Dynamic Causal Model for Steady State Responses
Rosalyn Moran
Wellcome Trust Centre for Neuroimaging
DCM for Steady State Responses
A dynamic causal model (DCM) of steady-state responses in electrophysiological data is summarised in terms of their cross-spectral density.
Where
These spectral data-features are generated by a biologically plausible, neural-mass model of coupled electromagnetic sources; where each source comprises three sub-populations.
Under linearity and stationarity assumptions, the model’s biophysical parameters (e.g., post-synaptic receptor density and time constants) prescribe the cross-spectral density of responses measured directly (e.g., local field potentials) or indirectlythrough some lead-field (e.g., electroencephalographic and magnetoencephalographicdata).
Inversion of the ensuing DCM provides conditional probabilities on the synaptic parameters of intrinsic and extrinsic connections in the underlying neuronal network
Overview
1. Data Features
2. The Generative Model in DCMs for Steady-State Responses - neural mass model
3. Bayesian Inversion: Parameter Estimates and Model Comparison
4. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation
Predicting Anaesthetic Depth
Overview
1. Data Features
2. The Generative Model in DCMs for Steady-State Responses - neural mass model
3. Bayesian Inversion: Parameter Estimates and Model Comparison
4. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation
Predicting Anaesthetic Depth
Steady State
Statistically:
A “Wide Sense Stationary” signal has 1st and 2nd moments that do not vary with respect to time
Dynamically:
A system in steady state has settled to some equilibrium after a transient
Data Feature:
Quasi-stationary signals that underlie: Spectral Densities in the Frequency Domain
Steady State
0 5 10 15 20 25 300
5
10
15
20
25
30
0 5 10 15 20 25 300
5
10
15
20
25
30
Frequency (Hz)
Frequency (Hz)
Pow
er (
uV2)
Pow
er (
uV2)
Source 2
Source 1
Cross Spectral Density E
EG
- M
EG
– L
FP
Tim
e S
eri
es
1
2
3
4
Cro
ss
Sp
ec
tral D
en
sity
1
1
2
2 3
3
4
4
Cross Spectral Density
Vector Auto-regression a p-order model:
Linear prediction formulas that attempt to predict an output y[n] of a system based on
the previous outputs
npnpnnn eyyyy ....2211
))(()( pAfg ij
ijijijij HHg )()()(
iwpijp
iwijiwijij eeeH
......
1)(
221
f 2
}{:)(....1 ccpAp Resulting in a matrices for c Channels
Cross Spectral Density for channels
i,j at frequencies
..)(
..)()(
12
1211
g
gg
Overview
1. Data Features
2. The Generative Model in DCMs for Steady-State Responses - neural mass model
3. Bayesian Inversion: Parameter Estimates and Model Comparison
4. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation
Predicting Anaesthetic Depth
),,( uxFx Neural state equation:
Electric/magneticforward model:
neural activityEEGMEGLFP
(linear)
Neural model:1 state variable per regionbilinear state equationno propagation delays
Neural model:8 state variables per region
nonlinear state equationpropagation delays
fMRIfMRI ERPsERPs
inputs
Hemodynamicforward model:neural activityBOLD(nonlinear)
DCM
),,( uxFx Neural state equation:
Electric/magneticforward model:
neural activityEEGMEGLFP
(linear)
Neural model:1 state variable per regionbilinear state equationno propagation delays
Neural model:8 state variables per region
nonlinear state equationpropagation delays
fMRIfMRI ERPs
ERPs
inputs
Hemodynamicforward model:neural activityBOLD(nonlinear)
Neural model:8-10 state variables per region
propagation delayslinearised model
modulation transfer function
SSRsSSRs
Electric/magneticforward model:
neural activityEEGMEGLFP
(linear)
DCM for SSRs
Neural Mass Model
Extrinsic Connections
neuronal (source) model
State equations ,,uxFx
spiny stellate cells
inhibitory interneurons
pyramidal cells
Intrinsic Connections
Internal Parameters
MEG/EEG/LFPsignal
MEG/EEG/LFPsignal
The state of a neuron comprises a number of attributes, membrane potentials, conductances etc. Modelling these states can become intractable. Mean field approximations summarise the statesin terms of their ensemble density. Neural mass models consider only point densities and describe the interaction of the means in the ensemble
Neural Mass Model
1. Synaptic Input Sigmoid Response Function
2. Synaptic Impulse Response Function Membrane Potential v
Firi
ng R
ate
Am
plitu
de
(E/IP
SP
)
Time msec (E/IPSP)
)exp()(
)()(
////
/
ieieieie
ie
ttHth
rthtv
)exp(1
1
))(exp(1
1
2121
vr
spiny stellate
cells
inhibitory interneurons
pyramidal cells
A F,L,B
BLFieie AHH ,,
2/1,54321 ,,,,,,,,,
Neural Mass Model
Backward connections
Lateral connections
Forward connections
inputs
4
4
3
3
1
1
2
2
u 12
4914
41
2))(( xxuaxsHx
xx
eeee
12
4914
41
2))(( xxuaxsHx
xx
eeee
5
5
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in infragranular layers
Inhibitory cells in supragranular layers
output
4
4
3
3
1
1
2
2
u
Intrinsicconnections
5
5
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in infragranular layers
Inhibitory cells in supragranular layers
),( uxfx
11812
102
1112511
1110
72
8938
87
2)(
2)()(
xxx
xxxSHx
xx
xxxSIBHx
xx
iiii
eeee
12
4914
41
2))()(( xxCuxSIFHx
xx
eeee
659
32
61246
63
22
51295
52
2)(
2))()((
xxx
xxxSHx
x
xxxSxBSHx
xx
iiii
eeee
constant input
ERP or Steady State Responses
Time Domain
Freq Domain
Time Domain
Freq Domain
Outputs Through Lead fieldc
3
c1
outputs1(t)
outputs2(t) output
s3(t)
neuronalstates
drivinginput u(t)
Freq DomainOutput
Freq DomainOutput
Freq DomainCortical InputFreq DomainCortical Input
/)( 21 fH
bf
aU
1)(
c2
+
Time DomainTime Domain
ERPOutputERP
Output
Pulse InputPulse Input
)(ty
Frequency Domain Generative Model(Perturbations about a fixed point)
Time Differential Equations
)(
)(
xly
Buxfx
State Space Characterisation
Cxy
BuAxx
Transfer FunctionFrequency Domain
BAsICsH )()(
Linearise
mV
• Transfer Function and Convolution Kernels
• First Order Volterra Series Expansion:
Exact Linear Impulse Response
Cross Spectral Density
dtetdtet tjkj
tjki
kij
)()()(
)(
)()(
tu
ts
k
iki
1
0
110 )()()(
dtuty
• By Definition, the Cross Spectral Density is given by
c3
c1
outputs1(t)
outputs2(t) output
s3(t)
c2
u1
Overview
1. Data Features
2. The Generative Model in DCMs for Steady-State Responses - a family of neural mass model
3. Bayesian Inversion: Parameter Estimates and Model Comparison
4. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation
Predicting Anaesthetic Depth
Bayesian Inversion
Time Domain
Freq Domain
Time Domain
Freq Domain
c3c1
NMM
NMM
NMM
Freq DomainOutput
Freq DomainOutput
Freq DomainCortical Input
Freq DomainCortical Input
)( fH
bf
aU
1)(
c2
+Frequency (Hz)
Po
wer
)(p)),|(||)(()|(ln mypqDmypF
Model Evidence
Approximate Posterior
)|(
),(),|(),|(
)(),|()|(
mGp
mpmGpmGp
dpmGpmGp
),(),(
))(),((),|(
Nmp
NmGp
Invert model
Make inferences
Define likelihood model
Specify priors
Neural Parameters
Observer function
Inference on models
Inference on parameters
Inversion
2/121
2 )|)((|)(
)()(
iii
iii
jH
G
LBFaieie AAAgHH ,,,,,,,,,,, ,,54321
Overview
1. Data Features
2. The Generative Model in DCMs for Steady-State Responses - a family of neural mass model
3. Bayesian Inversion: Parameter Estimates and Model Comparison
4. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation
Predicting Anaesthetic Depth
Glutamate & microdialysis
Schizophrenic: Rearing Models
Low GlutamateRegular Glutamate
Isolated mPFCControls mPFC
Low GlutamateRegular Glutamate
Isolated mPFCN=8
Controls mPFCN=7
mPFC-0.06
0
0.06
0.12
mV
mPFC EEG
-0.06
0
0.06
0.12
mV
Hypotheses
• Main findings from microdialysis:
– reduction in prefrontal glutamate levels of isolated group
→ sensitization of post-synaptic mechanisms (e.g. upregulation)
• Model parameters should reflect
amplitude of synaptic kernels
coupling parameters of glutamatergic connections
neuronal adaptation (i.e., 2)
Results
connections
Extrinsicforward
connections
4
1 2u
Intrinsic 5
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in infragranular layers
Extrinsicforward
connections
4 3
u
Intrinsic 5
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in infragranular layers
Inhibitory cells in supragranular layers
[161, 210]
[29,37]
[195, 233]
(0.4)
(0.37)(0. 13)
[3.8,6.3]
[4.6,3.9]
[0.76,1.34] (0.0003)
(0.17)
(0.04)eH
et
2
Control group estimates in blue Isolated animals in red with p values in parentheses.
In our simulation excitatory parameters were inferred with inhibitory connectivity (and impulse response) prior parameter variances set to zero.
sensitization of post-synaptic mechanisms
Increased neuronal adaption: decrease firing rate
Two-tailed paired t-test
Moran et al., NeuroImage, 2007
Model Fits
Overview
1. Data Features
2. The Generative Model in DCMs for Steady-State Responses - a family of neural mass model
3. Bayesian Inversion: Parameter Estimates and Model Comparison
4. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation
Predicting Anaesthetic Depth
Case Study: Depth of Anaesthesia
A1 A2
-0.06
0
0.06
0.12
mV
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
Trials:1: 1.4 Mg Isoflourane2: 1.8 Mg Isoflourane3: 2.4 Mg Isoflourane4: 2.8 Mg Isoflourane
(1 per condition)
30sec
A1
A2
Forward (Excitatory Connection)
Backward (Inhibitory Connection)
A1
A2
Forward (Excitatory Connection)
A1
A2
Forward (Excitatory Connection)
Lateral (Mixed Connection)
FB Model (1)
F Model (2)
L Model (3)
Lateral (Mixed Connection)
Models
1 2 3 40
50
100
150
200
250
300
A2 to A1: Modulatory
trial
stre
ngth
(%)
1 2 3 40
20
40
60
80
100
A1 to A2: Excitatory
trial
stre
ngth
(%)
Results
1 2 30
100
200
300
400
500
600
700
Log-
evid
ence
(rel
ative
)
Models
Bayesian Model Selection
1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Post
erio
r Mod
el P
roba
bility
Bayesian Model Selection
Models
Data identity has not been verified
A1
A2
Forward
Backward
FB Model (1)
Pathological Beta Rhythms in Parkinson’s
Traditional theory of negative motor symptoms induced by an
unbalance in the striatal outputs of direct ( ) /indirect ( ) pathways
Newer theory focused on pathological synchrony: STN
Beta oscillations correlate to disease state
20 Hz
Chronic loss Dopamine innervations in the Striatum
Pathological Beta Rhythms
DDD
Neuronal states:LFP model subsets
STN
Str
GPe
D
Ctx
GPiTh
GABA Glut
Pathological Beta Rhythms
STN
Str
GPe
Ctx
GPiTh
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
GPe to STN
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Str to GPe
Control
PD
Effects of Chronic
Dopamine Loss
Summary
• DCM is a generic framework for asking mechanistic questions of neuroimaging data
• Neural mass models parameterise intrinsic and extrinsic ensemble connections and synaptic measures
• DCM for SSR is a compact characterisation of multi- channel LFP or EEG data in the Frequency Domain
• Bayesian inversion provides parameter estimates and allows model comparison for competing hypothesised architectures
• Empirical results suggest valid physiological predictions