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)