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Pupil Control Systems

Pupil Control Systems. By: Darja Kalajdzievska, PhD-University of Manitoba & Parul Laul, PhD-University of Toronto Supervisor: Alex Potapov. Pupil Light Reflex of the Human Eye. Why does the pupil radius expand/contract?

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Pupil Control Systems

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  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. Introducing a Model 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:

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

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

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

  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

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

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

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

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

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

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

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

  15. Equal Amplitude Oscillations 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

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

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

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

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

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

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

  22. { 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:

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

  24. Further Questions/Model Flaws Questions • 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 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

  25. THANK YOU

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