on the intimate relationship between functional and effective connectivity

On the intimate relationship between functional and effective connectivity

Karl Friston, Wellcome Centre for Neuroimaging

The past decade has seen tremendous advances in characterising functional integration in the brain; especially in the resting state community. Much of this progress is set against the backdrop of a key dialectic between functional and effective connectivity. I hope to highlight the intimate relationship between functional and effective connectivity and how one informs the other. My talk will focus on the application of dynamic causal modelling to resting state timeseries or endogenous neuronal activity. I will survey recent (and rapid) developments in modelling distributed neuronal fluctuations (e.g., stochastic, spectral and symmetric DCM for fMRI) – and how this modelling rests upon functional connectivity. This survey concludes by looking at the circumstances under which functional and effective connectivity can be regarded as formally identical. I will try to contextualise these developments in terms of some historical distinctions that have shaped our approaches to connectivity in functional neuroimaging.

Dinner Speaking [edit]

The Dinner speech should not resort to the base forms of humor. The humor should be topical and relevant to the idea presented. This type of speech is found at the collegiate level and is typically eight to ten minutes long.

http://en.wikipedia.org/w/index.php?title=Individual_events_(speech)&action=edit&section=8

Circa 1993 Circa 2013

http://upload.wikimedia.org/wikipedia/commons/9/9a/Default_mode_network-WRNMMC.jpg

http://en.wikipedia.org/wiki/File:Default_Mode_Network_Connectivity.png

Michael D. Fox, MD, PhD

“Why did you guy’s drop the ball with functional connectivity?”

http://scholar.google.co.uk/citations?user=d9kqMj0AAAAJ&hl=en

http://scholar.google.co.uk/citations?user=d9kqMj0AAAAJ&hl=en

http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=294GzeQHeS9BOM&tbnid=pQTkVJ19Kln3dM:&ved=0CAUQjRw&url=http://f1socialdiary.com/2013/03/10/melbourne-the-f1-sd-guide-to-having-a-good-time/&ei=K735U5-BMMTkasyZgfAN&bvm=bv.73612305,d.ZGU&psig=AFQjCNGILTX5qFTucndOreKbfZPskExaTg&ust=1408962173516540

http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=WHq5_AZedD6S5M&tbnid=yHS3xjSU4teo8M:&ved=0CAUQjRw&url=http://www.cogneuro-lab.org/DynamicPage.aspx?u=12&ei=KL75U_yKO5TbavqIgPgG&bvm=bv.73612305,d.ZGU&psig=AFQjCNELFBGm4zfuceVx9kzW8853vn4tlw&ust=1408962418218160

( , , )x f x u

( ) ( , )y t g x e

( )u tThe forward (dynamic

causal) modelEndogenous fluctuations

( )

Effective connectivity

Functional connectivity

Observed timeseries

A connectivity reconstruction problem:

A degenerate (many-to-one) mapping between effective and functional connectivity

( , , )x f x u

( ) ( , )y t g x e



( )



Observed timeseries

( , , )x f x u

( )y t

( )u tBayesian model

inversionEndogenous fluctuations

( )



Observed timeseries

| , ( | )

ln | ( , )

p m q

p m F

argmin ( , )F

Posterior density

Log model evidence(Free energy)

Richard Feynman

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

model

prob

abilit

y

0 10 20 30 40 50 60-600

-500

-400

-300

-200

-100

0

100

model

log-

prob

abilit

y

ln ( | )p m ( | )p m

Bayesian model comparison

( , , )x f x u

( )y t

( )u tBayesian model

inversionEndogenous fluctuations

( )

| , ( | )

ln | ( , )

p m q

p m F

Posterior density

Log model evidence

Bayesian model averaging

| | , |m

p p m p m

Bayesian model inversion

| , ( | )

ln | ( , )

p m q

p m F

Posterior density

Log model evidence

Model evidence and Ockham’s principle

[ln ( | , )] [ ( | ) || ( | , )]q KLF E p m D q p m

Accuracy Complexity

ln |p m F

( , , ) ( )i iif x u A u B x Cu Complexity

fMRI models

EEG models

fMRI data

EEG data

Evidence is afforded by data …

Exogenous input

13

( )u t

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in infragranular layers

Inhibitory cells in supragranular layers

Endogenous output

3( )x t

31

2332

23 3 3 31 1 32 22 ( ) ( )e e ex x x x x

22 2 2 23 32 ( )i i ix x x x

21 1 1 13 32 ( ( ) )e e ex x x x u

| , ( | , )

: ( | ) 0i F

i F

i F i i

p y m p y mm m

p m

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

( | )i

i F FF

p mp y m p y m p y m d

p m

And the concept of reduced models

This means that we only have to invert the full model to score all reduced models; c.f., the Savage-Dickey density ratio

Armani, Calvin Klein and Versace design houses did not refuse this year to offer very brave and reduced models of the “Thong” and “Tango”. The designers consider that a man with the body of Apollo should not obscure the wonderful parts of his body.

Bayesian model reduction

Simulating the response of a four-node network

0 5 10 15-0.6

-0.4

-0.2

0

0.2

0.4True and MAP connections

0 10 20 30 40 50 60-600

-500

-400

-300

-200

-100

0

100Log-evidence

model

log-

prob

abilit

y

0 1 2 3 4 5 6-600

-500

-400

-300

-200

-100

0

100Log-evidence

graph size

log-

prob

abilit

y

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Model posterior

model

prob

abilit

y

And recovering (discovering) the true architecture

Complexity

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-400

-300

-200

-100

0

100Log- evidence

log-

prob

abilit

y

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Model posterior

model

prob

abilit

y

An empirical example (with six nodes)

0 200 400 600 800 1000 1200-4

-2

0

2vis: responses

0 200 400 600 800 1000 1200-5

0

5ag: responses

0 200 400 600 800 1000 1200-2

0

2

4sts: responses

0 200 400 600 800 1000 1200-4

-2

0

2ppc: responses

0 200 400 600 800 1000 1200-5

0

5fef: responses

0 200 400 600 800 1000 1200-5

0

5pfc: responses

time {seconds}

0.00 0.00 -0.57 -0.28 -0.17 -0.31 0.00 0.00 -0.34 0.00 -0.37 -0.42 0.57 0.34 0.00 -0.45 -0.43 -0.51 0.28 0.00 0.45 0.00 0.00 -0.25 0.17 0.37 0.43 0.00 0.00 -0.28 0.31 0.42 0.51 0.25 0.28 0.00

'vis' 'sts' 'pfc' 'ppc' 'ag' 'fef'

Differences in reciprocal connectivity

( , , )x f x u

( )y t



Observed timeseries

50 100 150 200 250-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

uu U

( )U t

argmin ( , )F y

Endogenous fluctuations

Deterministic DCM

( , , )x f x u

( )y t



Observed timeseries

( , )

( , )u u uD F y

D F y

Stochastic DCM

1 2 3 4 5 6 7 8 9-0.5

0


50 100 150 200 250-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Hidden states

time (bins)Extrinsic coupling parameter

1 2 3 4 5 6 7 8 9-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8 True and MAP connections

Extrinsic coupling parameter

50 100 150 200 250-1.5

-1

-0.5

0

0.5

1

1.5Signal and noise

time50 100 150 200 250

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Hidden states

time

50 100 150 200 250-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2 Hidden causes

time

Network or graph generating data

( )x t

Stochastic DCM

Deterministic DCM

Simulated responses of a three node network

( , , )x f x u

( )y t



Observed timeseriesSpectral DCM

( , , )x f x u

( , )u The forward (dynamic


( ) exp

( ) ( ) ( )x x

u e

g f

t t

Spectral DCM

( , , )x f x u

( , )ug The forward (dynamic causal) model

Endogenous fluctuations

†

( ) ( ( ))

( ) ( ) ( ) ( ) ( )u e

K t

g K g K g

FSpectral DCM

Complex cross-spectra

http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=Gyv-UBuMK2jUnM&tbnid=UOEzDqSXCTnPQM:&ved=0CAUQjRw&url=http://article.sapub.org/10.5923.j.ijis.20120204.04.html&ei=EA4DVIGeEoqUapGdgPgI&bvm=bv.74115972,d.d2k&psig=AFQjCNH9FU66c7LcGmunE3oRTBjfKO7dzQ&ust=1409572670514390

Network or graph generating data

-0.3

-0.20.4

0.2

0 50 100 150 200 250 300-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Endogenous fluctuations

time (seconds)

ampl

itude

0 50 100 150 200 250 300-0.1

-0.05

0

0.05

0.1Hidden states

time (seconds)

ampl

itude

Hemodynamic response and noise

0 50 100 150 200 250 300-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time (seconds)

ampl

itude

Region 1

Region 2

Region 3

0 50 100 150 200 250 300-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Frequency and time (bins)

real

Prediction and response

0 50 100 150 200 250 300-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Frequency and time (bins)

imag

inar

y

1 2 3 4 5 6 7 8 9-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


Simulated responses of a three node network

128 256 384 512 640 768 896 10240

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Session length (scans)

RMS

Root mean square error

1 2 3 4 5 6 7 8 9-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4True and MAP connections (BPA: 1024 scans)

1 2 3 4 5 6 7 8 9-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4True and MAP connections (BPA: 256 scans)

The effect of scan length:

128 256 384 512 640 768 896 10240

0.05

0.1

0.15

0.2

0.25

0.3

0.35

128 256 384 512 640 768 896 10240

0.05

0.1

0.15

0.2

0.25

Session length (scans)

0 5 10 15 20-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Strongest connection

Subjects

Post

erio

r exp

ecta

tion

(Hz)

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Stochastic

Subjects

Post

erio

r exp

ecta

tion

(Hz)

Spectral

stoc

hasti

c

spectral

EmpiricalSimulations

Accu

racy

Accu

racy

Comparing spectral and stochastic DCM

( ) ( , ) ( )

( ) ( , ) ( )

x t f x u t

y t g x e t

( ) ( ) ( ) ( )

( ) expx x

y t u t e t

g f

2| ( ) |( )

( ) ( )ij

ijii jj

gC

g g

†

†

( ) ( ) ( )

( ) ( )u e

g Y Y

K g K g

Dynamic causal model

Convolution kernel representationFunctional Taylor expansion

Spectral representationConvolution theorem

( ) ( ) ( ) ( )

( ) ( ( ))

Y K U E

K t

F

Cross-spectral density

Coherence

( )( )

(0) (0)ij

ij

ii jj

Cross-correlation

( ) ( ) ( )

( ) ( )

T

u e

y t y t

t 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( ) ( )Tiic I a I a

Auto-correlation

1

11[ , ]

T T

Tp

a y y y y

Auto-regression coefficients

1

1

( ) ( ) ( , ) ( ( , ))

( ) ( ) ( , ) ( ( , ))

Tu u

e e

u t u t g

t t g

F

F

Second-order data features (functional connectivity)

( , , )x f x u



( ) exp

( ) ( ) ( )x x

u e

g f

t t

x

T T

J f

J J V V

2Re( )

T

TTv

e

V V

V VV V

What if the connectivity was symmetrical?

Symmetrical DCM

( , , )x f x u



( ) exp

( ) ( ) ( )x x

u e

g f

t t

x

T T

J f

J J V V

2Re( )

T

TTv

e

V V

V VV V

Symmetrical DCM

( , , )x f x u



( ) exp

( ) ( ) ( )x x

u e

g f

t t

x

T T

J f

J J V V

1

2Re( )

T

Tv

x

V V

V VJ

Symmetrical DCM

In the absence of measurement noise, effective connectivity becomes the negative inverse functional connectivity

1 2 3 4 50

10

20

30

40

50

60

70Embedding (empirical)

Embedding dimension

Free

ene

rgy

1i

i

0 200 400 600 800 1000 1200-4

-2

0

2vis: responses

0 200 400 600 800 1000 1200-5

0

5ag: responses

0 200 400 600 800 1000 1200-2

0

2

4sts: responses

0 200 400 600 800 1000 1200-4

-2

0

2ppc: responses

0 200 400 600 800 1000 1200-5

0

5fef: responses

0 200 400 600 800 1000 1200-5

0

5pfc: responses

time {seconds}

The number of slow (unstable) modes and their time constants

( , , )x f x u



( ) exp

( ) ( ) ( )x x

u e

g f

t t

x

T Tm m m

J f

J J V J V

TV V

Large DCMs

Breaking the symmetry:

The forward (dynamic causal) model

Log evidence

Accuracy

Complexity

Number of modes (m)

Principal modes in the language system

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3566585/figure/f0005/

Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry.

chapter 1, “The Law of Gravitation,” p. 34

Richard Feynman

Thank you

And thanks to

Bharat BiswalChristian Büchel

CC ChenJean Daunizeau

Olivier David Marta GarridoSarah GregoryLee Harrison

Joshua KahanStefan Kiebel

Baojuan LiAndre MarreirosRosalyn MoranHae-Jeong Park

Will PennyAdeel Razi

Mohamed SeghierKlaas Stephan

And many others

on the intimate relationship between functional and effective connectivity

Documents

functional connectivity

functional neuroimaging

resting state community

state timeseries

neuronal fluctuations

modelling rests u

endogenous neuronal

rapid developments