Dynamic Causal Modelling - PowerPoint Presentation

Slide1

Dynamic Causal Modelling (DCM) for fMRI

Klaas Enno Stephan Laboratory for Social & Neural Systems Research (SNS) University of ZurichWellcome Trust Centre for NeuroimagingUniversity College London

SPM Course, FIL13 May 2011

Slide2

Structural, functional & effective connectivityanatomical/structural connectivity= presence of axonal connectionsfunctional connectivity = statistical dependencies between regional time series

effective connectivity = directed influences between neurons or neuronal populationsSporns 2007, Scholarpedia

Slide3

Some models of effective connectivity for fMRI dataStructural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000

regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997Volterra kernels Friston & Büchel 2000Time series models (e.g. MAR/VAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003Dynamic Causal Modelling (DCM)bilinear: Friston et al. 2003; nonlinear:

Stephan et al. 2008

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Dynamic causal modelling (DCM)DCM framework was introduced in 2003 for fMRI by Karl Friston, Lee Harrison and Will Penny (NeuroImage 19:1273-1302)part of the SPM software package

currently more than 160 published papers on DCM

Slide5

Neural state equation:

Electromagneticforward model:neural activityEEGMEGLFPDynamic Causal Modeling (DCM)simple neuronal model

complicated forward modelcomplicated neuronal model

simple forward model

fMRI

EEG/MEG

inputs

Hemodynamic

forward model:

neural activity

BOLD

Slide6

LG

leftLGrightRVF

LVF

FG

right

FG

left

LG = lingual gyrus

FG = fusiform gyrus

Visual input in the

- left (LVF)

- right (RVF)

visual field.

x

1

x

2

x

4

x

3

u

2

u

1

Example:

a linear

model of interacting visual regions

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Example: a linear model of interacting visual regionsLG = lingual gyrusFG = fusiform gyrusVisual input in the - left (LVF) - right (RVF)visual field.

state changeseffectiveconnectivity

externalinputs

system

state

input

parameters

LG

left

LG

right

RVF

LVF

FG

right

FG

left

x

1

x

2

x

4

x

3

u

2

u

1

Slide8

Extension: bilinear model

LGleft

LG

right

RVF

LVF

FG

right

FG

left

x

1

x

2

x

4

x

3

u

2

u

1

CONTEXT

u

3

Slide9

endogenous connectivity

direct inputsmodulation ofconnectivityNeural state equation

hemodynamic

model

λ

x

y

integration

BOLD

y

y

y

activity

x

1

(

t

)

activity

x

2

(

t

)

activity

x

3

(

t

)

neuronalstates

t

driving

input

u

1

(

t

)

modulatory

input

u

2

(

t

)

t







Slide10

Bilinear DCM

Bilinear state equation:

driving

input

modulation

Two-dimensional Taylor series (around x

0

=0, u

0

=0):

Slide11

DCM parameters = rate constants

The coupling parameter a thus describes the speed ofthe exponential change in x(t)Integration of a first-order linear differential equation gives anexponential function:

Coupling parameter a is inversely

proportional to the half life



of z(t):

Slide12

-x2

stimuliu1contextu2

x1

+

+

-

-

-

+

u

1

Z

1

u

2

Z

2

Example:

context-dependent decay

u

1

u

2

x

2

x

1

Penny et al. 2004,

NeuroImage

Slide13

The problem of hemodynamic convolutionGoebel et al. 2003, Magn. Res. Med.

Slide14

He

modynamic forward models are important for connectivity analyses of fMRI dataDavid et al. 2008, PLoS Biol.Granger causalityDCM

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stimulus functions

u

t

neural state

equation

hemodynamic

state

equations

Balloon model

BOLD signal change equation

The he

modynamic model in DCM

Stephan et al. 2007,

NeuroImage

Slide16

A

B

C



h

ε

How interdependent are neural and hemodynamic parameter estimates?

Stephan et al. 2007,

NeuroImage

Slide17

DCM is a Bayesian approachposterior 

likelihood ∙ priorBayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities.In DCM: empirical, principled & shrinkage priors.The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.

new data

prior knowledge

Slide18

stimulus function

umodelled BOLD response

observation model

hidden states

state equation

parameters

Combining the neural and h

e

modynamic

states gives the

complete

forward model

.

An

observation model

includes measurement

error

e

a

nd confounds

X

(e.g. drift).

Bayesian

inversion:

parameter

estimation

by means of

variational

EM under Laplace approximation

Result:

Gaussian a

posteriori

parameter distributions

, characterised by

mean

η

θ

|y

and

covariance

C

θ

|y

.

Overview:

parameter estimation

η

θ

|y

neural state

equation

Slide19

VB in a nutshell (mean-field approximation)

 Iterative updating of sufficient statistics of approx. posteriors by gradient ascent. Mean field approx. Neg. free-energy approx. to model evidence.

 Maximise neg. free energy wrt. q = minimise divergence, by maximising variational

energies

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Gaussian assumptions about the posterior distributions of the parametersposterior probability that a certain parameter (or contrast of parameters cT

ηθ|y) is above a chosen threshold γ:By default, γ

is chosen as zero ("does the effect exist?").

Inference about DCM parameters:

Bayesian single-subject analysis

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Bayesian single subject inference

LGleft

LG

right

RVF

stim.

LVFstim.

FG

right

FG

left

LD|RVF

LD|LVF

LD

LD

0.34



0.14

-0.08



0.16

0.13



0.19

0.01



0.17

0.44



0.14

0.29



0.14

Contrast:

Modulation LG right

 LG links by LD|LVF

vs.

modulation LG left

 LG right by LD|RVF

p(c

T

>0|y)

= 98.7%

Stephan et al. 2005,

Ann. N.Y. Acad. Sci.

Slide22

Likelihood distributions from different subjects are independent one can use the posterior from one subject as the prior for the next

Under Gaussian assumptions this is easy to compute:groupposterior covariance

individualposterior

covariances

group

posterior

mean

individual posterior

covariances and means

“Today’s posterior is tomorrow’s prior”

Inference about DCM parameters:

Bayesian parameter averaging (FFX group analysis)

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Inference about DCM parameters:RFX group analysis (frequentist)In analogy to “random effects” analyses in SPM,

2nd level analyses can be applied to DCM parameters:Separate fitting of identical models for each subjectSelection of (bilinear) parameters of interest

one-sample t-test:

parameter > 0 ?

paired t-test:

parameter 1 > parameter 2 ?

rmANOVA:

e.g. in case of multiple sessions per subject

Slide24

inference on

model structure

or inference on

model parameters?

inference on

individual models

or model space partition

?

comparison of model families using

FFX or RFX BMS

optimal model structure assumed to be identical across subjects?

FFX BMS

RFX BMS

yes

no

inference on

parameters of an optimal model

or

parameters of all models

?

BMA

definition of model space

FFX analysis of parameter estimates

(e.g. BPA)

RFX analysis of parameter estimates

(e.g. t-test, ANOVA)

optimal model structure assumed to be identical across subjects?

FFX BMS

yes

no

RFX BMS

Stephan et al. 2010,

NeuroImage

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Any design that is good for a GLM of fMRI data.What type of design is good for DCM?

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GLM vs. DCMDCM tries to model the same phenomena (i.e. local BOLD responses) as a GLM, just in a different way (via connectivity and its modulation).No activation detected by a GLM → no motivation to include this region in a deterministic DCM.However, a stochastic DCM could be applied despite the absence of a local activation.

Stephan 2004, J. Anat.

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Multifactorial design: explaining interactions with DCM

Task factorTask ATask BStim 1

Stim 2

Stimulus factor

T

A

/S1

T

B/S1

T

A

/S2

T

B/S2

X

1

X

2

Stim2/

Task A

Stim1/

Task A

Stim 1/

Task B

Stim 2/

Task B

GLM

X

1

X

2

Stim2

Stim1

Task A

Task B

DCM

Let’s assume that an SPM analysis shows a main effect of stimulus in X

1

and a stimulus



task interaction in X

2

.

How do we model this using DCM?

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Stim 1

Task A

Stim 2

Task A

Stim 1

Task B

Stim 2

Task B

Simulated data

X

1

X

2

+++

X

1

X

2

Stimulus 2

Stimulus 1

Task A

Task B

+

+++

+

+++

+

–

–

Stephan et al. 2007,

J. Biosci.

Slide29

Stim 1

Task A

Stim 2

Task A

Stim 1

Task B

Stim 2

Task B

plus added noise (SNR=1)

X

1

X

2

Slide30

DCM10 in SPM8DCM10 was released as part of SPM8 in July 2010 (version 4010).Introduced many new features, incl. two-state DCMs and stochastic DCMsThis led to various changes in model defaults, e.g.inputs mean-centredchanges in coupling priorsself-connections: separately estimated for each areaFor details, see: www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf

Further changes in version 4290 (released April 2011) to accommodate new developments and give users more choice (e.g. whether or not to mean-centre inputs).

Slide31

The evolution of DCM in SPMDCM is not one specific model, but a framework for Bayesian inversion of dynamic system modelsThe default implementation in SPM is evolving over timebetter numerical routines for inversionchange in priors to cover new variants (e.g., stochastic DCMs, endogenous DCMs etc.)To enable replication of your results, you should ideally state which SPM version you are using when publishing papers.

Slide32

Factorial structure of model specification in DCM10Three dimensions of model specification:bilinear vs. nonlinearsingle-state vs. two-state (per region)deterministic vs. stochasticSpecification via GUI.

Slide33

bilinear DCM

Bilinear state equation:

driving

input

modulation

driving

input

modulation

non-linear DCM

Two-dimensional Taylor series (around x

0

=0, u

0

=0):

Nonlinear state equation:

Slide34

Neural population activity

fMRI signal change (%)

x

1

x

2

x

3

Nonlinear dynamic causal model (DCM

)

Stephan et al. 2008,

NeuroImage

u

1

u

2

Slide35

V1

V5stim

PPC

attention

motion

1.25

0.13

0.46

0.39

0.26

0.50

0.26

0.10

MAP = 1.25

Stephan et al. 2008,

NeuroImage

Slide36

V1

V5PPCobserved

fitted

motion &attention

motion &

no attention

static

dots

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input

Single-state DCM

Intrinsic (within-region) coupling

Extrinsic (between-region) coupling

Two-state DCM

Two-state DCM

Marreiros et

al. 2008,

NeuroImage

Slide38

Stochastic DCM

Friston et al. (2008, 2011) NeuroImageDaunizeau et al. (2009) Physica Daccounts for stochastic neural fluctuationscan be fitted to resting state data

has unknown precision and smoothness  additional

hyperparameters

Li et

al. (2011) NeuroImage

Slide39

Thank you