Predicting Emergence from Cell’s Structure - Systematic Design for Emergence in Cellular Nonlinear Networks - page 126

among genes so that the CA will exhibit a desired behavior. Some interesting
circumstances occurred for the underlined genes in Table 7.1b. They can be re-
garded as anomalies since in those cases U classifies the behavior differently from
the pair

U p , . Actually the strongest result (with more confidence) is the one
based on the theoretically defined probabilistic exponent of growth. The experi-
mental determination has been done with only one instance of initial state. How-
ever there is a huge number of possibilities and quite often (this is actually the
case for the underlined genes) the resulted U is rather an exception than the rule. A
correct experimental result would be one where many initial states are considered
and all resulting U are averaged. This is actually the meaning of the probabilistic
exponent of growth. In fact simulations of the CAs confirmed the prediction based
on U in 100% of the cases, while initial states were randomly generated with a
probabilistic profile according to the above theory.

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7.6 Conclusions

The most significant result of this chapter is that a consistent theory for predicting
emergence analytically can be developed. As seen one needs to specify a initial
state probability profile and to associate various emergent properties (like pattern
explosion or implosion) with the evolution of probabilities for the cells within a
specified neighborhood. By using a probabilistic approach both the cell structure
and the neighborhood characteristic are captured into a consistent theory resem-
bling the previously developed “theory of local activity” with some new benefits:

(a) Unlike the “local activity theory” this new theory (of the probabilistic

exponents of growth ) considers not only the cell but also its neighbor-
hood. This makes possible to predict emergent phenomena with an
improved accuracy.

(b) It relies on information theory rather than circuits theory and there-

fore it is expected to be more general . It is known that in order to apply
the local activity theory one need to model the problem such that a cell
fits within the model of a multi-port, each cell being connected with
the others via a resistive grid. These are rather restrictive assumptions
limiting the applicability domain to Reaction-Diffusion cellular sys-
tems. The theory proposed herein is however applicable to any com-
plex system assuming that one has tools to compute or infer the
nonlinear functional mapping the probabilities of the inputs to the
probability of a nonlinear system output (the cell connected within a
specified neighborhood).

(c) The theory of probabilistic exponents of growth can be extended to

continuous state systems as well. Herein we exposed the theory for the
case of binary systems (where each variable is assigned a probability
of being in state “1”). This makes the calculus of the cells function of