Lieberman (2000 pages 522-3; 1993 pages 515-7) discussses a very simple “delta rule” which determines the change in the strength of the connection between a hypothetical stimulus neuron and a response neuron in the familiar example of a pairing of a tone with food.

As usual the idea is that the tone is neutral to start with, and does not elicit any salivation.

However, every time the tone is paired with food there is an increase in the link (connection strength) between the tone and the salivation response.

The delta rule says that the size of the increase is proportional to the difference between the maximum effect of the external influence of many pairings with food, and the current level of the conditioned internal link between the tone and salivation. In the numerical example given by Lieberman the proportion is set to 0.5 (producing a relatively steep learning curve) and the maximum possible conditioning effect is set at 10.

So at each trial the link between the tone and salivation increases by half of the remaining distance between the maximum and the current level :see the numerical example for the first 4 trials below. A diagrammatic version of a similar basic “learning curve” (with the proportion set to 0.1 rather than 0.5) was provided by Hull, 1943, and a version of this is reproduced here.

I = c (external input — current internal input)     trial 1
  = .5 (10 — 0) = .5 (10) = 5     
I = c (external input — current internal input)     trial 2
  = .5 (10 — 5) = .5 (5) = 2.5     
I = c (external input — current internal input)     trial 3
  = .5 (10 — 7.5) = .5 (2.5) = 1.25     
I = c (external input — current internal input)     trial 4
  = .5 (10 — 8.75) = .5 (1.25) = 0.625     
 

Lieberman (as above) points out that this simple rule for increasing the connection between a sensory neuron and an output neuron functions in exactly the same way as the formula introduced by Rescorla and Wagner (1972) for describing the behavioural effects of conditioning. (He discusses this formula much more extensively earlier: pp. 139-155, 2000; pp. 148-166, 1993).

Their formula is usually applied more generally to the difference between the maximum behavioural effects of a particular reinforcing outcome (the food) and the current level of the behavioral effect. It is sometimes informally applied to the difference between the maximum and the current level of anticipation of the expected outcome (e.g. Mackintosh, 1983; p190).

In either case, it models the "blocking" effect (Kamin, 1969): if a dog has been fully conditioned to salivate to the sound of a bell, adding a flashing light along with the bell does not lead to any association between the light and salivation (the effect of pairings with food has as it were already been used up by the bell)