Cellular and Natural Computing Models and Software Simulation - Systematic Design for Emergence in Cellular Nonlinear Networks - page 16

3.1.1 Genes

The cell dynamics, and its local functionality are prescribed by an unique gene [7]
vector

> @

G containing all parameters of the nonlinear
system modeling the cell. The analogy with biology is obvious, and the gene vec-
tor may be regarded as the equivalent of the information contained in the DNA
string.

a , a ,.. a ,.., b , b ,.. b ,...

1 2 n 1 1 2 n 2

3.1.2 Discrete and Continuous States and Outputs

In defining a cellular system one has to prescribe the variation domain of the state
and output variables. For example, one can use continuous state cells where the
outputs are defined within a bounded interval or one can also use discrete state
cells where the states or/and the outputs belong to a finite set of possible values.
A binary output cell implements Boolean functions, i.e. it provides a logical
“TRUE” or “FALSE” output for each of the 2 possible combinations of inputs.
Each input also represents a truth value. The cell can be again specified as a dis-
crete-time dynamical system but it can be also specified using a transition table or
a set of local rules . The last two modes of specifying a cell are specific to the cel-
lular automata formalism [41]. As we will show in the next chapter, a compact
piecewise linear description (i.e. a discrete dynamical system) can be found for
any Boolean or other type of input function:

Example 3: A Boolean cell

§ ·

¹

¨
©

. ) y ( t ) sign z

N

b u t

1 0 k k

k

Assuming that the set ^ is used to code the truth set ^ `

, the cell

1 FALSE , TRUE

equation (3.2) above defines a family of Boolean functions. For a particular Boolean

function one should specify the gene parameters. For example, the AND function
with three inputs is associated with the following gene:

> @ >

. G z , b , b , b 2

0 1 2 3

> @ >

If the gene is replaced with , the OR function with three inputs

G z , b , b , b 2

0 1 2 3

is implemented.

3.1.3 Boundary Conditions

In cellular automata the cells located on the boundary of a lattice have a special in
interconnectivity. One should define precisely the way in which these cells interact
with their neighbors. This is called a boundary condition . The choice of a particu-
lar boundary condition may have a great influence on the dynamics of the cellular
system. The following two are among the mostly used boundary conditions:
Periodic boundary conditions : Opposite borders of the lattice are connected. A
one-dimensional “line” becomes following that way a circle, a two-dimensional
lattice becomes a torus.