzig, Germany)