DCM: Advanced topics

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, November 2010

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Neural population activity

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x1 x2

x3

x1 x2

x3

CuxDxBuAdtdx n

j

jj

m

i

ii

1

)(

1

)(

u1

u2

),,( uxFdtdx

Neural state equation:

Electromagneticforward model:

neural activityEEGMEGLFP

Dynamic Causal Modeling (DCM)

simple neuronal modelcomplicated forward model

complicated neuronal modelsimple forward model

fMRI EEG/MEG

inputs

Hemodynamicforward model:neural activityBOLD

Overview

• Bayesian model selection (BMS) • Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• 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)

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

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. 2004a, NeuroImage

Approximations to the model evidence in DCM

No. of parameters

No. ofdata points

AIC favours more complex models,BIC favours simpler models.

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

)|(),(

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) 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.

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 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.

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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

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

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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

Stephan et al. 2009a, NeuroImage

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r1

p(r 1|y

)

p(r1>0.5 | y) = 0.986

Model space partitioning:

comparing model families

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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

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.

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

optimal model structure assumed to be identical across subjects?

FFX BMS RFX BMS

yes no

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

BMA

definition of model space

FFX analysis of parameter estimates

(e.g. BPA)

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


Overview


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

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

• These features necessitated various changes, e.g.– inputs mean-corrected– prior variance of coupling parameters no longer dependent on number of areas– simplified hemodynamic model– self-connections: separately estimated for each area

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

• Note: this is a different model from classical DCM!

• When publishing papers, you should state whether you are using DCM10 or classical DCM (now referred to as DCM8).

• If you don‘t know: DCM10 reveals its presence by printing a message to the command window.

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.

bilinear DCM

CuxDxBuAdtdx m

i

n

j

jj

ii

1 1

)()(CuxBuAdtdx m

i

ii

1

)(

Bilinear state equation:

driving input

modulation

driving input

modulationnon-linear DCM

...)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

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fMRI signal change (%)x1 x2

x3

CuxDxBuAdtdx n

j

jj

m

i

ii

1

)(

1

)(

Nonlinear dynamic causal model (DCM)


u1

u2

V1 V5stim

PPC

attention

motion

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%1.99)|0( 1,5 yDp PPCVV

1.25

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0.390.26

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0.10MAP = 1.25


V1V5PPC

observedfitted

motion &attention

motion &no attention

static dots

uinput

Single-state DCM

1x

Intrinsic (within-region)

coupling

Extrinsic (between-region)

coupling

NNNN

N

ijijij

x

xx

uBACuxx

1

1

111

Two-state DCM

Ex1

IN

EN

I

E

IINN

IENN

EENN

EENN

EEN

IIIE

EEN

EIEE

ijijijij

xx

xx

x

uBA

Cuxx

1

1

1

1111

11111

000

000

)exp(

Ix1

I

E

x

x

1

1

Two-state DCM

Marreiros et al. 2008, NeuroImage

Estimates of hidden causes and states(Generalised filtering)

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1inputs or causes - V2

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1.3hidden states - hemodynamic

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2predicted BOLD signal

time (seconds)

excitatorysignal

flowvolumedHb

observedpredicted

Stochastic DCM

( ) ( )

( )

( )j xjj

v

dx A u B x Cvdt

v u

Li et al., submitted

• all states are represented in generalised coordinates of motion

• random state fluctuations w(x) have unknown precision and smoothness two hyperparameters

• fluctuations w(v) induce uncertainty about how inputs influence neuronal activity inputs u effectively serve as priors on the hidden neuronal causes

Overview


Learning of dynamic audio-visual associations

CS Response

Time (ms)

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or

Target StimulusConditioning Stimulus

or

TS

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p(fa

ce)

trial

CS1

CS2

den Ouden et al. 2010, J. Neurosci.

Explaining RTs by different learning models

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p(F)

TrueBayes VolHMM fixedHMM learnRW

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Categoricalmodel

Bayesianlearner

HMM (fixed) HMM (learn) Rescorla-Wagner

Exce

edan

ce p

rob.

Bayesian model selection: hierarchical Bayesian model performs best

5 alternative learning models: • categorical probabilities• hierarchical Bayesian learner• Rescorla-Wagner• Hidden Markov models

(2 variants)

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RT

(ms)

p(outcome)

Reaction times

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

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p(F)

Putamen Premotor cortex

Stimulus-independent prediction error

p < 0.05 (SVC)

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

p(F) p(H)-2

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BO

LD re

sp. (

a.u.

)

p(F) p(H)-2

-1.5

-1

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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 current learning theories (e.g., FEP):synaptic plasticity during learning = PE dependent changes in connectivity

Plasticity of visuo-motor connections

• Modulation of visuo-motor connections by striatal prediction error activity

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

surprising stimuli – weaker for expected

stimuli


PPA FFA

PMd

Hierarchical Bayesian learning model

PUT

p = 0.010 p = 0.017

Prediction error in PMd: cause or effect?

Model 1 Model 2

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vide

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Model 1 minus Model 2

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r1p(

r 1|y)

p(r1>0.5 | y) = 0.991


Overview


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

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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 LG

FGFG

DCM

LGleft

LGright

FGright

FGleft

13 15.7%

34 6.5%

24 43.6%

12 34.2%

anatomical connectivity

probabilistictractography

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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

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1m24: a=0,b=-32

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1m26: a=0,b=-24

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1m28: a=0,b=-16

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1m29: a=0,b=-12

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1m30: a=0,b=-8

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1m31: a=0,b=-4

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1m32: a=0,b=0

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1m33: a=0,b=4

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1m34: a=0,b=8

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1m35: a=0,b=12

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1m36: a=0,b=16

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1m37: a=0,b=20

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1m38: a=0,b=24

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1m39: a=0,b=28

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1m40: a=0,b=32

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1m51: a=8,b=12

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1m59: a=12,b=28

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1m62: a=16,b=32

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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

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Models with anatomically informed priors (of an intuitive form)

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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 than anatomically uninformed ones: Bayes Factor >109

Key methods papers: DCM for fMRI and BMS – 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 (2011) 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.• Friston K, Stephan KE, Li B, Daunizeau J (2010) Generalised filtering. Mathematical Problems in

Engineering 2010: 621670.• 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. Submitted.• 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.

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. • 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.

Thank you

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DCM: Advanced topics