DCM: Advanced topics

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

Wellcome Trust Centre for NeuroimagingInstitute of NeurologyUniversity College London

Methods & models for fMRI data analysis in Neuroeconomics, April 2010

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 (%)

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 (%)

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

x1 x2

x3

CuxDxBuAdtdx n

j

jj

m

i

ii

1

)(

1

)(

u1

u2

Overview

• Bayesian model selection (BMS) • Nonlinear DCM for fMRI• Integrating tractography and DCM

Model comparison and selectionGiven 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

mypqKL

mpqKL

myp

dmpmypmyp

,|,|,

),|(log

)|(),|()|(

Model evidence:

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

Bayesian model selection (BMS)

)|()|(

2

1

mypmypBF

Model comparison via Bayes factor:accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

all possible datasets

y

p(y|

m)

Gharamani, 2004

Penny et al. 2004, NeuroImage Stephan et al. 2007, NeuroImage

pmypAIC ),|(log

Logarithm is a monotonic function

Maximizing log model evidence= Maximizing model evidence

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

mcomplexitymypmcomplexitymaccuracymyp

In SPM2 & SPM5, interface offers 2 approximations:

NpmypBIC log2

),|(log

Akaike Information Criterion:

Bayesian Information Criterion:

Log model evidence = balance between fit and complexity

Penny et al. 2004, NeuroImage

Approximations to the model evidence in DCM

No. of parameters

No. ofdata points

AIC favours more complex models,BIC favours simpler models.

Bayes factors

)|()|(

2

112 myp

mypB positive value, [0;[

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) Evidence1 to 3 50-75% weak

3 to 20 75-95% positive20 to 150 95-99% strong 150 99% Very strong

Kass & Raftery classification:

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

The negative free energy approximation

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

mypqKLF

mypqKLmpqKLmypmyp

,|,

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

mypqKLmypF ,|,)|(log

The complexity term in F• In contrast to AIC & BIC, the complexity term of the negative

free energy F accounts for parameter interdependencies.

• 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

Tyy CCC

mpqKL

|1

|| 21ln

21ln

21

)|(),(

V1 V5stim

PPCM2

attention

V1 V5stim

PPCM1

attention

V1 V5stim

PPCM3attention

V1 V5stim

PPCM4attention

BF 2966F = 7.995

M2 better than M1

BF 12F = 2.450

M3 better than M2

BF 23F = 3.144

M4 better than M3

M1 M2 M3 M4

BMS in SPM8: an example

Fixed effects BMS at group level

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

Average Bayes factor (ABF):

Problems:- blind with regard to group heterogeneity- sensitive to outliers

k

kijij BFGBF )(

( )kKij ij

k

ABF BF

)|(~ 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 population

Dirichlet distribution of model probabilities

Multinomial distribution of model labels

Measured data y

Model inversion by Variational Bayes (VB)

Stephan et al. 2009, NeuroImage

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

Subj

ects

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

m1m2

Stephan et al. 2009, NeuroImage

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

m1m2

1

1

11.884.3%r

2

2

2.215.7%r

%7.9921 rrp

0 2 4 6 8 10 12 145

10

15

20

25

30

1

F

Sampling ApproachVariational Bayes

1 11.8

Validation of VB estimates by sampling

Simulation study: sampling subjects from a heterogenous population• Population where 70% of all

subjects' data are generated by model m1 and 30% by model m2

• Random sampling of subjects from this population and generating synthetic data with observation noise

• Fitting both m1 and m2 to all data sets and performing BMS

MOG

LG LG

RVFstim.

LVFstim.

FGFG

LD|RVF

LD|LVF

LD LD

MOG

MOG

LG LG

RVFstim.

LVFstim.

FGFG

LD

LD

LD|RVF LD|LVF

MOG

m1

m2

Stephan et al. 2009, NeuroImage

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

100

200

300

400

500

600

700

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

2

4

6

8

10

12

14

16

18

m1 m2

m1 m2 m1 m2

log GBF12

<r>

true values:1=220.7=15.42=220.3=6.6

mean estimates:1=15.4, 2=6.6

true values:r1 = 0.7, r2=0.3

mean estimates:r1 = 0.7, r2=0.3

true values:1 = 1, 2=0

mean estimates:1 = 0.89, 2=0.11

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) = 0.986

Model space partitioning:

Nonlinear hemodynamic models

vs. linear ones

0

2

4

6

8

10

12

14

16

alph

a

0

2

4

6

8

10

12

alph

a

0

20

40

60

80

Sum

med

log

evid

ence

(rel

. to

RB

ML)

CBMN CBMN(ε) CBML CBML(ε)RBMN RBMN(ε) RBML RBML(ε)

CBMN CBMN(ε) CBML CBML(ε)RBMN RBMN(ε) RBML RBML(ε)

nonlinear models linear models

FFX

RFX

4

1

*1

kk

8

5

*2

kk

nonlinear linear

log GBF

Model space partitioning

1 73.5%r 2 26.5%r

1 2 98.6%p r r

m1 m2

m1m2

Stephan et al. 2009, NeuroImage

Overview

• Bayesian model selection (BMS) • Nonlinear DCM for fMRI• Integrating tractography and DCM

),,( uxFx Neural state equation:

Electric/magneticforward model:

neural activityEEGMEGLFP

(linear)

Dynamic causal modelling (DCM)

Neural model:1 state variable per regionbilinear state equationno propagation delays

Neural model:8 state variables per region

nonlinear state equationpropagation delays

fMRI ERPs

inputs

Hemodynamicforward model:neural activityBOLD(nonlinear)

intrinsic 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

Stephan & Friston (2007),Handbook of Brain Connectivity

bilinear DCM

CuxDxBuAdtdx m

i

n

j

jj

ii

1 1

)()(CuxBuAdtdx m

i

ii

1

)(

Bilinear state equation:

driving input

modulation

non-linear DCM

driving input

modulation

...)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

Nonlinear DCM: Attention to motion

V1 IFG

V5

SPC

Motion

Photic

Attention

.82(100%)

.42(100%)

.37(90%)

.69 (100%).47

(100%)

.65 (100%)

.52 (98%)

.56(99%)

Stimuli + Task

250 radially moving dots (4.7 °/s)Conditions:F – fixation onlyA – motion + attention (“detect changes”)N – motion without attentionS – stationary dots

Previous bilinear DCM

Friston et al. (2003)

Friston et al. (2003):attention modulates backward connections IFG→SPC and SPC→V5.

Q: Is a nonlinear mechanism (gain control) a better explanation of the data?

Büchel & Friston (1997)

modulation of back-ward or forward connection?

additional drivingeffect of attentionon PPC?

bilinear or nonlinearmodulation offorward connection?

V1 V5stim

PPCM2

attention

V1 V5stim

PPCM1

attention

V1 V5stim

PPCM3attention

V1 V5stim

PPCM4attention

BF = 2966M2 better than M1

M3 better than M2BF = 12

M4 better than M3BF = 23

Stephan et al. 2008, NeuroImage

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

Stephan et al. 2008, NeuroImage

V1V5PPC

observedfitted

motion &attention

motion &no attention

static dots

Learning of dynamic audio-visual associations

CS Response

Time (ms)

0 200 400 600 800 2000 ± 650

or

Target StimulusConditioning Stimulus

or

TS

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

p(fa

ce)

trial

CS1

CS2

den Ouden et al. 2010, J. Neurosci .

Hierarchical Bayesian learning model

observed events

probabilistic association

volatility

k

vt-1 vt

rt rt+1

ut ut+1

)exp(,~,|1 ttttt vrDirvrrp

)exp(,~,|1 kvNkvvp ttt

1kp

1: 1

1 1 1 1 1 1: 1 1 1

1:

1: 1

1: 1

prediction: , ,

, , , ,

update: , ,

, ,

, ,

t t t

t t t t t t t t t t

t t t

t t t t t

t t t t t t t

p r v K u

p r r v p v v K p r v K u dr dv

p r v K u

p r v K u p u r

p r v K u p u r dr dv dK

Behrens et al. 2007, Nat. Neurosci.

400 440 480 520 560 6000

0.2

0.4

0.6

0.8

1

Trial

p(F)

Comparison of different learning models

400 440 480 520 560 6000

0.2

0.4

0.6

0.8

1

Trial

p(F)

TrueBayes VolHMM fixedHMM learnRW

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Categoricalmodel

Bayesianlearner

HMM (fixed) HMM (learn) Rescorla-Wagner

Exce

edan

ce p

rob.

Bayesian model selection: hierarchical Bayesian learner performs best

Alternative learning models: True probabilitiesRescorla-WagnerHidden Markov models (2 variants)

0.1 0.3 0.5 0.7 0.9390

400

410

420

430

440

450

RT

(ms)

p(outcome)

Reaction times

den Ouden et al. 2010, J. Neurosci .

Putamen Premotor cortex

Stimulus-independent prediction error

p < 0.05 (SVC)

p < 0.05 (cluster-level whole- brain corrected)

p(F) p(H)-2

-1.5

-1

-0.5

0

BO

LD re

sp. (

a.u.

)

p(F) p(H)-2

-1.5

-1

-0.5

0

BO

LD re

sp. (

a.u.

)

den Ouden et al. 2010, J. Neurosci .

Prediction error (PE) activity in the putamen

PE during reinforcement learning

PE during incidental sensory learning

O'Doherty et al. 2004, Science

den Ouden et al. 2009, Cerebral Cortex

According to the FEP (and other learning theories):synaptic plasticity during learning = PE dependent changes in connectivity

PPA FFA

PMd

p(F)p(H)

PUT

d = 0.010 0.003p = 0.010

Prediction error gates visuo-motor connections

d = 0.011 0.004p = 0.017

• Modulation of visuo-motor connections by striatal PE activity

• Influence of visual areas on premotor cortex:– stronger for

surprising stimuli – weaker for

expected stimuli

den Ouden et al. 2010, J. Neurosci .

Overview

• Bayesian model selection (BMS) • Nonlinear DCM for fMRI• Integrating tractography and DCM

Diffusion-weighted imaging

Parker & Alexander, 2005, Phil. Trans. B

Probabilistic tractography: Kaden et al. 2007, NeuroImage

• computes local fibre orientation density by spherical deconvolution of the diffusion-weighted signal• estimates the spatial probability

distribution of connectivity from given seed regions• anatomical connectivity = proportion

of fibre pathways originating in a specific source region that intersect a target region • If the area or volume of the source

region approaches a point, this measure reduces to method by Behrens et al. (2003)

R2R1

R2R1

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

low probability of anatomical connection small prior variance of effective connectivity parameter

high probability of anatomical connection large prior variance of effective connectivity parameter

Integration of tractography and DCM

Stephan, Tittgemeyer et al. 2009, NeuroImage

LG(x1)

LG(x2)

RVFstim.

LVFstim.

FG(x4)

FG(x3)

LD|LVF

LD LD

BVFstim.

LD|RVF DCM structure

LGleft

LGright

FGright

FGleft

* 313

13

5.37 1015.7%

* 334

34

2.23 106.5%

* 224

24

1.50 1043.6%

* 212

12

1.17 1034.2%

anatomical connectivity

probabilistictractography

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

6.5%0.0384v

15.7%0.1070v

34.2%0.5268v

43.6%0.7746v

connection-specific priors for coupling parameters

0 0.5 10

0.5

1m1: a=-32,b=-32

0 0.5 10

0.5

1m2: a=-16,b=-32

0 0.5 10

0.5

1m3: a=-16,b=-28

0 0.5 10

0.5

1m4: a=-12,b=-32

0 0.5 10

0.5

1m5: a=-12,b=-28

0 0.5 10

0.5

1m6: a=-12,b=-24

0 0.5 10

0.5

1m7: a=-12,b=-20

0 0.5 10

0.5

1m8: a=-8,b=-32

0 0.5 10

0.5

1m9: a=-8,b=-28

0 0.5 10

0.5

1m10: a=-8,b=-24

0 0.5 10

0.5

1m11: a=-8,b=-20

0 0.5 10

0.5

1m12: a=-8,b=-16

0 0.5 10

0.5

1m13: a=-8,b=-12

0 0.5 10

0.5

1m14: a=-4,b=-32

0 0.5 10

0.5

1m15: a=-4,b=-28

0 0.5 10

0.5

1m16: a=-4,b=-24

0 0.5 10

0.5

1m17: a=-4,b=-20

0 0.5 10

0.5

1m18: a=-4,b=-16

0 0.5 10

0.5

1m19: a=-4,b=-12

0 0.5 10

0.5

1m20: a=-4,b=-8

0 0.5 10

0.5

1m21: a=-4,b=-4

0 0.5 10

0.5

1m22: a=-4,b=0

0 0.5 10

0.5

1m23: a=-4,b=4

0 0.5 10

0.5

1m24: a=0,b=-32

0 0.5 10

0.5

1m25: a=0,b=-28

0 0.5 10

0.5

1m26: a=0,b=-24

0 0.5 10

0.5

1m27: a=0,b=-20

0 0.5 10

0.5

1m28: a=0,b=-16

0 0.5 10

0.5

1m29: a=0,b=-12

0 0.5 10

0.5

1m30: a=0,b=-8

0 0.5 10

0.5

1m31: a=0,b=-4

0 0.5 10

0.5

1m32: a=0,b=0

0 0.5 10

0.5

1m33: a=0,b=4

0 0.5 10

0.5

1m34: a=0,b=8

0 0.5 10

0.5

1m35: a=0,b=12

0 0.5 10

0.5

1m36: a=0,b=16

0 0.5 10

0.5

1m37: a=0,b=20

0 0.5 10

0.5

1m38: a=0,b=24

0 0.5 10

0.5

1m39: a=0,b=28

0 0.5 10

0.5

1m40: a=0,b=32

0 0.5 10

0.5

1m41: a=4,b=-32

0 0.5 10

0.5

1m42: a=4,b=0

0 0.5 10

0.5

1m43: a=4,b=4

0 0.5 10

0.5

1m44: a=4,b=8

0 0.5 10

0.5

1m45: a=4,b=12

0 0.5 10

0.5

1m46: a=4,b=16

0 0.5 10

0.5

1m47: a=4,b=20

0 0.5 10

0.5

1m48: a=4,b=24

0 0.5 10

0.5

1m49: a=4,b=28

0 0.5 10

0.5

1m50: a=4,b=32

0 0.5 10

0.5

1m51: a=8,b=12

0 0.5 10

0.5

1m52: a=8,b=16

0 0.5 10

0.5

1m53: a=8,b=20

0 0.5 10

0.5

1m54: a=8,b=24

0 0.5 10

0.5

1m55: a=8,b=28

0 0.5 10

0.5

1m56: a=8,b=32

0 0.5 10

0.5

1m57: a=12,b=20

0 0.5 10

0.5

1m58: a=12,b=24

0 0.5 10

0.5

1m59: a=12,b=28

0 0.5 10

0.5

1m60: a=12,b=32

0 0.5 10

0.5

1m61: a=16,b=28

0 0.5 10

0.5

1m62: a=16,b=32

0 0.5 10

0.5

1m63 & m64

0

01 exp( )ijij

Connection-specific prior variance as a function of anatomical connection probability

• 64 different mappings by systematic search across hyper-parameters and

• yields anatomically informed (intuitive and counterintuitive) and uninformed priors

0 10 20 30 40 50 600

200

400

600

model

log

grou

p B

ayes

fact

or

0 10 20 30 40 50 60

680

685

690

695

700

model

log

grou

p B

ayes

fact

or

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

model

post

. mod

el p

rob.

0 0.5 10

0.5

1m1: a=-32,b=-32

0 0.5 10

0.5

1m2: a=-16,b=-32

0 0.5 10

0.5

1m3: a=-16,b=-28

0 0.5 10

0.5

1m4: a=-12,b=-32

0 0.5 10

0.5

1m5: a=-12,b=-28

0 0.5 10

0.5

1m6: a=-12,b=-24

0 0.5 10

0.5

1m7: a=-12,b=-20

0 0.5 10

0.5

1m8: a=-8,b=-32

0 0.5 10

0.5

1m9: a=-8,b=-28

0 0.5 10

0.5

1m10: a=-8,b=-24

0 0.5 10

0.5

1m11: a=-8,b=-20

0 0.5 10

0.5

1m12: a=-8,b=-16

0 0.5 10

0.5

1m13: a=-8,b=-12

0 0.5 10

0.5

1m14: a=-4,b=-32

0 0.5 10

0.5

1m15: a=-4,b=-28

0 0.5 10

0.5

1m16: a=-4,b=-24

0 0.5 10

0.5

1m17: a=-4,b=-20

0 0.5 10

0.5

1m18: a=-4,b=-16

0 0.5 10

0.5

1m19: a=-4,b=-12

0 0.5 10

0.5

1m20: a=-4,b=-8

0 0.5 10

0.5

1m21: a=-4,b=-4

0 0.5 10

0.5

1m22: a=-4,b=0

0 0.5 10

0.5

1m23: a=-4,b=4

0 0.5 10

0.5

1m24: a=0,b=-32

0 0.5 10

0.5

1m25: a=0,b=-28

0 0.5 10

0.5

1m26: a=0,b=-24

0 0.5 10

0.5

1m27: a=0,b=-20

0 0.5 10

0.5

1m28: a=0,b=-16

0 0.5 10

0.5

1m29: a=0,b=-12

0 0.5 10

0.5

1m30: a=0,b=-8

0 0.5 10

0.5

1m31: a=0,b=-4

0 0.5 10

0.5

1m32: a=0,b=0

0 0.5 10

0.5

1m33: a=0,b=4

0 0.5 10

0.5

1m34: a=0,b=8

0 0.5 10

0.5

1m35: a=0,b=12

0 0.5 10

0.5

1m36: a=0,b=16

0 0.5 10

0.5

1m37: a=0,b=20

0 0.5 10

0.5

1m38: a=0,b=24

0 0.5 10

0.5

1m39: a=0,b=28

0 0.5 10

0.5

1m40: a=0,b=32

0 0.5 10

0.5

1m41: a=4,b=-32

0 0.5 10

0.5

1m42: a=4,b=0

0 0.5 10

0.5

1m43: a=4,b=4

0 0.5 10

0.5

1m44: a=4,b=8

0 0.5 10

0.5

1m45: a=4,b=12

0 0.5 10

0.5

1m46: a=4,b=16

0 0.5 10

0.5

1m47: a=4,b=20

0 0.5 10

0.5

1m48: a=4,b=24

0 0.5 10

0.5

1m49: a=4,b=28

0 0.5 10

0.5

1m50: a=4,b=32

0 0.5 10

0.5

1m51: a=8,b=12

0 0.5 10

0.5

1m52: a=8,b=16

0 0.5 10

0.5

1m53: a=8,b=20

0 0.5 10

0.5

1m54: a=8,b=24

0 0.5 10

0.5

1m55: a=8,b=28

0 0.5 10

0.5

1m56: a=8,b=32

0 0.5 10

0.5

1m57: a=12,b=20

0 0.5 10

0.5

1m58: a=12,b=24

0 0.5 10

0.5

1m59: a=12,b=28

0 0.5 10

0.5

1m60: a=12,b=32

0 0.5 10

0.5

1m61: a=16,b=28

0 0.5 10

0.5

1m62: a=16,b=32

0 0.5 10

0.5

1m63 & m64

Stephan, Tittgemeyer et al. 2009, NeuroImage

Further reading: Methods papers on DCM for fMRI – part 1

• Chumbley JR, Friston KJ, Fearn T, Kiebel SJ (2007) A Metropolis-Hastings algorithm for dynamic causal models. Neuroimage 38:478-487.

• Daunizeau J, David, O, Stephan KE (2010) Dynamic Causal Modelling: A critical review of the biophysical and statistical foundations. NeuroImage, in press.

• Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuroimage 19:1273-1302.

• Kasess CH, Stephan KE, Weissenbacher A, Pezawas L, Moser E, Windischberger C (2010) Multi-Subject Analyses with Dynamic Causal Modeling. NeuroImage 49:3065-3074.

• Kiebel SJ, Kloppel S, Weiskopf N, Friston KJ (2007) Dynamic causal modeling: a generative model of slice timing in fMRI. Neuroimage 34:1487-1496.

• Marreiros AC, Kiebel SJ, Friston KJ (2008) Dynamic causal modelling for fMRI: a two-state model. Neuroimage 39:269-278.

• Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Comparing dynamic causal models. Neuroimage 22:1157-1172.

• Penny WD, Stephan KE, Mechelli A, Friston KJ (2004b) Modelling functional integration: a comparison of structural equation and dynamic causal models. Neuroimage 23 Suppl 1:S264-274.

Further reading: Methods papers on DCM for fMRI – part 2

• Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI responses. Curr Opin Neurobiol 14:629-635.

• Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM. Neuroimage 38:387-401.

• Stephan KE, Harrison LM, Kiebel SJ, David O, Penny WD, Friston KJ (2007) Dynamic causal models of neural system dynamics: current state and future extensions. J Biosci 32:129-144.

• Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM. Neuroimage 38:387-401.

• Stephan KE, Kasper L, Harrison LM, Daunizeau J, den Ouden HE, Breakspear M, Friston KJ (2008) Nonlinear dynamic causal models for fMRI. Neuroimage 42:649-662.

• Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009) Bayesian model selection for group studies. Neuroimage 46:1004-1017.

• Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009) Tractography-based priors for dynamic causal models. Neuroimage 47: 1628-1638.

• Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, Friston KJ (2010) Ten simple rules for Dynamic Causal Modelling. NeuroImage 49: 3099-3109.

),,( uxFx Neural state equation:

Electric/magneticforward model:

neural activityEEGMEGLFP

(linear)

Dynamic causal modelling (DCM)

Neural model:1 state variable per regionbilinear state equationno propagation delays

Neural model:8 state variables per region

nonlinear state equationpropagation delays

fMRI ERPs

inputs

Hemodynamicforward model:neural activityBOLD(nonlinear)

Take-home messages• Bayesian model selection (BMS):

generic approach to selecting an optimal model from a set of competing models

• random effects BMS for group studies:posterior model probabilities and exceedance probabilities

• nonlinear DCM:enables one to investigate synaptic gating processes via activity-dependent changes in connection strengths

• DCM & tractography:probabilities of anatomical connections can be used to inform the prior variance of DCM coupling parameters

• DCM implementations do not only exist for fMRI data, but also for electrophysiological data

Thank you