Upload
colleen-edwards
View
215
Download
1
Tags:
Embed Size (px)
Citation preview
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.
One ring to rule them all, one ring to find them, one ring to bring them all, and in the darkness bind them
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
Generative model and face validation
An empirical example
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
peristimulus time (ms)
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
peristimulus time (ms)
Induced response
peristimulus time (ms)
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
peristimulus time (ms)
Exogenous input
50 100 150 200 250 300 350 400 450 500
-60
-40
-20
0
peristimulus time (ms)
Hidden neuronal states (conductance and depolarisation)
Spectral density
peristimulus time (ms)
Hz
100 200 300 400 500
10
20
30
40
50
60
Coherence
peristimulus time (ms)
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)
100 200 300 400 500-60
-40
-20
0
20
40
60
source 1source 2
Predicted responses to sustained exogenous (stimulus) input
transfer function: 1 to 1
peristimulus time (ms)
Hz
100 200 300 400 500
10
20
30
40
50
60
transfer function: 2 to 1
peristimulus time (ms)
Hz
100 200 300 400 500
10
20
30
40
50
60
transfer function: 1 to 2
peristimulus time (ms)
Hz
100 200 300 400 500
10
20
30
40
50
60
transfer function: 2 to 2
peristimulus time (ms)
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)
Endogenous fluctuations
Endogenous fluctuations
100 200 300 400 500
-2
0
2
4
6evoked: source 1
peristimulus time (ms)
100 200 300 400 500
-4
-2
0
2
4
evoked: source 2
peristimulus time (ms)
100 200 300 400 500-4
-2
0
2
4
6
evoked: source 1
peristimulus time (ms)
100 200 300 400 500
-5
0
5
evoked: source 2
peristimulus time (ms)
Spectral density
peristimulus time (ms)
Hz
100 200 300 400 500
10
20
30
40
50
60
Coherence
peristimulus time (ms)
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
peristimulus time (ms)
Hz
100 200 300 400 500
10
20
30
40
50
60
Coherence
peristimulus time (ms)
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
Simulated responses(sample estimates
over16 trials)
Predicted responses(expectation under
known input)
Overview
The basic idea
Generative model and face validation
An empirical example
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
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
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
evoked: channel/source 2
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
0 100 200 300 400 500-4
-2
0
2
evoked: channel/source 3
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
0 100 200 300 400 500-3
-2
-1
0
1
evoked: channel/source 4
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
0 100 200 300 400 500
-2
0
2
evoked: channel/source 1
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
0 100 200 300 400 500
-5
0
5
evoked: channel/source 2
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
0 100 200 300 400 500
-4
-2
0
2
4
evoked: channel/source 3
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
0 100 200 300 400 500
-2
0
2
evoked: channel/source 4
peristimulus time (ms)
induced: condition 1
peristimulus time (ms)
Hz
-100 0 100 200 300 400 50010
15
20
25
30
35
Posterior predictions following inversion of induced responses
transfer function: 1 to 1
peristimulus time (ms)
Hz
0 200 400
10
20
30
transfer function: 3 to 1
peristimulus time (ms)
Hz
0 200 400
10
20
30
transfer function: 1 to 3
peristimulus time (ms)
Hz
0 200 400
10
20
30
transfer function: 3 to 3
peristimulus time (ms)
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