dynamic causal modelling of electromagnetic responses karl friston - wellcome trust centre for...

Dynamic causal modelling of electromagnetic responses

Karl Friston - Wellcome Trust Centre for Neuroimaging, Institute of Neurology, UCL

In recent years, dynamic causal modelling has become established in the analysis of invasive and non-invasive electromagnetic signals. In this talk, I will briefly review the basic idea behind dynamic causal modelling – namely to equip a standard electromagnetic forward model, used in source reconstruction, with a neural mass or field model that embodies interactions within and between sources. A key point here is that the resulting forward or generative models can predict a large variety of data features – such as event or induced responses, or indeed their complex cross spectral density – using the same underlying neuronal model.

Dynamic causal modelling brings a new perspective to characterising event and induced responses – empirical response components, previously reified as objects of study in their own right (such as the mismatch negativity or P300) now become data features that have to be explained in terms of neuronal dynamics and changes in distributed connectivity. In other words, dynamic causal modelling emphasises the neurobiological mechanisms that underlie responses – over all channels and peristimulus time – without particular regard for the phenomenology of classical response components. My hope is to incite some discussion of this shift in perspective and its implications.

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One ring to rule them all, one ring to find them, one ring to bring them all, and in the darkness bind them

http://payingattentiontothesky.files.wordpress.com/2012/06/lord_of_the_rings_one_ring.jpg

Overview

The basic idea (functional and effective connectivity)

Generative model and face validation

An empirical example

( , , , )x f x u

( )u t

( , ) ( ) [ ]

[ ] [ ( , , , )] ( [ ], ,0, )

t G E x

E x E f x u f E x u

y

1( ) ( , )kt

( , , , )x f x u

( )u t

1( ) ( , )kt

Generative model

Exogenous and endogenous fluctuations

Neuronal dynamics

Evoked responses Induced responses

2( , , ) ( , , ) ( , ) ( , , )

( , , ) ( ) exp ( [ ]) ( [ ])v

x

t h K t g K t

k t G f E x f E x

g

( , )u

( , )y g

2( , , ) ( , , ) ( , ) ( , , )

( , , ) ( ) exp ( [ ]) ( [ ])v

x

t h K t g K t

k t G f E x f E x

g

( , , , )x f x u

( )u t

( ) [ ( )]t E y ty ( , ) [ ( , ) ( , ) ]t E s t s t g

1( ) ( , )kt

Model inversion

Exogenous and endogenous fluctuations

( )

( , , )uy g

( , ) ( ) [ ]

[ ] [ ( , , , )] ( [ ], ,0, )

t G E x

E x E f x u f E x u

y

Bayesian model inversion and parameter averaging

Invert model of induced responses

( ) argmin ( , , )qq F q g y

( | , , ) ( )

ln ( , | ) ( , , )

p m q

p m F q

y g

y g g yInference on parameters

and models

Invert model of evoked responses

( ) argmin ( , )qq F q y y

Update priors( | , ) ( )p m q yy

We seek the posterior conditioned on both evoked and induced responses. Using Bayes rule we have:

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

( | , ) ( | , ) ( | )

( | , ) ( | , )

p m p m p m

p m p m p m

p m p m

y g y g

g y

g y

Giving the likelihood and prior for induced responses

( ) ( , ) ( )

( ) ( . ) ( )

x t f x u v t

y t g x u w t

( ) ( ) ( ) ( )

( ) exp( )x x

y t v t w t

k g f

2| ( ) |( )

( ) ( )ij

ijii jj

gC

g g

( ) [ ( ) ( ) ]

( ) ( )

t

v w

g E Y Y

K g K g

State-space model

Convolution kernel representationFunctional Taylor expansion

Spectral representationConvolution theorem

( ) ( ) ( ) ( )

( ) [ ( )]

Y K V W

K t

F

Cross-spectral density

Coherence

( )( )

(0) (0)ij

ij

ii jj

Cross-correlation

( ) [ ( ) ( ) ]

( ) ( )

Tt

v w

E y t y t

k t k t

Cross-covariance

F

1F

1( ) ( ) ( )

p

iiy t a y t i z t

y a z

2| ( ) |( ) ln 1

( )ij

ijii

SG

g

1

( ) ( ) ( )

( ) ( ( ))

Y S Z

S I A

Autoregressive representationYule Walker equations

Spectral representationConvolution theorem

1

( ) ( ) ( ) ( )

( ) [ , , ]p

Y A Y Z

A a a

F

Directed transfer functions

Granger causality

1 1( ) ( )Tii I a I a

Auto-correlation

1

11

( )

[ , , ]

T T

Tp

a y y y y

Auto-regression coefficients

1

1

[ ( ) ( ) ] ( ) ( ( ))

[ ( ) ( ) ] ( ) ( ( ))

Tt v v

Tt w w

E v t v t g

E w t w t g

F

F

1

2 1

0

0

0

yy

y y

Measures of functional connectivity or statistical dependence among observed responses

Models of effective connectivity among hidden states causing observed responses

1

2 1

1

0

0

0

( )p i j

jj

aa

a a

A a e

Effective connectivity

Cross spectral density

Volterra kernels

Cross covariance functions

Spectral factorsAuto regression

coefficients

Director transfer functions

Granger causality nonparametricparametric

0 0( ) ( , ) exp ( , )x xk t g x t f x

1( ) [ ( )]t g F

( ) [ ( )]g t F

( ) ( ) ( )g

( ) ( ) ( )v wg K g K g

1

*

1

( ) ( ) (0)

(0) (0)

[ ( )]

z

S

F

1a C

1

1

( ) ( ( ))

(0)

( ) [ ]

Tz

S I A

C

A a

F

2 2| ( ) |( ) ln 1

( )zij ij

ij ziizjj ii

SG

g

v wk k

Modulation transfer functions

( ) [ ( )]K k t F

1( ) [ ( )]k t K F

Spectral measures

Overview

The basic idea



Early source (1)Higher source (2)

Endogenous fluctuations

Infragranular layer

supragranular layer

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( , , )

( , , , )

( ) ( ) ( )

( ( , ) )

i i i iE I

i i i ikj u

i i i i i i iL L E E I I u i

i i j j ik k kj R k

j

x V g g

CV g V V g V V g V V u v

g V V g

( )iEj

Deep pyramidal cells

Inhibitory interneurons

Superficial pyramidal cells

Spiny stellate cells

Forward extrinsic connections

Backward extrinsic connections

Intrinsic connectionsGranular layer( )iEj

Generative (conductance based neural mass) model based on the canonical microcircuit

( )iIj

Inhibitory connections: k = EExcitatory connections: k = I

( )ikj

( )u t( )t

Exogenous (subcortical) input

( )tEndogenous fluctuations

4

8

8

1

322

32

4

8

4

0 10 20 30 40 50 60 700

1

2

3

4

frequency {Hz}

Intrinsic backward connections) Transfer functions

Freq

uenc

y

(log) parameter scaling-2 -1 0 1 2

10

20

30

40

50

60

0 10 20 30 40 50 60 700

1

2

3

4

5

frequency {Hz}

Self-inhibition of superficial cells Transfer functions

Freq

uenc

y

(log) parameter scaling-2 -1 0 1 2

10

20

30

40

50

60

The effect of parameters on transfer functions – contribution analysis

100 200 300 400 500

-2

0

2

4

6Evoked response: source 1

peristimulus time (ms)

Induced response


Hz

100 200 300 400 500

10

20

30

40

50

60

100 200 300 400 500

-4

-2

0

2

4

Evoked response: source 2


Induced response


Hz

100 200 300 400 500

10

20

30

40

50

60

50 100 150 200 250 300 350 400 450 5000

2

4

6

8


Exogenous input

50 100 150 200 250 300 350 400 450 500

-60

-40

-20

0


Hidden neuronal states (conductance and depolarisation)

Spectral density


Hz

100 200 300 400 500

10

20

30

40

50

60

Coherence


Hz

100 200 300 400 500

10

20

30

40

50

60

Spectral density


Hz

100 200 300 400 500

10

20

30

40

50

60

Cross-covariance

peristimulus (ms)

lag

(ms)

100 200 300 400 500-60

-40

-20

0

20

40

60

source 1source 2

Predicted responses to sustained exogenous (stimulus) input

transfer function: 1 to 1


Hz

100 200 300 400 500

10

20

30

40

50

60



Hz

100 200 300 400 500

10

20

30

40

50

60



Hz

100 200 300 400 500

10

20

30

40

50

60



Hz

100 200 300 400 500

10

20

30

40

50

60

0 20 40 60 80 100 120 140-1.5

-1

-0.5

0

0.5

1kernel: 1 to 1

lag (ms)0 20 40 60 80 100 120 140

-0.02

-0.01

0

0.01

0.02kernel: 2 to 1

lag (ms)

0 20 40 60 80 100 120 140-4

-2

0

2

4kernel: 1 to 2

lag (ms)0 20 40 60 80 100 120 140

-1

-0.5

0

0.5

1kernel: 2 to 2

lag (ms)

Transfer functions and spectral asymmetries

Forward connections (gamma)

backward connections (beta)



100 200 300 400 500

-2

0

2

4

6evoked: source 1


100 200 300 400 500

-4

-2

0

2

4

evoked: source 2


100 200 300 400 500-4

-2

0

2

4

6

evoked: source 1


100 200 300 400 500

-5

0

5

evoked: source 2


Spectral density


Hz

100 200 300 400 500

10

20

30

40

50

60

Coherence


Hz

100 200 300 400 500

10

20

30

40

50

60

Spectral density

peristimulus time (ms)Hz

100 200 300 400 500

10

20

30

40

50

60

Cross-covariance

peristimulus (ms)

lag

(ms)

100 200 300 400 500

-50

0

50

Spectral density


Hz

100 200 300 400 500

10

20

30

40

50

60

Coherence


Hz

100 200 300 400 500

10

20

30

40

50

60

Spectral density


Hz

100 200 300 400 500

10

20

30

40

50

60

Cross-covariance

peristimulus (ms)

lag

(ms)

100 200 300 400 500

-50

0

50

Simulated responses(sample estimates

over16 trials)

Predicted responses(expectation under

known input)

Overview

The basic idea



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

-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 Backward

Freq

uenc

y (H

z)

LV RV

RFLF

input

FLBL FNBL FLBN FNBN

-70000

-60000

-50000

-40000

-30000

-20000

-10000

0

Functional asymmetries in forward and backward connections Phenomenological DCM for induced responses (Chen et al 2008)

LV RV

RFLF

input

Posterior predictions following inversion of event responses

Estimates of dipole orientation

Sensor level observations

and DCM predictions

channelstim

e (m

s)

Observed (adjusted) 1

50 100 150 200 250

-100

0

100

200

300

400

500

channels

time

(ms)

Predicted

50 100 150 200 250

-100

0

100

200

300

400

500

0 100 200 300 400 500

-2

0

2

evoked: channel/source 1


induced: condition 1


Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500-8

-6

-4

-2

0

2

4





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500-4

-2

0

2





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500-3

-2

-1

0

1





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500

-2

0

2





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500

-5

0

5





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500

-4

-2

0

2

4





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

0 100 200 300 400 500

-2

0

2





Hz

-100 0 100 200 300 400 50010

15

20

25

30

35

Posterior predictions following inversion of induced responses



Hz

0 200 400

10

20

30



Hz

0 200 400

10

20

30



Hz

0 200 400

10

20

30



Hz

0 200 400

10

20

30

0 100-0.5

0

0.5 kernel: 1 to 1

lag (ms)

0 100-2

0

2 kernel: 3 to 1

lag (ms)

0 100-10

0

10 kernel: 1 to 3

lag (ms)0 100

-0.5

0

0.5 kernel: 3 to 3

lag (ms)

Forward connections (gamma?)

LV RV

RFLF

input

LV RV

RFLF

input

Backward connections (beta)

Transfer functions and spectral asymmetries

Thank you

And thanks to

Gareth BarnesAndre Bastos

CC ChenJean Daunizeau Marta Garrido

Lee HarrisonMartin Havlicek

Stefan KiebelMarco Leite

Vladimir LitvakAndre MarreirosRosalyn Moran

Will PennyDimitris Pinotsis

Krish SinghKlaas Stephan

Bernadette van Wijk

And many others

Table 1a: state space model

State space model Random fluctuations Convolution kernels

( ) ( , ) ( )

( ) ( , ) ( )

x t f x v t

y t g x w t

[ ( ) ( ) ] ( , )

[ ( ) ( ) ] ( , )

Tv

Tw

E v t v t

E w t w t

0 0

( ) ( ) ( ) ( )

( ) ( , ) exp ( , )x x

y t k t v t w t

k t g x t f x

Table 1b: second-order dependencies

Cross covariance Cross spectral density Autoregression coefficients Directed transfer functions

Cross covariance v wk k 1( ) [ ( )]t g F 1( ) ( )( ) TzC I a I I a 1( ) [ ( ) ( ) ]zt S S F

Spectral density ( ) [ ( )]g t F ( ) ( ) ( )

( ) [ ( )]v wg K g K g

K k t

F

*

1

( ) ( ) ( )

( ) ( [ ])

zg S S

S I a

F

( ) ( ) ( )

( ) ( )zg S S

Autoregression coefficients 1

( )ij ij

a C

C toeplitz

1

1( ) [ ( )]

a C

t g

F

1

*

1

[ ] [ ]

[ ] (0) (0)

(0)

T T

Tz

T

y y a z

a y y y y

z z

C

E E

E

1

1

1

1

[ ( )]

( ) ( )

( ) ( ) (0)

[ ( )]

a A

A I S

S

F

F

Directed transfer functions 1 1( ) ( )S I C 1 1

1

( ) ( [ ])

( ) [ ( )]

S I C

t g

F

F

1( ) ( ( ))

( ) [ ]

S I A

A a

F

( ) ( ) ( )

( ) ( ) ( )

Y S Z

A Y Z

Table 1c: normalized measures

Cross correlation Coherence Granger causality Normalised directed transfer functions

( )( )

(0) (0)ij

ij

ii jj

2| ( ) |( )

( ) ( )ij

ijii jj

g

g g

2 2| ( ) |( ) ln 1

( )zij ij

ij zjjzii ii

SG

g

| ( ) |

( )| ( ) |

ijij

ii

SD

S

11 21 1

12 11 22 21 2

0 0 0(0) ( ) (1)

0 0 0[ ] : [ ] :

0 0 0( ) (0) ( 1)

ij ij ijijT T

ij ij ijij ij

ij ij ij

py y a

C y y C y y y ay y y y a a

p p

E E

superficial

deep

( )ix

( )ix

( )ix

( )iv

( )iv

( )iv

( 1)iv

( ) ( ) ( ) ( ) ( 1)

( ) ( ) ( ) ( )

i i i i iv v v v

i i i ix x x

D

D

( )i

Errors (superficial pyramidal cells)

Expectations (deep pyramidal cells)

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

( ) ( ) ( ) ( ) ( ) ( )

( ( , ))

( ( , ))

i i i i i iv v v x v

i i i i i ix x x x v

g

f

D

( )i

2 2( ) ( 1)2

1( ) ( )i i

v v

2( 1) ( )iv

2

1

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100 1200

1

2

frequency (Hz)

0 20 40 60 80 1000

2

4

6

8

10

12

14

spec

tral

pow

er

Forward transfer function

0 20 40 60 80 1000

1

2

3

4

5

6

frequency (Hz)sp

ectr

al p

ower

Backward transfer function

Andre Bastos

V4 V1

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dynamic causal modelling of electromagnetic responses karl friston - wellcome trust centre for...

Documents

field model

basic idea generative

underlying neuronal

basic idea functional

complex cross spectral

generative models

distributed connectivity

empirical example early