In response to a flash of light, the voltage across the membrane of a photoreceptor (see the beginning of Section 4.2) will first increase and then decrease. The peak of this flash response tells Typical experimental data can be seen in Figs. 17.1 and 17.12.

the brain how bright the flash was. However, since the visual system adapts to a background light level, the response to a light flash will be smaller in the presence of background light than it is in the dark. This is called adaptation.

Fig. 4.9 shows how the sensitivity of a photoreceptor (a salamander rod in this case) decreases as the background light level increases. This is just another way of saying that the photoreceptor adapts to the background light.

The peak, V, of the photoreceptor response, and the intensity, I, of the light flash are related by the Naka–Rushton equation, V(I) =

VmaxI I + σ

, where σ depends on the background light level. Experimental Note that the form of the Naka–

Rushton equation is just the same as that of the Michaelis–Menten equation that we use often throughout this book. This is another example of how simple mathematical relationships occur many times in quite different scientific concepts.

data from a turtle photoreceptor are shown in Fig. 4.15.

a. Plot the Naka–Rushton equation for a variety of values of Vmax and σ. What doesVmax control? What does σ control?

b. Set Vmax = 1 and plot the Naka–Rushton equation for

σ = 1, σ = 2 and σ = 3. What happens to the graph as σ

increases?

c. Show analytically (i.e., not just by drawing a picture, but with a mathematical argument) that doubling σ has the same effect as dividing I by 2 (and thus the curve is "stretched" by a factor of 2).

d. Write V as a function of log10 I. That is, if x = log10 I, write V = V(x).

e. Set Vmax = 1 and plot the Naka–Rushton equation as a function of x = log10 I for σ = 1, σ = 2 and σ = 3. What happens to the graph as σ increases?

f. Show analytically (i.e., not just by drawing a picture, but with a mathematical argument) that, when V is plotted as a function of log10 I, doubling σ has the same effect as shifting the curve to the right by log10 2.
g. Finally (mostly just for fun), find values of Vmax and σ that reproduce the data shown in Fig. 4.15.