J. Daunizeau

ICM, Paris, France

ETH, Zurich, Switzerland

Dynamic Causal Modellingof fMRI timeseries

Overview

1 DCM: introduction

2 Dynamical systems theory

4 Bayesian inference

5 Conclusion

Overview

1 DCM: introduction

2 Dynamical systems theory

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

localizing brain activity:functional segregation

Introductionfrom functional segregation to functional integration

« Where, in the brain, didmy experimental manipulation

have an effect? »

A B

u2u1

A B

u2u1

effective connectivity analysis:functional integration

« How did my experimental manipulationpropagate through the network? »

?

1 2

31 2

31 2

3

time

0 ?tx

t

0t

t t

t t

t

Introductiondynamical 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

• realistic neuronal model• linear observation model

EEG/MEGEEG/MEG

inputs

IntroductionDCM: evolution and observation mappings

• agnostic neuronal model• realistic observation model

fMRIfMRI

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

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

Lebreton et al., 2011

IntroductionDCM for fMRI: assessing mimetic desire in the brain

Overview

1 DCM: introduction

2 Dynamical systems theory

4 Bayesian inference

5 Conclusion

Dynamical systems theorysystem’s stability

-0.5 0 0.5 1 1.5 2

-20

-10

0

10

20

time (sec)

x(t)

a=1.2

-0.5 0 0.5 1 1.5 2

-20

-10

0

10

20

time (sec)

x(t)

a=-1.2

fixed point = stable fixed point = unstable

 .

a<0 a>0

Dynamical systems theorydynamical modes in ND

Dynamical systems theorydamped oscillations: spirals

x1

x2

Dynamical systems theorydamped oscillations: states’ correlation structure

Dynamical systems theoryimpulse response functions: convolution kernels

u

0 20 40 60 80 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

u(t

)

input u

0 20 40 60 80 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

x(t)

output xu

Dynamical systems theorysummary

• Motivation: modelling reciprocal influences (feedback loops)

• Dynamical repertoire depend on the system’s dimension (and nonlinearities):o D>0: fixed pointso D>1: spiralso D>1: limit cycles (e.g., action potentials)o D>2: metastability (e.g., winnerless competition)

• Linear dynamical systems can be described in terms of their impulse response

limit cycle (Vand Der Pol) strange attractor (Lorenz)

( )

1

mi

ii

x A u B x Cu

bilinear state equation:

a24

c1

4

13

driving input

b12

2

d24 gating effect

u1

u2modulatory effect

2 2 2

0 2

0

( , ) ,0 ...2

f f f f xx f x u f x x u ux

x u x u x

( ) ( )

1 1

m ni j

i ji j

x A u B x D x Cu

nonlinear state equation:

Stephan et al., 2008

Dynamical systems theoryagnostic neural dynamics

experimentally controlledstimulus

u ( ) ( )

1 1

m ni j

i ji j

x A u B x D x Cu

tneural states dynamics

(rCBF) flow induction

f s

s

v

v

q1

0 0( ) /

changes in dHb

q f E f,E E v q / v 1/

changes in volume

v f v

f

q

( 1)

vasodilatory signal

s x s f s

f

Balloon model

hemodynamic states dynamics

BOLD signal change observation

0 1 2 30

1 0 0

2 0 0

3

( , ) 1 1 1

4.3

1

S qq v V k q k k v

S v

k E TE

k r E TE

k

0{ , , , , , }h E

( ) ( ){ , , , }n i jA B C D

Friston et al., 2003

Dynamical systems theorythe neuro-vascular coupling

Overview

1 DCM: introduction

2 Dynamical systems theory

4 Bayesian inference

5 Conclusion

Bayesian inferenceforward and inverse problems

,p y m

forward problem

likelihood

,p y m

inverse problem

posterior distribution

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

1exp

2p

4 0.05P

→ Distribution of data, given fixed parameters:

2

2

1exp

2p y y f

f

Likelihood:

Prior:

Bayes rule:

Bayesian paradigmlikelihood, priors and the model evidence

generative model m

Bayesian paradigmthe likelihood function of an alpha kernel

holding the parameters fixed holding the data fixed

0 20 40 60 80 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

u(t

)input u

0 20 40 60 80 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

x(t)

output x

Bayesian inferencetype, role and impact of priors

• Types of priors:

Explicit priors on model parameters (e.g., connection strengths)

Implicit priors on model functional form (e.g., system dynamics)

Choice of “interesting” data features (e.g., ERP vs phase data)

• Impact of priors:

On parameter posterior distributions (cf. “shrinkage to the mean” effect)

On model evidence (cf. “Occam’s razor”)

On free-energy landscape (cf. Laplace approximation)

• Role of priors (on model parameters):

Resolving the ill-posedness of the inverse problem

Avoiding overfitting (cf. generalization error)

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

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

ren

ces

in lo

g- m

odel

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

Overview

1 DCM: introduction

2 Dynamical systems theory

4 Bayesian inference

5 Conclusion

Conclusionsummary

• Functional integration

→ connections are recruited in a context-dependent fashion

→ which connections are modulated by experimental factors?

• Dynamical system theory

→ DCM uses it to model feedback loops

→ linear systems have a unique impulse response function

• Bayesian inference

→ parameter estimation and model comparison/selection

→ types, roles and impacts of priors

ConclusionDCM for fMRI: variants

stochastic DCM

two-states DCM

2 2 2

2 2

f f f f xx x u ux

x u x u x

0, xt N Q

Ex1

Ix1

11 11exp IE IEA uBIEx ,

1

time (s)

x 1 (

A.U

.) ( )p x t

ConclusionDCM for fMRI: validation

activation deactivation

David et al., 2008

• Suitable experimental design:– any design that is suitable for a GLM (including multifactorial designs)– include rest periods (cf. build-up and decay dynamics)– re-write the experimental manipulation in terms of:

• driving inputs (e.g., presence/absence of visual stimulation) • modulatory inputs (e.g., presence/absence of motion in visual inputs)

• Hypothesis and model:– Identify specific a priori hypotheses (≠ functional segregation)– which models are relevant to test this hypothesis?– check existence of effect on data features of interest– formal methods for optimizing the experimental design w.r.t. DCM [Daunizeau et al., PLoS Comp. Biol., 2011]

Conclusionplanning a compatible DCM study

References

Daunizeau et al. 2012: Stochastic Dynamic Causal Modelling of fMRI data: should we care about neural noise? Neuroimage 62: 464-481.

Schmidt et al., 2012: Neural mechanisms underlying motivation of mental versus physical effort. PLoS Biol. 10(2): e1001266.

Daunizeau et al., 2011: Optimizing experimental design for comparing models of brain function. PLoS Comp. Biol. 7(11): e1002280

Daunizeau et al., 2011: Dynamic Causal Modelling: a critical review of the biophysical and statistical foundations. Neuroimage, 58: 312-322.

Den Ouden et al., 2010: Striatal prediction error modulates cortical coupling. J. Neurosci, 30: 3210-3219.

Stephan et al., 2009: Bayesian model selection for group studies. Neuroimage 46: 1004-1017.

David et al., 2008: Identifying Neural Drivers with Functional MRI: An Electrophysiological Validation. PloS Biol. 6: e315.

Stephan et al., 2008: Nonlinear dynamic causal models for fMRI. Neuroimage, 42: 649-662.

Friston et al., 2007: Variational Free Energy and the Laplace approximation. Neuroimage, 34: 220-234.

Sporns O., 2007: Brain connectivity. Scholarpedia 2(10): 1695.

David O., 2006: Dynamic causal modeling of evoked responses in EEG and MEG. Neuroimage, 30: 1255-1272.

Friston et al., 2003: Dynamic Causal Modelling. Neuroimage 19: 1273-1302.

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)