Abstract

This tutorial is about the inversion of dynamic input-state-output systems. Identification of the systems parameters proceeds in a Bayesian framework given known, deterministic inputs and observed responses of the [neuronal] system.

We develop this approach for the analysis of effective connectivity or coupling in the brain, using experimentally designed inputs and fMRI and EEG responses. In this context, the parameters correspond to effective connectivity and, in particular, bilinear parameters reflect the changes in connectivity induced by inputs. The ensuing framework allows one to characterise experiments, conceptually, as an experimental manipulation of integration among brain regions (by contextual or trial-free inputs, like time or attentional set) that is perturbed or probed using evoked responses (to trial-bound inputs like stimuli).

As with previous analyses of effective connectivity, the focus is on experimentally induced changes in coupling ( c.f. psychophysiologic interactions). However, unlike previous approaches to connectivity in neuroimaging, the causal model ascribes responses to designed deterministic inputs, as opposed to treating inputs as unknown and stochastic.

Imaging Clinic Tuesday 26th October: 10AM-4.30PM; Building 26, room 135; Clayton Campus

Dynamic Causal Modelling (tutorial)Karl Friston, Wellcome Centre for Neuroimaging, UCL

Dynamic Causal ModellingState and observation equationsModel inversion

DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models

DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connections

DCMs for LFPSteady-state responses

V1

V4

BA37

STG

BA39

Structural perturbationsStimulus-free - u

e.g., attention, time

Dynamic perturbationsStimuli-bound u

e.g., visual words

Functional integration and the enabling of specific pathways

y

y y

y

y

measurement

neuronal network

Observed data

)(tu

ix

input

( , , )x f x u

),(xgyForward model (measurement)

Model inversion

Forward models and their inversion

Forward model (neuronal)

( | , , , )p y x u m ( , | , , )p x y u m

Model specification and inversion

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),,(

xgy

uxfx

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

)(),|()|(

mypmpmypmyp

dpmypmyp

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

NN

p y m gp m

Invert model

Inference

Define likelihood model

Specify priors

Neural dynamics

Observer function

Design experimental inputs)(tu

Inference on models

Inference on parameters

Dynamic Causal ModellingState and observation equationsModel inversion

DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models

DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connectionsInduced responses

DCMs for LFPSteady-state responses

1y 3y2y

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uxfB

xfA

CBA

2

},,{

The bilinear (neuronal) model

23b

12a

1c

averageconnectivity

exogenous causes

bilinearconnectivity

CuxuBAuxfx

)(),,(

Input )(tu

1x 3x2x

Dynamic perturbation

Structural perturbation

f s

1( ) / ατq f E f v q v 1/ ατv f v

( 1)s x s γ f

0 1 2 3( ) ( ( )) ( (1 ) (1 ) (1 ))y t g x t V k q k q v k v

Output: a mixture of intra- and extravascular signal

)(tx

0 8 16 24 sec

ix

Hemodynamic models for fMRI

basically, a convolution

1y

signal

flow

dHbvolume

The plumbing

Neural population activity

BOLD signal change (%)

x1 x2u1

x3

u2

– –

11 12 1 111

21 22 23 2 21 22

32 33 3 32

0 0 0 0 00 0 0 0

0 0 0 0 0

a a x cu

x a a a u b xu

a a x c

A toy example

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

Stimuli 250 radially moving dots at 4.7 degrees/s

Pre-Scanning5 x 30s trials with 5 speed changes (reducing to 1%)Task: detect change in radial velocity

Scanning (no speed changes)4 100 scan sessions;each comprising 10 scans of 4 conditions

F A F N F A F N S .................

F - fixation point A - motion stimuli with attention (detect changes)N - motion stimuli without attentionS - no motion

Buchel et al 1999

V5+

PPC

An fMRI study of attention

V1 IFG

V5

Photic

Attention

.92

.43

.62

.40

.53.35

.73

.49

.53

3) Attentional modulation of prefrontal connections

sufficient to explain regionally specific attentional effects2) Segregation of

motion information to V5

1) Hierarchical architecture

Friston et al 1999

SPC

Motion

FFA PPA

MFG

-0.80

-0.31

faces houses faces houses

rivalry non-rivalry

1.05 0.08

0.300.51

2.43 2.41

0.04 -0.03 0.02 0.06

0.02 -0.03

FFAPPAMFG

time (s)

Stephan et al 2008

( ) ( )( )i ii ix A u B x D x Cu

Nonlinear DCM: modulation of connections in inferotemporal cortex under binocular rivalry

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

Modeling excitatory and inhibitory dynamics

Andre Marreiros et al

IISPSP

IESPSP

EISPSP

EESPSP

EESPV

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EEVV

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5555515

1111

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IISPSP

IESPSP

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EESPSP

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5555

5555515

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IISPSP

IESPSP

EISPSP

EESPSP

EESPV

IIVV

IEVV

EESPV

EIVV

EEVV

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IIVV

IEVV

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EIVV

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5555

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511111

Model comparison: where is attention mediated?

Model comparison

Andre Marreiros et al

Hierarchical connections in the brain and laminar specificity

Dynamic Causal ModellingState and observation equationsModel inversion

DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models

DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connectionsInduced responses

DCMs for LFPSteady-state responses

neuronal mass models of distributed sources

State equations

( , , ) x f x u

Output equation

(3)( , ) y g x LV

Exogenous input

E13

( )u t

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in infragranular layers

Inhibitory cells in supragranular layers

Measured response

)( )3(Vg

(1) (1) (1) (1)

(1) (3) (3) (1)13

( ) ( )

( ( , ) )L L E E V

EE E V R E E

CV g V V g V V u

g V g

E31

IIRVI

II

EERVE

EE

VIIEELL

gVg

gVg

VVgVVgVVgVC

)),((

)),((

)()()(

)2()2()2(22

)2(

)2()3()3(23

)2(

)2()2()2()2()2()2(

IIRVI

II

EERVE

EE

VIIEELL

gVg

gVg

VVgVVgVVgVC

)),((

)),((

)()()(

)3()2()2(32

)3(

)3()1()1(31

)3(

)3()3()3()3()3()3(

E23I

32 12I

uinput

x

ERPs

Comparing models (with and without backward connections)

A1 A1

STG

input

STG

IFG

FB

A1 A1

STG

input

STG

IFG

F

0 200 400

0

0 200 400

0

FB vs. F

without with

A1A1

STGSTG

IFG

Garrido et al 2007

log-evidence

ln ( | ) Fp y m

The MMN and perceptual learning

MMN

standards deviants

ERP standardsERP deviantsdeviants - standards

Garrido et al 2008

Model comparison:Changes in forward and backward connections

A1 A1

STG STG

ForwardBackward

Lateral

input

A1 A1

STG STG

ForwardBackward

Lateral

input

A1 A1

STG

ForwardBackward

Lateral

input

-

STG

IFGIFGIFG

Forward (F) Backward (B) Forward and Backward (FB)

Garrido et al 2009

A1A1

STGSTG

IFGA1 A1

STG STG

ForwardBackward

Lateral

input

A1 A1

STG STG

ForwardBackward

Lateral

input

A1 A1

STG

ForwardBackward

Lateral

input

-

STG

IFGIFGIFG

Forward (F) Backward (B) Forward and Backward (FB)

FFB

log

evid

ence

Bayesian model comparison

subjects

Forward (F)

Backward (B)

Forward and Backward (FB)

Two subgroups

Garrido et al 2008

1 2 3 4 5 1 2 3 4 5

A1 A1

STG

subcortical input

STG

repetition effects

monotonic phasic

1 2 3 4 50

20

40

60

80

100

120

140

160

180

200

1 2 3 4 50

50

100

150

200

250

Intrinsic connections

Extrinsic connections

number of presentations

The dynamics of plasticity:Repetition suppression

Garrido et al 2009

K frequency modes in j-th source

KKij

Kij

Kijij

ij

AA

AAA

1

111

Nonlinear (between-frequency) coupling

Linear (within-frequency) coupling

Extrinsic (between-source) coupling

Neuronal model for spectral features

)()()(1

1

1111

tuC

Ctg

AA

AA

g

gtg

JJJJ

J

J

Data in channel space

12

( ) ( )

( , )( ) ( ( ))

( , )

j

j j

j K

x t L d t

g tg t FT x t

g t

)(td

Inversion of electromagnetic model L

)(tu

klijA

jg

input

Intrinsic (within-source) coupling

),( tgi

DCM for induced responses – a different sort of data feature

CC Chen et al 2008

LV RV

RFLF

input

LV RV

RFLF

input

Frequency-specific coupling during face-processing

CC Chen et al 2008

From 32 Hz (gamma) to 10 Hz (alpha) t = 4.72; p = 0.002

4 12 20 28 36 44

44

36

28

20

12

4

SPM t df 72; FWHM 7.8 x 6.5 Hz

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Right hemisphereLeft hemisphere

Forward Backward Forward BackwardFr

eque

ncy

(Hz)

LV RV

RFLF

input

FLBL FNBL FLBN FNBN

-59890

-16308 -16306 -11895

-70000

-60000

-50000

-40000

-30000

-20000

-10000

0

Functional asymmetries in forward and backward connections

CC Chen et al 2008

Dynamic Causal ModellingState and observation equationsModel inversion

DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models

DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connections

DCMs for LFPSteady-state responses

Glutamatergic stellate cells

GABAergic cells

Glutamatergic Projection cells

Data

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

0 20 400

5

CortexGPe

StriatumSTN

Cortex GPeStriatum STN

DCMs for steady-state responses:characterizing coupling parameters Cross-spectral data features

6-OHDA lesion model of Parkinsonism

Moran et al

1. Cortex

2. Striatum

3. External globus pallidus (GPe)

4. Subthalamic Nucleus (STN)

6. Thalamus

5. Entopeduncular Nucleus (EPN)

Changes in the basal ganglia-cortical circuits

Moran et al

Control 6-OHDA Lesioned

1

2

3

4

6

4.25 ± 0.17

1.44 ± 0.18

5.24 ± 0.16

6. 91 ± 0.190.90

± 0

.21

1.43 ± 0.38

0.29 ± 0.31

0.85 ± 0.36

5

0.72 ± 0.44

1

2

3

4

5

3.43 ± 0.16

3.07 ± 0.17

5.00 ± 0.15

2.33 ± 0.21 1.0

4 ±

0.20

1.18 ± 0.33

1.03 ± 0.356

0.74 ± 0.28

MAP estimates

EPN

to T

halam

us

Thala

mus

to C

tx

Ctx t

o St

riatu

m

Ctx t

o ST

N

Stria

tum

to G

Pe

Stria

tum

to E

PN

STN

to E

PN

STN

to G

Pe

GPe

to S

TN

012345678

**

Thank you

And thanks to

CC ChenJean Daunizeau Marta GarridoLee HarrisonStefan Kiebel

Andre MarreirosRosalyn Moran

Will PennyKlaas Stephan

And many others