Since I
is finite, J may be a uniformly random set. But uniformly random
may be too extreme a concept for biological modeling. Any other measure, representing
a range in the activity levels of the automata in I, may be used.
Any measure at this level may be extended to a product>measure on the set
of all possible sequences <J1,
J2,..., Ji,...>
of sets of active automata.
Given a schedule
<J1, J2,...,
Ji,...> of sets of active
automata, a run <S0, S1,
S2,..., Si,...>
of global states results from each initial global state S0.
The schedules and runs are elements of product spaces with natural measures.
The product measure on the space of schedules defined an interesting measure
on the space of all runs. Then, an analysis and 0,1-laws on runs may be derived.
Using this theory, the expected global behavior of an irregular and asynchronous
network can be computed from a description of basics. The required basics
are (1) the automaton (Q,...), (2) the communications function
M (which greatly reduces the information
This model was
originally based on skin tissue. In the skin, cells communicate with neighbors
through channels connecting the cytoplasms of neighboring cells -- these channels
are known as connexions or gap junctions. It has been the basis of successful
research into classical systems (e.g., M. Fischer’s WakeUp Problem, Hopfield’s
Neural Networks, Dijkstra’s Self Stabilizing systems), biological systems
(e.g., A.Turing’s Leopards’ Spots Problem), and packet-switching network.
Classical results are greatly simplified by using the tools designed for this
model. Many of these results have appeared in refereed journals -- see my
bibliography.