Dynamic Causal Modelling (DCM)

Marta I. Garrido

migarrido@ucla.edu

Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros, Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner

Motivation

Functional specialisationFunctional specialisation

Analysis of regionally specific effects

Analysis of regionally specific effects

Functional integrationFunctional integration

Interactions between distant regions

Interactions between distant regions

Varela et al. 2001, Nature Rev Neuroscience

Functional Connectivity

• Correlations between activity in spatially remote regions

• independent of how the dependencies are caused

MODEL-FREE MODEL-DRIVEN

Effective Connectivity

• The influence one neuronal system exerts over another

• Requires a mechanism or a generative model of measured brain responses

Outline

I. DCM: the neuronal and the hemodynamic models

II. Estimation and Bayesian inference

III. Application: Attention to motion in the visual system

IV. Extensions for fMRI and EEG data

I. DCM: the basic idea

• Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI).

• The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ).

λ

z

y

The aim of DCM is to estimate and make inferences about the coupling among brain areas, and how that coupling is influences by changes in the experimental contex. (Friston et al. 2003, Neuroimage)

intrinsic connectivity

direct inputs

modulation ofconnectivity

Neuronal state equation CuzBuAz jj ++= ∑ )( )(&

u

zC

z

z

uB

z

zA

j

j

∂∂

=

∂∂

∂∂

=

∂∂

=

&

&

&

)(

hemodynamicmodelλ

z

y

integration

t

drivinginput u1(t)

modulatoryinput u2(t)

t

BOLDy

y

yactivity

z2(t)

activityz1(t)

activityz3(t)

direct inputs

c1

b23a12

I. Conceptual overview

Stephan & Friston 2007, Handbook of Connectivity

sf

tionflow induc

=&

s

v

f

v

q q/vvf,Efqô /á

dHbchanges in

1)( −= ρρ&/ávfvô

volumechanges in

1−=&

f

q

)1( −−−= fãszs ry signalvasodilato

κ&

},,,,{ ρατγκθ =h},,,,{ ρατγκθ =h

( ) ,)(

signal BOLD

qvty λ=

I. The hemodynamic “Balloon” model

)(

input neuronal

tz

5 hemodynamic parameters:

Buxton et al. 1998Mandeville et al. 1999Friston et al. 2000, NeuroImage

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

)(

)(

)(1

tz

tz

tz

n

M• State vector

– Changes with time

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

),,...(

),,...(

1

1111

nnn

n

n uzzf

uzzf

z

z

θ

θ

M

&

M

&• Rate of change of state vector

– Interactions between elements

– External inputs, u

( , , )z f z u θ=&• System parameters θ

I. Elements of a dynamic neuronal system

11

dzsz

dt=−

Decay function

Half-life τ:

Generic solution to the ODEs in DCM:

ln 2 /s τ=

10.5 (0)z

τ

1 1

1

( ) 0.5 (0)

(0)exp( )

z z

z s

ττ

== −

1 1 1( ) (0)exp( ), (0) 1z t z st z= − =

I. Connectivity parameters = rate constants

Coupling parameter describes the speed ofthe exponential decay

( )1 1

2 21 1 2

1

2

1

2 21

21

(0) 1

(0) 0

( ) exp( )

( ) exp( )

0

z sz

z s a z z

z

z

z t st

z t sa t st

a

= −

= −

=

=

= −

= −

>

&

&

1;4 21 == as

2;4 21 == as

1;8 21 == as

z2

21a

z1

s

s

z1 sa21t z2

I. Linear dynamics: 2 nodes

u2

u1

z1

z2

activity in z2 is coupled to z1 via coefficient a21

u1

21a

001

01211

2

1

212

1 >⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−=⎥

⎦

⎤⎢⎣

⎡au

czz

as

zz&

&

z1

z2

I. Neurodynamics: 2 nodes with input

Stimulus function

000

00

1

01 2211

2

1221

22

1

212

1 >⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−=⎥

⎦

⎤⎢⎣

⎡bu

czz

bu

zz

as

zz&

&

u2

u1

z1

z2

modulatory input u2 activity through the coupling a21

u1

u2

index, not squared

z1

z2

I. Neurodynamics: modulatory effect

21a

0,,00

00

1

1 22121121

2

1221

22

1

21

12

2

1 >⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−=⎥

⎦

⎤⎢⎣

⎡baau

czz

bu

zz

aa

szz&

&

u2

u1

z1

z2

reciprocal connection

disclosed by u2

u1

u2 z1

z2

I. Neurodynamics: reciprocal connections

21a12a

0 20 40 60

0

2

4

0 20 40 60

0

2

4

seconds

blue: neuronal activity

red: bold response

h1

h2

u1

u2 z1

z2

h(u,θ) represents the BOLD response (balloon model) to input

BOLD(no noise)

I. Hemodynamics

0 20 40 60

0

2

4

0 20 40 60

0

2

4

seconds

BOLDnoise addedy1

y2

u1

u2 z1

z2

euhy += ),( θy represents simulated observation of BOLD response, i.e. includes noise

blue: neuronal activity

red: bold response

I. Hemodynamics (with noise)

I. Bilinear state equation in DCM for fMRI

state changes

latentconnectivity

drivinginputs

state vector

CuzBuAzm

j

jj ++= ∑

=

)(1

&

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∑=

mnmn

m

n

m

j jnn

jn

jn

j

j

nnn

n

n u

u

cc

cc

z

z

bb

bb

u

aa

aa

z

z

M

L

MOM

L

M

L

MOM

L

L

MOM

L

&

M

& 1

1

1111

11

111

1

1111

induced connectivity

n regions m drv inputsm modulatory inputs

context-dependent

Constraints on•Hemodynamic parameters•Connections

Models of•Hemodynamics in a single region•Neuronal interactions

Bayesian estimation

)(θp

)()|()|( θθθ pypyp ∝

)|( θyp

posterior

priorlikelihood term

II. Estimation: Bayesian framework

Mp

p-1

Mpost

post-1

d-1

Md θ

γηθ|y

probability that a parameter (or

contrast of parameters cT ηθ|y) is

above a chosen threshold γ

sf

tionflow induc

=&

s

f

vq/vvf,Efqô /á

dHbchanges in

1)( −= ρρ&/ávfvô

volumechanges in

1−=&

f

)1( −−−= fãszs ry signalvasodilato

κ&

II. Parameter estimation

• Specify model (neuronal and hemodynamic level)

• Make it an observation model by adding measurement error e and confounds X (e.g. drift).

• Bayesian parameter estimation using expectation-maximization.

• Result:(Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y.

ηθ|y

v

stimulus function u

modeled BOLD response

q

( , , )h x u θ ( , , )y h x u X eθ β= + +

observation model

hidden states},,,,{ qvfszx =

state equation( , , )x F x u θ=&

parameters

},{

},...,{

},,,,{1

nh

mn

h

CBBA

θθθ

θ

ρατγκθ

=

=

=

neuronal stateequation∑ ++= CuzBuAz j

j )(&

Given competing hypotheses, which model is the best?

Pitt & Miyung 2002, TICS

)(

)()|(log

mcomplexity

maccuracymyp −=

)|(

)|(

jmyp

imypBij =

==

II. Bayesian model comparison

V1

V5

SPC

Motion

Photic

Attention

0.85

0.57 -0.02

1.360.70

0.84

0.23

Model 1:attentional modulationof V1→V5

V1

V5

SPC

Motion

PhoticAttention

0.86

0.56 -0.02

1.42

0.550.75

0.89

Model 2:attentional modulationof SPC→V5

1 2log ( | ) log ( | )p y m p y m>>

III. Application: Attention to motion in the visual system

Büchel & Friston

• potential timing problem in DCM:

temporal shift between regional time series because of multi-slice acquisition

• Solution:– Modelling of (known) slice timing of each area.

1

2

slic

e ac

quis

ition

visualinput

Slice timing extension now allows for any slice timing differences

Long TRs (> 2 sec) no longer a limitation.

Kiebel et al. 2007, Neuroimage

IV. Extensions: Slice timing model

)(tu

ijij uBA +

input

Single-state DCM

1x

Intrinsic (within-region) coupling

Extrinsic (between-region) coupling

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

NNNN

N

x

x

tx

AA

AA

A M

K

MOM

L 1

1

111

)(

Two-state DCM

Ex1

)exp( ijij uBA +

Ix1

11 11exp( )IE IEA uB+IEx ,1

IN

EN

I

E

AA

AAA

AA

AAA

xx

xx

tx

eeeee

eeeee

A

IINN

IENN

EINN

EENNN

IIIE

NEIEE

M

L

MOM

L

1

1

)(

000

000

1

1111

11111

IV. Extensions: Two-state model

Marreiros et al. 2008, Neuroimage

bilinear DCM

CuxDxBuAdt

dx m

i

n

j

jj

ii +⎟⎟

⎠

⎞⎜⎜⎝

⎛++= ∑ ∑

= =1 1

)()(CuxBuA

dt

dx m

i

ii +⎟

⎠

⎞⎜⎝

⎛ += ∑=1

)(

Bilinear state equation:

driving input

modulation

non-linear DCM

driving input

modulation

Two-dimensional Taylor series (around x0=0, u0=0):

Nonlinear state equation:

...2

)0,(),(2

2

22

0 +∂∂

+∂∂

∂+

∂∂

+∂∂

+≈=x

xf

uxuxf

uuf

xxf

xfuxfdtdx

Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas.

IV. Extensions: Nonlinear DCM

Stephan et al. 2008, Neuroimage

Jansen and Rit 1995

David et al. 2006, Kiebel et al. 2006, Neuroimage

€

x.

= f (x,u,θ)

IV. Extensions: DCM for ERPs

Forward

Backward

Lateral

with backward connections and without

a b c

A1 A1

STG

input

STG

IFG

FB

A1 A1

STG

input

STG

IFG

F

rIFG

rSTG

rA1lA1

lSTG

IV. Extensions: DCM for ERPs

Garrido et al. 2007, PNAS

standardsdeviants

Grand mean ERPsa b

c

ERP oddball

model inversion from 0 to t where t = 120:10:400 ms for F and FB

128 EEG electrodes

Garrido et al. 2007, PNAS

IV. Extensions: DCM for ERPs

IV. Extensions: DCM for ERPs

Garrido et al. 2008, Neuroimage

The DCM cycle

Design a study to investigatethat system

Extraction of time seriesfrom SPMs

Parameter estimationfor all DCMs considered

Bayesian modelselection of optimal DCM

Statistical test on parameters

of optimal model

Hypotheses abouta neural system

DCMs specificationmodels the system

Data acquisition

fMRI data

Posterior densities of parameters

Neuronal dynamics Hemodynamics

Model comparison

DCM roadmap

Model inversion usingExpectation-Maximization

State space Model

Priors

Dynamic Causal Modelling (DCM)

Marta I. Garrido

migarrido@ucla.edu

Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros, Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner