Exponents of Growth - Systematic Design for Emergence in Cellular Nonlinear Networks - page 82

5.2 Mutations for Continuous State Cellular Systems

The above characterization based on exponents of growth may be also used for the
discovery of interesting emergent phenomena in continuous-state cellular auto-
mata. One major advantage of representing the cell as nonlinear piecewise-linear
function rather than as a Boolean table is the easiness to achieve continuously val-
ued mutations. This is done simply by replacing the hard sign() nonlinearity from
the definition of the Boolean piecewise-linear cell with the smooth (linear satu-
rated) nonlinearity fy . In order to get interesting emergent

behaviors it is advisable to start with Boolean cells exhibiting rich complex dy-
namics when connected in various CA topologies. By tuning a parameter of the
piecewise-linear representation (see Chap. 3) of the original Boolean cell within a
H

x 0 x 1 x 1

certain range (e.g. adding r

within the resulting continuous state cellular nonlinear network. These regimes
are separated by bifurcation points in the space of the variable parameter. Experi-
mentally it was found than such behaviors are likely to be complex and interesting
when starting from with a set of parameters

in (3.4)), one can identify different emergent regimes

tb defining a piecewise-linear
Boolean cell assigned to class “d” (unstable near the edge) after the measurement
of the exponent of growth.

,

1 ,..

t

m 1

Fig. 5.11. Mutations of gene 471: A linear saturated output for the cell is chosen instead of
the hard limiter and H 1 33 . The result is a slow growing random cluster