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In Rushton's paper on the Principle of Univariance, he states:

Thus, though the rod input has two variables, wave-length and energy, the output differs only in one respect, namely 'brightness'.

However, as far as I understand, a photon can only vary according to one dimension, namely wavelength (vibrational energy), because the wavelength and energy of a photon are interdependent.

This suggests then that the input to a cone cell, for brightness to be perceived (i.e. the intensity of light), must be a beam of photons, so that the input can vary in wavelength and amplitude (does this equate to the number of quanta being absorbed by the photopigment per unit of time?).

My question, then, is how can a discrete event (the changing of shape of a photopigment) encode a continuous value (i.e. intensity)?

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Indeed, brightness is a matter of photon count.

In any given rod, there are numerous photopigment molecules. There are about 1 billion rhodopsin molecules in any given rod, so although a change in conformation of any single rhodopsin molecule is a discrete event, you can consider the fraction of 11-cis- versus all-trans retinal (a photoreceptor's initial readout of brightness) to be effectively continuous, even more so because there is a <1 probability of a photon of light being absorbed by any rhodopsin molecule, so the discrete measure itself is an approximation of the 'actual' brightness/photon count.

Although it isn't perfectly analogous, because here we are talking about proportions rather than actual bitwise encoding, consider that computers are also able to represent fairly continuous values, but they are still always discrete approximations to those continuous values. A 24-bit computer monitor, for example, can actually only display 256 different intensities for each R, G, and B color stream (8 bits each).

The principle being discussed in the paper you describe is that once a rod (or cone) absorbs a photon, it loses any information about what the wavelength was, only that it was sufficient energy to activate the rhodopsin in that photoreceptor. There are several other questions and answers here on Biology.SE that discuss how wavelength (i.e. color) information can be extracted by differential activation of different cone types, for example:

Why can cones detect color but rods can't?

How do our eyes detect light at different frequencies?

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