Assumptions The light has constant intensity and is shone constantly into the pupil The light is shone on the edge of the pupil and the location of the light does not change Direct relation between retinal neuron firing and iris muscles Parameters (Rmin, Rmax, R1, R2) for models 2 and 3
MODEL 1 Equations: –Firing rate is dependent on: pupil radius »amount of light that has contact with pupil df/dt= a*G(r)*Intensity Change in radius of pupil Firing rate dr/dt= -b*f
WRONG AND A DAY LOST!!! a*G(r)*Intensity is not a rate; it is actually the number of neurons firing Assuming instantaneous results; no delay rate -The model does not yield oscillating results; get decaying results -Delay: 200 ms Different constants for pupil constriction and dilation
Introducing A Delay – Model 2 & 3 F0 – Optimal number of neurons firing F1 – Number of neurons that will fire as a result of incident light. k – scaling parameter g(r) – Fraction of F1 actually making it through iris at radius r.
Model 2 Assume a form for g(r) Some analysis -Analytical solution? yup -only oscillatory solutions -prediction for delay
Model 3 Assume a form for g(r) Analysis -Analytical solution? no -Can find linear stability