The system: Pupil

As stated earlier, Size of pupil aperture determines the amount of light allowed to the retina. m When light intensity is higher than desired, the pupil light reflex causes a contraction to reduce the amount of admitted radiation.


When light intensity is lower, aperture enlarges to allow more light to the retina.


Suppose the level of illumination is suddenly changed. Let the illumination be unit step function of time.
u(t) = 0 t < 0
= 1 t > 0

This demands a change in the pupil size. The pupil cannot change its size instantaneously. But it approximates it with an exponential function.

Thus, pupil response to step response of change in illumination consists of two parts:
· Exponential delay: As explained above, signal shape is changed from the step function to exponential function.
· Pure delay: The signal is shifted and there is no shape change in the step function. The pure delay occurs because of the time taken by the signal for to and fro conduction on nerve fiber from eye to brain. Time for each synapse varies from 0.3 to several milliseconds. Synapse is the junction between two neurons.

The response of the pupil to the step change in illumination. (a)The response is a combination of a pure delayed response and an exponential delayed response. Taking the response apart (b)a pure delay and (c) an exponential lag.


Transfer function for exponential delay component:

Let the input function given to the pupil be a unit step function.
Input: Unit step function
u(t) = 0 t < 0
= 1 t > 0

Therefore as discussed earlier, the response of the pupil is an exponentially delayed output response.
Output response=
t=0 output response=0
when t=large output response=1
Laplace transform of input =
Laplace transform of output =
The transfer function is the ratio Laplace transforms for the input and output. Hence,the transfer function is given by,

Thus the transfer function which characterizes the output response to the input function is of the form where k is called gain constant and a is called the speed of an exponential response.