In a frequency-of-seeing experiment like that of Hecht, Shlaer and Pirenne

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Vision and photoreception: counting photons

Problem

In a frequency-of-seeing experiment like that of Hecht, Shlaer and Pirenne, a subject is dark-adapted and shown many brief flashes of fixed mean intensity. For a flash whose mean number of photons absorbed by rods is

\bar a = 4

, the subject reports seeing it on

21\%

of trials. For a brighter flash with

\bar a = 9

, the subject reports seeing it on

88\%

of trials. Assume that the number of photons absorbed on any one trial is Poisson-distributed with mean

\bar a

, and that the subject reports "seen" whenever at least

n

photons are absorbed, where

n

is a fixed integer property of the visual system. ((a)) Starting from the Poisson probability of absorbing exactly

k

photons, derive the general expression for

P_{\rm see}(n,\bar a)

, the probability of absorbing

n

or more photons. ((b)) Evaluate

P_{\rm see}(n,\bar a)

\bar a = 4

and at

\bar a = 9

for the three candidate thresholds

n = 5

n=6

and

n=7

, and hence determine which value of

n

is consistent with the two percentages quoted above. ((c)) In a real experiment only the numbers of photons at the cornea are known, here

40

and

90

, and the fraction

f

\bar a=fI

is unknown. Taking

n=6

, find the

f

that makes the

40

-photon flash seen on

21\%

of trials and predict the fraction seen for the

90

-photon flash. Explain why, with

f

free, the two percentages test

n

less sharply than in (b).

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