Dynamic Causal Modelling (DCM ) for fMRI

Dynamic Causal Modelling (DCM) for fMRI

Rosalyn MoranVirginia Tech Carilion Research Institute

With thanks to the FIL Methods Groupfor slides and images

SPM Course, UCL May 2013

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 DCM250

Overview• Dynamic causal models (DCMs)

– Basic idea– Neural level– Hemodynamic level– Parameter estimation, priors & inference

• Applications of DCM to fMRI data - Attention to Motion - The Status Quo Bias




Functional vs Effective Connectivity

Functional connectivity is defined in terms of statistical dependencies, it is an operational concept that underlies the detection of (inference about) a functional connection, without any commitment to how that connection was caused

- Assessing mutual information & testing for significant departures from zero- Simple assessment: patterns of correlations- Undirected or Directed Functional Connectivity eg. Granger Connectivity

Effective connectivity is defined at the level of hidden neuronal states generating measurements. Effective connectivity is always directed and rests on an explicit (parameterised) model of causal influences — usually expressed in terms of difference (discrete time) or differential (continuous time) equations.

- Eg. DCM- causality is inherent in the form of the model ie. fluctuations in hidden neuronal states

cause changes in others: for example, changes in postsynaptic potentials in one area are caused by inputs from other areas.

),,( uxFdtdx

Neural state equation:

Electromagneticforward model:

neural activityEEGMEGLFP

Dynamic Causal Modeling (DCM)

simple neuronal modelcomplicated forward model

complicated neuronal modelsimple forward model

fMRI EEG/MEG

inputs

Hemodynamicforward model:neural activityBOLD

Deterministic DCM

u1 A(1,1)

A(2,1)

A(1,2)

A(2,2)

x1u2

B(1,2)

H{1}

y

H{2}

y

x2

C(1)




LGleft

LGright

RVF LVF

FGright

FGleft

Visual input in the - left (LVF) - right (RVF)visual field.

LG = lingual gyrusFG = fusiform gyrusx1 x2

x4x3

u2 u1

Example: a linear model of interacting visual regions

LGleft

LGright

RVF LVF

FGright

FGleft

LG = lingual gyrusFG = fusiform gyrus

Visual input in the - left (LVF) - right (RVF)visual field.x1 x2

x4x3

u2 u1



LG = lingual gyrusFG = fusiform gyrus

Visual input in the - left (LVF) - right (RVF)visual field.

state changes

effectiveconnectivity

externalinputs

systemstate

inputparameters

},{ CA

LGleft

LGright

RVF LVF

FGright

FGleft

x1 x2

x4x3

u2 u1

Extension: bilinear model

LGleft

LGright

RVF LVF

FGright

FGleft

x1 x2

x4x3

u2 u1

CONTEXTu3

Vanilla DCM:Deterministic Bilinear DCM

CuxBuAdtdx m

i

ii

1

)(

Bilinear state equation:

driving input

modulation

...)0,(),(2

0

uxuxfu

ufx

xfxfuxf

dtdx

Simply a two-dimensional taylor expansion (around x0=0, u0=0):

-

x2

stimuliu1

contextu2

x1

+

+

-

-

-+ u1

Z1

u2

Z2

Example: context-dependent decay

u1

u2

x2

x1

Penny et al. 2004, NeuroImage

+

bilinear DCM

CuxDxBuAdtdx m

i

n

j

jj

ii

1 1

)()(CuxBuAdtdx m

i

ii

1

)(

Bilinear state equation:

driving input

modulation

driving input

modulationnon-linear DCM

...)0,(),(2

0

uxuxfu

ufx

xfxfuxf

dtdx

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

Nonlinear state equation:

...2

)0,(),(2

2

22

0

x

xfux

uxfu

ufx

xfxfuxf

dtdx

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

Neural population activity

0 10 20 30 40 50 60 70 80 90 100

0

1

2

3

0 10 20 30 40 50 60 70 80 90 100-1

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100

0

1

2

3

fMRI signal change (%)x1 x2

x3

CuxDxBuAdtdx n

j

jj

m

i

ii

1

)(

1

)(

Nonlinear dynamic causal model (DCM)

Stephan et al. 2008, NeuroImage

u1

u2

endogenous connectivity

direct inputs

modulation ofconnectivity

Neural state equation CuxBuAx jj )( )(

uxC

xx

uB

xxA

j

j

)(

hemodynamicmodelλ

x

y

integration

BOLDyyy

activityx1(t)

activityx2(t) activity

x3(t)

neuronalstates

t

drivinginput u1(t)

modulatoryinput u2(t)

t




• Cognitive system is modelled at its underlying neuronal level (not directly accessible for fMRI).

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

• Overcoming Regional variability of the haemodynamic response

• ie DCM not based on temporal precedence at the measurement level

λ

x

y

Basics of DCM: Neuronal and BOLD level

λ

x

y

Basics of DCM: Neuronal and BOLD level

“Connectivity analysis applied directly on fMRI signals failed because hemodynamics varied between regions, rendering termporal precedence irrelevant” ….The neural driver was identified using DCM, where these effects are accounted for…

},,,,,{ h

important for model fitting, but of no interest for statistical inference

The hemodynamic model

)(activity

tx

• 6 hemodynamic parameters:

• Computed separately for each area region-specific HRFs!

sf

tionflow induc

(rCBF)

s

v

vq q/vvEf,EEfqτ /α

dHbchanges in1

00 )( /αvfvτ

volumechanges in1

f

q

s

f

Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage

stimulus functionsut

neural state equation

hemodynamic state equations

Estimated BOLD response

},,,,,{ h

important for model fitting, but of no interest for statistical inference

The hemodynamic model

)(activity

tx

• 6 hemodynamic parameters:

• Computed separately for each area region-specific HRFs!

sf

tionflow induc

(rCBF)

s

v

vq q/vvEf,EEfqτ /α

dHbchanges in1

00 )( /αvfvτ

volumechanges in1

f

q

s

f

Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage

stimulus functionsut

neural state equation

hemodynamic state equations

Estimated BOLD response

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

A

B

C

h

ε

How interdependent are neural and hemodynamic parameter estimates?





DCM is a Bayesian approach

)()|()|( pypyp posterior likelihood ∙ prior

)|( yp )(p

Bayes 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

sf (rCBF)induction -flow

s

v

f

stimulus function u

modelled BOLD response

vq q/vvf,Efqτ /α1)(

dHbin changes

/αvfvτ 1

in volume changes

f

q

)1(signalry vasodilatodependent -activity

fγszs

s

)(xy eXuhy ),(

observation model

hidden states

state equation

parameters

},{

},...,{

},,,,{1

nh

mn

h

CBBA

• Combining the neural and hemodynamic states gives the complete forward model.

• An observation model includes measurement error e and 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 ηθ|y and covariance Cθ|y.

Overview:parameter estimation

ηθ|y

neural stateequation( )j

jx A u B x Cu

VB in a nutshell (mean-field approximation)

, | ,p y q q q

( )

( )

exp exp ln , ,

exp exp ln , ,

q

q

q I p y

q I p y

Iterative updating of sufficient statistics of approx. posteriors by gradient ascent.

ln | , , , |

ln , , , , , |q

p y m F KL q p y

F p y KL q p m

Mean field approx.

Neg. free-energy approx. to model evidence.

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

Bayesian Inversion

Regional responsesSpecify generative forward model

(with prior distributions of parameters)

Variational 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

• Gaussian assumptions about the posterior distributions of the parameters

• posterior probability that a certain parameter (or contrast of parameters) is above a chosen threshold γ:

• By default, γ is chosen as zero – the prior ("does the effect exist?").

Inference about DCM parameters:Bayesian single-subject analysis

Likelihood distributions from different subjects are independent

NiiN

iN

yy

N

iyyyy

N

iyyy

CC

CC

,...,|1

|1|,...,|

1

1|

1,...,|

11

1

Under Gaussian assumptions this is easy to compute:

groupposterior covariance

individualposterior covariances

groupposterior mean

individual posterior covariances and means

Inference about DCM parameters: Bayesian parameter averaging (FFX group analysis)

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 subject

Selection of 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

Model evidence:Approximation: Free Energy

kk

mypmypBF )(ln)(ln 212,1

Fixed Effects Model selection via log Group Bayes factor:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

)|( imyp

)],|(),([)|(ln GpqKLmypF i

( | , )p r y

Random Effects Model selectionvia Model probability:

)( 1 Kkqkr

Inference about Model Architecture, Bayesian Model Selection




DCM – Attention to MotionParadigm

What connection in the network mediates attention ?

4 conditions- fixation only baseline

- observe static dots + photic- observe moving dots + motion

- attend to moving dots

Bayesian Model Selection

Results

Büchel & Friston 1997, Cereb. CortexBüchel et al. 1998, Brain

V5+

SPCV3A

Attention – No attention

m1 m2

V1 V5stim

PPC

ModulationBy attention

V1 V5Externalstim

PPC


m3

V1 V5stim

PPC


m4

V1 V5stim

PPC


V1 V5stim

PPC

attention

1.25

0.13

0.46

0.39 0.26

0.26

0.10

estimatedeffective synaptic strengths

for best model (m4)

[Stephan et al., Neuroimage, 2008]

models marginal likelihood

ln p y m

Bayesian Model Selection

V1 V5stim

PPC

attention

motion

-2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

%1.99)|0( 1,5 yDp PPCVV

1.25

0.13

0.46

0.390.26

0.50

0.26

0.10MAP = 1.25


Parameter Inference

V1V5PPC

observedfitted

motion &attention

motion &no attention

static dots

Data Fits




Overcoming status quo bias in the human brain Fleming et al PNAS 2010

Decision Accept Reject

Diff

icul

tyLo

w

H

igh


Main effect of difficulty in medial frontal and right inferior frontal cortex


Diff

icul

tyLo

w

H

igh


Interaction of decision and difficulty in region of subthalamic nucleus:Greater activity in STN when default is rejected in difficult trials


Diff

icul

tyLo

w

H

igh


DCM: “aim was to establish a possible mechanistic explanation for the interaction effect seen in the STN. Whether rejecting the default option is reflected in a modulation of connection strength from rIFC to STN, from MFC to STN, or both “…

MFC

rIFC

STN


MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Difficulty

RejectRejectRejectReject

Example:Overcoming status quo bias in the human brain Fleming et al PNAS 2010

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Reject

MFC

rIFC

STN

Difficulty

Difficulty

RejectRejectRejectReject


The summary statistic approachEffects across subjects consistently greater than zero P < 0.01 *P < 0.001 **

The evolution of DCM in SPM• DCM is not one specific model, but a framework for Bayesian inversion of

dynamic system models

• The default implementation in SPM is evolving over time– better numerical routines for inversion– change 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.

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.

Thank you

Documents

Dynamic Causal Modelling (DCM ) for fMRI