Rosalyn Moran Virginia Tech Carilion Research Institute Dynamic Causal Modelling for Cross Spectral...
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Transcript of Rosalyn Moran Virginia Tech Carilion Research Institute Dynamic Causal Modelling for Cross Spectral...
Rosalyn Moran
Virginia Tech Carilion Research Institute
Dynamic Causal Modelling for Cross Spectral Densities
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
simple neuronal model
Slow time scale
fMRI
complicated neuronal model
Fast time scale
EEG/MEG
),,( uxFdt
dx
Neural state equation:
Hemodynamicforward model:neural activity BOLD
Time Domain Data
Electromagneticforward model:neural activity EEGMEGLFP
Time Domain ERP DataPhase Domain DataTime Frequency DataSteady State Frequency DataCross Spectral Densities (Frequency Domain)
Dynamic Causal Modelling: Generic Framework
simple neuronal model
Slow time scale
fMRI
complicated neuronal model
Fast time scale
EEG/MEG
),,( uxFdt
dx
Neural state equation:
Electromagneticforward model:
neural activity EEGMEGLFP
CSDs
Hemodynamicforward model:neural activity BOLD
Time Domain Data
Frequency (Hz)
Pow
er (m
V2 )
“theta”
Dynamic Causal Modelling: Framework
simple neuronal model
fMRIfMRI
complicated neuronal model
EEG/MEGEEG/MEG
),,( uxFdt
dx
Neural state equation:
Electromagneticforward model:
Hemodynamicforward model:
Generative M
odel
Baye
sian
Inve
rsio
n
Empirical Data
Model Structure/ Model Parameters
Inference on models
Dynamic Causal Modelling: FrameworkBa
yesi
an In
vers
ion
)|(
)|(),|(),|(
myp
mpmypmyp
Bayes’ rules:
Model 1Model 2 Model 1
Free Energy: )),()(()(ln mypqDmypF max
-2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
%1.99)|0( yconnp
Inference on parameters
)|(
)|(
2
1
myp
mypBF
Model comparison via Bayes factor:
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
Inference on models Inference on parameters
Dynamic Causal Modelling: FrameworkBa
yesi
an In
vers
ion
)|(
)|(
2
1
myp
mypBF
Model comparison via Bayes factor:
)|(
)|(),|(),|(
myp
mpmypmyp
Bayes’ rules:
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
Model 1Model 2 Model 1
Free Energy: )),()(()(ln mypqDmypF max
-2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
),()( mypq
%1.99)|0( yconnp
Dynamic Causal Modelling: Neural Mass Model
neuronal (source) model
State equations
Extrinsic Connections
,,uxFx
spiny stellate cells
inhibitory interneurons
PyramidalCells
Intrinsic Connections
Internal Parameters
EEG/MEG/LFPsignal
EEG/MEG/LFPsignal
Properties of tens of thousands of neurons approximated by their average response
Dynamic equations mimic physiology and produce electrophysiological responses
A Neural Mass Model (6) layer cortical regions)State equations: A dynamical systems description
of anatomy and physiology ,,uxFx
Extrinsic Connections
spiny stellate
cells
Supragranular Pyramidal
Cells + inhibitory
interneurons
Deep Pyramidal
Cells + inhibitory
interneurons
Intrinsic Connections
Internal ParametersEg.Time constants of Sodium ion channels
GABAa receptors
AMPA receptors
Neurotransmitters: Glu/GABA
Dynamics mimicked at AMPA and GABA receptors
AP generation zone
synapses
eH
e
Cortico-cortical connection
GABAa receptors
AMPA receptors
Neurotransmitters: Glu/GABA
AP generation zone
1 Intrinsic Connection
Cortico-cortical connection
GranularLayer:Excitatory Cells
SupragranularLayer:Inhibitory Cells
3
InfragranularLayer:Pyramidal Cells
Parameters quantify contributions at AMPA and GABA receptors
synapses
eH
e
Cortico-cortical connection
GABAa receptors
AMPA receptors
Neurotransmitters: Glu/GABA
AP generation zone
1 Intrinsic ConnectionGranularLayer:Excitatory Cells
SupragranularLayer:Inhibitory Cells
3
InfragranularLayer:Pyramidal Cells
Extrinsicforward
connections
4 3
214
014
41
2))()((
ee
LF
e
e xxCuxSIAA
Hx
xx
1 2)( 0xSAF
)( 0xSAL
)( 0xSABExtrinsic backward connections
Intrinsic connections
Extrinsic lateral connections
,,uxfx
0x
278
038
87
2))()((
ee
LB
e
e xxxSIAA
Hx
xx
236
746
63
225
1205
52
650
2)(
2))()()((
iii
i
ee
LB
e
e
xxxS
Hx
xx
xxxSxSAA
Hx
xx
xxx
spiny stellate
cells
inhibitory interneurons
pyramidal cells
State equations in a 6 layer cortical model
Time Differential Equations
)(
)(
xly
Buxfx
State Space Characterisation
Cxy
BuAxx
Transfer FunctionFrequency Domain
BAsICsH )()(
Linearise
mV
State equations to Spectra
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007) A neural mass model of spectral responses in electrophysiology. NeuroImage
Predicted response (Pyramidal Cell Depolarization)
Given an empirical recording: estimate parameters of the model
4g 3g
2g1
24914
41
2))(( xxuaxsHxxx
eeee kkgk
&&
Excitatory spiny cells in granular layers
3g
1g 2g
5g
AMPA receptor density
GABAa receptor density
Glutamaterelease
0 2 4 6 8 10 12 14 16 180.7
0.8
0.9
1
1.1
1.2
1.3
1.4
800
2
4
6
8
10
12
14
16
Frequency (Hz)
Fre
qu
ency
(H
z)
No
rmal
ised
Po
wer
(a
.u.)
L-Dopa
Placebo
Frequency (Hz)
No
rmal
ised
Po
wer
(a
.u.)
0 6 11 16
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Measurement
AMPA time constant
GABArelease
Bayesian Inversion
Increased activity at GABAreceptors in supragranular layers
Superficial layers
Granular layers
Deep layers
GABAa TC
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
Neuromodulators: Acetylcholine/Dopamine
E13 E
31
E23
I32
EERVE
EE
VEELL
gVg
IVVgVVgVC
)),((
)()()1()3()3(
13)1(
)1()1()1()1(
NMDANMDARVI
INMDA gVg )),(( )2()2()2(23
)2(
fMg
= ))(exp(1
1)3(V
NMDANMDARVI
INMDA
IIRVI
II
EERVE
EE
gVg
gVg
gVg
)),((
)),((
)),((
)2()2()2(31
)2(
)2()2()2(22
)2(
)2()3()3(23
)2(
Superficial layers
Granular layers
Deep layers
IIRVI
II
EERVE
EE
VIINMDAMGNMDA
EELL
gVg
gVg
VVgVVfg
VVgVVgVC
)),((
)),((
)()(
)()(
)2()2()2(22
)2(
)2()3()3(23
)2(
)2()2()2()2(
)2()2()2()2(
Sodium Channel
Chloride Channel
Potassium Channel
Depolarization dependentCalcium Channel
NMDA mediated switch
Neurotransmitters: Glu/GABA
A conductance model offers more biological plausibility
Moran, Stephan, Dolan, Friston (2011) Consistent Spectral Predictors for Dynamic Causal Models of Steady State Responses. NeuroImage
0 2 4 6 8 10 12 14 16 180.7
0.8
0.9
1
1.1
1.2
1.3
1.4
800
2
4
6
8
Frequency (Hz)
Fre
qu
ency
(H
z)
No
rmal
ised
Po
wer
(a
.u.)
Frequency (Hz)N
orm
alis
ed P
ow
er
(a.u
.)
AMPA/NMDA Ratio higher in Prefrontal Regions than Parietal Regions
0 6 11 160.7
0.8
0.9
1
1.1
1.2
1.31.4
Roadmap
Specify model
Extract Data Features
Maximise the model evidence
(~-F)
Test models or MAP parameters
Find your experimental data
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50
60 70 80 90 100
Prediction Prediction
Prediction
Summary:DCM for Steady State Responses
Cortical Macrocolumns and free parameters
dx/dt = Ax + B
| H1(ω) . H1*(ω) |
| H1(ω) . H2*(ω) |
| H2(ω) . H2*(ω) |
Generative Model
Frequency (Hz)
Prediction Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Prediction Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Prediction Response
Pow
er (m
V2 )
0 10 20 30 40 50
60 70 80 90 100
Cortical Macrocolumns and free parameters
dx/dt = Ax + B
| H1(ω) . H1*(ω) |
| H1(ω) . H2*(ω) |
)exp()(
)()()(
////
/
ieieieie
ie
ttHth
tuthtv
| H2(ω) . H2*(ω) |
Model Inversion
Summary:DCM for Steady State Responses
Outline
• DCM & Spectral Data Features (the Basics)• DCM for CSD vs DCM for SSR
• DCM for CSD Example
Time to Frequency DomainLinearise around a stable fixed point or LC
DCM for SSR
DCM for CSD
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50
60 70 80 90 100
Prediction Prediction
Prediction
DCM for Cross Spectral Densities
Cortical Macrocolumns and free parameters
dx/dt = Ax + B H1(ω) . H2*(ω)
)exp()(
)()()(
////
/
ieieieie
ie
ttHth
tuthtv
Generative Model
Spectra and Phase lagCoherence
Cross Correlations
H1(ω) . H1*(ω)
H2(ω) . H2*(ω)
Cortical Macrocolumns and free parameters
dx/dt = Ax + B
)exp()(
)()()(
////
/
ieieieie
ie
ttHth
tuthtv
Model Inversion using
full complex signal
Spectra and Phase lagCoherence
Cross Correlations
H1(ω) . H2*(ω)
H2(ω) . H2*(ω)
H1(ω) . H1*(ω)
DCM for Cross Spectral Densities Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Response
Pow
er (m
V2 )
0 10 20 30 40 50
60 70 80 90 100
Prediction Prediction
Prediction
Accommodating Imaginary Numbers
F
ln2
1ln2
1
2
1
2ln2
ln2
1
2
1
)),()(()ln(
11
T
T n
ypqDyF
E:
)( 11
11
T
T
G
GG
M:
11 )()(
))((2
1)(
))((2
1)(
II
PPGGPtrI
GGPtrI
jiTT
ij
TTi
Real and imaginary errors
Real and imaginary derivatives wrt fx, G
1. Interface Additions2. New CSD routines, similar to SSR3. SPM_NLSI_GN accommodates imag numbers, slopes, curvatures 4. A host of new results features, in channel and source space!
Roadmap
Specify model
Extract Data Features
Maximise the model evidence
(~-F)
Test models or MAP parameters
Find your experimental data And also report
phase lags coherence &
delaysIn channel or source space
PFC
Hipp
Conditional Estimates: Spectral Power
10 20 30 400
2
4
6
8
10
12
14
16
18
mode 1 to 1
frequency Hz10 20 30 40
0
2
4
6
8
10
12
14
16
18
mode 2 to 1
frequency Hz
10 20 30 400
5
10
15
Spectral density over modes(in channel-space)
frequency (Hz)
abs(
CSD
)
10 20 30 400
2
4
6
8
10
12
14
16
18
mode 2 to 2
frequency Hz
predicted: trial 1observed: trial 1
predicted: mode 1observed: mode 1predicted: mode 2observed: mode 2
Abs(H1(ω) . H1*(ω)) Abs(H1(ω) . H2*(ω))
Abs(H2(ω) . H1*(ω)) Abs(H2(ω) . H2*(ω))
Pow
er
PFC
Hipp
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Channels: 2 to 1
frequency Hz
predicted: trial 1
observed: trial 1
0 10 20 30 40 501
1
1
1
1
1
1
1
Coh: pfc to hipp
frequency Hz
Conditional Estimates: Coherence
|(H1(ω).H2*(ω))|2
______________________
{(H1(ω).H1*(ω)) + (H2(ω).H2*(ω))}
PFC
Hipp
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 -50 0 50 100
-0.05
0
0.05
0.1
0.15
0.2mode 1 to 1
lag (ms)-100 -50 0 50 100
-0.05
0
0.05
0.1
0.15
0.2mode 2 to 1
lag (ms)
-100 -50 0 50 100
-0.05
0
0.05
0.1
0.15
0.2
Auto-covariance(in channel-space)
Lag (ms)
auto
-cov
aria
nce
-100 -50 0 50 100
-0.05
0
0.05
0.1
0.15
0.2
mode 2 to 2
lag (ms)
trial 1
channel 1channel 2
Conditional Estimates: Covariance
PFC
Hipp
arg(H1(ω).H2*(ω))
____________ ω
0 10 20 30 40 50-25
-20
-15
-10
-5
0
5Delay (ms) 2 to 1
frequency Hz
predicted: trial 1observed: trial 1
0 10 20 30 40 50-25
-20
-15
-10
-5
0
5
10
Delay (ms) PfC to Hipp
Frequency Hz
trial 1
Conditional Estimates: Delays
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
-0.06
0
0.06
0.12
mV
A2
LFP
-0.06
0
0.06
0.12
mV
-0.06
0
0.06
0.12
mV
-0.06
0
0.06
0.12
mV
A1
1.4 % Isoflurane
1.8 % Isoflurane
2.4 % Isoflurane
2.8 % Isoflurane
SummaryDCM 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
The FIL Methods Group
Karl FristonDimitris PinotsisMarco LeiteVladimir LitvakJean DaunizeauStephan KiebelWill PennyKlaas StephanAndre BastosPascal Fries
Acknowledgments