models of effective connectivity & dynamic causal ... · handbook of brain connectivity . v1 v4...

Models of effective connectivity &Dynamic Causal Modelling (DCM)

Slides obtained from:

Karl Friston Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London

Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich

Overview

•  Brain connectivity: types & definitions –  anatomical connectivity –  functional connectivity –  effective connectivity

•  Dynamic causal models (DCMs) –  DCM for fMRI: Neural and hemodynamic levels –  Parameter estimation & inference

•  Applications of DCM to fMRI data –  Design of experiments and models –  Some empirical examples and simulations

Connectivity

A central property of any system

Communication systems Social networks (internet) (Canberra, Australia)

FIgs. by Stephen Eick and A. Klovdahl;see http://www.nd.edu/~networks/gallery.htm

Structural, functional & effective connectivity

•  anatomical/structural connectivity = presence of axonal connections

•  functional connectivity = statistical dependencies between regional time series

•  effective connectivity = causal (directed) influences between neurons or

neuronal populations

Sporns 2007, Scholarpedia

Anatomical connectivity

•  neuronal communication via synaptic contacts

•  visualisation by tracing techniques

•  long-range axons “association fibres”

Different approaches to analysing functional connectivity

•  Seed voxel correlation analysis

•  Eigen-decomposition (PCA, SVD)

•  Independent component analysis (ICA)

•  any other technique describing statistical dependencies amongst regional time series

Does functional connectivity not simply correspond to co-activation in SPMs?

No, it does not - see the fictitious example on the right: Here both areas A1 and A2 are correlated identically to task T, yet they have zero correlation among themselves: r(A1,T) = r(A2,T) = 0.71 but r(A1,A2) = 0 !

task T regional response A2 regional response A1

Stephan 2004, J. Anat.

Pros & Cons of functional connectivity analyses

•  Pros: –  useful when we have no experimental control over the

system of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.)

•  Cons: –  interpretation of resulting patterns is difficult / arbitrary –  no mechanistic insight into the neural system of interest –  usually suboptimal for situations where we have a priori

knowledge and experimental control about the system of interest

models of effective connectivity necessary

For understanding brain function mechanistically, we need models of effective connectivity, i.e.

models of causal interactions among neuronal

populations.

Some models for computing effective connectivity from fMRI data

•  Structural 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. 1997

•  Volterra kernels Friston & Büchel 2000

•  Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003

•  Dynamic Causal Modelling (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

Overview




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 100 published papers on DCM

GLM DCM

Inference Within voxels Among ROIs

On BOLD signal Neuronal Activation

Answering Where the stimulus produced activation.

How the stimulus activated the system of interconnected ROIs.

Using Frequentist Estimation Bayesian Estimation

DCM vs GLM: Cheat Sheet

),,( θuxFdtdx =

Neural state equation:

Electromagnetic forward model:

neural activity, EEGMEG LFP

Dynamic Causal Modeling (DCM)

simple neuronal model complicated forward model

complicated neuronal model simple forward model

fMRI EEG/MEG

inputs

Hemodynamicforward model:neural activity, BOLD

Stephan & Friston 2007, Handbook of Brain Connectivity

V1

V4

BA37

STG

BA39

Structural perturbations Stimulus-free - u

e.g., attention, time

Dynamic perturbations Stimuli-bound u

e.g., visual words

Functional integration and the enabling of specific pathways

y

y y

y

y

measurement

neuronal network

Observed data

)(tu

ix

input

x = f (x,u,θ ) +ω

εθ += ),(xgyForward model (measurement)

Model inversion

Forward models and their inversion

Forward model (neuronal)

( | , , , )p y x u mθ ( , | , , )p x y u mθ

θ

Model specification and inversion

εθ

θ

+=

=

),(

),,(

xgy

uxfx

)|(),(),|(),|(

)(),|()|(

mypmpmypmyp

dpmypmyp

θθθ

θθθ

=

= ∫

( | , ) ( ( ), ( ))( , ) ( , )

NN

p y m gp m θ θ

θ θ θθ µ

= Σ= Σ

Invert model

Inference

Define likelihood model

Specify priors

Neural dynamics

Observer function

Design experimental inputs )(tu

Inference on models

Inference on parameters

f = s

q = f E( f ) ρ − v1/α q vv = f − v1/α

s = x −κ s − ( f −1)

0 1 2 3( ) ( ( )) ( (1 ) (1 ) (1 ))y t g x t V k q k q v k v= = − + − + −

Output: a mixture of intra- and extravascular signal

)(tx

0 8 16 24 sec

ix

Hemodynamic models for fMRI basically, a convolution

1y

signal

flow

dHb volume

The plumbing

Neural population activity

BOLD signal change (%) x1 x2 u1

x3

u2

– –

x =

a11 a12 0

a21 a22 a230 a32 a33

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+ u2

0 0 0b21 0 0

0 0 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

x1x2x3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+

c1100

00c32

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

u1u2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

A toy example

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

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 90 100 0

0.2 0.4 0.6

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

0.1 0.2 0.3

DCM for fMRI: 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 parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.

intrinsic connectivity

direct inputs

modulation of connectivity

Neural state equation CuxBuAx jj ++= ∑ )( )(

uxC

xx

uB

xxA

j

j

∂∂=

∂∂

∂∂=

∂∂=

)(

hemodynamic model λ

x

y

integration

BOLD y y y

activity x1(t)

activity x2(t) activity

x3(t)

neuronal states

t

driving input u1(t)

modulatory input u2(t)

t

Stephan & Friston (2007), Handbook of Brain Connectivity

LG left

LG right

RVF LVF

FG right

FG left

LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x1 x2

x4 x3

u2 u1

x1 = a11x1 + a12x2 + a13x3 + c12u2x2 = a21x1 + a22x2 + a24x4 + c21u1x3 = a31x1 + a33x3 + a34x4x4 = a42x2 + a43x3 + a44x4

Example: a linear system of dynamics in visual cortex

Example: a linear system of dynamics in visual cortex

LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field.

state changes

effective connectivity

externalinputs

systemstate

input parameters

x1

x2

x3

x4

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

a11 a12 a13 0

a21 a22 0 a24

a31 0 a33 a34

0 a42 a43 a44

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

x1

x2

x3

x4

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+

0c21

00

c12

000

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

u1

u2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

x = Ax +Cu

},{ CA=θ

LG left

LG right

RVF LVF

FG right

FG left

x1 x2

x4 x3

u2 u1

Extension: bilinear dynamic system

LG left

LG right

RVF LVF

FG right

FG left

x1 x2

x4 x3

u2 u1

CONTEXT u3

x = (A+ u jB( j )

j=1

m

∑ )x +Cu

x1

x2

x3

x4

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

a11 a12 a13 0

a21 a22 0 a24

a31 0 a33 a34

0 a42 a43 a44

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+ u3

0 b12(3) 0 0

0 0 0 00 0 0 b34

(3)

0 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

x1

x2

x3

x4

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+

0c21

00

c12

000

0000

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

u1

u2

u3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Bilinear DCM

CuxBuAdtdx m

i

ii +⎟

⎠⎞⎜

⎝⎛ += ∑

=1

)(

Bilinear state equation:

driving input

modulation

...)0,(),(2

0 +∂∂

∂+∂∂+

∂∂+≈= ux

uxfu

ufx

xfxfuxf

dtdx

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

DCM parameters = rate constants

dx axdt

= 0( ) exp( )x t x at=

The coupling parameter a thus describes the speed of the exponential change in x(t)

0

0

( ) 0.5exp( )

x xx a

ττ

==

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

a = ln2 /τ

00.5x

a/2ln=τ

Coupling parameter a is inversely proportional to the half life of z(t): τ

-

x2

stimuli u1

context u2

x1

+

+

-

-

- +

u 1

Z 1

u 2

Z 2

x = Ax + u2B(2)x +Cu1

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= σ a12

a21 σ⎡

⎣⎢⎢

⎤

⎦⎥⎥x + u2 11

2

b 0

022

2

b

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥x + c1 0

0 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

u1u2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Example: context-dependent decay u1

u2

x2

x1

Penny, Stephan, Mechelli, Friston NeuroImage (2004)

Interpretation of DCM parameters •  Dynamic model (differential equations)

neural parameters correspond to rate constants (inverse of time constants = Hz!)

speed at which effects take place

•  Identical temporal scaling in all areas by factorising A and B with σ: all connection strengths are relative to the self-connections.

•  Each parameter is characterised by the mean (ηθ|y) and covariance of its a posteriori distribution. Its mean can be compared statistically against a chosen threshold γ.

γ ηθ|y

θ n = {A, B,C,σ}

p = ln2 /τ p

A →σA =σ−1 a12

a21 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The problem of hemodynamic convolution

Goebel et al. 2003, Magn. Res. Med.

sf

tionflow induc

= (rCBF)

s

v

stimulus functions

vq q/vvEf,EEfqτ /α

dHbchanges in1

00 )( −=/αvfvτ volumechanges in

1−=

f

q

)1(

−−−= fγsxssignalryvasodilato

κ

u

s

CuxBuAdtdx m

j

jj +⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑

=1

)(

t

neural state equation

( ) ( )

εε

ϑ

λ

−===

⎥⎦⎤

⎢⎣⎡ −+⎟

⎠⎞⎜

⎝⎛ −+−≈Δ=

1

3.4

111),(

3

002

001

32100

kTEErkTEEk

vkvqkqkV

SSvq

hemodynamic state equations f

Balloon model

BOLD signal change equation

},,,,,{ ερατγκθ =h

important for model fitting, but (usually) of no interest for statistical inference

•  6 hemodynamic parameters:

•  Computed separately for each area (like the neural parameters) region-specific HRFs!

The hemodynamic model in DCM

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

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?

Stephan et al. 2007, NeuroImage

sf =(rCBF)induction -flow

s

v

f

stimulus function u

modelled BOLD response

vq

changes in dHb q = f E( f,ρ) ρ − v1/q / v/αvfvτ 1

in volume changes−=

f

q

)1(signalry vasodilatodependent -activity

−−−= fγszs κ

s

)(xy λ= eXuhy ++= βθ ),(observation model

hidden states { , , , , }z x s f v q=

state equation z = F (x,u,θ )

parameters

},{},...,{},,,,{

1

nh

mn

h

CBBAθθθ

θρατγκθ

===

•  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 parameter estimation by means of a Levenberg-Marquardt gradient ascent, embedded into an EM algorithm.

•  Result:Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.

Overview:parameter estimation

ηθ|y

neural state equation x = (A+ u jB

j )x +Cu∑

p-value: probability of getting the observed data in the effect’s absence. If small, reject null hypothesis that there is no effect.

0

0

: 0( | )Hp y H

θ =

Limitations:  One can never accept the null hypothesis  Given enough data, one can always demonstrate a significant effect  Correction for multiple comparisons necessary

Solution: infer posterior probability of the effect

Probability of observing the data y, given no effect.

)|( yp θ

Problems of classical (frequentist) statistics

Probability of the effect, given the observed data

Bayes‘ Theorem

Reverend Thomas Bayes 1702 - 1761

“Bayes‘ Theorem describes, how an ideally rational person processes information." Wikipedia

)()()|()|(

yppypyP θθθ =

Likelihood Prior

Evidence

Posterior

)()()|()|(

yppypyP θθθ =

Given data y and parameters, the conditional probabilities are:

p(θ | y) = p(y,θ )p(y) )(

),()|(θθθ

pypyp =

Eliminating gives Bayes’ rule:

Likelihood Prior

Evidence

Posterior

Bayes’ Theorem

p(y,θ )

Bayesian statistics

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

)|( θyp )(θp

Bayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities.

Priors can be of different sorts:empirical, principled or 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

y  Observation of data

Likelihood

prior distribution

 Formulation of a generative model

 Update of beliefs based upon observations, given a prior state of knowledge

p(θ | y)∝ p(y |θ )p(θ )

Principles of Bayesian inference

p(y |θ )p(θ )

LG left

LG right

RVF LVF

FG right

FG left

Example: modelled BOLD signal Underlying model (modulatory inputs not shown)

LG = lingual gyrus Visual input in the FG = fusiform gyrus - left (LVF)

- right (RVF) visual field.

blue: observed BOLD signal red: modelled BOLD signal (DCM)

left LG

right LG

•  needed for Bayesian estimation, embody constraints on parameter estimation

•  express our prior knowledge or “belief” about parameters of the model

•  hemodynamic parameters: empirical priors

•  temporal scaling: principled prior

•  coupling parameters: shrinkage priors

Priors in DCM

posterior likelihood · prior

Bayes Theorem

p(θ | y)∝ p(y |θ ) ⋅ p(θ )

Priors in DCM •  system stability:

in the absence of input, the neuronal states must return to a stable mode → constraints on prior variance

of intrinsic connections (A) → probability <0.001 of obtaining

a non-negative Lyapunov exponent (largest real eigenvalue of the intrinsic coupling matrix)

•  shrinkage priors for coupling parameters (η=0) → conservative estimates!

•  self-inhibition: ensured by priors on the decay rate constant σ (ησ=1, Cσ=0.105) → these allow for neural

transients with a half life in the range of 300 ms to 2 seconds NB: a single rate constant for all

regions! •  Identical temporal scaling in all areas by

factorising A and B with σ: all connection strengths are relative to the self-connections.

θ =

σaij

bijk

cik

θ h

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

, ηθ =

1000ηθ

h

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

, Cθ =

0.105 0CA

CB

10 Ch

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

A →σA =σ−1 a12

a21 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Shrinkage Priors Small & variable effect Large & variable effect

Small but clear effect Large & clear effect

Parameter estimation in DCM •  Combining the neural and

hemodynamic states gives the complete forward model:

•  The observation model includes measurement error and confounds X (e.g. drift):

•  Bayesian parameter estimation under Gaussian assumptions by means of EM and gradient ascent.

•  Result: Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.

ηθ|y

x = {z, s, f ,v,q}θ =θ n +θ h

x = f (x,u,θ )y = λ(x) = h(u,θ )

y = h(u,θ )+ Xβ + ε

y − h(u,ηΘ|y )→ min

EM and gradient ascent

•  Bayesian parameter estimation by means of expectation maximisation (EM) –  E-step:

gradient ascent (Fisher scoring & Levenberg-Marquardt regularisation) on the log posterior to compute

•  (i) the conditional mean ηθ|y (= expansion point of gradient ascent), •  (ii) the conditional covariance Cθ|y

–  M-step: ReML estimates of hyperparameters i for error covariance components Qi:

•  Note: Gaussian assumptions about the posterior (Laplace approximation)

Ce = λiQi∑

•  Gaussian assumptions about the posterior distributions of the parameters

•  Use of the cumulative normal distribution to test the 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

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −=

cCc

cp

yT

yT

N

θ

θ γηφ

Bayesian single subject inference

LG left

LG right

RVF stim.

LVF stim.

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(cTηθ|y>0|y) = 98.7%

Stephan et al. 2005, Ann. N.Y. Acad. Sci.

Inference about DCM parameters: Bayesian fixed-effects group analysis

Because the likelihood distributions from different subjects are independent, one can combine their posterior densities by using the posterior of one subject as the prior for the next:

)|()...|()|(),...,|(...

)|()|( )()|()|(),|(

)()|( )|(

111

12

1221

11

ypypypyyp

ypyppypypyyp

pypyp

NNN θθθθ

θθθθθθ

θθθ

−∝

∝∝∝

1,...,|

1|

1|,...,|

1

1|

1,...,|

11

1

−

=

−

=

−−

⎟⎠⎞⎜

⎝⎛=

=

∑

∑

NiiN

iN

yy

N

iyyyy

N

iyyy

CC

CC

θθθθ

θθ

ηη

Under Gaussian assumptions this is easy to compute:

group posterior covariance

individual posterior covariances

group posterior mean

individual posterior covariances and means

Overview




Bayesian model selection (BMS) Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?

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

Which model represents thebest balance between model fit and model complexity?

Pitt & Miyung (2002), TICS

θθθ dmpmypmyp )|(),|()|( ∫ ⋅=Model evidence:

Various approximations, e.g.: -  negative free energy -  AIC -  BIC

Penny et al. (2004) NeuroImage

Bayesian model selection (BMS)

)|()|(

2

1

mypmypBF =

Model comparison via Bayes factor:

)|()|(),|(),|(

mypmpmypmyp θθθ =Bayes’ rules:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

Bayes factors

)|()|(

2

112 myp

mypB =

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

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

To compare two models, we can just compare their log evidences.

B12 p(m1|y) Evidence 1 to 3 50-75% weak

3 to 20 75-95% positive 20 to 150 95-99% strong

> 150 > 99% Very strong Kass & Raftery classification:

Kass & Raftery 1995, J. Am. Stat. Assoc.

Any design that is good for a GLM of fMRI data.

What type of design is good for DCM?

GLM vs. DCM

DCM tries to model the same phenomena as a GLM, just in a different way:

It is a model, based on connectivity and its modulation, for explaining experimentally controlled variance in local responses.

No activation detected by a GLM → inclusion of this region in a DCM is useless!

Stephan 2004, J. Anat.

Multifactorial design: explaining interactions with DCM

Task factor Task A Task B

Stim

1

Stim

2

Stim

ulus

fact

or

TA/S1 TB/S1

TA/S2 TB/S2

X1 X2

Stim2/Task A

Stim1/ Task A

Stim 1/ Task B

Stim 2/ Task B

GLM

X1 X2

Stim2

Stim1

Task A Task B

DCM Let’s assume that an SPM analysis shows a main effect of stimulus in X1 and a stimulus task interaction in X2.

How do we model this using DCM?

Stim 1Task A

Stim 2Task A

Stim 1Task B

Stim 2Task B

Simulated data

X1

X2

+++ X1 X2

Stimulus 2

Stimulus 1

Task A Task B

+ +++ +

+++ +

– –

Stephan et al. 2007, J. Biosci.

Stim 1Task A

Stim 2Task A

Stim 1Task B

Stim 2Task B

plus added noise (SNR=1)

X1

X2

Example studies of DCM for fMRI •  DCM now an

established tool for fMRI & M/EEG analysis

•  >100 studies published, incl. high-profile journals

•  combinations of DCM with computational models

Thank you

models of effective connectivity & dynamic causal ... · handbook of brain connectivity . v1 v4...

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