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DCM: Advanced Topics 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 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 N euralpopulation 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 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 fM R Isignalchange (% ) x 1 x 2 x 3 x 1 x 2 x 3 Cu x D x B u A dt dx n j j j m i i i 1 ) ( 1 ) ( u 1 u 2 Klaas Enno Stephan Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering, University of Zurich & ETH Zurich Laboratory for Social & Neural Systems Research (SNS), University of Zurich Wellcome Trust Centre for Neuroimaging, University College London Methods & models for fMRI data analysis November 2012

DCM: Advanced Topics

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Page 1: DCM:   Advanced Topics

DCM: Advanced Topics

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fMRI signal change (%)

x1 x2

x3

x1 x2

x3

CuxDxBuAdt

dx n

j

jj

m

i

ii

1

)(

1

)(

u1

u2

Klaas Enno Stephan

Translational Neuromodeling Unit (TNU)Institute for Biomedical Engineering, University of Zurich & ETH Zurich

Laboratory for Social & Neural Systems Research (SNS), University of Zurich

Wellcome Trust Centre for Neuroimaging, University College London

Methods & models for fMRI data analysisNovember 2012

Page 2: DCM:   Advanced Topics

Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions

Page 3: DCM:   Advanced Topics

),,( uxFdt

dx

Neural state equation:

Electromagneticforward model:

neural activityEEGMEGLFP

Dynamic Causal Modeling (DCM)

simple neuronal modelcomplicated forward model

complicated neuronal modelsimple forward model

fMRIfMRI EEG/MEGEEG/MEG

inputs

Hemodynamicforward model:neural activityBOLD

Page 4: DCM:   Advanced Topics

Generative models & model selection

• any DCM = a particular generative model of how the data (may) have been caused

• modelling = comparing competing hypotheses about the mechanisms underlying observed data

a priori definition of hypothesis set (model space) is crucial

determine the most plausible hypothesis (model), given the data

• model selection model validation!

model validation requires external criteria (external to the measured data)

Page 5: DCM:   Advanced Topics

Model comparison and selection

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

Page 6: DCM:   Advanced Topics

( | ) ( | , ) ( | ) p y m p y m p m d

Model evidence:

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

Bayesian model selection (BMS)

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

McKay 1992, Neural Comput.Penny et al. 2004a, NeuroImage

Page 7: DCM:   Advanced Topics

pmypAIC ),|(log

Logarithm is a monotonic function

Maximizing log model evidence= Maximizing model evidence

)(),|(log

)()( )|(log

mcomplexitymyp

mcomplexitymaccuracymyp

SPM2 & SPM5 offered 2 approximations:

Np

mypBIC log2

),|(log

Akaike Information Criterion:

Bayesian Information Criterion:

Log model evidence = balance between fit and complexity

Penny et al. 2004a, NeuroImagePenny 2012, NeuroImage

Approximations to the model evidence in DCM

No. of parameters

No. ofdata points

Page 8: DCM:   Advanced Topics

The (negative) free energy approximation

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

log ( | )

log ( | , ) , | , | ,

p y m

p y m KL q p m KL q p y m

log ( | ) , | ,

log ( | , ) , |accuracy complexity

F p y m KL q p y m

p y m KL q p m

Page 9: DCM:   Advanced Topics

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: Since SPM8, only F is used for model selection !

y

Tyy CCC

mpqKL

|1

|| 2

1ln

2

1ln

2

1

)|(),(

Page 10: DCM:   Advanced Topics

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 could 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.

Page 11: DCM:   Advanced Topics

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

Page 12: DCM:   Advanced Topics

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

Page 13: DCM:   Advanced Topics

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

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

)|(~ pmpm kk

);(~ rDirr

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

Random effects BMS for heterogeneous groups

Dirichlet parameters = “occurrences” of models in the population

Dirichlet distribution of model probabilities r

Multinomial distribution of model labels m

Measured data y

Model inversion by Variational Bayes (VB) or MCMC

Stephan et al. 2009a, NeuroImagePenny et al. 2010, PLoS Comp. Biol.

Page 14: DCM:   Advanced Topics

-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

m1m2

Data: Stephan et al. 2003, ScienceModels: Stephan et al. 2007, J. Neurosci.

Page 15: DCM:   Advanced Topics

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

11.8

84.3%r

2

2

2.2

15.7%r

%7.9921 rrp

Stephan et al. 2009a, NeuroImage

Page 16: DCM:   Advanced Topics

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:

comparing model families

0

2

4

6

8

10

12

14

16

alp

ha

0

2

4

6

8

10

12

alp

ha

0

20

40

60

80

Su

mm

ed

log

ev

ide

nc

e (

rel.

to R

BM

L)

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

Page 17: DCM:   Advanced Topics

Comparing model families – a second example

• data from Leff et al. 2008, J. Neurosci

• one driving input, one modulatory input

• 26 = 64 possible modulations

• 23 – 1 input patterns

• 764 = 448 models

• integrate out uncertainty about modulatory patterns and ask where auditory input enters

Penny et al. 2010, PLoS Comput. Biol.

Page 18: DCM:   Advanced Topics

Bayesian Model Averaging (BMA)

• abandons dependence of parameter inference on a single model

• uses the entire model space considered (or an optimal family of models)

• computes average of each parameter, weighted by posterior model probabilities

• represents a particularly useful alternative– when none of the models (or model

subspaces) considered clearly outperforms all others

– when comparing groups for which the optimal model differs

1..

1..

|

| , |

n N

n n Nm

p y

p y m p m y

NB: p(m|y1..N) can be obtained by either FFX or RFX BMS

Penny et al. 2010, PLoS Comput. Biol.

Page 19: DCM:   Advanced Topics

inference on model structure or inference on model parameters?

inference on individual models or model space partition?

comparison of model families using

FFX or RFX BMS

comparison of model families using

FFX or RFX BMS

optimal model structure assumed to be identical across subjects?

FFX BMSFFX BMS RFX BMSRFX BMS

yes no

inference on parameters of an optimal model or parameters of all models?

BMABMA

definition of model spacedefinition of model space

FFX analysis of parameter estimates

(e.g. BPA)

FFX analysis of parameter estimates

(e.g. BPA)

RFX analysis of parameter estimates(e.g. t-test, ANOVA)

RFX analysis of parameter estimates(e.g. t-test, ANOVA)

optimal model structure assumed to be identical across subjects?

FFX BMS

yes no

RFX BMS

Stephan et al. 2010, NeuroImage

Page 20: DCM:   Advanced Topics

Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions

Page 21: DCM:   Advanced Topics

DCM10 in SPM8

• DCM10 was released as part of SPM8 in July 2010 (version 4010).

• Introduced many new features, incl. two-state DCMs and stochastic DCMs

• This led to various changes in model defaults, e.g.– inputs mean-centred– changes in coupling priors– self-connections estimated separately for each area

• For details, see: www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf

• Further changes in version 4290 (released April 2011) to accommodate new developments and give users more choice (e.g., whether or not to mean-centre inputs).

Page 22: DCM:   Advanced Topics

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– improvements of numerical routines (e.g., 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 (release number) you are using when publishing papers.

In the next SPM version, the release number will be stored in the DCM.mat.

Page 23: DCM:   Advanced Topics

endogenous connectivity

direct inputs

modulation ofconnectivity

Neural state equation CuxBuAx jj )( )(

u

xC

x

x

uB

x

xA

j

j

)(

hemodynamicmodelλ

x

y

integration

BOLDyyy

activityx1(t)

activityx2(t) activity

x3(t)

neuronalstates

t

drivinginput u1(t)

modulatoryinput u2(t)

t

The classical DCM:a deterministic, one-state, bilinear model

Page 24: DCM:   Advanced Topics

Factorial structure of model specification in DCM10• Three dimensions of model specification:

– bilinear vs. nonlinear

– single-state vs. two-state (per region)

– deterministic vs. stochastic

• Specification via GUI.

Page 25: DCM:   Advanced Topics

bilinear DCM

CuxDxBuAdt

dx m

i

n

j

jj

ii

1 1

)()(CuxBuA

dt

dx m

i

ii

1

)(

Bilinear state equation:

driving input

modulation

driving input

modulation

non-linear DCM

...)0,(),(2

0

uxux

fu

u

fx

x

fxfuxf

dt

dx

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

Nonlinear state equation:

...2

)0,(),(2

2

22

0

x

x

fux

ux

fu

u

fx

x

fxfuxf

dt

dx

Page 26: DCM:   Advanced Topics

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

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

CuxDxBuAdt

dx n

j

jj

m

i

ii

1

)(

1

)(

Nonlinear dynamic causal model (DCM)

Stephan et al. 2008, NeuroImage

u1

u2

Page 27: DCM:   Advanced Topics

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

0.39

0.26

0.50

0.26

0.10MAP = 1.25

Stephan et al. 2008, NeuroImage

Page 28: DCM:   Advanced Topics

uinput

Single-state DCM

1x

Intrinsic (within-region)

coupling

Extrinsic (between-region)

coupling

NNNN

N

ijijij

x

x

x

uBA

Cuxx

1

1

111

Two-state DCM

Ex1

IN

EN

I

E

IINN

IENN

EENN

EENN

EEN

IIIE

EEN

EIEE

ijijijij

x

x

x

x

x

uBA

Cuxx

1

1

1

1111

11111

00

0

00

0

)exp(

Ix1

I

E

x

x

1

1

Two-state DCM

Marreiros et al. 2008, NeuroImage

Page 29: DCM:   Advanced Topics

Estimates of hidden causes and states(Generalised filtering)

0 200 400 600 800 1000 1200-1

-0.5

0

0.5

1inputs or causes - V2

0 200 400 600 800 1000 1200-0.1

-0.05

0

0.05

0.1hidden states - neuronal

0 200 400 600 800 1000 12000.8

0.9

1

1.1

1.2

1.3hidden states - hemodynamic

0 200 400 600 800 1000 1200-3

-2

-1

0

1

2predicted BOLD signal

time (seconds)

excitatorysignal

flowvolumedHb

observedpredicted

Stochastic DCM

( ) ( )

( )

( )j xjj

v

dxA u B x Cv

dt

v u

Li et al. 2011, NeuroImage

• all states are represented in generalised coordinates of motion

• random state fluctuations w(x) account for endogenous fluctuations,have unknown precision and smoothness two hyperparameters

• fluctuations w(v) induce uncertainty about how inputs influence neuronal activity

• can be fitted to resting state data

Page 30: DCM:   Advanced Topics

Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions

Page 31: DCM:   Advanced Topics

Prediction errors drive synaptic plasticity

McLaren 1989

x1 x2

x3

PE(t)

01

t

t kk

w w PE

0

0

( )t

tw a PE d

0aR

synaptic plasticity during learning = f (prediction error)

Page 32: DCM:   Advanced Topics

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(f

ace)

trial

CS1

CS2

den Ouden et al. 2010, J. Neurosci.

Page 33: DCM:   Advanced Topics

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

Behrens et al. 2007, Nat. Neurosci.

prior on volatility

Page 34: DCM:   Advanced Topics

Explaining RTs by 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

Exc

eed

ance

pro

b.

Bayesian model selection:

hierarchical Bayesian model performs best

5 alternative learning models:

• categorical probabilities

• hierarchical Bayesian learner

• Rescorla-Wagner

• Hidden Markov models (2 variants)

0.1 0.3 0.5 0.7 0.9390

400

410

420

430

440

450

RT

(m

s)

p(outcome)

Reaction times

den Ouden et al. 2010, J. Neurosci.

Page 35: DCM:   Advanced Topics

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 .

Page 36: DCM:   Advanced Topics

Prediction error (PE) activity in the putamen

PE during reinforcement learning

PE during incidentalsensory learning

O'Doherty et al. 2004, Science

den Ouden et al. 2009, Cerebral Cortex

Could the putamen be regulating trial-by-trial changes of task-relevant connections?

PE = “teaching signal” for synaptic plasticity during

learning

p < 0.05 (SVC)

PE during activesensory learning

Page 37: DCM:   Advanced Topics

Prediction errors control plasticity during adaptive cognition

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

surprising stimuli – weaker for expected

stimuli

den Ouden et al. 2010, J. Neurosci .

PPA FFA

PMd

Hierarchical Bayesian learning model

PUT

p = 0.010 p = 0.017

ongoing pharmacological and genetic studies

Page 38: DCM:   Advanced Topics

events in the world

association

volatility

Hierarchical variational Bayesian learning

sensory stimuli

11kx

1kx

12kx

2kx

3kx1

3kx

1ku ku

1 13 3 3| , ~ ,exp( )k k kp x x N x

1p

1 12 2 3 2 3| , ~ ,exp( )k k k k kp x x x N x x

1 2 2| ~k k kp x x Bernoulli S x

1 1 1| ~ 0, 1k k k kp u x MoG x x

Mean-field decomposition

1.. 11 2 3, , |k k k k k kp u x u q x q x q x q

Mathys et al. (2011), Front. Hum. Neurosci.

Page 39: DCM:   Advanced Topics

Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions

Page 40: DCM:   Advanced Topics

Diffusion-weighted imaging

Parker & Alexander, 2005, Phil. Trans. B

Page 41: DCM:   Advanced Topics

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

Page 42: DCM:   Advanced Topics

LG LG

FGFG

DCM

LGleft

LGright

FGright

FGleft

13 15.7%

34 6.5%

24 43.6%

12 34.2%

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

Proof of concept study

connection-specific priors for coupling parameters

Stephan, Tittgemeyer et al. 2009, NeuroImage

Page 43: DCM:   Advanced Topics

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

Page 44: DCM:   Advanced Topics

0 10 20 30 40 50 600

200

400

600

model

log

gro

up

Bay

es f

acto

r

0 10 20 30 40 50 60

680

685

690

695

700

model

log

gro

up

Bay

es f

acto

r

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

model

po

st.

mo

del

pro

b.

Models with anatomically informed priors (of an intuitive form)

Page 45: DCM:   Advanced Topics

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

Models with anatomically informed priors (of an intuitive form) were clearly superior to anatomically uninformed ones: Bayes Factor >109

Page 46: DCM:   Advanced Topics

Overview

• Bayesian model selection (BMS)

• Extended DCM for fMRI: nonlinear, two-state, stochastic

• Embedding computational models in DCMs

• Integrating tractography and DCM

• Applications of DCM to clinical questions

Page 47: DCM:   Advanced Topics

model structure

Model-based predictions for single patients

set of parameter estimates

BMS

model-based decoding

Page 48: DCM:   Advanced Topics

BMS: Parkison‘s disease and treatment

Rowe et al. 2010,NeuroImage

Age-matched controls

PD patientson medication

PD patientsoff medication

DA-dependent functional disconnection of the SMA

Selection of action modulates connections between PFC and SMA

Page 49: DCM:   Advanced Topics

Model-based decoding by generative embedding

Brodersen et al. 2011, PLoS Comput. Biol.

step 2 —kernel construction

step 1 —model inversion

measurements from an individual subject

subject-specificinverted generative model

subject representation in the generative score space

A → B

A → C

B → B

B → C

A

CB

step 3 —support vector classification

separating hyperplane fitted to discriminate between groups

A

CB

jointly discriminativemodel parameters

step 4 —interpretation

Page 50: DCM:   Advanced Topics

Discovering remote or “hidden” brain lesions

Page 51: DCM:   Advanced Topics

Discovering remote or “hidden” brain lesions

detect “down-stream” network changes altered synaptic coupling among non-lesioned regions

Page 52: DCM:   Advanced Topics

Model-based decoding of disease status: mildly aphasic patients (N=11) vs. controls (N=26)

Connectional fingerprints from a 6-region DCM of auditory areas during speech perception

Brodersen et al. 2011, PLoS Comput. Biol.

MGB

PT

HG (A1)

S

MGB

PT

HG (A1)

S

Page 53: DCM:   Advanced Topics

Model-based decoding of disease status: aphasic patients (N=11) vs. controls (N=26)

Classification accuracy

Brodersen et al. 2011, PLoS Comput. Biol.

MGB

PT

HG(A1)

MGB

PT

HG(A1)

auditory stimuli

Page 54: DCM:   Advanced Topics

-10

0

10

-0.5

0

0.5

-0.1

0

0.1

0.2

0.3

0.4

-0.4-0.2

0 -0.5

0

0.5-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-10

0

10

-0.5

0

0.5

-0.1

0

0.1

0.2

0.3

0.4

-0.4-0.2

0 -0.5

0

0.5-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

gen

erative em

bedd

ing

L.H

G

L.H

G

Vo

xel (

64,-

24,4

) mm

L.MGB L.MGBVoxel (-42,-26,10) mmVoxel (-56,-20,10) mm R.HG L.HG

controlspatients

Voxel-based feature space Generative score space

Multivariate searchlightclassification analysis

Generative embedding using DCM

Page 55: DCM:   Advanced Topics

F

D

EC

B

A

Definition of ROIs

Are regions of interest definedanatomically or functionally?

anatomically functionally

Functional contrasts

Are the functional contrasts definedacross all subjects or between groups?

1 ROI definitionand nmodel inversions

unbiased estimate

Repeat n times:1 ROI definition and nmodel inversions

unbiased estimate

1 ROI definition and nmodel inversions

slightly optimistic estimate:voxel selection for training set and test set based on test data

Repeat n times:1 ROI definition and 1 model inversion

slightly optimistic estimate:voxel selection for training set based on test data and test labels

Repeat n times:1 ROI definition and nmodel inversions

unbiased estimate

1 ROI definition and nmodel inversions

highly optimistic estimate:voxel selection for training set and test set based on test data and test labels

across subjects

between groups

Brodersen et al. 2011, PLoS Comput. Biol.

Page 56: DCM:   Advanced Topics

Key methods papers: DCM for fMRI and BMS – part 1

• Brodersen KH, Schofield TM, Leff AP, Ong CS, Lomakina EI, Buhmann JM, Stephan KE (2011) Generative embedding for model-based classification of fMRI data. PLoS Computational Biology 7: e1002079.

• Daunizeau J, David, O, Stephan KE (2011) Dynamic Causal Modelling: A critical review of the biophysical and statistical foundations. NeuroImage 58: 312-322.

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

• Friston K, Stephan KE, Li B, Daunizeau J (2010) Generalised filtering. Mathematical Problems in Engineering 2010: 621670.

• Friston KJ, Li B, Daunizeau J, Stephan KE (2011) Network discovery with DCM. NeuroImage 56: 1202–1221.

• Friston K, Penny W (2011) Post hoc Bayesian model selection. Neuroimage 56: 2089-2099.

• 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.

• Li B, Daunizeau J, Stephan KE, Penny WD, Friston KJ (2011). Stochastic DCM and generalised filtering. NeuroImage 58: 442-457

• 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.

Page 57: DCM:   Advanced Topics

Key methods papers: DCM for fMRI and BMS – part 2

• Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing Families of Dynamic Causal Models. PLoS Computational Biology 6: e1000709.

• Penny WD (2012) Comparing dynamic causal models using AIC, BIC and free energy. Neuroimage 59: 319-330.

• 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 (2009a) Bayesian model selection for group studies. NeuroImage 46:1004-1017.

• Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009b) 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.

Page 58: DCM:   Advanced Topics

Thank you