1. Dynamic Causal Model for Steady State Responses
Rosalyn Moran
Wellcome Trust Centre for Neuroimaging

2. 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 indirectly
through some lead-field (e.g., electroencephalographic and magnetoencephalographic
data).
Inversion of the ensuing DCM provides conditional probabilities on the synaptic parameters of intrinsic and extrinsic connections in the underlying neuronal network

3. Overview Data Features
The Generative Model in DCMs for Steady-State Responses - neural mass model
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

4. Overview Data Features
The Generative Model in DCMs for Steady-State Responses - neural mass model
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

5. 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

6. Steady State Power (uV2) Source 2 Frequency (Hz) Source 1 Power (uV2)30 25
Power (uV2)
Source 2 20 15 10 5 5 10 15 20 25 30
Frequency (Hz) 5 10 15 20 25 30
Source 1
Power (uV2)
Frequency (Hz)

7. Cross Spectral Density1
EEG - MEG – LFP Time Series 2
Cross Spectral Density 3 1 2 4 3 1 2 3 4 4

8. 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
Resulting in a matrices for c Channels
Cross Spectral Density for channels i,j at frequencies

9. Overview Data Features
The Generative Model in DCMs for Steady-State Responses - neural mass model
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

10. DCM fMRI ERPs Neural state equation: inputs
Hemodynamic forward model: neural activityBOLD
(nonlinear)
Electric/magnetic
forward model: neural activityEEG MEG
LFP
(linear)
Neural state equation:
fMRI
ERPs
Neural model:
1 state variable per region
bilinear state equation
no propagation delays
Neural model:
8 state variables per region
nonlinear state equation
propagation delays
inputs

11. DCM for SSRs Electric/magnetic forward model: neural activityEEG MEG
LFP
(linear)
Hemodynamic forward model: neural activityBOLD
(nonlinear)
Electric/magnetic
forward model: neural activityEEG MEG
LFP
(linear)
Neural state equation:
SSRs
fMRI
ERPs
Neural model:
8-10 state variables per region
propagation delays
linearised model
modulation transfer function
Neural model:
1 state variable per region
bilinear state equation
no propagation delays
Neural model:
8 state variables per region
nonlinear state equation
propagation delays
inputs

12. Neural Mass Model State equations neuronal (source) model
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 states
in terms of their ensemble density. Neural mass models consider only point densities and describe the interaction of the means in the ensemble
MEG/EEG/LFP
signal
Extrinsic Connections
Intrinsic Connections
spiny stellate cells
inhibitory interneurons
pyramidal cells
neuronal (source) model
Internal Parameters
State equations

13. inhibitory interneurons
Neural Mass Model
A F,L,B
inhibitory interneurons
1. Synaptic Input Sigmoid Response Function
spiny stellate cells
Firing Rate
pyramidal cells
2. Synaptic Impulse Response Function
Membrane Potential v
Amplitude (E/IPSP)
Time msec (E/IPSP)

14. Neural Mass Model u u ) , ( u x f = & g g g g g g g g g g g g g g g g
Lateral connections
inputs
Intrinsic
connections
Inhibitory cells in supragranular layers
Inhibitory cells in supragranular layers g g g g 11 8 12 10 2 5 7 9 3 ) ( x S H I B i e - = +
& k g 5 5 5 5
Backward connections g g g g g g g g 4 4 4 4 3 3 3 3
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers
constant input ) , ( u x f =
& 1 2 4 9 ) ( (( x Cu S I F H e k g - + =
& x x
&
& = = x x 1 1 4 4 u u x x
&
& = = k k H H ( ( g g s s ( ( x x - - a a ) ) + + u u ) ) - - 2 2 k k x x - - k k 2 2 x x 4 4 e e e e 1 1 9 9 e e 4 4 e e 1 1 g g g g g g g g 1 1 1 1 2 2 2 2
Excitatory pyramidal cells in infragranular layers
Excitatory pyramidal cells in infragranular layers 6 5 9 3 2 12 4 1 ) ( )) x S H BS i e - = +
& k g m
Forward connections
output

15. ERP or Steady State Responses+
Freq Domain
Output
ERP
Output
Outputs Through
Lead field c1 c2 c3
Time Domain
neuronal
states
output
s2(t)
Time Domain
Freq Domain
output
s3(t)
output
s1(t)
Pulse Input
Freq Domain
Cortical Input
driving
input u(t)

16. Frequency Domain Generative Model (Perturbations about a fixed point)
Time Differential
Equations
State Space
Characterisation
Transfer Function
Frequency Domain
Linearise mV

17. Cross Spectral Density
Transfer Function and Convolution Kernels
First Order Volterra Series Expansion:
Exact Linear Impulse Response
output
s2(t)
output
s3(t)
output
s1(t) u1
By Definition, the Cross Spectral Density is given by

18. Overview Data Features
The Generative Model in DCMs for Steady-State Responses - a family of neural mass model
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

19. Bayesian Inversion + Output c3 c1 c2 Time Domain Freq Domain NMM
Frequency (Hz)
Power
Time Domain
Freq Domain c3 c1
NMM
Output
Cortical Input c2 +
Model Evidence
Approximate Posterior

20. Define likelihood model
Inversion
Neural Parameters
Define likelihood model
Observer function
Specify priors
Inference on models
Invert model
Inference on parameters
Make inferences

21. Overview Data Features
The Generative Model in DCMs for Steady-State Responses - a family of neural mass model
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

22. Glutamate & microdialysis
Schizophrenic: Rearing Models
Controls
Controls
mPFC
mPFC
N=7
Isolated
Isolated
mPFC
N=8
mPFC
Regular Glutamate
Regular Glutamate
Low Glutamate
Low Glutamate
mPFC
EEG
0.12
0.12
0.06
0.06 mV mV - -
0.06
0.06
mPFC

23. 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)
because: amplitude of EPSP → activation of voltage-sensitive Ca2+ channels → intracellular Ca2+ → Ca-dependent K+ currents → IAHP → adaptation

24. Results u u g g g g g g g [3.8,6.3] [4.6,3.9] [0.76,1.34] [29,37]
sensitization of post-synaptic mechanisms
Intrinsic
Intrinsic 5 g 5 g
connections
Inhibitory cells in supragranular layers
[3.8,6.3]
(0.04) 4 g 3 g
[29,37]
(0.4)
[4.6,3.9] 4 g
(0.17)
Extrinsic
Extrinsic
forward
forward
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers
connections
connections u u 1 g 2 g
[195, 233]
[161, 210]
(0. 13)
(0.37)
Excitatory pyramidal cells in infragranular layers
Excitatory pyramidal cells in infragranular layers
Control group estimates in blue
Isolated animals in red with p values in parentheses.
[0.76,1.34]
(0.0003)
Increased neuronal adaption: decrease firing rate
In our simulation excitatory parameters were inferred with inhibitory connectivity (and impulse response) prior parameter variances set to zero.
Two-tailed paired t-test
Moran et al., NeuroImage, 2007

25. Model Fits

26. Overview Data Features
The Generative Model in DCMs for Steady-State Responses - a family of neural mass model
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

27. Case Study: Depth of Anaesthesia
LFP
0.12
0.12
Trials:
1: 1.4 Mg Isoflourane
2: 1.8 Mg Isoflourane
3: 2.4 Mg Isoflourane
4: 2.8 Mg Isoflourane
(1 per condition)
0.06
0.06 mV mV - -
0.06
0.06
30sec
0.12
0.12
0.06
0.06 mV mV - -
0.06
0.06

28. Models FB Model (1) A1 Forward (Excitatory Connection) A2 F Model (2)
Backward (Inhibitory Connection)
F Model (2) A1
Forward (Excitatory Connection)
Forward (Excitatory Connection) A2
L Model (3)
Lateral (Mixed Connection) A1 A2
Lateral (Mixed Connection)

29. Results FB Model (1) A1 Forward A2 Backward 1 2 3 4 20 40 60 80 10020 40 60 80
100
A1 to A2: Excitatory
trial
strength (%)
Results
FB Model (1) A1 A2
Forward
Backward 1 2 3 4 50
100
150
200
250
300
A2 to A1: Modulatory
trial
strength (%)

30. Pathological Beta Rhythms in Parkinson’s
Chronic loss Dopamine innervations in the Striatum
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

31. Pathological Beta RhythmsD
Neuronal states:
LFP model subsets
STN
Str
GPe
Ctx
GPi Th
GABA Glut

32. Pathological Beta Rhythms
Ctx
Str
Effects of Chronic
Dopamine Loss
GPe
STN
GPi Th
Control PD
0.9
1.6
0.8
1.4
0.7
1.2
0.6 1
0.5
0.8
0.4
0.6
0.3
0.2
0.4
0.1
0.2
GPe to STN
Str to GPe

33. 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