J. Daunizeau Motivation, Brain and Behaviour group, ICM, Paris, France Wellcome Trust Centre for...
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J. Daunizeau
Motivation, Brain and Behaviour group, ICM, Paris, FranceWellcome 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
fMRI EEG/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
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
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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 ?txt
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
''
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)
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Neural ensembles dynamics DCM for M/EEG: synaptic dynamics
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
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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
Sx
S
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Neural ensembles dynamics DCM for M/EEG: 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
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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 Sx
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
1exp2
p
4 0.05P
→ Distribution of data, given fixed parameters:
2
2
1exp2
p 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” :
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
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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
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
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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
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
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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)