1. Why we sleep

2. Overview Estimates: brain consists of 100 billions of neurons that are connected with about 10 14 of synapses Function of the brain is based on interaction between highly networked neurons by means of electrical impulses Typically neurons connect to at least a thousand other neurons

3. Neurons are typically composed of a soma, a dendritic tree and an axon The axon extends away from the cell body and is the main conducting unit for carrying signals to other neurons. Signals flow in only one direction

4. About 1000 times a night, billions of neurons undergo a synchronous one- second burst of non-REM electrical activity. Throughout the night, the bursts become smaller. The bursts disappear completely just before waking The longer a person has been sleep- deprivated, the bigger the initial burst

5. Classical Interpretation

6. The waves of brain activity during deep sleep reactivate neurons strengthen neuronal connection The bursts let the brain slowly reinforce synaptic connections that already exist.  We sleep to remember

7. New Interpretation By Giulio Tononi, a neuroscientist at the University of Wisconsin

8. “Going up and down, up and down, basically all the neurons fire and then all are silent – it’s a wonderful way for the brain to tell the synapses to get weaker” The progressive weakening allows only the strong connections to survive.

9. Tononi’s theory Without paring unneeded information, our brains would face “space crunch” By proportionally weakening synapses, the brain ensures that they retain the same strength relative to each other.

10. Development of the model discrete model for strength of the synapses during our sleep simulate two interpretations: 1. brain bursts cause strengthening 2. brain bursts cause weakening of synapses include influence of neighbouring synapses

11. General equations syn (i,j),t+1 = syn (i,j),t + α * (g(t) + f(t)); g(t) = β * (syn (i-1,j),t + syn (i+1,j),t + syn (i,j-1),t + syn (i,j+1),t ) f(t) = -c(n) * (t - n) * (t - n - 1);  Parabola with negative coefficient in front of t 2  ]0, 1[, ]3, 4[, ]6, 7[,...  Maximum at t = n + 0.5 c(n) = μ * (n + 0.5) (-0.1)  strictly decreasing

12. Graph of f and c

13. Choice of signs and parameters leads to different interpretations classical interpretation: syn (i,j),t+1 = syn (i,j),t + α * (g(t) + f(t)), α = 0.01 new interpretation: syn (i,j),t+1 = syn (i,j),t - α * (g(t) + f(t)), α = 0.01 our interpretation: syn (i,j),t+1 = syn (i,j),t + α * (g(t) + f(t)) for syn (i,j),t ≥ threshold syn (i,j),t+1 = syn (i,j),t + α * (g(t) - f(t)), α = 0.01 for syn (i,j),t < threshold

14. Realisation of function g(t) of neighbours Assumptions: cell represents neuron with synapses strength of synapses is proportional to strength of neuron  focus on synapses simplification: neuron sends signals only to one neighbour, but can be reached by 0 to 4 neighbours (von Neumann neighbourhood)

17. Simulations – part I

19. Classical syn (i,j),t+1 = syn (i,j),t + 0.01 * (g(t) + f(t)); g(t) = 0.001* (syn (i-1,j),t + syn (i+1,j),t + syn (i,j-1),t + syn (i,j+1),t ) f(t) = -c(n) * (t - n) * (t - n - 1), c(n) = (n + 0.5) (-0.1)

21. New syn (i,j),t+1 = syn (i,j),t - 0.01 * (g(t) + f(t)); g(t) = 0.001* (syn (i-1,j),t + syn (i+1,j),t + syn (i,j-1),t + syn (i,j+1),t ) f(t) = -c(n) * (t - n) * (t - n - 1), c(n) = (n + 0.5) (-0.1)

23. Threshold syn (i,j),t+1 = syn (i,j),t + 0.01 * (g(t) + f(t)) for syn (i,j),t ≥ threshold syn (i,j),t+1 = syn (i,j),t + 0.01 * (g(t) - f(t)) for syn (i,j),t < threshold syn thres = 0.5

25. Comparison: initial conditions vs. model with threshold

26. Simulations – part II: Bigger bursts

27. Classical syn (i,j),t+1 = syn (i,j),t + 0.01 * (g(t) + f(t)); g(t) = 0.001* (syn (i-1,j),t + syn (i+1,j),t + syn (i,j-1),t + syn (i,j+1),t ) f(t) = -c(n) * (t - n) * (t - n - 1), c(n) = 1.5 * (n + 0.5) (-0.1)

29. Comparison: higher vs. lower initial bursts (old interpretation)

30. New syn (i,j),t+1 = syn (i,j),t - 0.01 * (g(t) + f(t)); g(t) = 0.001* (syn (i-1,j),t + syn (i+1,j),t + syn (i,j-1),t + syn (i,j+1),t ) f(t) = -c(n) * (t - n) * (t - n - 1), c(n) = 1.5 * (n + 0.5) (-0.1)

32. Comparison: higher vs. lower initial bursts (new interpretation)

33. Threshold syn (i,j),t+1 = syn (i,j),t + 0.01 * (g(t) + f(t)) for syn (i,j),t ≥ threshold syn (i,j),t+1 = syn (i,j),t + 0.01 * (g(t) – f(t)) for syn (i,j),t < threshold syn thres = 0.5

35. Comparison: higher vs. lower initial bursts (with threshold)

36. Why do we sleep? Conservation of metabolic energy, higher mental function, heat retention, learning and memory? highly simplifying assumptions ideas which you could base further models on Classical interpretation: 170% (200%) of original strength after a few iterations  every memory reinforces New interpretation: significant weakening of synapses, only the initially strongest survive  principally we forget

37. Bigger bursts cause stronger synapses at the end (classical), more vanishing (new), both for our model  after sleep loss our brain has to process more data, more extreme results Our model: growing and diminishing synapses, depending on the initial conditions  “strong” memories persist and reinforce, unimportant ones disappear nobody can retain every cognition sleep as the brain’s selection of the most important things to retain (new interpretation or our model?)