A SOM is formed of neurons located on a regular, usually 1- or
2-dimensional grid. Each neuron i of the SOM is represented by an
n-dimensional weight or reference vector 
, where n is equal to the dimension of the input
vectors. Higher dimensional grids can but they are not generally used
since their visualization is much more problematic. Usually the map
topology is a rectangle but also toroidal topologies have been used
succesfully.
The neurons of the map are connected to adjacent neurons by a
neighborhood relation dictating the structure of the map. Immediate
neighbors, the neurons that are adjacent, belong to the 1-neighborhood
of the neuron i. In the 2-dimensional case the neurons of
the map can be arranged either on a rectangular or a hexagonal lattice.
Neighborhoods of different sizes in rectangular and hexagonal lattices
are illustrated in figure 2.1. The number of neurons
determines the granularity of the resulting mapping, which affects the
accuracy and the generalization capability of the SOM.
Figure 2.1: Neighborhoods (size 1, 2 and 3) of
the unit marked with
black dot: (a) hexagonal lattice, (b) rectangular lattice.
The innermost polygon corresponds to 1-neighborhood, the second
to the 2-neighborhood and the biggest to the 3-neighborhood.
