10_DCM_CSD

Dynamic Causal Modelling for
Cross Spectral Densities
Rosalyn Moran
Virginia Tech Carilion Research Institute
Outline
• DCM & Spectral Data Features (the Basics)
• DCM for CSD vs DCM for SSR
• DCM for CSD Example
Outline
• DCM & Spectral Data Features (the Basics)
• DCM for CSD vs DCM for SSR
• DCM for CSD Example
Dynamic Causal Modelling: Generic Framework
Electromagnetic
forward model:
neural activity EEG
MEG
LFP
Hemodynamic
forward model:
neural activity BOLD
Time Domain Data
Time Domain ERP Data
Phase Domain Data
Time Frequency Data
SteadySpectral
State Frequency
Cross
Densities Data
(Frequency Domain)
dx
 F ( x, u, )
dt
Neural state equation:
fMRI
simple neuronal model
Slow time scale
EEG/MEG
complicated neuronal model
Fast time scale
Electromagnetic
forward model:
neural activity EEG
MEG
LFP
CSDs
Hemodynamic
forward model:
neural activity BOLD
Time Domain Data
“theta”
Power (mV2)
Dynamic Causal Modelling: Generic Framework
Frequency (Hz)
dx
 F ( x, u, )
dt
Neural state equation:
fMRI
simple neuronal model
Slow time scale
EEG/MEG
complicated neuronal model
Fast time scale
Dynamic Causal Modelling: Framework
Empirical Data
Generative Model
Bayesian Inversion
Hemodynamic
forward model:
Model Structure/ Model
Parameters
Electromagnetic
forward model:
Neural state equation:
dx
fMRI
simple neuronal model
dt
 F ( x, u, )
EEG/MEG
complicated neuronal model
Dynamic Causal Modelling: Framework
Bayes’ rules:
p ( | y , m ) 
p ( y |  , m ) p ( | m )
p( y | m)
Bayesian Inversion
Free Energy:
F  ln p ( y m )  D ( q ( ) p ( y , m ) )
max
Inference on models
Inference on parameters
Model 1
Model 2
Model 1
0.8
Model comparison via Bayes factor:
p ( y | m1 )
BF 
p( y | m2 )
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
-1
0
1
2
3
4
5
p ( conn  0 | y )  99 . 1 %
Dynamic Causal Modelling: Framework
Bayes’ rules:
p ( | y , m ) 
p ( y |  , m ) p ( | m )
p( y | m)
Bayesian Inversion
Free Energy:
F  ln p ( y m )  D ( q ( ) p ( y , m ) )
max
Inference on parameters
Inference on models
Model 1
Model 2
Model 1
0.8
Model comparison via Bayes factor:
p ( y | m1 )
BF 
p( y | m2 )
0.7
q ( )  p ( y , m )
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
-1
0
1
2
3
4
5
p ( conn  0 | y )  99 . 1 %
Dynamic Causal Modelling: Neural Mass Model
EEG/MEG/LFP
signal
Properties of tens of thousands of
neurons approximated by their average
response
Intrinsic
Connections
inhibitory
interneurons
neuronal (source) model
spiny stellate
cells
Internal
Parameters
x  F  x , u ,  
Pyramidal
Cells
Extrinsic
Connections
State equations
Dynamic equations mimic physiology and produce
electrophysiological responses
Neurotransmitters: Glu/GABA
AMPA receptors
A Neural Mass Model (6) layer cortical regions)
x  F  x , u ,  
State equations: A dynamical systems description
of anatomy and physiology
Intrinsic
Connections
Supragranular
Pyramidal Cells
+ inhibitory
interneurons
Internal
Parameters
Eg.
Time constants of
Sodium ion channels
spiny stellate
cells
Deep Pyramidal
Cells + inhibitory
interneurons
Extrinsic
Connections
GABAa receptors
Dynamics mimicked at AMPA and GABA receptors
Neurotransmitters: Glu/GABA
AMPA receptors
Cortico-cortical connection
3
AP generation zone
1
GABAa receptors
Supragranular
Layer:
Inhibitory Cells
Intrinsic Connection
Granular
Layer:
Excitatory Cells
synapses
He

Infragranular
Layer:
Pyramidal Cells
AP generation zone
e
Cortico-cortical connection
Parameters quantify contributions at AMPA and
Neurotransmitters: Glu/GABA
GABA receptors
AMPA receptors
3
AP generation zone
1
Supragranular
Layer:
Inhibitory Cells
Intrinsic Connection
Granular
Layer:
Excitatory Cells
synapses
He

Infragranular
Layer:
Pyramidal Cells
e
Cortico-cortical connection
GABAa receptors
State equations in a 6 layer cortical model
x  f  x , u , 

x 7  x 8
x 8 

inhibitory
interneurons
Extrinsic
forward
connections
spiny
stellate
cells
Extrinsic
L
A S ( x0 )
lateral
connections
F
He
e
(( A  A   3 I ) S ( x 0 )) 
B
L
e

x7
e
2
3
4
x1  x 4
x 4 
A S (x0 )
He
e
(( A  A   1 I ) S ( x 0 )  Cu ) 
F
1
pyramidal
cells
2 x8
L
Intrinsic
connections

2 x4
e

x1
e
2
2
x 0  x 5  x 6
x0
x 2  x 5
x 5 
He
e
(( A  A ) S ( x 0 )   2 S ( x1 )) 
B
L
x 3  x 6
x 6 
Hi
i
 4 S ( x7 ) 
2 x6
i

x3
i
2
2 x5
e

x2
e
2
Extrinsic
B
A S ( x0 )
backward
connections
State equations to Spectra
Time Differential
Equations
State Space
Characterisation
x  f ( x )  Bu
x  Ax  Bu
y  l( x)
y  Cx
Transfer Function
Frequency Domain
H ( s )  C ( sI  A ) B
Linearise
mV
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007) A neural mass model of spectral responses in electrophysiology. NeuroImage
Given an empirical recording: estimate parameters of the
model
GABAa receptor
density
AMPA receptor
density
Superficial layers
v4 = g 4
g 4 = k e H e AB S(v6region 2 ) + k e H eg 3S (v6 ) - 2k e g 4 - k e2 v4
g5 = k i H ig 5 S (v7 ) - 2k i g5 - k i2 v5
v7 = g 4 - g5
3
Granular layers
v1 = g1 spiny cells in granular layers
Excitatory
1 H
x4 AF S(v region2 ) + k H g S(v ) - 2k g - k 2 v
g1 = xk
e 1
e 1
x4 ekeeHe (1s(x9 6a) u) 2keex4 e k1e2x1 6
v2
1
=g
Deep layers
1.4
1.31.4
1.21.3
Glutamate
release
i2 = k e H e A S(v
region 2
6
) + k e H eg 2 S(v1 ) - 2k e g 2 - k v
2
e 2
v3 = g 3
1.1
1 1
0.8
0.8
0.7
0.7
0 2
4
6
8
g3 = k i H ig 4 S(v7 ) - 2k i g3 - k v
AMPA time
constant
6
8
6
4
10 12 14 16 18
2
Frequency (Hz)
0
2
i 3
v6 = g 2 - g3
1.2
1.1
0.90.9
2
B
GABAa
TC
2
Increased
activity at
GABA
receptors
in
16
L-Dopa
14
supragranular
12
Placebo
layers 10
Frequency (Hz)
4
Normalised Power (a.u.)
Normalised Power (a.u.)
GABA
release
5
Measurement
v5 = g5
800
11
Frequency (Hz)
Bayesian Inversion
Predicted response
(Pyramidal Cell
Depolarization)
Moran, Stephan, Seidenbecher, Pape, Dolan, Friston (2009) Dynamic Causal Model of Steady State Responses. NeuroImage
Friston, Bastos, Litvak, Stephan, Fries, Moran (2012) DCM for complex data: cross-spectra, coherence and phase-delays. NeuroImage
16
A conductance model offers more biological plausibility
Neurotransmitters: Glu/GABA
Superficial layers
C V
(2)
Sodium Channel
(2)
)  g E (V E  V
(2)
 g NMDA f MG (V NMDA  V
(2)
Chloride Channel
I
 32
 g L (V L  V
g
g
(2)
E
 k E (  ( 
g
(2)
I
 k I (  ( 
(2)
NMDA
E
23
I
22
 k I (
I
23
(3)
V
(2)
V
 (
(2)
 VR , 
(3)
 VR , 
(2)
(2)
V
 VR , 
(2)
Neuromodulators:
Acetylcholine/Dopamine
)
)  g I (V I  V
(2)
) g
(2)
E
)  E
) g
(2)
I
)  I
(2)
(2)
)  V
)  g NMDA )   NMDA
(2)
Granular layers

Potassium ChannelC V
(1 )
 g L (V L  V
(1 )
)  g E (V E  V
(1 )
(1 )

E
23
)  I  V
(1 )
E
(3)
(3)
(1 )
g E  k E (  13  (  V  V R ,  )  g E )   E
(2)
E
(3)
(3)
(2)
g E  k E (  23  (  V  V R ,  )  g E )   E
(2)
I
(2)
(2)
(2)
g I  k I (  22  (  V  V R ,  )  g I )   I
Depolarization dependent
1.4
Calcium Channel
1.4
(2)
I
(2)
(2)
(2)
g NMDA  k I (  31 (  V  V R ,  )  g NMDA )   NMDA
f Mg 
1  exp(   (   V
1.31.3
1.21.2
1.1
1.1
1
(3)
))
AMPA/NMDA
Ratio higher in
Prefrontal
Regions
than Parietal
Regions
NMDA mediated switch
Frequency (Hz)
Deep layers
E
 31
Normalised Power (a.u.)
Normalised Power (a.u.)
 13E
1
1 0.9
0.90.8
0.7
0.8
0.7
0
0
2
4
6
8
10
12
14
16
Frequency (Hz)
Moran, Stephan, Dolan, Friston (2011) Consistent Spectral Predictors for Dynamic Causal Models of Steady State Responses. NeuroImage
6
4
18
2
Frequency (Hz)
6
8
800
11
16
Roadmap
Extract Data
Features
Specify model
Find your
experimental data
Maximise the
model evidence
(~-F)
Test models or
MAP parameters
0
10
20
30
40
50
60
70
Response
Prediction
Power (mV2)
Response
Prediction
Power (mV2)
Summary:
DCM for Steady
State Responses
80
90
100
0
10
20
30
40
60
5
70
80
90
100
0
Frequency (Hz)
Frequency (Hz)
Power
(mV2)
Prediction
Response
0
10
20
30
40
50
Frequency (Hz)
| H2(ω) . H2*(ω) |
| H1(ω) . H1*(ω) |
| H1(ω) . H2*(ω) |
Cortical Macrocolumns and free parameters
dx/dt = Ax + B
60
70
80
90
100
0
10
20
30
40
50
60
70
Prediction
Response
Power (mV2)
Prediction
Response
Power (mV2)
Summary:
DCM for Steady
State Responses
80
90
100
0
10
20
30
40
60
5
Power (mV2)
10
20
30
40
50
Frequency (Hz)
| H2(ω) . H2*(ω) |
| H1(ω) . H1*(ω) |
dx/dt = Ax + B
v ( t )  he / i ( t )  u ( t )
Cortical Macrocolumns and free parameters
80
90
100
Prediction
Response
0
| H1(ω) . H2*(ω) |
70
0
Frequency (Hz)
Frequency (Hz)
he / i ( t )  H e / ik e / i t exp(  t k e / i )
60
70
80
90
100
Outline
• DCM & Spectral Data Features (the Basics)
• DCM for CSD vs DCM for SSR
• DCM for CSD Example
Time to Frequency Domain
Linearise around a stable fixed point or LC
DCM for SSR
DCM for CSD
Power (mV2)
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
60
5
70
80
90
100
0
Frequency (Hz)
Frequency (Hz)
Prediction
Response
Power
(mV2)
DCM for
Cross Spectral
Densities
Response
Prediction
Power (mV2)
Response
Prediction
0
10
20
30
40
50
60
70
80
90
Frequency (Hz)
H2(ω) . H2*(ω)
H1(ω) . H1*(ω)
Spectra and Phase lag
Coherence
Cross Correlations
H1(ω) . H2*(ω)
dx/dt = Ax + B
v ( t )  he / i ( t )  u ( t )
Cortical Macrocolumns and free parameters
he / i ( t )  H e / ik e / i t exp(  t k e / i )
100
Power (mV2)
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
60
5
70
80
90
100
0
Frequency (Hz)
Frequency (Hz)
Prediction
Response
Power
(mV2)
DCM for
Cross Spectral
Densities
Response
Prediction
Power (mV2)
Response
Prediction
0
10
20
30
40
50
60
70
80
90
Frequency (Hz)
H2(ω) . H2*(ω)
H1(ω) . H1*(ω)
Spectra and Phase lag
Coherence
Cross Correlations
H1(ω) . H2*(ω)
dx/dt = Ax + B
v ( t )  he / i ( t )  u ( t )
Cortical Macrocolumns and free parameters
he / i ( t )  H e / ik e / i t exp(  t k e / i )
100
Accommodating Imaginary Numbers
F
Real and imaginary
errors
F  ln( y  )  D ( q ( ) p ( y ,  ) )

1
1
   
T
2
Real and imaginary
derivatives wrt fx, G

1
ln 
1

2

T

 

2
E:
1

n
ln 2 
2
1
ln 

2

1

ln 
2
M:

 1


1
 G  G  
T
 
 1
1
I ( )  



(G      )
T
1
2
I (  )   


tr ( Pi ( 
1
2
T
tr ( Pij ( 
1
1
  I (  )  I (  ) 

   G   G  ))
T
T

   G   G  )  Pi  P j  )
T
Roadmap
Extract Data
Features
Specify model
Find your
experimental data
1.
2.
3.
4.
And also report
phase lags
coherence &
delays
In channel or
source space
Maximise the
model evidence
(~-F)
Test models or
MAP parameters
Interface Additions
New CSD routines, similar to SSR
SPM_NLSI_GN accommodates imag numbers, slopes, curvatures
A host of new results features, in channel and source space!
Conditional Estimates:
Spectral Power
mode 2 to 1
Power
mode 1 to 1
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
10
20
30
40
frequency Hz
Spectral density over modes
(in channel-space)
predicted: mode 1
observed: mode 1
predicted: mode 2
observed: mode 2
15
abs(CSD)
0
10
20
30
0
predicted: trial 1
observed: trial 1
18
16
Hipp
14
12
PFC
4
20
30
40
frequency (Hz)
Abs(H2(ω) . H2*(ω))
mode 2 to 2
6
10
Abs(H2(ω) . H1*(ω))
40
8
5
Abs(H1(ω) . H2*(ω))
frequency Hz
10
10
Abs(H1(ω) . H1*(ω))
2
0
10
20
30
40
frequency Hz
Conditional Estimates:
Coherence
Coh: pfc to hipp
1
1
|(H1(ω).H2*(ω))|2
______________________
Channels: 2 to 1
0.9
1
0.8
1
{(H1(ω).H1*(ω)) + (H2(ω).H2*(ω))}
0.7
0.6
1
0.5
1
0.4
0.3
1
0.2
1
1
0
0.1
10
20
30
frequency Hz
40
50
0
0
10
20
30
frequency Hz
predicted: trial 1
observed: trial 1
40
50
Hipp
PFC
Conditional Estimates:
Covariance
mode 1 to 1
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-100
-50
0
50
100
mode 2 to 1
-100
-50
0
50
F-1(H1(ω).H1*(ω))
F-1(H1(ω).H2*(ω))
F-1(H2(ω).H1*(ω))
F-1(H2(ω).H2*(ω))
100
lag (ms)
lag (ms)
mode 2 to 2
Auto-covariance
(in channel-space)
0.2
channel 1
trial 1
0.15
channel 2
0.15
auto-covariance
0.2
Hipp
0.1
0.1
0.05
PFC
0.05
0
0
-0.05
-100
-50
0
Lag (ms)
50
100
-0.05
-100
-50
0
50
lag (ms)
100
Conditional Estimates:
Delays
Delay (ms) PfC to Hipp
arg(H1(ω).H2*(ω))
____________
ω
Delay (ms) 2 to 1
10
5
trial 1
5
predicted: trial 1
observed: trial 1
0
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
0
10
20
30
Frequency Hz
40
50
-25
0
10
20
30
40
50
frequency Hz
Hipp
PFC
Outline
• DCM & Spectral Data Features (the Basics)
• DCM for CSD vs DCM for SSR
• DCM for CSD Examples
Pharmacological Manipulation of Glutamate and GABA
- 4 levels of anaesthesia: each successively decreasing glutamate and increasing
GABA
(Larsen et al Brain Research 1994; Lingamaneni et al Anesthesiology 2001; Caraiscos et al J Neurosci 2004 ; de Sousa et
al Anesthesiology 2000 )
- LFP recordings from primary auditory cortex (A1) & posterior auditory field
(PAF)
- White noise stimulus & Silence
1.4 %
Isoflurane
1.8 %
Isoflurane
2.4 %
Isoflurane
2.8 %
Isoflurane
A1
LFP
0.12
0.06
mV 0
-0.06
A2
0.12
0.06
mV 0
-0.06
Summary
DCM for CSD:
Suitable for long time series with trial-specific spectral features eg
pronounced beta
Fits complex spectral data features
Offers similar connectivity estimates to DCM for ERPs
With estimates of frequency specific delays and coherence
Can be used with all biophysical, Neural Mass Models (CMC, LFP
etc.)
Thank You
Acknowledgments
The FIL Methods Group
Karl Friston
Dimitris Pinotsis
Marco Leite
Vladimir Litvak
Jean Daunizeau
Stephan Kiebel
Will Penny
Klaas Stephan
Andre Bastos
Pascal Fries

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