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of cells that are directly interacting with the central cell. The neighborhood is
represented by the set
N
and a
unique index k
is chosen to identify the neighboring
cell in a particular neighborhood. The assignment of different integer values to
k
is
conventional and some examples are provided in Fig. 3.1 where indexes are
depicted in white for each type of neighborhood.
The cells are in fact
nonlinear dynamic systems
. The cell dynamics can be
con-
tinuous in time
and mathematically described by ordinary differential equations
ODEs or
discrete in time
and then their dynamics is described by
difference
equa-
tions.
defines the dynamics of the output
Example 1
:
y
(
t
)
a
y
t
1
of a
y
1
t
1
k
k
k
N
discrete time autonomous
cell defined by a weighted summation of all neighbor-
ing cell outputs at the previous time clock. The discrete time variable
t
is here an
integer
.
The above example represents one of the simplest possible cells. It is autono-
mous since there is no external input to drive the dynamics of the cell. In the gen-
eral case a cell is described by the following variables:
Inputs
- usually denoted by variable
u
(scalar) or
u
(vector of inputs);
States
- usually denoted by variable
x
(scalar) or
x
(vector of states);
Initial states
-
a particular state variable at the initial moment,
t=0
.
Outputs
- usually denoted by variable
y
(scalar) or
y
(vector of outputs).
In terms of a computing machine,
inputs
and
initial states
are used to supply the
cellular system with the
input data
(to be processed) while the result or
output
data
is available in the form of a pattern (spatial, temporal or spatio-temporal) of
the
outputs.
For simpler cells sometimes outputs and states may coincide, other-
wise the outputs represent a nonlinear function applied to states.
Example 2:
The “standard cellular neural network” cell
The ODE defining the standard cellular neural network [27] is:
x
x
a
y
b
u
z
(3.1)
¦
¦
1
1
k
k
k
k
k
N
k
N
1
where
,
y
x
1
x
1
k
k
k
2
and the initial states
x
are specified for all cells in the neighborhood
N
.
Observe that this is a more complex,
continuous time
, dynamic system with a
nonlinearity induced by the relationship between states and outputs.
0
In general (nonstandard CNN cell) (3.1) is replaced with:
(3.1')
,
x
x
f
G
,
y
k
u
,
1
1
k
where
f
is a nonlinear function and
G
is a gene as defined next.
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