1. the rule is changed by an arbitrary external function of time;
2. the rule change is changed by using the aver- age activity of a cell array and a threshold for
each rule for example, a rule is switched, or reversed to its opposite, on the average activity
over the threshold;
3. hybrid of manners 1 and 2; 4. multigate method in which switch-on regions
and switch-off regions are prepared for a rule with respect to the average activity.
We call this kind of CA
2 3
‘‘ruledynamics on CA
2 3
’’. We also call a series of temporal develop- ment of cell states of CA
2 3
a pattern dynamics. In the generalized sense, the ruledynamics is the
temporal change of rules which governs the tem- poral development of quantities in a system. It is
an expandable concept. For ruledynamics on CA
2 3
, one can see that the change of rules over time gives rise to more complicated temporal pat-
terns, some of which cannot be seen in Wolfram approach to CA
2 3
. These patterns, however, may be found out in higher neighborhood cases. The
neighborhood referred to here means the con- tributed cells to determine the state of each cell at
the next point in time. Then, the ruledynamics on CA
2 3
can realized a subgroup of multi-neighbor- hood cellular automata of two states. Most of our
works concerned with ruledynamics were pub- lished in Japanese. A review of the ruledynamics
is therefore given in Section 2.
Since the ruledynamics on CA
2 3
is a model only, by using a computer program, we sought how the
rule switching brings about physically and ob- served what system can actually perform the rule-
dynamics. We discovered the physical reality of ruledynamics through the consideration of neural
networks consisting of McCulloch – Pitts type neu- rons, each of which is an abstracted nerve cell
with two states firing, resting. The state change of an element in any two-state system can be
expressed by a Boolean function. The rule change therefore means making the choice of a Boolean
function. This is the same in a McCulloch – Pitts neuron network. The control of rule selection is
carried out automatically by the activities of in- hibitory neurons connecting to each other and to
excitatory neurons. This is the fundamental recog- nition of ruledynamics and we understand that
the concept of ruledynamics can be applicable to actual neuronal networks in the brain. These
points are discussed in Section 3. In our standing point, neuronal cording exists in relationships re-
duced from the temporal development of cell ar- ray states.
Although we already knew the equivalence of the ruledynamics on CA
2 3
to McCulloch – Pitts neuron network, Nagai and Aizawa 1995, 1997,
it was difficult to derive directly a set of equations of ruledynamics on CA
2 3
from those of McCul- loch – Pitts neuron network. We therefore assert
that the ruledynamics on CA
2 3
can produce the same pattern dynamics generated by the McCul-
loch – Pitts neuron network. Here, the equivalence of a ruledynamics on CA
2 3
to a McCulloch – Pitts neuron network means that a ruledynamics on
CA
2 3
reproduces the same relationship of pattern dynamics generated by a McCulloch – Pitts neuron
network. We speculate that an extended ruledy- namics of site-dependent rule change can imitate
McCulloch – Pitts neuron networks. An actual demonstration for this speculation is carried out
by using a ring form of eight McCulloch – Pitts neurons. These are shown in Section 4.
2. Ruledynamics on two state cellular automata of neighborhood-three
In this section, we provide a concise review of our previous works for ruledynamics on CA
2 3
and then we expand the ruledynamics on CA
2 3
to the ruledynamics of site-dependent rule changes. The
requirement will be realized when we recreate any binary system by using the ruledynamics on CA
2 3
. A ruledynamics was realized on the cellular
automata. Following the approach taken in the study by Aizawa and Nishikawa 1986, we ex-
tend pattern dynamics of the two states cellular automata of neighborhood-three CA
2 3
, Wolfram 1983, 1986, to ruledynamics, Aizawa and Nagai
1987, by changing the coefficients for the combi- nation of basic five rules temporally. Aizawa and
Nishikawa concluded that most of temporal pat- terns of cell states on CA
2 3
can be reproduced by a set of 32 rules, which are given by the combina-
tion in a fundamental basis of five rules {g
1
, g
2
, g
3
, g
4
, g
5
}. The fundamental rule basis is the follow- ing five rules:
g
1
= f
= S
i − 1
+ S
i
+ S
i + 1
, 2.1
g
2
= f
1
= S
i − 1
· S
i
+ S
i
· S
i + 1
+ S
i + 1
· S
i − 1
, g
3
= f
2
= S
i − 1
· S
i
· S
i + 1
, g
4
= p = S
i
− S
i − 1
· S
i
− S
i + 1
, g
5
= f
1
· p mod 2 for all,
where f , f
1
, f
2
, and p are the notations used in Aizawa and Nishikawa 1986 and S
i
takes the value 0 or 1. The notation to represent the func-
tions of rules for CA
2 3
is followed as by Aizawa and Nishikawa 1986. Aizawa and Nishikawa’s
rule basis is modified from Wolfram 1983, origi- nally. Wolfram 1986 also presented the represen-
tation of rules by Boolean functions. But the simple representation is given by the form of
modulo 2. For Aizawa and Nishikawa’s rule ba- sis, only two basic rules become simple Boolean
functions. The g
1
is equivalent to the logic exclu- sive-OR and g
3
is the logic AND. The other three are not simple. So, we use the form of modulo 2
for rules of CA
2 3
. The ruledynamics becomes the temporal devel-
opment of each element cell state in the array of n cells given by the following recurrence equation:
S
i
t + 1 =
k
o
k
t · g
k
S
i − 1
t,S
i
t,S
i + 1
t mod2,
2.2 where o
k
t denotes the coefficient for kth basis rule at time t and changes the value 1 or 0
temporally. Self-sustaining pattern development in the ruledynamics appears when a quantity con-
cerned with pattern is fed back to coefficients {o
k
t}. An example is shown in Fig. 1 in which the coefficients of fundamental rules are deter-
mined using the average activity of the cell array at the previous time. The temporal development
of ith cell state for this case is expressed as follows, Aizawa and Nagai 1989:
S
i
t + 1 =
k
u h
k
S
i
t
− T
k
· g
k
S
i − 1
t,S
i
t,S
i + 1
t mod2,
2.3 where B S
i
\
t
denotes the average activity at time t, T
k
means a switching threshold for rule k, u
x is the Heaviside step function, h
k
signifies the sign for switching manner, namely, + 1 denotes
switch-on if B S
i
\
t
takes a value greater than the threshold T
k
and − 1 switch-on if B S
i
\
t
takes lower than the threshold T
k
. This type of ruledynamics on CA
3 2
has the average activity
Fig. 1. An example of the ruledynamics on CA
3 2
: a temporal pattern of cell states, b temporal change of rules and c time course of average activity.
Fig. 2. Relationship between state vectors at time t and t + 1 for CA
3 2
. System size is 7. Aizawa – Nishikawa 32 rules are depicted.
synchronization behavior investigated by Kim and Aizawa 1999. The average synchronization
implies that a given temporal rule change attracts the temporal pattern changes of cell states for any
initial configuration of cell states in the average activity sense. In other words, the pattern dynam-
ics for the given temporal rule change become similar to each other. This is a kind of robustness
of the ruledynamics on CA
3 2
. Usually, the time course of average activity behaves like that of
electro encepharo gram EEG, Aizawa and Na- gai 1989 as shown in Fig. 1c. The ruledynamics
brings a large scale fluctuation in the temporal behavior of the average activities if rule switching
occurs at a certain value of the average activity, since each rule has the characteristic that the
average activity for considered rule converges to a certain value within normal fluctuation, Gutowitz
et al., 1987.
In order to see any two state system can be imitated by a certain ruledynamics, we extend the
rule-change method, making it more general. The most generalized case is described by following
expression, i.e.,
S
i
t + 1 = f
i,t
S
i − 1
t,S
i
t,S
i + 1
t mod 2,
2.4 where f
i,t
{ f , f
1
, f
2
,..., f
m
} is a rule in a rule space that is selected for each site of the cell array and
each step of time development. In the general situation, it is suitable to describe the temporal
development of the system using the state vector of the system St = S
1
t, S
2
t,..., S
n
t at time t. The time development of the state vector is
given by the following formal recurrence equa- tion, namely:
St + 1 = F
t t + 1
St, 2.5
where F
t t + 1
is the formal expression of a global rule. The global rule means the relationship be-
tween the states of a whole system from time t to time t + 1. The assignment of state vectors to
rational numbers yields a representation of the global rule. The examples of global relation of
state vectors at time t and t + 1 are shown in Fig. 2, where the rational number assignment is taken
as jt = S2
i
S
i
t.
3. Rule-dynamical aspect of McCulloch – Pitts neuron network
