J. Daunizeau

Motivation, Brain and Behaviour group, ICM, Paris, FranceWellcome Trust Centre for Neuroimaging, London, UK

Dynamic Causal Modelling for EEG/MEG:principles

Overview

1 DCM: introduction

2 Dynamical systems theory

3 Neural states dynamics

4 Bayesian inference

5 Conclusion

Overview

1 DCM: introduction

2 Dynamical systems theory

3 Neural states dynamics

4 Bayesian inference

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? »

?

( , , )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

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

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

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

Overview

1 DCM: introduction

2 Dynamical systems theory

3 Neural states dynamics

4 Bayesian inference

5 Conclusion

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

Dynamical systems theoryexponentials

Dynamical systems theoryinitial values and fixed points

Dynamical systems theorytime constants

Dynamical systems theorymatrix exponential

Dynamical systems theoryeigendecomposition of the Jacobian

Dynamical systems theorydynamical modes

Dynamical systems theoryspirals

Dynamical systems theoryspirals

Dynamical systems theoryspiral state-space

Dynamical systems theoryembedding

Dynamical systems theorykernels and convolution

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)

Overview

1 DCM: introduction

2 Dynamical systems theory

3 Neural states dynamics

4 Bayesian inference

5 Conclusion

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

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)

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

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

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

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

Overview

1 DCM: introduction

2 Dynamical systems theory

3 Neural states dynamics

4 Bayesian inference

5 Conclusion

Bayesian inferenceforward and inverse problems

,p y m

forward problem

likelihood

,p y m

inverse problem

posterior distribution

( ) ( ) ( )0

i i ijj

i j

t t t y L w

Bayesian inferencethe electromagnetic forward problem

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

Likelihood:

Prior:

Bayes rule:

Bayesian paradigmlikelihood, priors and the model evidence

generative model m

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

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 Dynamical systems theory

3 Neural states dynamics

4 Bayesian inference

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

• 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

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)