1. Pupil Control Systems By: Darja Kalajdzievska, PhD-University of Manitoba & Parul Laul, PhD-University of Toronto Supervisor: Alex Potapov

2. Pupil Light Reflex of the Human Eye Why does the pupil radius expand/contract? The size of the pupil controls the amount of light let in to the eye As intensity of light increases, pupil contracts As intensity of light decreases, pupil dilates The pupil cannot react instantaneously to light disturbances, there is a delay in reaction time

3. First we must determine the optimal light intensity for the eye, We consider intensity as a function of area (radius R): We assume that there is an equilibrium amount of light (which corresponds to an optimal area for that light intensity) that the eye prefers-A* and that the pupil will expand/contract until it allows this amount of light in: Here, A represents the area of light on the pupil, which is also the intensity amount, We determine area to be: Introducing a Model

4. We know that the change in radius is proportional to light intensity: If A(R) is the region of intersection of light and the eye: So we have the ODE: Lettingand then Formulating the Model

5. Solving this ODE with initial condition R(0)=Ro, we get an expression for R as a function of time: The Final Model – Instantaneous Reaction

6. Instantaneous Reaction Radius of Pupil versus Time with Fixed Light Intensity Time (sec) Pupil radius (mm) Parameters

7. Delay – Intuitive Idea Assumptions Intensity - normalized between 0 (low intensity) and 1 (high intensity) Radius size fluctuates between 2 and 4 mm, R_o =4mm Time Delay – 0.18 ms Instantaneous change in radius after delay

9. We can change our previous ODE to incorporate a delay in reaction time: Now we use the Taylor Series Expansion to turn our 1 st order ODE to a 2 nd order ODE: To get rid of the inhomogeneous term-c, we let r = R-c Refining the Model – Introducing Delay

10. If we let and We get an ODE which is simple to solve:

11. We look for solutions of the form: Since we need a solution that exhibits oscillations, we are looking for the case of complex roots: Plugging this back into the ODE:

12. This gives a solution: Delay Reaction...Almost There

13. To determine We use the initial condition and the fact that the initial velocity =0 due to the delay Our solution is now: The Final Model – Delay Reaction

14. Delay – Growth of Pupil Radius Time (sec) Pupil radius (mm) Parameters Radius of Pupil versus Time with Fixed Light Intensity

15. Delay – Decay of Pupil Radius Time (sec) Pupil radius (mm) Parameters Radius of Pupil versus Time with Fixed Light Intensity

16. In this case, we want To obtain equal amplitude oscillations, we fix values for all other parameters and try to determine this value of In our experiment: Oscillations grow Oscillations decay Equal Amplitude Oscillations

17. Delay – Equilibrium Oscillations Time (sec) Pupil radius (mm) Parameters Radius of Pupil versus Time with Fixed Light Intensity

18. Numerical Approximations – Delay Model Recall: Converting to Discrete Time, we obtain:

19. Numerical Approximations – Delay Model 0 { Measured data from first M steps

20. Numerical Simulations – Growth Time (sec) Pupil radius (mm) Radius of Pupil versus Time with Fixed Light Intensity

21. Numerical Simulations – Decay Time (sec) Pupil radius (mm) Radius of Pupil versus Time with Fixed Light Intensity

22. Numerical Simulations – Equilibrium Oscillations Time (sec) Pupil radius (mm) Radius of Pupil versus Time with Fixed Light Intensity Plot illustrates approximate equilibrium oscillations

23. Numerical Simulations – Incorporating Non-Linearity To account for the fact that the radius of the eye is bounded, we define: { Converting to the Discrete Model, we obtain:

24. Numerical Simulations – Non-linear Time (sec) Pupil radius (mm) Radius of Pupil versus Time with Fixed Light Intensity

25. Further Questions/Model Flaws Human error / human disturbances Assume that the area of the light on the eye changes to head movements, fatigue of eye muscles, etc. These disturbances will increase with time, so let distance Questions Flaws In reality, the diameter of the pupil can neither increase or decrease unboundedly, so a nonlinear model should be introduced to regulate oscillations Height of the area of intersection of light and pupil was assumed to be constant (h), in reality, this height and indeed the entire area of intersection would be a function of light intensity Experimentation should be done to find more accurate parameter values

26. THANK YOU