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

2. DCM for Steady State Responses
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).

3. Overview Data Features
The Generative Model in DCMs for Steady-State Responses – a family
of neural mass models
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 - a family of neural mass models
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 Density: The Data1 2 3 4
EEG - MEG – LFP Time Series 1 2 3 4
A few LFP channels or EEG/MEG spatial modes

8. Cross Spectral Density: The data from a time series
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 - a family of neural mass models
Bayesian Inversion: Parameter Estimates and Model Comparison
Example. DCM for Steady State Responses:
Glutamate with Microdialysis validation
Predicting Anaesthetic Depth

10. Dynamic Causal Modelling: Generic Framework
Electromagnetic
forward model: neural activityEEG MEG
LFP
Time Domain ERP Data
Phase Domain Data
Time Frequency Data
Steady State Frequency Data
Hemodynamic forward model: neural activityBOLD
Time Domain Data
Neural state equation:
EEG/MEG
fMRI
complicated neuronal model
Fast time scale
simple neuronal model
Slow time scale

11. Dynamic Causal Modelling: Generic Framework
Frequency (Hz)
Power (mV2)
Hemodynamic forward model: neural activityBOLD
Time Domain Data
Electromagnetic
forward model: neural activityEEG MEG
LFP
Steady State Frequency Data
“theta”
Neural state equation:
EEG/MEG
fMRI
complicated neuronal model
Fast time scale
simple neuronal model
Slow time scale

12. Model Structure/ Model Parameters
Dynamic Causal Modelling: Framework
Empirical Data
simple neuronal model
fMRI
complicated neuronal model
EEG/MEG
Neural state equation:
Electromagnetic
forward model: neural activityEEG MEG
LFP
Hemodynamic forward model: neural activityBOLD
Bayesian Inversion
Generative Model
Model Structure/ Model Parameters

13. Model Structure/ Model Parameters
Dynamic Causal Modelling: Framework
Empirical Data
simple neuronal model
fMRI
complicated neuronal model
EEG/MEG
Neural state equation:
Electromagnetic
forward model: neural activityEEG MEG
LFP
Hemodynamic forward model: neural activityBOLD
Bayesian Inversion
Generative Model
Model Structure/ Model Parameters

14. Neural Mass Model State equations neuronal (source) model
Extrinsic Connections
spiny stellate cells
inhibitory interneurons
Pyramidal
Cells
Intrinsic Connections
Internal Parameters
EEG/MEG/LFP
signal
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
neuronal (source) model
State equations

15. Neural Mass Model g g g g g g g g g Intrinsic connections 5
Inhibitory cells in agranular layers 4 g 4 g 3 g 3 g
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
& 1 g 1 g 2 g 2 g
Excitatory pyramidal cells in agranular layers
Extrinsic Connections:
Forward
Backward
Lateral
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

16. Neural Mass Model g g g g g g g g g Synaptic ‘alpha’ kernel
Intrinsic
connections 5 g
Inhibitory cells in agranular layers 4 g 4 g 3 g 3 g
Synaptic ‘alpha’ kernel
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
& 1 g 1 g 2 g 2 g
Sigmoid function
Excitatory pyramidal cells in agranular layers
Extrinsic Connections:
Forward
Backward
Lateral
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

17. Neural Mass Model g g g g g g g g g Synaptic ‘alpha’ kernel
Intrinsic
connections 5 g
Inhibitory cells in agranular layers 4 g 4 g 3 g 3 g
Synaptic ‘alpha’ kernel
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
& 1 g 1 g 2 g 2 g
Sigmoid function
Excitatory pyramidal cells in agranular layers
Extrinsic Connections:
Forward
Backward
Lateral
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

18. Neural Mass Model g g g g g g g g g Synaptic ‘alpha’ kernel
Intrinsic
connections 5 g
Inhibitory cells in agranular layers
: Receptor Density 4 g 4 g 3 g 3 g
Synaptic ‘alpha’ kernel
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
& 1 g 1 g 2 g 2 g
Sigmoid function
Excitatory pyramidal cells in agranular layers
Extrinsic Connections:
Forward
Backward
Lateral
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

19. Neural Mass Model g g g g g g g g g Synaptic ‘alpha’ kernel
Intrinsic
connections 5 g
Inhibitory cells in agranular layers
: Receptor Density 4 g 4 g 3 g 3 g
Synaptic ‘alpha’ kernel
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
& 1 g 1 g 2 g 2 g
Sigmoid function
Excitatory pyramidal cells in agranular layers
: Firing Rate
Extrinsic Connections:
Forward
Backward
Lateral
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

20. Neural Mass Model g g g g g g g g g Synaptic ‘alpha’ kernel
Intrinsic
connections 5 g
Inhibitory cells in agranular layers
: Receptor Density 4 g 4 g 3 g 3 g
Synaptic ‘alpha’ kernel
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
& 1 g 1 g 2 g 2 g
Sigmoid function
Excitatory pyramidal cells in agranular layers
: Firing Rate
Extrinsic Connections:
Forward
Backward
Lateral
Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

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

22. Dynamic Causal Modelling: Steady State Responses
Transfer Function
Frequency Domain

23. Dynamic Causal Modelling: Steady State Responses
Transfer Function
Frequency Domain
Spectrum channel/mode 1
Cross-spectrum modes 1& 2
Transfer Function
Frequency Domain
Power (mV2)
Power (mV2)
Frequency (Hz)
Frequency (Hz)
Power (mV2)
Transfer Function
Frequency Domain
Frequency (Hz)
Spectrum mode 2

24. 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)

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

26. Model Structure/ Model Parameters
Bayesian Inversion
Frequency (Hz)
Power
Empirical Data
Bayesian Inversion
Generative Model
Time Domain
Freq Domain c3 c1
NMM
Output
Cortical Input c2 +
Model Structure/ Model Parameters

27. Bayesian Inversion Model 1 Model 1 Model 2 Bayes’ rules: Free Energy:
max
Inference on parameters
Model 1
Inference on models
Model 1
Model 2
Model comparison via Bayes factor:
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model

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

29. Glutamatergic processing and microdialysis
Microdialysis measurements of glutamate
Two groups of rats with different rearing conditions
LFP recordings from mPFC
Controls
Controls
mPFC
mPFC
Isolated
Isolated
mPFC
mPFC
N=7
N=8
Regular Glutamate
Low Glutamate
4.2 ± 1.4μM
1.5 ± 0.8μM (36%)
mPFC
EEG
0.12
0.12
0.06
0.06
mPFC mV mV - -
0.06
0.06

30. Glutamatergic processing and microdialysis Experimental data
Oscillations from 10 mins : one area (mPFC)
blue: control animals
red: isolated animals
* p<0.05, Bonferroni-corrected

31. Predictions about expected parameter estimates from the microdialysis measurements
upregulation of AMPA receptors
amplitude of synaptic kernels
( He)
chronic reduction in extracellular glutamate levels
SFA
()
 EPSPs
 activation of voltage-sensitive Ca2+ channels → intracellular Ca2+ → Ca-dependent K+ currents
→ IAHP
sensitisation of postsynaptic mechanisms
Van den Pool et al. 1996, Neuroscience
Sanchez-Vives et al. 2000, J. Neurosci.

32. Glutamatergic processing and microdialysis Hypotheses
mPFC
Increased EPSP
Increased adaptation
Excitatory spiny cells
in granular layers
Excitatory pyramidal cells
in agranular layers
Inhibitory cells in
agranular layers
Synaptic ‘alpha’ kernel
Decreased Sigmoid Firing

33. Glutamatergic processing and microdialysis
Results 5 g 5 g
Inhibitory cells in supragranular layers
[ ]
(0.04) 4 g 3 g
[29,37]
(0.4) 4 g
Extrinsic
Extrinsic
Excitatory spiny cells in granular layers
forward
forward
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
[0.76,1.34]
(0.0003)
Control group estimates in blue,
isolated animals in red,
p values in parentheses.
Moran, Stephan, Kiebel, Rombach, O’Connor, Murphy, Reilly, Friston (2008)

34. 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

35. Depth of Anaesthesia Trials: 1: 1.4 Mg Isoflourane
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
(White Noise and Silent Auditory Stimulation)
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

36. Models Ln GBF FB Model (1) A1 Forward (Excitatory Connection)
Backward (Modulatory Connection)
BF Model (2)
Backward (Modulatory Connection) A1
Model 1
Model 2 5 10 15 20 25 30 35
Ln GBF A2
Forward (Excitatory Connection)

37. Model Fits: Model 1

38. Results He: maxEPSP Hi: maxIPSP Isoflurane Isoflurane A1 A2 Isoflurane

39. 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

40. inhibitory interneurons
-100
-50 50
0.2
0.4
0.6
0.8 1
pyramidal cells
spiny stellate cells
fMG
pyramidal cells
pyramidal cells
Membrane Potential (mV)
Exogenous Input (I) 40