BioSystems 58 2000 177 – 185
Rule-dynamical generalization of McCulloch – Pitts neuron networks
Yoshinori Nagai
a,b,
, Yoji Aizawa
c
a
Center for Information Science, School of Political and Economic Sciences, Kokushikan Uni6ersity,
4
-
28
-
1
Setagaya, Setagaya-ku, Tokyo
154
-
8515
, Japan
b
Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National Uni6ersity, Canberra, ACT
0200
, Australia
c
Department of Applied Physics, School of Science and Engineering, Waseda Uni6ersity,
3
-
4
-
1
Okubo, Shinjuku-ku, Tokyo
169
-
8555
, Japan
Abstract
A new aspect for neuronal networks is presented. The aspect is based on the concept of ruledynamics which was originally proposed by one of the authors, Aizawa. The concept of ruledynamics were modeled on the two states
cellular automata of neighborhood-three CA
2 3
. A brief review of ruledynamics is also presented, because most publications of the authors so far have been in Japanese. Our concise assertion in the present paper is that a neuronal
network realizes a kind of ruledynamics. This assertion is a speculation on the comparison of McCulloch – Pitts neuron networks with ruledynamics on CA
2 3
. A trial is originally shown to demonstrate that a McCulloch – Pitts neuron network can be imitated by an extended version of ruledynamics on CA
2 3
. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords
:
Ruledynamics; Rule space; Cellular automata; McCulloch – Pitts neuron network; Threshold change www.elsevier.comlocatebiosystems
1. Introduction
Aizawa proposed
the concept
of a
new paradigm for an extended dynamics, that was
called ‘‘Ruledynamics’’,
Aizawa and
Nagai 1987. We actually realized a ruledynamics apply-
ing the concept for the two states cellular au- tomata
of neighborhood-three
which was
intensively investigated by Wolfram 1983, 1986. Then, we constructed a rule-dynamical system on
the two states cellular automata of neighborhood- three CA
2 3
, based on the work by Aizawa and Nishikawa 1986. They asserted that the pattern
dynamics of CA
2 3
, except for travelling patterns, can be reproduced by 32 rules which are given by
a combination of the five fundamental rules. Then, we made a new kind of two states cellular
automata of neighborhood-three CA
2 3
in which the rule to govern the change of each cell state
also varies in some manner, step-by-step, over time. We designed four types of rule change,
Aizawa and Nagai 1987, namely:
Corresponding author. 0303-264700 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved.
PII: S 0 3 0 3 - 2 6 4 7 0 0 0 0 1 2 1 - 0
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
