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Page 1: DCM: Advanced issues

DCM: Advanced issues

Klaas Enno Stephan

Centre for the Study of Social & Neural SystemsInstitute for Empirical Research in EconomicsUniversity of Zurich

Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

SPM Course 2008Zurich

Page 2: DCM: Advanced issues

intrinsic connectivity

direct inputs

modulation ofconnectivity

Neural state equation CuzBuAz jj )( )(

u

zC

z

z

uB

z

zA

j

j

)(

hemodynamicmodelλ

z

y

integration

BOLDyyy

activityz1(t)

activityz2(t) activity

z3(t)

neuronalstates

t

drivinginput u1(t)

modulatoryinput u2(t)

t

Stephan & Friston (2007),Handbook of Connectivity

Page 3: DCM: Advanced issues

Overview

• Bayesian model selection (BMS)

• Timing errors & sampling accuracy

• The hemodynamic model in DCM

• Advanced DCM formulations for fMRI

– two-state DCMs

– nonlinear DCMs

• An outlook to the future

Page 4: DCM: Advanced issues

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 5: DCM: Advanced issues

dmpmypmyp )|(),|()|( Model evidence:

Bayesian model selection (BMS)

)|(

)|(),|(),|(

myp

mpmypmyp

Bayes’ rule:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability)of the model

integral usually not analytically solvable, approximations necessary (e.g. AIC or BIC)

Page 6: DCM: Advanced issues

dmpmypmyp )|(),|()|(

Model evidence p(y|m)Gharamani, 2004

p(y

|m

)

all possible datasets y

a specific y

Balance between fit and complexity

Generalisability of the model

Model evidence: probability of generating data y from parameters that are randomly sampled from the prior p(m).

Maximum likelihood: probability of the data y for the specific parameter vector that maximises p(y|,m).

Page 7: DCM: Advanced issues

pmypAIC ),|(log

Logarithm is a monotonic function

Maximizing log model evidence= Maximizing model evidence

)(),|(log

)()( )|(log

mcomplexitymyp

mcomplexitymaccuracymyp

At the moment, two approximations available in SPM interface:

Np

mypBIC 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.

Page 8: DCM: Advanced issues

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) Evidence

1 to 3 50-75 weak

3 to 20 75-95 positive

20 to 150 95-99 strong

150 99 Very strong

Raftery classification:

Page 9: DCM: Advanced issues

AIC:

BF = 3.3

BIC:

BF = 3.3

BMS result:

BF = 3.3

Two models with identical numbers of parameters

Page 10: DCM: Advanced issues

AIC:

BF = 0.1

BIC:

BF = 0.7

BMS result:

BF = 0.7

Two models with different numbers of parameters

&

compatible AIC/BIC based decisions about models

Page 11: DCM: Advanced issues

AIC:

BF = 0.3

BIC:

BF = 2.2

BMS result:

“AIC and BIC disagree about which model is superior - no decision can be made.”

Two models with different numbers of parameters

&

incompatible AIC/BIC based decisions about models

Page 12: DCM: Advanced issues

Further reading on BMS of DCMs

• Theoretical papers:– Penny et al. (2004) Comparing dynamic causal models. NeuroImage 22:

1157-1172.

– Stephan et al. (2007) Comparing hemodynamic models with DCM. NeuroImage 38: 387-401.

• Applications of BMS & DCM (selection):– Grol et al. (2007) Parieto-frontal connectivity during visually-guided

grasping. J. Neurosci. 27: 11877-11887.

– Kumar et al. (2007) Hierarchical processing of auditory objects in humans. PLoS Computat. Biol. 3: e100.

– Smith et al. (2006) Task and content modulate amygdala-hippocampal connectivity in emotional retrieval. Neuron 49: 631-638.

– Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27: 3512-3522.

Page 13: DCM: Advanced issues

Overview

• Bayesian model selection (BMS)

• Timing errors & sampling accuracy

• The hemodynamic model in DCM

• Advanced DCM formulations for fMRI

– two-state DCMs

– nonlinear DCMs

• An outlook to the future

Page 14: DCM: Advanced issues

Timing problems at long TRs/TAs

• 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 TAs)– 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

Page 15: DCM: Advanced issues

Slice timing in DCM: three-level model

),,( hhzzgv

),( Tvhx

),,( uzfz n

3rd level

2nd level

1st level

sampled BOLD response

BOLD response

neuronal response

z = neuronal states u = inputszh = hemodynamic states v = BOLD responsesn, h = neuronal and hemodynamic parameters T = sampling time points

Kiebel et al. 2007, NeuroImage

Page 16: DCM: Advanced issues

Slice timing in DCM: an example

t

1 TR 2 TR 3 TR 4 TR 5 TR

t

1 TR 2 TR 3 TR 4 TR 5 TR

OriginalDCM

PresentDCM

1T

2T1T

2T1T

2T1T

2T1T

2T

1T 1T 1T 1T 1T2T 2T 2T 2T 2T

Page 17: DCM: Advanced issues

Overview

• Bayesian model selection (BMS)

• Timing errors & sampling accuracy

• The hemodynamic model in DCM

• Advanced DCM formulations for fMRI

– two-state DCMs

– nonlinear DCMs

• An outlook to the future

Page 18: DCM: Advanced issues

LGleft

LGright

RVF LVF

FGright

FGleft

Example: BOLD signal modelled with DCM

black: measured BOLD signalred: predicted BOLD signal

Page 19: DCM: Advanced issues

sf

tionflow induc

(rCBF)

s

v

stimulus functions

v

q q/vvEf,EEfqτ /α

dHbchanges in

100 )( /αvfvτ

volumechanges in

1

f

q

)1(

fγsxs

signalryvasodilato

u

s

CuxBuAdt

dx m

j

jj

1

)(

t

neural state equation

1

3.4

111),(

3

002

001

32100

k

TEErk

TEEk

vkv

qkqkV

S

Svq

hemodynamic state equationsf

Balloon model

BOLD signal change equation

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

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

• 6 hemodynamic parameters:

• Empirically determineda priori distributions.

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

The hemodynamic model in DCM

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

Page 20: DCM: Advanced issues

Recent changes in the hemodynamic model

(Stephan et al. 2007, NeuroImage)

• new output non-linearity, based on new exp. data and mathematical derivations

less problematic to apply DCM to high-field fMRI data

• field-dependency of output coefficients is handled better, e.g. by estimating intra-/extravascular BOLD signal ratio

BMS indicates that new model performs better than original Buxton model

Page 21: DCM: Advanced issues

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

r,Br,A r,C

A

B

C

h

ε

How independent are our neural and hemodynamic parameter estimates?

Stephan et al. 2007, NeuroImage

Page 22: DCM: Advanced issues

Overview

• Bayesian model selection (BMS)

• Timing errors & sampling accuracy

• The hemodynamic model in DCM

• Advanced DCM formulations for fMRI

– two-state DCMs

– nonlinear DCMs

• An outlook to the future

Page 23: DCM: Advanced issues

)(tu

ijij uBA

input

Single-state DCM

1x

Intrinsic (within-region) coupling

Extrinsic (between-region) coupling

NNNN

N

x

x

tx

AA

AA

A

CuxuBAt

x

1

1

111

)(

)(

Two-state DCM

Ex1

IN

EN

I

E

AA

AAA

AA

AAA

u

x

x

x

x

tx

ee

eee

ee

eee

A

CuxABt

x

IINN

IENN

EINN

EENNN

IIIE

NEIEE

1

1

)(

00

0

00

0

)(

1

1111

11111

)exp( ijij uBA

Ix1

IEx ,1

Marreiros et al. 2008, NeuroImage

Page 24: DCM: Advanced issues

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

non-linear DCM

driving input

modulation

...)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 25: DCM: Advanced issues

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

BOLD signal change (%)

x1 x2u1

x3

u2

– –

++

++++

+++

+++

+

2

1

32

11

3

2

1)3(

213

3332

232221

1211

0

0

0

0

000

00

000

0

0

u

u

c

c

x

x

x

dx

aa

aaa

aa

dt

dx

Neuronal state equation:

Stephan et al., submitted

Page 26: DCM: Advanced issues

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 = 2966

M2 better than M1

M3 better than M2

BF = 12

M4 better than M3

BF = 23

Stephan et al., submitted

Page 27: DCM: Advanced issues

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

A B

Stephan et al., submitted

Page 28: DCM: Advanced issues

V1

V5PPC

observedfitted

motion &attention

motion &no attention

static dots

Stephan et al., submitted

Page 29: DCM: Advanced issues

Overview

• Bayesian model selection (BMS)

• Timing errors & sampling accuracy

• The hemodynamic model in DCM

• Advanced DCM formulations for fMRI

– two-state DCMs

– nonlinear DCMs

• An outlook to the future

Page 30: DCM: Advanced issues

),,( uxFx Neural state equation:

Electric/magneticforward model:

neural activityEEGMEGLFP

(linear)

DCM: generative model for fMRI and ERPs

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

Neural model:8 state variables per region

nonlinear state equationpropagation delays

fMRIfMRI ERPsERPs

inputs

Hemodynamicforward model:neural activityBOLD(nonlinear)

Page 31: DCM: Advanced issues

Neural mass model of a cortical macrocolumn

ExcitatoryInterneurons

He, e

PyramidalCells

He, e

InhibitoryInterneurons

Hi, e

Extrinsic inputs

Excitatory connection

Inhibitory connection

e, i : synaptic time constant (excitatory and inhibitory) He, Hi: synaptic efficacy (excitatory and inhibitory) 1,…,: intrinsic connection strengths propagation delays

21

43

MEG/EEGsignal

MEG/EEGsignal

Parameters:

Parameters:

Jansen & Rit (1995) Biol. Cybern.David et al. (2006) NeuroImage

mean firing rate

mean postsynaptic

potential (PSP)

mean PSP

mean firing rate

Page 32: DCM: Advanced issues

236

746

63

225

1205

52

650

214

014

41

278

038

87

2)(

2))()()((

2))()((

2))()((

iii

i

ee

LB

e

e

ee

LF

e

e

ee

LB

e

e

xxxS

Hx

xx

xxxSxSAA

Hx

xx

xxx

xxCuxSIAA

Hx

xx

xxxSIAA

Hx

xx

spiny stellate

cells

inhibitory interneurons

pyramidal cells

4 3

1 2)( 0xSAF

)( 0xSAL

)( 0xSAB

Extrinsicforward

connections

Extrinsic backward connections

Intrinsic connections

neuronal (source) model

Extrinsic lateral connections

State equations ,,uxFx

DCM for ERPs: neural state equations

David et al. (2006) NeuroImage

MEG/EEGsignal

MEG/EEGsignal

mV

Inhibitory cells in supra/infragranular layers

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in supra/infragranular layers

activity

Page 33: DCM: Advanced issues

DCM for LFPs

• extended neural mass models that can be fitted to LFP data (both frequency spectra and ERPs)

• explicit model of spike-frequency adaptation (SFA)

• current validation work to establish the sensitivity of various parameters wrt. specific neurotransmitter effects

• validation of this model by LFP recordings in rats, combined with pharmacological manipulations

Moran et al. (2007, 2008) NeuroImage

standards deviants

A1

A2