Cellular and Natural Computing Models and Software Simulation - Systematic Design for Emergence in Cellular Nonlinear Networks - page 18

conferences as well as in numerous journal papers or books. Some recent tutorials
about CNNs are [7,29,31,45].

Several CNN simulators are freely available and the reader may check [6], and
[7] for an easy to use, computing platform independent CNN simulator. A wide
range of simulators as well as news about the progress in the CNN research can be
accessed from [ ]. The SCNN simulator from the pplied hysics Department of

the oethe niversity in Frankfurt am ain can be used on nix platforms [].

3.2.2 The Generalized Cellular Automata

In [7], the idea of a generalied cellular automata (C) was introduced as an ex-
tension of the CNN so that a CA include s Cs as a special case. The main idea
of the CA is to use a CNN in a discrete-time loop.
In other words, for each period of the clock the CNN system evolves until it
eventually reaches a steady output (for some CNN genes it is possible to have
complex oscillatory dynamics) starting from a given initial condition and from
inputs variables that are copies of the CA neighboring cells outputs at the end of
the previous time cycle.
The additional CNN loop is thus given by the discrete time equation
1

. There are two cases of interest:

u ( t ) y ( t

k k

x hen the CNN cell is uncoupled (i.e. all coefficients for in (1)) it a 0 k t 2

k

was proved that the nonlinear dynamic system (3.1') converges towards a steady

state output solution and therefore aft enough period of time T the cell can be
described as a simple nonlinear equation of the following form:

where F is a nonlinear function and G is a gene (i.e. tunable
parameters). In this case the CA can emulate any C, provided that there ex-
ist a method to map the local rules or transition table into the nonlinear function
F . s it can be easily observed such CA can also implement discrete-time but
continuous state cellular automata.

y t F G , u t

1 k

x hen the CNN cell is coupled , one should first determine a set of proper genes
such that the same steady state behavior occurs during the clock time T . This is
often not a trivial task. Then, the behavior of the resulting CA is more com-

plex than that of a normal C. The reason is that an emergent computation

already takes place during the period of time T in the CNN, therefore at the dis-

crete time moment when the output of a cell is sampled, it does not represent
only the contribution of the neighboring cells as in the case of a “classic” cellu-
lar automata but rather the contribution of all CNN cells. In some sense one can
say that employing a continuous time CNN during the clock time is equivalent
to artificially increasing the neighborhood to the whole cellular space. This be-
havior may imply interesting computational consequences.