Associative-Projective Neural Networks (APNNs) - Neural Networks and Micromechanics - page 79

One such method is the direct enumeration of the neurons that belong to this field.
Another method of definition is the introduction of functional dependence, which
describes the concrete subsets. Then the receptive connections from these fields can
be described:

R ð x jiþr u j m ; jjþs u j n ! y ij ð y ÞÞ; (5.6)

where x ij are the neurons belonging to the receptive field; y ij are the neurons
switched on by the receptive connections from the receptive fields; r u , s u are the
parameters that determine the position of neurons in the receptive field of the retina;
y

is the threshold of the neuron; and m , n are the sizes of the retina. After
determining the set of pairs ( r u , s u ), we determine the geometry of the receptive
field, so the formula would be valid for all points. The toroidal closing is done on
the boundaries of the neural fields (retina).
Let us consider several examples of the receptive connection definition.
Figure 5.6 gives an example of receptive fields. In Fig. 5.6a , the receptive field is
defined as:
r 0 =0, s 0 =0,
r 1 =1, s 1 =0,
r 2 =2, s 2 =0.
In Fig. 5.6b , the receptive field is described as:
r 0 =0, s 0 =0,
r 1 =0, s 1 = -1,
r 2 =1, s 2 =0,
r 3 =1, s 3 = -1.
The receptive connections R reflect the fields defined above into the neurons that
belong to the buffer field B 2 . The receptive fields can be defined with the aid of the
formulas. Thus, for example, if the receptive field is the circle of the specific radius,
then it is possible to define it as follows: all neurons for which r u 2 r 2 belong

to the receptive field, where r is the defined circle radius. If the receptive field is a
ring that has the average radius r and width 2 * d , then its description is:

s u 2 þ <

2

r u þ s u < ð

2

ð r d Þ < r þ d Þ : (5.7)

The hierarchical organization of associative-projective neural networks .
The associative-projective neural networks were constructed to be used in
systems of knowledge representation, an important element of which is their ability
to work with the information of different generalization levels. The associative and
buffer fields described above are used to create the hierarchicial system. Projective
and receptive connections are used to transfer information from one level to another
or between the fields of one level. An example of a hierarchical associative-
projective structure is given in Fig. 5.7 . The structure includes two levels, L 1 and
L 2 . The first level consists of the associative field A 1 and the buffer fields B 1 , B 2 .