1. Sleep Models – A synaptic settling!

2. Why do we sleep? Traditionally, psychologists and sleep scientists believed that we sleep to remember (reinforce memories) Tononi’s theory proposes that we sleep to forget (to cull excess memories, and only reinforce very strong synaptic connections)

3. Sleep Models Sleep models are typically based on two processes, Process S and Process C Process S is the homeostatic process Process C is the circadian process

4. Tononi’s Hypothesis Tononi’s hypothesis suggests that the strength of synaptic weight is proportional to the homeostatic Process S (Tononi also incorporates the idea of “Slow Wave Activity”, which describes brainwave activity during deep sleep )

5. Negative Feedback Therefore, when and, when and.

6. Biological & Mathematical Assumptions Please note: Exclusively qualitative analysis The model for synaptic strength was derived from the already given model for Process S (homeostatic process) – without it we would not have been able to measure the idea of memory storage due to lack of empirical data. In our models we assume that we can neglect the influence of Process C (the effect of Circadian rhythm)

7. MODEL 1a – Constant SWA y(t)=Synaptic strength α=Threshold value for the change in y, and x= Amplitude of SWA a1= min value for Amplitude a2= max value for Amplitude t1= awake time t2= asleep time t= time T= period of y

8. MODEL 1a – Mathematical Analysis Case 1:Case 2: Conditions: and

9. MODEL 1a VS. Process S

10. MODEL 1b – Varying SWA y(t)= Synaptic strength x(t)= Amplitude of SWA α=Threshold value for the change in y γ=Threshold value for the change in t= Time c= Translation parameter of Amplitude

11. MODEL 1b VS. Process S Our function for x(t) modeled the amplitude more naturally but we lost the shape of Process S by changing it. This made us realize the common error with models 1a and 1b: we were assuming the shape of the amplitude – thus leading us to model 2a where the amplitude is a consequence of the interaction between amplitude and synaptic strength.

12. MODEL 2a – Feedback loop x(t)= Amplitude of Slow Wave Activity (SWA) y(t)= Synaptic strength α=Threshold value for Amplitude (x) OR β= Threshold value for Synaptic strength (y) OR (This is an application of the Lotka- Volterra Predator-Prey model.)

13. MODEL 2a – Mathematical Analysis Steady States must satisfy both and Therefore, our two steady states are: and The Jacobian Matrix is

14. MODEL 2a – Mathematical Analysis We evaluate the Jacobian at, in order to determine the nature of the trivial steady state: Eigenvalues are λ=α,-β Therefore the trivial steady state (0,0) is a saddle node. Next, we evaluate the Jacobian at in order to determine the nature of the trivial steady state: Eigenvalues are (purely imaginary) Therefore the non-trivial steady state is a neutral centre.

15. MODEL 2a – Mathematical Analysis

16. Criticism of Model 2a We achieved a natural model for the amplitude of the slow wave activity. However, this does not necessarily run on the same scale as the circadian rhythm. (And in reality, the circadian rhythm does affect the amplitude of our brainwaves.)

17. Implications, modifications, limitations and future directions Time constraint – our biggest limitation Limited mathematical research and empirical data (another limitation) Exclusively qualitative analysis (limitation) As a possible modification, it may be beneficial to include the effects of the Circadian Rhythm (Process C) Hence, Model 2b (possible future direction)

18. MODEL 2b – Oscillating SWA x(t)= Amplitude of Slow Wave Activity (SWA) y(t)= Synaptic strength α= Threshold value for Amplitude (x) β= Threshold value for Synaptic strength (y) κ= Constant (causes a sudden impulse in Amplitude, caused by circadian rhythm) t= Time T= Period of y Amplitude just after Amplitude just before Amplitude just after Amplitude just before t1= Awake time t2= Asleep time

19. The End By: Jane & Allysa Many thanks to: Gustavo & Frederic As well as all the Profs and TA’s!