1. The sleep-wake cycle: constraining steady states by electroencephalogram analysis  Modelling neurons and the brain  EEG and stability analysis  Constraints on sleep-wake states  Physiology and the sleep-wake cycle Anthony L. Krensel

2. The brain: Cortex: “higher order” functions... Thalamus: filters information to cortex Brainstem: functions include arousal Continuum Modelling:  Modelling neural populations by average properties and significant connections

3. Neuronal signalling Receives signal Propagates down axon Synapse: Terminals transmit neurochemicals, onto next neuron Response to Signal

4. Neuronal signalling Receives signal Propagates down axon Synapse: Terminals transmit neurochemicals, onto next neuron Response to Signal

5. The EEG S2 Spectrum EC Spectrum Awake, eyes closed (EC) Enhancement at low f (gold) 1/f behaviour (green) Strong alpha peak (red) Small beta peak (orange) Sleep, stage 2 (S2) Enhancement at low f (gold) 1/f behaviour (green) Spindle peak/peaks (blue) 3

6. The EIRS Model Cortex: ● Excitatory (e) ● Inhibitory (i) Thalamus: ● Reticular nucleus (r) ● Relay nuclei (s) Subthalamic Input (n) Can find steady state firing rates

7. Steady State Steady state firing rates  d/dt = 0 so,,, can be computed. Nonlinear relationship: and The linear gains are

8. The EEG spectrum and Stability Instability = power in single frequency diverges First approximation to spectrum: given by squared modulus of transfer function:

9. Stability Analysis InstabilityDispersion relation Most unstable case: k = 0 

10. EEG analysis  Examine very low frequency regime,,,

11. A constrained parameter space (EC)

12. Constraining EC

13. Constrained parameter space (S2)

14. Constraining S2

15. Qualitative Results Firing rates change as expected Reticular sleep-wake switch: active in sleep Relay nuclei inputs increase Strong intra-cortical connectivity increase

16. Arousal projection to the cortex

17. Inputs to the model

18. Summary and the Future Constrained EC/S2 states in x,y,z Demonstrated links to physiology of sleep-wake Found a thalamic reticular sleep-wake switch Extend this work to encompass sleep cycle Results guide modelling of sleep-wake inputs Ultimately link EIRS model to existing brainstem models: e.g. the Phillips-Robinson model, the Circadian Oscillator, etc.

19. Acknowledgements  Prof. Peter Robinson  Dr. Peter Drysdale

20. Extra equations (basic model equations)

21. More equations