Weber-Fechner and power/lognormal laws

The Weber-Fechner relations (from the 19th century) state that the neural representations in the brain of sensory stimuli, objects and time perception vary logarithmically with the intensity of the stimuli, the number of objects and the length of the time interval respectively.

Said mathematically, dP (change in perception) is proportional to dS/S (change in stimulus divided by stimulus magnitude) leading to P ~ O(ln S).

Some examples to illustrate what this means:

  • The higher the image resolution, the higher the delta further needed to make difference perceptible at all
  • The perceived costs/benefits of information production is proportional to the log of the amount of information that already exists (hence why we feel much more excited to work in green areas rather than beaten-up ones).

It so also happens that maximization of information content (entropy) under the above condition generates a power law distribution of the frequency and size of information when the information is dependent on one dimension (such as images that depend on resolution), or a log-normal distribution when the information is dependent on two dimensions (e.g., video/audio that depend on both resolution and time). This is explained in this paper.

Interestingly, if it were just the economic costs involved (and not the neurophysiological), then the distributions would be expected to be exponential, not power or lognormal.