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3.7 Kohonen's Self-Organizing Map 39

retinotopic map in the primary visual cortex (e.g. Obermayer et al. 1990).
Fig. 3.5 shows the basic operation of the Kohonen feature map. The
map is built by a m (usually two) dimensional lattice A of formal neurons.

Each neuron is labeled by an index a A, and has reference vectors w a
attached, projecting into the input space X (for more details, see Kohonen
1984; Kohonen 1990; Ritter et al. 1992).

a *

x

w
a *
Array of
Neurons a

Input Space X

Figure 3.5: The “Self-Organizing Map” (“SOM”) is formed by an array of pro-
cessing units, called formal neurons. Here the usual case, a two-dimensional array
is illustrated at the right side. Each neuron has a reference vector w a attached,
which is a point in the embedding input space X. A presented input x will se-
lect that neuron with w a closest to it. This competitive mechanism tessellates the
input space in discrete patches - the so-called Voronoi cells.

The response of a SOM to an input vector x is determined by the ref-

erence vector w a of the discrete “best-match” node a . The “winner”

neuron a is defined as the node which has its reference vector w a closest
to the given input

a argmin kw a xk (3.9)

a A
This competition among neurons can be biologically interpreted as a result
of a lateral inhibition in the neural layer. The distribution of the reference
vectors, or “weights” w a, is iteratively developed by a sequence of training

vectors x. After finding the best-match neuron a all reference vectors are