1. Study of the Sleep Stages from a Physical Point of View Mostafa M. Dini

2. Viscoelasticity and Stress-Strain correlation Viscous material return back to the origin position linear with time. In purely viscous materials strain lags stress by a 90 degree phase lag. An example for the viscous materials is honey. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Purely elastic materials have stress and strain in phase so that the response of one caused by the other is immediate. Mechanical Behaviour of Materials; Meyers and Chawla; 1999

3. Brain is a Viscoelastic Material Viscoelastic materials have elements of both viscous and elastic properties Viscoelasticity is a property which exhibit both viscous and elastic characteristics when under stress. Viscoelastic materials exhibit a behaviour in between the viscous and elastic materials regarding the time lag in strain Ref.: Wiki

4. Brain (a viscoelastic material) characteristics during waking 1) During the day the brain stiffness increases with the rate of the stress loads exposed. 2) The normal type of stress loads is cyclic loading and the phase lag between the stress and strain conclude dissipation of energy which if could not be transmitted, temporary will be remains in the location and the viscous effect will increases by new loads 3) Consequently, the reconnection of the synapses following a strain impact is not completely as before, communication effectiveness will be decreased, 4) Sleep is essential to remove the accumulated confined energy and regenerate the brain functionality

5. Brain (a viscoelastic material) characteristics during sleep 1) By sleeping the input stress is held constant. The strains start to increase with time (creep) and cause the synapses configuration changes; 2) reconfiguration of the synapses keeps the strain to remain constant and to decrease the stress with time (relaxation); Ref.: Wiki

6. properties of the brain similar to other viscoelastic materials : hysteresis is seen in the stress-strain curve hysteresis stress relaxation occurs: step constant strain causes decreasing stress stress relaxation creep occurs: step constant stress causes increasing strain creep Ref.: Wiki

7. Viscoelastic creep for a viscoelastic material When subjected to a step constant stress viscoelastic materials experience a time-dependent increase in strain This phenomenon is known as viscoelastic creep At a time t_0 a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period The material responds to the stress with a strain that increases until the material ultimately fails When the stress is maintained for a shorter time period the material undergoes an initial strain until a time t_1 after which the strain immediately decreases (discontinuity) then gradually decreases at times t > t_1 to a residual strainexperiencetimematerial timematerial timematerialtime

8. Operating and Equilibrium Lines Below its critical stress the viscoelastic creep modulus is independent of stress applied A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied stresses are below the material's critical stress value time

9. Brain as a Closed System during the Sleep There is no input into or output from the brain during the sleep. Therefore the sleeping brain is a closed system. In a mechanical closed system, the waste energy increases the system entropy. In a sleeping brain, the confined energy changes the synapses configurations; and then the configurations will be normalized and established over the layer.

10. Varying Operating Line; Varying Equilibrium Line During the waking the operating line will shift to higher constant strain. The operating line approaches the equilibrium line which at such a condition layer is saturated by the confined energy and cannot absorb more if remains protected from injury. The tissue`s mechanical strength protect the tissue from rupturing within normal loads.

11. Energy Between Operating and Equilibrium Lines The energy between the two curves is the confined energy accumulated by high strains during the waking time. The confined energy will be consumed during stress relaxation and creep steps.

12. Reconfiguration In Brain; Waste Heat in Mechanical System In a mechanical closed system, the confined energy (energy between the operating and relaxation lines) converts into heat as a waste energy. In the sleeping brain, the confined energy (energy between the operating and relaxation lines) is consumed in synapses reconfigurations.

13. NREM a Constant Stress Stage; REM a Constant Strain Stage stress relaxation occurs: step constant strain causes decreasing stress stress relaxation creep occurs: step constant stress causes increasing strain creep Ref.: Wiki

14. Number of Stages to Recover the Operability Stress relaxation and creep steps will recover the equilibrium state of the brain layers as the layers that can operate efficiently. The recovery will be fulfilled by sequence reconfiguration (creep) and normalization (stress relaxation) steps.

15. Typical Sleep Stages for a Child, an Adult and an Old Person

16. Discussion for Measuring viscoelasticity of the Brain viscoelasticity Though there are many instruments that test the mechanical and viscoelastic response of materials broadband viscoelastic spectroscopy (BVS) and resonant spectroscopy] (RUS) are more commonly used to test viscoelastic behavior because they can be used above and below ambient temperatures and are more specific to testing viscoelasticity These two instruments employ a damping mechanism at various frequencies and time ranges with no appeal to time- temperature superposition Rod Lakes (1998): Viscoelastic solids CRC Press Using BVS and RUS to study the mechanical properties of materials is important to understanding how a material exhibiting viscoelasticity will perform. spectroscopybehaviorviscoelasticitytimematerialviscoelasticity Ref.: Wiki For the brain tissues in different areas this measurement has been done and the stress-strain curves are plotted. However, for the brain vital tissues the data will be different and changes by time during the functioning and regeneration. The only way is the preparation of indirect measurements by counter checking with function measurements.

17. Discussion of Sleep Numbers correctness The stress-strain curve can be plot for a pathway during one unique mentation, The stress-strain curve can be plot for a layer, The stress-strain curve can be plot for one specific region or lobe, The stress-strain curve can be plot for the whole brain for a cycle of waking-sleep. The sub-steps of NREM stages should be a summation of the involved areas stress-strain curves.