DCM Advanced, Part II

Will Penny (Klaas Stephan)

Wellcome Trust Centre for NeuroimagingInstitute of NeurologyUniversity College London

SPM Course 2014 @ FIL

Overview

• Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Clinical Applications

endogenous 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

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

Factorial structure of model specification in DCM• 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

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

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

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)

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

dx A u B x Cvdt

v u

Li et al. 2011, NeuroImage

• random state fluctuations w(x) account for endogenous fluctuations,

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

• can be fitted to resting state data

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

dx A u B x Cvdt

v u

• Good working knowledge of dDCM

• sDCMs (esp. for nonlinear models) can have richer dynamics than dDCM

• Model selection may be easier than with dDCM

• See Daunizeau et al. ‘sDCM: Should we care about neuronal noise ?’, Neuroimage, 2012

Overview

• Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Clinical Applications

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

Behrens et al. 2007, Nat. Neurosci.

prior on volatility

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

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)

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 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

Prediction errors control plasticity during adaptive cognition

• 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

den Ouden et al. 2010, J. Neurosci .

PPA FFA

PMd

Hierarchical Bayesian learning model

PUT

p = 0.010 p = 0.017

Overview

• Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Clinical Applications

model structure

Model-based predictions for single patients

set of parameter estimates

BMS

model-based decoding

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

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 → BA → CB → BB → C

A

CB

step 3 —support vector classification

separating hyperplane fitted to discriminate between groups

A

CB

jointly discriminativemodel parameters

step 4 —interpretation

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

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

-10

0

10

-0.50

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

-0.50

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

generative em

bedding

L.H

G

L.H

G

Voxe

l (64

,-24,

4) m

m

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

Summary

• Model Selection• Extended DCM for fMRI: nonlinear, two-state, stochastic • Embedding computational models in DCMs• Clinical Applications