J. Daunizeau Wellcome Trust Centre for Neuroimaging, London, UK UZH – Foundations of Human Social...

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

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



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





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





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



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



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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