Functional Integration& Dynamic Causal Modelling (DCM)

Klaas Enno Stephan &Lee Harrison

Functional Imaging LabWellcome Dept. of Imaging NeuroscienceUniversity College London

ICN SPM CourseMay 6th, 2005

Overview

• Classical approaches to functional & effective connectivity

• Generic concepts of system analysis

• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation

– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example

• DCM for ERPs

System analyses in functional neuroimaging

System analyses in functional neuroimaging

Functional integrationAnalyses of Analyses of inter-regional effectsinter-regional effects: : what are the interactions what are the interactions between the elements of a given between the elements of a given neuronal system?neuronal system?

Functional integrationAnalyses of Analyses of inter-regional effectsinter-regional effects: : what are the interactions what are the interactions between the elements of a given between the elements of a given neuronal system?neuronal system?

Functional connectivity= the temporal correlation = the temporal correlation betweenbetween spatially remote spatially remote neurophysiologicalneurophysiological events events

Functional connectivity= the temporal correlation = the temporal correlation betweenbetween spatially remote spatially remote neurophysiologicalneurophysiological events events

Effective connectivity= the influence that the elements = the influence that the elements

of a neuronal system exert over of a neuronal system exert over anotheranother

Effective connectivity= the influence that the elements = the influence that the elements

of a neuronal system exert over of a neuronal system exert over anotheranother

Functional specialisationAnalyses of Analyses of regionally specific regionally specific effectseffects: which areas constitute a : which areas constitute a neuronal system?neuronal system?

Functional specialisationAnalyses of Analyses of regionally specific regionally specific effectseffects: which areas constitute a : which areas constitute a neuronal system?neuronal system?

MECHANISM-FREE MECHANISTIC MODEL

Functional connectivity: methods

• Seed-voxel correlation analyses

• Eigenimage analysis– Principal Components Analysis (PCA)– Singular Value Decomposition (SVD)– Partial Least Squares (PLS)

• Independent Component Analysis (ICA)

Definition of eigenvector e and eigenvalue :

Ae = e

Geometric meaning:

Eigenvectors are only scaled by Y; their direction is unchanged principal axes of the matrix operation

PCA: Y* = EY

Re-expresses Y in terms of a new orthonormal basis E, the eigenvectors of the covariance matrix YYT. The new axes account for extreme and orthogonal components of the covariance structure.

Eigen-decomposition: some general maths

e e

SVD: Y = USVT

A general method for eigen-decomposition of matrices. If dealing with spatio-temporal data, SVD re-expresses Y in terms of the eigenvectors of the spatial and temporal covariance matrices.

Y (DATA

)

m x n matrix

n voxels

m

scan

s

Y = USVT = s1u1v1T + s2u2v2

T + ... + srurvr

T

APPROX. OF Y

u1

= APPROX. OF Y

APPROX. OF Y

+ s2 + sr+ ...s1

u2

ur

v1T

SVD applied to spatio-temporal data

v2T vr

T

r = rank of Y

U = eigenvector decomposition of YYT (temporal covariance), m x m matrix.

V = eigenvector decomposition of YTY (spatial covariance), n x n matrix.

S = diagonal matrix of singular values (square root of eigenvalues), m x n matrixNote: YTY and YYT have thesame non-zero eigenvalues!

ui, vi = i-th column of U and V

A time-series of 1D images:128 scans of 40 “voxels”(note: for display reasons, the transpose of the data matrix is shown)

Eigenvariates U:Temporal expression of the first three eigenimages over time

Singular values S and eigenimages V ("spatial modes")

The time-series reconstructed from the first three eigenimages

SVD: an example using simulated data

YT

u1…u3

diag(S)

40

voxels

128 scans

Y* = s1u1v1T + s2u2v2

T + s3u3v3T

v1…v3

Pros & Cons of functional connectivity

• Pros:– useful when we have no model of what

caused the data (e.g. sleep, hallucinatons, etc.)

• Cons:– no mechanistic insight into the neural

system of interest– inappropriate for situations where we have

a priori knowledge and experimental control about the system of interest

models of effective connectivity necessary

Models of effective connectivity

• Structural Equation Modelling (SEM)

• Psycho-physiological interactions (PPI)

• Multivariate autoregressive models (MAR)& Granger causality techniques

• Kalman filtering

• Volterra series

• Dynamic Causal Modelling (DCM)

now

later

Psycho-physiological interaction (PPI)

e

βSSTT

βSS

TT y

BA

BA

321

221

1

)( )(

)(

)(

e

βSSTT

βSS

TT y

BA

BA

321

221

1

)( )(

)(

)(

Task factorTask A Task B

Sti

m 1

Sti

m 2

Sti

mu

lus

fact

or

TA/S1

GLM of a 2x2 factorial design:

We can replace one main effect in the GLM by the time series of an area that shows this main effect.

E.g. let's replace the main effect of stimulus type by the time series of V1:

main effectof task TB/S1

TA/S2 TB/S2

e

βVTT

βV

TT y

BA

BA

3

2

1

1 )(

1

)(

e

βVTT

βV

TT y

BA

BA

3

2

1

1 )(

1

)(

main effectof stim. type

interaction

main effectof taskV1 time series main effectof stim. typepsycho-physiologicalinteraction

PPI example: attentional modulation of V1→V5

V1V1

attention

no attention

V1 activity

V5 a

ctiv

ity

SPM{Z}

time

V5 a

ctiv

ity

Friston et al. 1997, NeuroImage 6:218-229Büchel & Friston 1997, Cereb. Cortex 7:768-778

V1 x Att.V1 x Att.

=

V5V5

V5V5

Attention

PPI: interpretation

Two possible interpretations of the PPI

term:

V1

Modulation of V1V5 by attention

Modulation of the impact of attention on V5 by V1.

V1V1 V5V5 V1

V5V5

attentionattention

V1V1

attentionattention

e

βVTT

βV

TT y

BA

BA

3

2

1

1 )(

1

)(

e

βVTT

βV

TT y

BA

BA

3

2

1

1 )(

1

)(

Pros & Cons of PPIs• Pros:

– given a single source region, we can test for its context-dependent connectivity across the entire brain

• Cons:– very simplistic model:

only allows to model contributions from a single area – ignores time-series properties of data– not easily used with event-related data– operates at the level of BOLD time series

limited causal interpretability in neural terms,more powerful models needed

DCM!

Overview

• Classical approaches to functional & effective connectivity

• Generic concepts of system analysis

• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation

– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example

• DCM for ERPs

Models of effective connectivity = system models.But what precisely is a system?

• System = set of elements which interact in a spatially and temporally specific fashion.

• System dynamics = change of state vector in time

• Causal effects in the system:– interactions between elements– external inputs u

• System parameters :specify the nature of the interactions

• general state equation for non-autonomous systems

)(

)(

)(1

tz

tz

tz

n

),,...(

),,...(

1

1111

nnn

n

n uzzf

uzzf

z

z

),,( uzFz

overall system staterepresented by state variables

nz

z

zdt

dz

1

change ofstate vectorin time

Example: linear dynamic system

LGleft

LGright

RVF LVF

FGright

FGleft

LG = lingual gyrusFG = fusiform gyrus

Visual input in the - left (LVF) - right (RVF)visual field.z1 z2

z4z3

u2 u1

state changes

effectiveconnectivity

externalinputs

systemstate

inputparameters

2

1

12

21

4

3

2

1

444342

343331

242221

131211

4

3

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u

u

c

c

z

z

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z

aaa

aaa

aaa

aaa

z

z

z

z

CuAzz

},{ CA

Extension:

bilinear dynamic system

LGleft

LGright

RVF LVF

FGright

FGleft

z1 z2

z4z3

u2 u1

CONTEXTu3

CuzBuAzm

j

jj

)(1

3

2

112

21

4

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334

312

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444342

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u

u

uc

c

z

z

z

z

b

b

u

aaa

aaa

aaa

aaa

z

z

z

z

Bilinear state equation in DCM

state changes

intrinsicconnectivity

m externalinputs

systemstate

direct inputs

CuzBuAzm

j

jj

)(1

},...,{ 1 CBBA mn

mnmn

m

n

m

j jnn

jn

jn

j

j

nnn

n

n u

u

cc

cc

z

z

bb

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u

aa

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z

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1

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1111

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1111

modulation ofconnectivity

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.

BOLDy

y

y

hemodynamicmodel

Inputu(t)

activityz2(t)

activityz1(t)

activityz3(t)

effective connectivity

direct inputs

modulation ofconnectivity

The bilinear model CuzBuAz jj )(

c1 b23

a12

neuronalstates

λ

z

y

integration

Neural state equation ),,( nuzFz Conceptual overview

Friston et al. 2003,NeuroImage

u

z

u

FC

z

z

uuz

FB

z

z

z

FA

jj

j

2

-

Z2

stimuliu1

contextu2

Z1

+

+

-

-

-+

u1

Z1

u2

Z2

2

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12

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1

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u

uczuz

a

a

z

z

CuzBuAzz

bb

Example: generated neural data

u1

u2

z2

z1

sf

tionflow induc

s

v

f

v

q q/vvf,Efqτ /α

dHbchanges in

1)( /αvfvτ

volumechanges in

1

f

q

)1( fγszs

ry signalvasodilato

},,,,{ h},,,,{ h

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

signal BOLD

qvty ,)(

The hemodynamic “Balloon” model

)(tz

activity • 5 hemodynamic parameters:

• Empirically determineda priori distributions.

• Computed separately for each area (like the neural parameters).

LGleft

LGright

RVF LVF

FGright

FGleft

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 signalred: modelled BOLD signal (DCM)

left LG

right LG

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 )(xy eXuhy ),(

observation model

hidden states},,,,{ qvfszx

state equation),,( uxFx

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 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 stateequation CuzBuAz j

j )( CuzBuAz jj )(

• 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

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

Bayes Theorem

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 priorsfor coupling parameters (η=0)→ conservative estimates!

h

B

A

hhik

kij

ij

C

C

C

C

c

b

a

0

1

0105.0

,

0

0

0

1

,

h

B

A

hhik

kij

ij

C

C

C

C

c

b

a

0

1

0105.0

,

0

0

0

1

,

• 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 secondsNB: 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.

1

1

21

12

a

a

AA

1

1

21

12

a

a

AA

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):

),()(

),,(

},,,,{

uhxy

uxfx

qvfszxhn

),()(

),,(

},,,,{

uhxy

uxfx

qvfszxhn

Xuhy ),( Xuhy ),(

• 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

min ),( θ|yuhy min ),( θ|yuhy

Bayesian estimation: univariate Gaussian case

Relative precision weighting

Normal densities

exy

ppe

yy

pey

yx

x

222||

22

2

2|

1

11

x

Univariatelinear model

),;()( 2ppNp

),;()|( 2exyNyp

),;()|( 2|| yyNyp

p

y|

(1)θ

(2)θOne step if Ce is known.

GeneralLinear Model

Bayesian estimation: multivariate Gaussian case

Normal densities eXθy ),;()( ppNp Cηθθ

),;()|( eNp CXθyθy

),;()|( || yyNyp Cηθθ

ppeT

yy

peT

y

ηCyCXCη

CXCXC1

||

111|

Bayesian estimation: nonlinear case

Friston (2002) NeuroImage, 16: 513-530.

Gradient ascent (Fisher scoring) with priors

iyppe

Tiy

iy

iy

peTi

y

|111

||1

|

1111| )(

ηηCrCJCηη

CJCJC

Local linearization by 1st order Taylor:

r

e

h

h

hhh

h

iy

i

iy

iy

iy

iyi

y

θJ

ηyr

ηθθ

θ

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ηθθ

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)(

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)(

)()(

)()(

)(

|

|

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||

|

Current estimatesi

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y || , Cη

),;Δ()Δ(

),;()(

| pi

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Np

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Cηθθ

Prior density

),Δ;()Δ|(

)),(;()|(

e

e

Np

hNp

CθJrθr

Cθyθy

Likelihood

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)

iie QC

E-step: Fisher scoring

)()(

)(

|

1

|2

2

|1

|

|2

21

|

1

iy

iy

iy

iy

iy

iy

ff

f

ηθ

ηθ

ηη

ηθ

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);(log);,|(log

);,|(log

uθuλθy

uλyθ

pp

pf

For DCM, the Fisher scoring scheme can be derived by taking partial derivatives of the log posterior wrt. to the parameters:

11

|2

2

|11

|

)(

)(

peT

y

yppeT

y

fd

fd

CJCJηθ

ηηCrCJηθ

Generally, Fisher scoring is identical to the Gauss-Newton algorithm (except that the expectation of the Hessian is taken): θθ

θθ

ffii

1

2

21

This results in equations equivalent to those derived for posterior mode estimation in non-linear cases, using local linearisation (see previous slides) :

E-step: Levenberg-Marquardt regularisationFor updating the parameters in the E-step, an LM regularisation ensures a good balance between steepest gradient ascent and Gauss-Newton ascent.

θθθ

fii 1

θθθθ

ffii

1

2

21

θI

θθθ

ffii

1

2

21

θθθθθ

ffdiag

fii

1

2

2

2

21

steepest gradient ascentlarge step for large gradient risk of overshooting a maximumsmall steps for small gradients slow progress in “uninteresting” areas

Gauss-Newton algorithmderived from 2nd order Taylor expansion quadratic assumption about local properties of objective function

Levenberg algorithmdynamically adjusts : small Gauss-Newtonlarge steepest gradient ascent

Levenberg-Marquardt algorithm:makes use of curvature even if is high:scales each component of the gradient according to the curvature

Parameter estimation (new):output in command window

E-Step: 1 F: -1.514001e+003 dp: 8.299907e-002

E-Step: 2 F: -1.200724e+003 dp: 9.638851e-001

E-Step: 3 F: -1.115951e+003 dp: 2.703493e-001

E-Step: 4 F: -1.077757e+003 dp: 2.002973e-002

E-Step: 5 F: -1.075699e+003 dp: 4.219233e-003

E-Step: 6 F: -1.075663e+003 dp: 1.030322e-003

E-Step: 7 F: -1.075661e+003 dp: 3.595806e-004

E-Step: 8 F: -1.075661e+003 dp: 2.273264e-006

2θdp

2θdp

ypepypT

pyeT hhF ||

1|

1 logloglog)()())(())((2

1 CCCθθCθθθyCθy ypepyp

Tpye

T hhF ||1

|1 logloglog)()())(())((

2

1 CCCθθCθθθyCθy

objective function

Change of the norm of the parameter vector(= magnitude of update)

Overview

• Classical approaches to functional & effective connectivity

• Generic concepts of system analysis

• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation

– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example

• DCM for ERPs

DCM parameters = rate constants

azdt

dz cattz )exp()(

)exp()( 0 atztz Decay function:

Half-life:

)exp(

5.0)(

0

0

az

zz

Generic solution to the ODEs in DCM:

/2lna

05.0 z

Inference about DCM parameters:Bayesian single-subject analysis

• Bayesian parameter estimation in DCM: Gaussian assumptions about the posterior distributions of the parameters

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

• γ can be chosen as zero ("does the effect exist?") or as a function of the expected half life τ of the neural process: γ = ln 2 / τ

cCc

cp

yT

yT

N

cCc

cp

yT

yT

N

ηθ|y

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

ypyp

pypypyyp

pypyp

NNN

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

...

)|()|(

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

)()|( )|(

111

12

1221

11

ypypypyyp

ypyp

pypypyyp

pypyp

NNN

1,...,|

1|

1|,...,|

1

1|

1,...,|

11

1

NiiN

iN

yy

N

iyyyy

N

iyyy

CC

CC

1,...,|

1|

1|,...,|

1

1|

1,...,|

11

1

NiiN

iN

yy

N

iyyyy

N

iyyy

CC

CC

Under Gaussian assumptions this is easy to compute:

groupposterior covariance

individualposterior covariances

groupposterior mean

individual posterior covariances and means

See:spm_dcm_average.m Neumann & Lohmann, NeuroImage 2003

Inference about DCM parameters:group analysis (classical)

• 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

Separate fitting of identical models for each subject

Selection of bilinear parameters of interestSelection of bilinear

parameters of interest

one-sample t-test:

parameter > 0 ?

one-sample t-test:

parameter > 0 ?

paired t-test: parameter 1 > parameter 2 ?

paired t-test: parameter 1 > parameter 2 ?

rmANOVA: e.g. in case of

multiple sessions per subject

rmANOVA: e.g. in case of

multiple sessions per subject

Model comparison and selection

Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?

Which model represents the best balance between model fit and model complexity?

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

Pitt & Miyung (2002), TICS

Bayesian Model Selection

Bayes theorem:

Model evidence:

The log model evidence can be represented as:

Bayes factor:

)|(

)|(),|(),|(

myp

mpmypmyp

dmpmypmyp )|(),|()|(

)(

)()|(log

mcomplexity

maccuracymyp

)|(

)|(

jmyp

imypBij

Penny et al. 2004, NeuroImage

Approximations to model evidence

Laplace approximation:

Akaike information criterion (AIC):

Bayesian information criterion (BIC):

pmaccuracymyAIC )()|(

Penny et al. 2004, NeuroImage

yppypT

py

eT

e hh

mcomplexitymaccuracyF

||1

|

1

log2

1log

2

1)()(

2

1

))(())((2

1log

2

1

)()(

CCθθCθθ

θyCθyC

yppypT

py

eT

e hh

mcomplexitymaccuracyF

||1

|

1

log2

1log

2

1)()(

2

1

))(())((2

1log

2

1

)()(

CCθθCθθ

θyCθyC

SNp

maccuracymyBIC log2

)()|(

Unfortunately, the complexity term depends on the prior density, which is determined individually for each model to ensure stability. Therefore, we need other approximations to the model evidence.

The DCM cycle

Design a study thatallows 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

Hypothesis abouta neural system

Definition of DCMs as systemmodels

Data acquisition

Planning a DCM-compatible study• Suitable experimental design:

– preferably multi-factorial (e.g. 2 x 2)– e.g. one factor that varies the driving (sensory) input– and one factor that varies the contextual input

• TR:– as short as possible (optimal: < 2 s)

• Hypothesis and model:– define specific a priori hypothesis– which parameters are relevant to test this hypothesis?– ensure that intended model is suitable to test this hypothesis → simulations before experiment– define criteria for inference

Timing problems at long TRs• Two potential timing problems

in DCM:1. wrong timing of inputs2. temporal shift between

regional time series because of multi-slice acquisition• DCM is robust against timing errors up to approx. ± 1 s

– compensatory changes of σ and θh

• Possible corrections:– slice-timing (not for long TRs)– restriction of the model to neighbouring regions– in both cases: adjust temporal reference bin in SPM

defaults (defaults.stats.fmri.t0)

1

2

slic

e a

cquis

itio

n

visualinput

Practical steps of a DCM study - I

1. Conventional SPM analysis (subject-specific)• DCMs are fitted separately for each session

→ consider concatenation of sessions or adequate 2nd level analysis

2. Definition of the model (on paper!)• Structure: which areas, connections and inputs?• Which parameters represent my hypothesis?• How can I demonstrate the specificity of my results?• What are the alternative models to test?

3. Defining criteria for inference:• single-subject analysis: stat. threshold? contrast?• group analysis: which 2nd-level model?

Practical steps of a DCM study - II

4. Extraction of time series, e.g. via VOI tool in SPM• cave: anatomical & functional standardisation

important for group analyses!

5. Possibly definition of a new design matrix, if the “normal” design matrix does not represent the inputs appropriately.• NB: DCM only reads timing information of each input

from the design matrix, no parameter estimation necessary.

6. Definition of model• via DCM-GUI or directly

in MATLAB

7. DCM parameter estimation• cave: models with many regions & scans can

crash MATLAB!

8. Model comparison and selection:• Which of all models considered is the optimal

one? Bayesian model selection tool

9. Testing the hypothesis Statistical test onthe relevant parametersof the optimal model

Practical steps of a DCM study - III

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 different conditions

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

F - fixation point onlyA - motion stimuli with attention (detect changes)N - motion stimuli without attentionS - no motion

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

V5+

PPCV3A

Attention – No attention

Attention to motion in the visual system

V1

IFG

V5

SPC

Motion

Photic

Attention

0.88

0.48

0.37

0.72

0.42

0.66

0.56

0.55•Visual inputs drive V1, Visual inputs drive V1,

activity then spreads to activity then spreads to hierarchically arranged hierarchically arranged visual areas.visual areas.

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

•The intrinsic connection The intrinsic connection V1→V5 is insignificant in V1→V5 is insignificant in the absence of motion the absence of motion (a(a2121=-0.05).=-0.05).

•Attention increases the Attention increases the backward-connections backward-connections IFGIFG→SPC and SPC→V5. →SPC and SPC→V5.

A simple DCM of the visual system

0.26

-0.05

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

V1

V5

SPC

Motion

Photic

Attention

0.85

0.57 -0.02

1.360.70

0.84

0.23

V1

V5

SPC

Motion

PhoticAttention

0.86

0.56 -0.02

1.42

0.550.75

0.89V1

V5

SPC

Motion

PhoticAttention

0.85

0.57 -0.02

1.36

0.030.70

0.85

Attention0.23

Model 1:attentional modulationof V1→V5

Model 2:attentional modulationof SPC→V5

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

Comparison of three simple models

Bayesian model selection: Model 1 better than model 2,

model 1 and model 3 equal

→ Decision for model 1: in this experiment, attention

primarily modulates V1→V5

ATT_MOT

V1

V5

VIS STIM

V5

V1+

X

DCM Neurophysiology

Psycho-physiologicalinteraction = bilinear effect

Physio-physiologicalinteractions in DCMThe parameter estimation scheme can currently not deal with physio-physiologicalinteractions.

We aim to implement this in the near future, however.

V1

V5

VIS STIM

V5

V1+

SPCSPC

CuzBuAz jj )( CuzBuAz j

j )(

CuDzzAzz T CuDzzAzz T

Physio-physiologicalinteraction = quadratic effect

Overview

• Classical approaches to functional & effective connectivity

• Generic concepts of system analysis

• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation

– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example

• DCM for ERPs (slides provided by Stefan Kiebel)

Neural mass modelNeuronal assembly

Time [ms] v [mV]

Mean firing rate m(t)

Mean firing rate m(t)

Mean membrane potential

v(t)

Mean membrane potential

v(t)

Mean firing rate m(t)

Mean firing rate m(t)

mh

Jansen‘s model for a cortical areaExcitatory

InterneuronsHe, e

PyramidalCellsHe, e

InhibitoryInterneurons

Hi, i

Extrinsic inputs

Excitatory connection

Inhibitory connection

e, i : synaptic time constant (excitatory and inhibitory) He, Hi: synaptic efficacy (excitatory and inhibitory) 1,…,: connectivity constants

21

43

MEG/EEGsignal

MEG/EEGsignal

Parameters:Parameters:

Jansen & Rit, Biol. Cybern., 1995Jansen & Rit, Biol. Cybern., 1995

11

121211

12))((

xv

vxvSpH

xeee

e

22

222322

12)(

xv

vxvSH

xiii

i

Output : y(t)=v1-v2

MEG/EEG signal = dendritic signal of pyramidal cells

33

3232133

12)(

xv

vxvvSH

xeee

e

1v

2v 3v

Input : p(t) cortical noise

Jansen‘s model for a cortical area

Jansen & Rit, Biol. Cybern., 1995Jansen & Rit, Biol. Cybern., 1995

Connectivity between areas

1 2

1 2 1 2 1 2

Cor

tex

Bottom-up Top-Down Lateral

Supra granularSupra granular

Layer IVLayer IV

Infra granularInfra granular

Felleman & Van Essen, Cereb. Cortex, 1991Felleman & Van Essen, Cereb. Cortex, 1991

Inh.Inter.

Inh.Inter.

Exc.Inter.

abu

Exc.Inter.

Pyr.Cells

Inh.Inter.

Pyr.Cells

Inh.Inter.

Exc.Inter.

atd

Exc.Inter.

Pyr.Cells

Inh.Inter.

Pyr.Cells

Exc.Inter.

ala

Exc.Inter.

Pyr.Cells

Inh.Inter.

Pyr.Cells

Pyramidal cellsInhibitory interneurons

Excitatory interneurons

Pyramidal cellsInhibitory interneurons

Area 1 Area 2 Area 1 Area 2 Area 1 Area 2

Connectivity between areas

Cor

tex

Supra granularSupra granular

Layer IVLayer IV

Infra granularInfra granular

Bottom-up Top-Down Lateral

David et al., NeuroImage, in revisionDavid et al., NeuroImage, in revision

Network of areas MEG/EEG scalp data

Input (Stimuli)

Estimates of model parameters

Modulation of connectivityModulation of connectivity differences between ERP/ERFsdifferences between ERP/ERFs

Dynamic causal modelling

Spatial model

Depolarisation ofpyramidal cells

Forward model

Scalp data

)(fV

K)(gKVY

Forward model

Estimation of model parameters

)(p

)()|()|( pypyp

)|( yp

Parameters• Neurodynamics

• Connections (stability)

)|(maxargˆ yp

Known parameters: Source locations Network connectionsGain matrix K

Source locations Network connectionsGain matrix K

Unknown parameters:

Synaptic time constants and efficaciesCoupling parametersPropagation delays between areasInput parameters

Synaptic time constants and efficaciesCoupling parametersPropagation delays between areasInput parameters

Bayesian estimationModel components:•Neural mass•Spatial forward model

Model components:•Neural mass•Spatial forward model

Model components:•Neurodynamics•Connections

Model components:•Neurodynamics•Connections

Friston et al., NeuroImage, 2003Friston et al., NeuroImage, 2003

Expectation/Maximization