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
In this section, we discuss the network consist- ing of McCulloch – Pitts neurons, McCulloch and
Pitts 1943. We call this network a McCulloch – Pitts neuron network. In the McCulloch – Pitts
neuron network, each neuron takes two states firing, resting only. The state vector for McCul-
loch – Pitts neuron network is the binary digit array of system size n. The temporal development
of each neuron state is governed by the following recurrence equation:
S
i
t + 1 = u W
ij
S
j
t − T
i
, 3.1
where S
i
t is the state of ith neuron at time t, W
ij
denotes the connection weight from jth neuron to ith neuron, T
i
means the threshold of ith neuron, and u... signifies the Heaviside step function,
i.e., ux = 1, for x ` 0, and ux = 0 for x B 0. Firstly, we consider a simple network whose
temporal development is described by the follow- ing directly equivalent equation to CA
2 3
, namely, S
i
t + 1 = uS
i − 1
t + S
i
t + S
i + 1
t − T
i
. 3.2
It is obvious from Eq. 3.2 that Boolean func- tion like relation between S
i
t + 1 and S
i − 1
t, S
i
t, S
i − 1
t is changed when the threshold T
i
is varied. The result of Eq. 3.2 for the threshold
values is expressed as follows: S
i
t + 1 = 1 for T
i
0, S
i
t + 1 =
S
i − 1
t OR S
i
t OR S
i + 1
t for 0 B T
i
1, 3.3
S
i
t + 1 = S
i − 1
t AND S
i
t AND S
i + 1
t OR S
i − 1
t AND S
i
t OR S
i + 1
t for 1 B T
i
2, S
i
t + 1 = S
i − 1
t AND S
i
t AND S
i + 1
t for 2 B T
i
3, and S
i
t + 1 = 0 for 3 B T
i
, where Boolean logics NOT, OR, and AND are
reprented by , OR, and AND, respectively. The fact referred to here was investigated by many
researchers for more general case described by the equation similar to Eq. 3.1Hu, 1965; Lewis II
and Coates, 1967; Sheng, 1969; Muroga, 1971. This research field is Threshold logic. In the dif-
ferent value case of connection weights {W
ij
}, many more types of Boolean functions appear in
the relationship between S
i
t + 1 and S
i − 1
t, S
i
t, S
i + 1
t than the simple case, as seen in the textbook by Muroga 1971. Hence, it is obvious
that the change of threshold value brings a change in the Boolean relationship. Since the temporal
development of state in a McCulloch – Pitts neuron network is governed by threshold logic
processes, the rule-dynamical property of McCul- loch – Pitts neuron network realizes the change of
threshold over the time. To consider the rule-dynamical property of Mc-
Culloch – Pitts neuron network, we rewrite the Eq. 3.1 for the temporal development of McCul-
loch – Pitts neuron network as follows:
S
i +
t + 1 = u W
ij ++
S
j +
t − T
i
t, 3.4
S
k −
t + 1 = u W
kj − +
S
j +
t − T
k
t, T
k
t = W
kl −−
S
l −
t + T
k
, T
i
t = W
il + −
S
l −
t + T
i
, where S
i +
t or S
j +
t denotes the state of ith or jth excitatory neurons at time t, S
k −
t or S
l −
t that of kth or lth inhibitory neurons at time t,
W
++
the connection weight from excitatory neu- ron to excitatory one, W
− +
that from excitatory neuron to inhibitory one,
W
+ −
the absolute value of connection weight from inhibitory neu-
ron to excitatory one, and W
−−
that from inhibitory neuron to inhibitory neuron. The
rewritten equations explicitly express the temporal change of the threshold caused by the activities of
inhibitory neurons. Therefore, it is obvious that the McCulloch – Pitts neuron network becomes a
kind of rule-dynamical system proposed by us if
the network
has inhibitory
neurons in
connections. The above considerations are still valid for real
neuronal networks. Some kinds of neurons such as pyramidal cells show spontaneous activities
and they usually have the feature of bifurcation phenomena in the temporal change of membrane
potential, Hayashi et al., 1985; Hayashi and Ishizuka,
1992, which
undergo deterministic
chaos, with respect to the bifurcation parameter of stimulus current. In the actual situation of a
neuron in the brain, stimulus current is the spatial summation of post-synaptic currents. The effec-
tive synaptic current contributing to neuronal firing activities is regulated by the synaptic current
from inhibitory neurons. Therefore, the rule change in the sense of ruledynamics is caused by
the activities of inhibitory neurons. In the real
neuronal network, the rule corresponds to the orbital manifold in the phase space, the shape of
which is deformed by the stimulus current. A neuronal activity is just an orbit on the manifold.
The change in the manifold shape therefore produces a change of orbit, namely, varies
temporal shape of action potential. In this sense, we can establish the concept of rule for the
electric potential activities in real nervous systems.
4. Correspondence of ruledynamics on two state cellular
