Cellular and Natural Computing Models and Software Simulation - Systematic Design for Emergence in Cellular Nonlinear Networks - page 20

j

where is the state variable of cell k (using the neighborhood index as in Fig. 3.1

x
with the central cell indexed as k= 1) in layer j. The gene is also distributed among
layers as

G 1 and in addition each layer is characteried by the scalar diffu-
sion coefficient D corresponding physically to the conductance of the resistors in
the resistive grid in Fig. 3.2.
D-CNNs have the interesting property that emergent behaviors can be pre-
dicted by carrying out a local activity analysis of the isolated cell (ignoring the
coupling). A detailed analysis based on the local activity [7,] reveals regions of
emergence in the cell parameter space [-11,1] called an “edge of chaos”. So far,
an analytic method to draw the boundaries of the “edge of chaos” regions in the
cell parameter space exists only for reaction-sifussion cellular systems. Though,

,...G

m

within this chapter certain algorithmic methods for detecting emergence regions

within the cell parameter space of arbitrary cellular systems are given.

3.3 Matlab Simulation of Generalized Cellular Automata
and Cellular Neural Networks

It is convenient to simulate various types of cellular system using the program-
ming environment atlab (http:/www.math works.com/products/matlab) produced
by athworks. e will assume here the reader has some basic knowledge of atlab.
For readers unfamiliar with this language, a good starting point might be the
primers [0] or [1].

3.3.1 Uncoupled GCAs

In order to simulate an uncoupled CA one should first write the function
CCE.M which implements the CA cell. It is in fact a simple descrip-
tion of the nonlinear function

:

y t F G , u t

1 k

function y=gca_u_cell(u1,u2,u3,u4,u5,u6,u7,u8,u9)
Z=[-1,-2,-4,-8,-7];
B=[1 1 1 1 1 1 1 1 1];
s=-1;
sigm=B(1)*u1+B(2)*u2+B(3)*u3+B(4)*u4+B(5)*u5+B(6)*u6+B(7)*u7+B(8)*u8+B(9)*u9;
w=Z(1)+abs(Z(2)+abs(Z(3)+abs(Z(4)+abs(Z(5)+sigm))));
y=s*sign(w);
%y=0.5*(abs(w+1)-abs(w-1));
%y=w;

The nine inputs of this function can be a scalar, a vector or an array. The above
program implements the arity9 local logic function. owever, by simply changing