Perceptrons, BPN and Hopfield type networks engage in supervised learning in the sense that they require an explicit teaching (or setting) phase requiring foreknowledge of what is ``right'' and what is ``wrong'' behaviour. This permits the definition of ``error'' in behaviour, and thence some form of gradient descent.
It is interesting to enquire whether unsupervised learning is a possibility in such networks, particularly in view of the fact that human learning is largely unsupervised.
Tuevo Kohonen has been active in the study of neural architectures and
associative memories for many years, and is interested in building
unsupervised learning systems.
He models a biological neuron by noting that its output is an oscillation
at a frequency
y that changes in time between two saturation limits in response to an
input 
;
Persistent
behaviour involves 
;
if 
exists we can write
Note here that if 
is the familiar sigmoid, and the (matrix
of) functions 
are taken as simple scalar multiples 
,
this equation reduces to the generalised perceptron update rule, providing
a slightly more stable base for that model of neural activity.
In biological systems this effect is augmented by an excitation due to very close active neurons, and an inhibition due to active neurons that are slightly more remote. This effect may be modelled by convolving a ``Mexican hat'' function h into the response function
Picturing the neural collection as a 2D array, we observe that if the convolution kernel is of appropriate size, activity tends to stabilise via a relaxation process into one ``bubble'', whose size and location are dependent on the input.
Kohonen models learning in such arrays by Hebbian reinforcement when input and output are both high, together with a forgetting term to permit bidirectional weight change
-- this procedure is applicable at ``active'' units, which will be those
within the excitation bubble.
This equation can be simplified if we permit
only to be 0 or 1, and
to be 0 or 
.
It may be shown that the ``activity bubble'' centres around the unit for which the familiar quantity
is maximal. For this unit, and those active within the bubble, we will perform (from equation 7;
During learning, 
and the bubble radius are reduced.
