Mohamed Seghier

Wellcome Trust Centre for Neuroimaging,University College London, UK

DCM: Dynamic CausalModelling for fMRI

Wellcome Trust Centre for Neuroimaging

SPM-Course Edinburgh, April 2010

DCM [default] implementation:

Deterministic Stochastic [Daunizeau et al. 2009]

Bilinear Nonlinear [Stephan et al. 2008]

The one-state neuronal The two-state [Marreiros et al. 2008]

DCM is a generative model= a quantitative / mechanistic description of how observed data are generated.

Key features:1- Dynamic2- Causal3- Neuro-physiologically motivated4- Operate at hidden neuronal interactions5- Bayesian in all aspects.

The hemodynamics Deterministic dynamical systems

[Friston et al. 2000 Neuroimage] [Friston 2002 Neuroimage]

[Friston et al. 2003 Neuroimage]

“DCM is used to test the specific hypothesis thatmotivated the experimental design. It is not an exploratorytechnique […]; the results are specific to the tasks andstimuli employed during the experiment.”

“The central idea behind dynamic causal modelling(DCM) is to treat the brain as a deterministicnonlinear dynamic system that is subject to inputsand produces outputs.”

“DCM assumes the responses are driven by designedchanges in inputs.”

[Friston et al. 2003 Neuroimage]

Input u(t)

connectivity parameters

System state z(t)State changes of a systemare dependent on:

– the current state

– external inputs

– its connectivity

– time constants & delays

System =a set of elements whichinteract in a spatially andtemporally specific fashion

),,( uzFdt

dz

What is a system?

(evolution equation)

Basic idea of DCM for fMRI

λ

z

y

♣ Effective connectivity is parameterised in terms of coupling amongunobserved brain states (e.g., neuronal activity in different regions).The objective is to estimate these parameters by perturbing thesystem and measuring the response.

♣ A cognitive system is modelled as a bilinear model of neuralpopulation dynamics (z).

♣ The modelled neuronal dynamics (z) is transformed into area-

specific BOLD signals (y) by a hemodynamic forward model (λ).

Aim: to estimate the parameters of a reasonablyrealistic neural model such that the predictedregional blood oxygen level dependent (BOLD)signals, correspond as closely as possible to theobserved BOLD signals.

Neurodynamics: 2 nodes with input

u2

u1

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Neurodynamics: positive modulation

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modulatory input u2 activitythrough the coupling 21a

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Neurodynamics: reciprocal connections

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reciprocalconnectiondisclosed by u2

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bilineardynamicsystem R1

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z1 z2

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u2 u1CONTEXT

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Bilinear state equation in DCM for fMRI

statechanges

connectivityexternalinputs

statevector

directinputs

CuzBuAzm

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)(1

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modulation ofconnectivity

n regions m inputs (driv.)m inputs (mod.)

The neural state equation

“C”, the direct or driving effects:- extrinsic influences of inputs on neuronal activity.

“A”, the intrinsic coupling or the latent connectivity:- fixed or endogenous effective connectivity;- first order connectivity among the regions in the absence of input.

“B”, the bilinear term, modulatory effects, or the induced connectivity:- context-dependent change in connectivity;- eq. a second-order interaction between the input and activity in a sourceregion when causing a response in a target region.

[Units]: rates, [Hz];Strong connection = an effect that is influenced quicklyor with a small time constant.

CuzBuAzm

j

jj

)(1

DCM parameters = rate constants

dxax

dt 0( ) exp( )x t x at

The coupling parameter athus describes the speed of

the exponential change in x(t)0

0

( ) 0.5

exp( )

x x

x a

Integration of a first-order linear differential equation gives anexponential function:

/2lna

00.5x

a/2ln

Coupling parameter is inversely

proportional to the half life of x(t):

If AB is 0.10 s-1 this means that, per unit time, the increase in activity in Bcorresponds to 10% of the activity in A

a

hemodynamicmodelλ

z

y

integration

BOLDyyy

activityx1(t)

activityx2(t) activity

x3(t)

neuronalstates

t

drivinginput u1(t)

modulatoryinput u2(t)

t

[Stephan & Friston (2007),Handbook of Brain Connectivity]

endogenousconnectivity

direct inputs

modulation ofconnectivity

Neural state equation CuzBuAz jj )( )(

u

zC

z

z

uB

z

zA

j

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)(

hemodynamicmodel ??

λ

z

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integration

BOLDyyy

activityx1(t)

activityx2(t) activity

x3(t)

neuronalstates

t

drivinginput u1(t)

modulatoryinput u2(t)

t

endogenousconnectivity

direct inputs

modulation ofconnectivity

Neural state equation CuzBuAz jj )( )(

u

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[Stephan & Friston (2007),Handbook of Brain Connectivity]

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Hemodynamics: the indirect link

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The hemodynamic model

State Equations

[Friston 2000 Neuroimage]

Output function: a mixture of intra- and extra-vascular signal

Flow component:s : activity-dependent signal;

f : flow inducing signal

Balloon component:v : the rate of change of volume;

q : the change in deoxyhemoglobin

sf

tionflow induc

(rCBF)

s

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inputs

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q q/vvEf,EEfqτ /α

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1

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hemodynamicstate equationsf

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BOLD signalchange equation

},,,,,{ h

important for model fitting,but of no interest forstatistical inference

• 6 hemodynamic parameters:

• Empirically determineda priori distributions.

• Area-specific estimates(like neural parameters) region-specific HRFs!

The hemodynamic model

[Friston et al. 2000, NeuroImage][Stephan et al. 2007, NeuroImage]

R1left

R2right

u2 u1

R4right

R3left

Example: modelled BOLD signal

black: observed BOLD signal

red: modelled BOLD signal

CuzBuAzm

j

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)(1

Multiple-input multiple-output system

Priors & parameter estimation

Bayesian statistics (inversion)

)()|()|( pypyp

posterior likelihood ∙ prior

)|( yp )(p

Bayes theorem allows us to express our prior knowledgeor “belief” about parameters of the model.

The posterior probability of the parametersgiven the data is an optimal combination ofprior knowledge and new data, weighted bytheir relative precision.

new data prior knowledge

Priors in DCM

- hemodynamic parameters: empirical priors

- coupling parameters of self-connections: principled priors

- coupling parameters other connections: shrinkage priors

Constraints on parameter estimation:

Inference about DCM parameters:Bayesian inversion

• Gaussian assumptions about the posterior distributions of theparameters

• Use of the cumulative normal distribution to test the probability thata certain parameter (or contrast of parameters cT ηθ|y) is above achosen threshold γ:

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

cCc

cp

y

T

y

T

N

ηθ|y

Bayesian parameter estimation by means of expectation-maximisation (EM)

[Friston 2002 Neuroimage]

yy

BOLD

DCM: practical stepsSelect areas you want to model

• Extract timeseries of these areas(x(t))

• Specify at neuronal level

– what drives areas (c)

– how areas interact (a)

– what modulates interactions (b)

• State-space model with 2 levels:

– Hidden neural dynamics

– Predicted BOLD response

• Estimate model parameters:

Gaussian a posteriori parameterdistributions, characterised bymean ηθ|y and covariance Cθ|y.

neuronalstates activity

x1(t) a12 activityx2(t)

c2

c1

Driving input(e.g. sensory stim)

Modulatory input(e.g. context/learning/drugs)

b12

ηθ|y

Stimuli 250 radially moving dots at 4.7 degrees/s

Pre-Scanning

5 x 30s trials with 5 speed changes (reducing to 1%)

Task - detect change in radial velocity

Scanning (no speed changes)

6 normal subjects, 4 x 100 scan sessions;

each session comprising 10 scans of 4 differentconditions

F A F N F A F N S .................

F - fixation point only

A - motion stimuli with attention (detect changes)

N - motion stimuli without attention

S - no motion

[Büchel & Friston 1997, Cereb. Cortex][Büchel et al. 1998, Brain]

Attention – No attention

Attention to motion in the visual system

V5

SPC

Attention – No attention

How we can interpret, mechanistically,the increase in activity of area V5 byattention when motion is physicallyunchanged.

Choice of areas and time series extraction. Three ROIs: V1, V5, and SPC.

Definition of driving inputs. All visual stimuli/conditions (photic: A N S)

Definition of modulatory inputs. The effects of motion and attention (A N)

Building the model:1- how to connect regions (intrinsic connections “A”);2- how the driving inputs enter the system (extrinsic effects “C”);3- define the context-dependent connections (modulatory effects “B”).

V1

V5

SPC

Motion

Photic

Attention• Visual inputs drive V1.

• Activity then spreads tohierarchically arranged visualareas.

• Motion modulates the strength ofthe V1→V5 forward connection.

• Attention modualtes the strengthof the SPC→V5 backwardconnection.

Re-analysis of data from[Friston et al., 2003 NeuroImage]

• Motion modulates thestrength of the V1→V5forward connection.

• The intrinsic connectionV1→V5 is insignificant in theabsence of motion (a21=-0.05 Hz).

• Attention increases thebackward-connectionSPC→V5.

V1

V5

SPC

Motion

Photic

Attention

0.88

0.48

0.37

0.42

0.66

0.56

-0.05

Re-analysis of data fromFriston et al., NeuroImage 2003

After DCM estimation:

Are there otherplausible/alternative models?

V1

V5

SPC

Motion

PhoticAttention

0.86

0.56 -0.02

1.42

0.55

0.75

0.89

Model 1:attentional modulationof V1→V5

V1

V5

SPC

Motion

Photic

Attention

0.85

0.57 -0.02

1.360.70

0.84

0.23

Model 2:attentional modulationof SPC→V5

V1

V5

SPC

Motion

PhoticAttention

0.85

0.57 -0.02

1.36

0.03

0.70

0.85

Attention

0.23

Model 3:attentional modulationof V1→V5 and SPC→V5

How we can compare between competing hypotheses? BMS (Bayesian Model Selection)

Alternative models (hypothesis-driven approach):

Model evidence and selection

Given competing hypotheseson functional mechanisms ofa system, which model is thebest?

For which model m does p(y|m)become maximal?

Which model represents thebest balance between modelfit and model complexity?

[Pitt and Miyung 2002 TICS]

dmpmypmyp )|(),|()|( Model evidence:

Bayesian model selection (BMS)

)|(

)|(),|(),|(

myp

mpmypmyp

Bayes’ rule:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

integral usually not analytically solvable, approximations necessary

Model evidence: probability of generating data y from parameters that arerandomly sampled from the prior p(m).

Maximum likelihood: probability of the data y for the specific parameter vector that maximises p(y|,m).

Logarithm is amonotonic function

Maximizing log model evidence

= Maximizing model evidence

)(),|(log

)()()|(log

mcomplexitymyp

mcomplexitymaccuracymyp

Log model evidence = balance between fit and complexity

[Penny et al. 2004, NeuroImage][Penny et al. 2010, PLoS Comp Biol]

Approximations to the model evidence in DCM

The negative variotional free energy (F) approximation

Under Gaussian assumptions about the posterior (Laplace approximation),the negative free energy F is a lower bound on the log model evidence:

mypqKLmypF ,|,)|(log Kullback-Leibler (KL) divergence

The complexity term in F

• The negative free energy F accounts for parameterinterdependencies.

• The complexity term of F is higher

– the more independent the prior parameters ( effective DFs)

– the more dependent the posterior parameters

– the more the posterior mean deviates from the prior mean

• NB: SPM8 only uses F for model selection !

y

T

yy CCC

mpqKL

|1

||2

1

2

1

2

1

)|(),([Penny et al. 2004 Neuroimage][Stephan et al. 2009 Neuroimage]

Bayes factors

)|(

)|(

2

112

myp

mypBF

positive value, [0;[

But: the log evidence is just some number – not very intuitive!

A more intuitive interpretation of model comparisons is madepossible by Bayes factors:

To compare two models, we can just compare their logevidences.

Very strong 99% 150

strong95-99%20 to 150

positive75-95%3 to 20

weak50-75%1 to 3

Evidencep(m1|y)BF12

Kass & Raftery classification:[Kass & Raftery 1995, J. Am. Stat. Assoc.]

Bayesian Model Selection in group studies.

Fixed effects BMS at group level

Group Bayes factor (GBF) for 1...K subjects:

Average Bayes factor (ABF):

k

kijij BFGBF )(

( )kKij ij

k

ABF BF

)|(

)|(

j

iij

myp

mypBF

Problems:► blind with regard to group heterogeneity;► sensitive to outliers.

)|(~ 111 mypy)|(~ 111 mypy

)|(~ 222 mypy)|(~ 111 mypy

)|(~ pmpm kk

);(~ rDirr

)|(~ pmpm kk )|(~ pmpm kk),1;(~1 rmMultm

Random effects BMS for group studies

Dirichlet parameters= “occurrences” of models in the populations

Dirichlet distribution of model probabilities

Multinomial distribution of subject-specificmodels

Measured data

[Stephan et al. 2009, Neuroimage]

-5 -4 -3 -2 -1 0 1 2 3 4 5

Sim

ula

ted

data

sets

Log model evidence differences

x1 x2u1

x3

u2

x1 x2u1

x3

u2

incorrect model (m2) correct model (m1)

m2 m1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

r1

p(r

1|y

)

p(r1>0.5 | y) = 1.000

1 2

/

0.934, 0.066

k k kk

r

r r

Exceedance probability

1 220.537, 1.463 Estimates of Dirichlet parameters

Post. expectations of modelprobabilities

%100

|211

yrrp

[Stephan et al. 2009, Neuroimage]

-35 -30 -25 -20 -15 -10 -5 0 5

Su

bje

cts

Log model evidence differences

MOG

LG LG

RVFstim.

LVFstim.

FGFG

LD|RVF

LD|LVF

LD LD

MOGMOG

LG LG

RVFstim.

LVFstim.

FGFG

LD

LD

LD|RVF LD|LVF

MOG

m2 m1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r1

p(r

1|y

)

p(r1>0.5 | y) = 0.997

157.0,843.0

194.2,806.11

21

21

rr

[Stephan et al. 2009, Neuroimage]

Interface in SPM8

k

kkkr

Post. expectations ofmodel probabilities

yrrp

kKj

jkk |

:\}...1{

Exceedance probability

Levels of inference: Group/population level

-- Family level ---- System/model level --

-- Parameter/connection level --

♣ Family level:- Useful when no clear winning model // models have common characteristics .Models assigned to subsets (families) with shared parametersInference: a class of models that best explains the data.

FFX: subjects assumed to use identical systems.RFX: optimal models vary across subjects.

Variational Bayes: fast/accurate Nmod < N_sub.Gibbs sampling: optimal N_mod >> N_sub.

♣ System level:- Useful when a clear winning model can be identified (BMS).Inference: the best combination of inputs+connections that explains the data.

♣ Connection level:- Useful if interested in connectivity parameters (e.g. modulations).Inference: Bayesian parameters averaging (BPA) or t-test on DCM parameters.

Inference: BMA on the winning family (or the whole model space).

[Penny et al. 2010, PLoS Comp Biol]

Fig. 1. This schematic summarizes the typical sequence of analysis in DCM, depending on the question of interest. Abbreviations: FFX=fixedeffects, RFX=random effects, BMS=Bayesian model selection, BPA=Bayesian parameter averaging, BMA=Bayesian model averaging,ANOVA=analysis of variance.

Stephan et al. (2010). Ten Simple Rules for DCM. NeuroImage

It is helpful to constrain your DCM model space.

number of ROIs limited to 8 in SPM8 (GUI).

(e.g., 5 ROIs, fully connected, 1 Billion alternatives for modulations!).

Define sets of models that are plausible, in a systematic way, given priorknowledge about the system (e.g. anatomical, TMS, previous studies).

Bad models will affect your BMS results (BMS = a “relative” space)!

BMS has nothing to say about the “true” models.find the most plausible (useful) model, given a set of alternatives.Best model = best balance between accuracy and complexity.

BMS cannot be applied to models fitted to different data!(Only models with the same ROIs can be compared using BMS).

Extensions in DCM for fMRI (SPM8):

• Bayesian Model Selection BMS [Penny et al. 2004 Neuroimage].

• Slice specific sampling [Kiebel et al. 2007 Neuroimage].

• Refined hemodynamic model [Stephan et al. 2007 Neuroimage].

• The two-state DCM [Marreiros et al. 2008 Neuroimage].

• The non-linear DCM [Stephan et al. 2008 Neuroimage].

• Random-effects BMS (VB) [Stephan et al. 2009 Neuroimage].

• Random-effects BMS (Gibbs) [Penny et al. 2010 PLoS Comp Biol].

• Stochastic DCM [Daunizeau et al. 2009 Physica D].

• Anatomical-based priors for DCM [Stephan et al. 2009 Neuroimage].

• Family level inference BMS [Penny et al. 2010 PLoS Comp Biol].

• Bayesian model averaging BMA [Penny et al. 2010 PLoS Comp Biol].

Design a study thatallows to investigatethat system

Extraction ofROI time

series

Parameter estimationfor all DCMs considered

Bayesian modelselection ofoptimal DCM

Inference on thefamily, model,

connection level

Hypothesis abouta neural system

Data analysis (SPM)

Define plausibleDCMs

The DCM cycle

Theoritical reviews:

Stephan et al. (2010). Ten Simple Rules for DCM. NeuroImage

Daunizeau et al. (2010). DCM: a critical review of the biophysical and statistical foundations. NeuroImage

Friston (2009). Causal modelling and brain connectivity in fMRI. PLoS Biol

Applications: (recent examples of DCM-fMRI at the FIL)

- Word reading via the putamen:Seghier and Price (2010) Cerebral Cortex.

- Intelligible speech perception:Leff et al. (2008) J Neurosci.

- Associative learning and prediction error:den Ouden et al. (2009) Cerebral Cortex.