On the intimate relationship between functional and effective connectivity
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Transcript of On the intimate relationship between functional and effective connectivity
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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.
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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.
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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.
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Circa 1993 Circa 2013
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Michael D. Fox, MD, PhD
“Why did you guy’s drop the ball with functional connectivity?”
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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.
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( , , )x f x u
( ) ( , )y t g x e
( )u tThe forward (dynamic
causal) modelEndogenous fluctuations
( )
Effective connectivity
Functional connectivity
Observed timeseries
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A connectivity reconstruction problem:
A degenerate (many-to-one) mapping between effective and functional connectivity
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( , , )x f x u
( ) ( , )y t g x e
( )u tThe forward (dynamic
causal) modelEndogenous fluctuations
( )
Effective connectivity
Functional connectivity
Observed timeseries
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( , , )x f x u
( )y t
( )u tBayesian model
inversionEndogenous fluctuations
( )
Effective connectivity
Functional connectivity
Observed timeseries
| , ( | )
ln | ( , )
p m q
p m F
argmin ( , )F
Posterior density
Log model evidence(Free energy)
Richard Feynman
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0 10 20 30 40 50 600
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log-
prob
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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
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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
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| , ( | , )
: ( | ) 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
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Simulating the response of a four-node network
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model
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graph size
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1Model posterior
model
prob
abilit
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And recovering (discovering) the true architecture
Complexity
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0 0.5 1 1.5 2 2.5 3 3.5
x 104
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log-
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x 104
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model
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An empirical example (with six nodes)
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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
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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.
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( , , )x f x u
( )y t
( )u tThe forward (dynamic
causal) modelEndogenous fluctuations
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
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( , , )x f x u
( )y t
( )u tThe forward (dynamic
causal) modelEndogenous fluctuations
Observed timeseries
( , )
( , )u u uD F y
D F y
Stochastic DCM
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1 2 3 4 5 6 7 8 9-0.5
0
0.5True and MAP connections
50 100 150 200 250-0.2
-0.15
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time (bins)Extrinsic coupling parameter
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time50 100 150 200 250
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time
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time
Network or graph generating data
( )x t
Stochastic DCM
Deterministic DCM
Simulated responses of a three node network
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( , , )x f x u
( )y t
( )u tThe forward (dynamic
causal) modelEndogenous fluctuations
Observed timeseriesSpectral DCM
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( , , )x f x u
( , )u The forward (dynamic
causal) modelEndogenous fluctuations
( ) exp
( ) ( ) ( )x x
u e
g f
t t
Spectral DCM
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( , , )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
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Network or graph generating data
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time (seconds)
ampl
itude
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ampl
itude
Hemodynamic response and noise
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Region 1
Region 2
Region 3
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Frequency and time (bins)
real
Prediction and response
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Frequency and time (bins)
imag
inar
y
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0.1
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Simulated responses of a three node network
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128 256 384 512 640 768 896 10240
0.05
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0.25
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0.35
Session length (scans)
RMS
Root mean square error
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1 2 3 4 5 6 7 8 9-0.3
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0.4True and MAP connections (BPA: 256 scans)
The effect of scan length:
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128 256 384 512 640 768 896 10240
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Session length (scans)
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Strongest connection
Subjects
Post
erio
r exp
ecta
tion
(Hz)
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Stochastic
Subjects
Post
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r exp
ecta
tion
(Hz)
Spectral
stoc
hasti
c
spectral
EmpiricalSimulations
Accu
racy
Accu
racy
Comparing spectral and stochastic DCM
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( ) ( , ) ( )
( ) ( , ) ( )
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
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1F
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y a z
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( )ij
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( ) ( ( ))
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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
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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)
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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.
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( , , )x f x u
( , )u The forward (dynamic
causal) modelEndogenous fluctuations
( ) 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
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( , , )x f x u
( , )u The forward (dynamic
causal) modelEndogenous fluctuations
( ) 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
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( , , )x f x u
( , )u The forward (dynamic
causal) modelEndogenous fluctuations
( ) 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
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1 2 3 4 50
10
20
30
40
50
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70Embedding (empirical)
Embedding dimension
Free
ene
rgy
1i
i
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time {seconds}
The number of slow (unstable) modes and their time constants
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( , , )x f x u
( , )u The forward (dynamic
causal) modelEndogenous fluctuations
( ) 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:
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The forward (dynamic causal) model
Log evidence
Accuracy
Complexity
Number of modes (m)
Principal modes in the language system
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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
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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