Biomedical Engineering Reference
In-Depth Information
Table 5.4
W
11
W
18
W
24
W
27
W
34
W
37
W
42
W
43
W
53
W
58
W
62
W
65
W
73
W
77
W
83
W
88
Table 5.5
block 1
block 2
block 3
block 4
xx xx
xx
xx
xx xx xx xx
xx xx xx xx
xx xx xx xx
module
M
13
M
14
*
M
11
M
12
M
13
Table 5.6
M
21
*
M
22
M
23
M
24
M
31
M
32
M
33
M
34
M
43
*
M
41
M
42
M
4
the complete matrix. Below, we will consider the method developed and utilized by
us for neural networks having a large quantity of neurons [
7
].
5.2.4.2 Constructing modular neural networks
To construct a modular neural network, the entire set of
n
neurons is divided into
k
blocks, and each of these blocks is divided on
k
modules. Table
5.5
gives an
example of the partition when
k
= 4. (In Table
5.5
, x is the position of a neuron).
Let us designate the modules through
M
ij
(
i
is the number of the block,
j
is the
number of the module inside the block) and place them in the rectangular matrix
(see Table
5.6
). The modules of one block are located in the matrix row. Let us
connect the outputs of the neurons belonging to the modules of the first matrix
column with the inputs of the neurons belonging to the modules of the first matrix
row. We say that module
M
ij
is connected with module
M
rs
if each neuron from
M
ij
is connected with all neurons from
M
rs
.
Let us connect the outputs of the second column with the inputs of the second
row, and so forth. We will call this neural network structure the modular structure.
Let us consider some of its properties. It is clear that each block contains
n
/
k
neurons, and each module contains
n
/(
k
k
) neurons. In accordance with
connection rules, the output of each neuron is connected with the inputs of all
neurons of one block. Thus, each neuron has
n
/
k
connections. The total quantity of
connections will be respectively equal (
n
n
)/
k
. If increasing the number
of neurons
n
proportionally increases a quantity of blocks
k
, then the total quantity
of connections will grow linearly depending on the number of neurons. The
limitation is that when no more than one neuron remains in each module, further
