Dynamic Causal Model for Steady State Responses

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. 30 25 20 15 10 5 0 0 5 10 15 20 25 30 Steady State 30 25 Source 2 Power (uV2) 20 15 10 5 0 0 5 10 15 20 25 30 Frequency (Hz) Source 1 Power (uV2) Frequency (Hz)

7. Cross Spectral Density: The Data 1 EEG - MEG – LFP Time Series 2 Cross Spectral Density 3 1 2 4 1 2 3 4 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EEGMEG LFP Time Domain ERP Data Phase Domain Data Time Frequency Data Steady State Frequency Data Hemodynamicforward 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 Electromagnetic forward model:neural activityEEGMEG LFP Steady State Frequency Data “theta” Hemodynamicforward model:neural activityBOLD Time Domain Data Power (mV2) Frequency (Hz) Neural state equation: EEG/MEG fMRI complicated neuronal model Fast time scale simple neuronal model Slow time scale

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

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

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

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

16. Neural Mass Model g 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Inhibitory cells in agranular layers Synaptic ‘alpha’ kernel Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers 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 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Inhibitory cells in agranular layers Synaptic ‘alpha’ kernel Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers 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 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Inhibitory cells in agranular layers : Receptor Density Synaptic ‘alpha’ kernel Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers 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 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Inhibitory cells in agranular layers : Receptor Density Synaptic ‘alpha’ kernel Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers 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 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Inhibitory cells in agranular layers : Receptor Density Synaptic ‘alpha’ kernel Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers 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) Transfer Function Frequency Domain State Space Characterisation Time Differential Equations Linearise mV

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

23. Dynamic Causal Modelling: Steady State Responses Transfer Function Frequency Domain Cross-spectrum modes 1& 2 Spectrum channel/mode 1 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 Time Domain Freq Domain output s2(t) 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. Power Frequency (Hz) + Freq Domain Output c3 c2 c1 Time Domain Freq Domain NMM NMM NMM Freq Domain Cortical Input Bayesian Inversion Empirical Data Generative Model Bayesian Inversion Model Structure/ Model Parameters

27. Inference on parameters Model 1 Bayes’ rules: Free Energy: max Inference on models Bayesian Inversion 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 1.5 ± 0.8μM (36%) 4.2 ± 1.4μM mPFC EEG 0.12 0.12 0.06 0.06 mPFC 0 0 mV mV - - 0.06 0.06

30. Glutamatergic processing and microdialysisExperimental 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 Inhibitory cells in agranular layers Increased adaptation Excitatory spiny cells in granular layers Excitatory pyramidal cells in agranular layers Synaptic ‘alpha’ kernel DecreasedSigmoidFiring

33. g g 5 5 g g g 4 4 3 g g 1 2 Glutamatergic processing and microdialysis Results Inhibitory cells in supragranular layers [3.86.3] (0.04) [29,37] (0.4) Extrinsic Extrinsic Excitatory spiny cells in granular layers forward forward Excitatory spiny cells in granular layers connections connections u u [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 A1 A2 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 0 0 mV mV - - 0.06 0.06 30sec 0.12 0.12 0.06 0.06 0 0 mV mV - - 0.06 0.06

36. Models FB Model (1) A1 Forward (Excitatory Connection) Backward (Modulatory Connection) A2 BF Model (2) Backward (Modulatory Connection) A1 35 30 A2 25 Forward (Excitatory Connection) Ln GBF 20 15 10 5 0 Model 1 Model 2

37. Model Fits: Model 1

38. Results He: maxEPSP Isoflurane Isoflurane A1 A2 Hi: maxIPSP Isoflurane 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 pyramidal cells fMG spiny stellate cells 1 pyramidal cells pyramidal cells 0.8 Membrane Potential (mV) 0.6 0.4 Exogenous Input (I) 0.2 0 -100 -50 0 50 40