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)