Dynamical systems and operators associated with a single neuronic equation

MATHEMATICAL

Dynamical Neuronic

BIOSCIENCES

18, 191-244

Systems and Operators

(1973)

Associated with a Single

Equation

TOSIO KITAGAWA Research Institute of Fundamental Information Kyushu University, Fukuoka, Japan

Science,

ABSTRACT The paper deals with dynamical behaviors of the solutions of a single neuronic n-l equation: x(t+l) = l[ 2 akx(t-k)-01, where l[u] = I for u>O and = 0 for ~50, in k=O

order to have a systematic insight into a certain family of transition phenomena among 2” state configurations. For this purpose we are concerned with the specific operators of the six kinds: dpo, Zp,, ZLpcII, Yp,r, Zfll and .Z?B~(I = 0, 1, 2, , n-1). In Sections 5 and 6 we discuss reverberation phenomena for =Yz,. A regular articular representation of each state configuration is used to find out all the types and the total number of reverberation cycles under the applications of the operator LY%,,.Section 7 deals with the applications of Z’zPa,_ t, while Sections 8 and 9 deal with 3~~. Section 10 gives some ideas on the use of our family of operators for the representation of the linear threshold functions associated with the single neuronic equation.

1. INTRODUCTION The mathematical model of a neural [4-61 consists of three parts : I. neuronic II. mnemonic III. adiabatic

(or decision)

equations

(or evolution) learning

developed

by Caianiello

(NE),

equations

hypothesis

According to E. R. Caianiello explained as follows :

network

(ME), and

(or rule) (ALH). [6], the roles of these equations

are

“I. A NN (neural network) is a system composed of non-linear elements (nodes of NN) which interact through couplings (characterizing the meshes of the NN) in a manner described by NE; these give its instantaneous behavior: conventionally, a solution of the NE may be termed a “thought” of the NN. II. The strengths of the couplings may vary themselves as a consequence of the activity of the MN: such changes, when they occur, are described by ME. III. The rate of change typical of II is “secular” with respect of the times typical of I.” 0 American Elsevier Publishing Company, Inc., I973

192

TOSIO KITAGAWA

The purpose of this paper is to give some fundamental observations on the solutions of the functional equations regarding NE proposed by Caianiello [4] and discussed by him [5, 61 and his colleagues working at the Laboratorio di Cibernetica de1 C.N.R., Napoli, including de Luca [8, 91, Aiello, Burattini, and Caianiello [l], Caianiello, de Luca and Ricciardi [7], as well as many Japanese engineers and mathematicians, such as Nagumo and Sato [l 11,Amari [2, 31, Suzuki, Katsuno, and Matano [12], Ishihara [IO], and Yamaguchi [14]. The NE functional equations are given by N L(i) x,(t + 1) = 1 c c @xj(t - k) - tii

1)

j=l k=O

for i = 1, 2, . . ., N, where l[u] =

1 0 1

for u > 0, for u < 0.

(1-l) (1.2)

In Sec. 2 we shall pick up some of the main features of the functional equations given by (1 .l). Some general considerations with specific emphasis on reverberation phenomena have been given by various authors mentioned above. In contrast to these considerations, our attitude for discussing the system of functional equations is to take into consideration all the possible behavior of dynamical systems defined by (1.1). Due to this attitude, a systematic description of the dynamical systems is indispensable, and we shall discuss not only each individual graph but also a family of graphs in which transitions among graphs are one of our main concerns. In accordance with such a research approach we shall propose to discuss a certain family of transition phenomena: (i) convergences (ii) convergences (iii) disintegration figurations.

to stable configurations, to reverberation of reverberation

cycles, and cycles into

other

types

of con-

We are also interested in a detailed discussion of various types of reverberation cycles. We shall, therefore, introduce several notions that are useful in discussing the dynamical behavior of the solutions of the functional equations. It is to be noted here that, although our ultimate goal is to give a set of universal observations on the dynamical behavior of the solutions of functional equations including (1. l), we shall start with a specific type of the equation : n-l x(t + 1) = 1 c a,x(t - k) - E) (1.3) [ k=O which is an NE for a single neuron.

1,

ON NEURONIC

EQUATIONS

193

It is indeed our approach to start with the simplest equation and to discuss all the possible features of the dynamical behavior of all possible solutions of the functional equations and then to proceed to more complicated equations. By proceeding in this way, we are expecting to prepare a transition map of various configurations. Such a transition map will give us meaningful observations on evolution equations and learning processes when we enter into the consideration of ME and ALH in the sense of Caianiello [4, 61. In Sec. 2, characteristic features of the neuronic functional equations are summarized from mathematical points of view. Since, in this paper, we confine ourselves to a single neuronic equation (1.3), there is a certain set of conventions that simplify the notation given in Sec. 3. Section 3 also explains the principal attitude of our research in three assertions (a), (b), and (c) with an illustration of the special case of (1.3) when n - 1 = 2. Section 4 is devoted to the introduction of concurrence functions and allied operators defined over the whole space of 2” state configurations. We are interested in various features of the transition of state configurations under sequential applications of each of these operators. In this paper we are concerned the specific operators of the six kinds: pw, Tz, Zm,, Ye,, Yp,, and 2~~ where I = 0, 1, 2, . . ., n - 1. In Sec. 5, reverberation cycles for Ztp,, are discussed. It is immediately clear that any state configuration belongs to one and only one reverberation cycle under applications of Ya,. Proposition 5.2 shows that any state configuration has a regular articular representation by which it can be directly observed that it belongs to a reverberation cycle of specific length. A regular articular representation of each state configuration is also useful to find all the types and the total number of reverberation cycles for n-dimensional state configurations under application of the operator Ta,. It is shown in Proposition 6.10 that the total number N(Z) of reverberation cycles having the exact length I can be obtained through a certain set of difference operators. Section 7 deals with reverberation cycles of n-dimensional state configurations under applications of $Pd,,_,. In this case there exists a set of state configurations that are not a member of any reverberation cycle but are transitory state configurations in the sense that repeated applications of Y,,_, will carry each of them into some state configuration, which is a member of some reverberation cycle called the #“‘*-reverberation cycle. Here again an articular representation is equally important as in the case of Za,, in Sec. 6, however, with more delicate complications. In Sets. 8 and 9 we turn to a family of transition operators called 2& (I = 0, 1, 2, . . .) n - 1). Preparatory considerations on Yp, show that a reverberation cycle under application of YE0 can be disintegrated

194

TOSIO KITAGAWA

into a set of state configurations consisting of transitory state configurations and those belonging to a reverberation cycle(s) under application of dpp,. It is noted that S,,, and YZ, can be considered to be Y;“aOand Zp,, respectively. We proceed to introduce Yppl (1 = 2, 3,. . ., n - l), and some introductory observations on the mutual relationship among these operators are given with reference to simple examples. In order to have a summarized feature associated with transition of state configurations induced by application of Y,-,, we shall introduce a new coordinate system suited to Yp, for each fixed 1, where 1 s I 5 n - 1. Dynamical behavior induced by application of each YF~_ j (j = 1, 2, 3, 4, 5) is discussed in some detail. After these preparations, we turn, in Sec. 9, to a discussion of reverberation cycles under application of P’p,, where the uses of the new coordinate system introduced in Sec. 8 play an important role. Throughout these eight Sections we have been substantially concerned with the specific family of operators PO, Z’,, P’,,, YZo, Yb,, and 28, (I = 1, 2, . . .) n - 1). In fact, although our explanations have been limited to Yii, and ZD,, situations are simpler for the cases of Ya and PG, and analogous observations are valid between P’@, (= YD,,) and ZMO (= Yb,), and also between Ts, and 55’~~respectively. At this stage we wish to make clear the conditions imposed on the set of coefficients 0 or even the {a,_,_,> (k = 0, 1,2, . f ., n - 1) and the parameter function 0(t) by which the translatable functional operator defined in (1.3) becomes equivalent to one member of the operator family consisting of J!?~, YG, YO,, and Yp, (I = 0, 1, 2, . . ., n - 1). This topic will be discussed in a future paper [l]. It will be shown that the uses of the operator family have a certain significance provided that specific considerations are given in composing these operators. In this paper, Sec. 10, an illustration of the systematic uses of the operator family is given with regard to the special case of (1.3) when n - 1 = 1. It is also noted that there is a need for introducing the notion of conditional operator representations of l[z@)]. A few remarks on the change of operator representations in connection with the functional equation when 0 in (1.3) is a function of t. 2. CHARACTERISTIC FEATURES FUNCTIONAL EQUATIONS

OF THE

NEURONIC

values Let xi(t) (i = 1, 2, . . ., IV) be N functions of all the nonnegative oft which takes either 1 or 0 for each t. We assume that a set of functional equations (1.1) hold where {a!$)} (i,j = 1, 2, . . ., N; k = 1, 2, . . ., Y(i)) of t. and {Oi> (i = 1, 2, . . ., N) are assigned sets of constants independent

ON NEURONIC

195

EQUATIONS

The characteristic features of a system of functional can be summarized in the following five aspects.

equations

(1.1)

(1) Threshold functions induced by the function (1.2) involve nonlinearity of the functional equations and also reduce discrete activities of N neurons. (2) Networks equations (1.1).

among

N neurons

are defined by a system of functional

(3) Translations with time are our concern with respect to a system of functional equations (I) which defines a dynamical system of Ndimensional vector X(t) = [xi(t) x2(t), . . ., x~(t)]. (4) The functional

operator

I Y(t) defined by

JJ,(t + 1) - Al { Y(f)> I{ Y(t)} =

y,(t + 1) - My(t))

)

. . . . . . . . .

with f&{r(t)} = 1

[

i

;

k=O

j=l

a$‘yj(t

- k) - ei

translatable, i.e. commutative with all translations where T,Z(t) E Z(t + c(), in the sense that we have T,(I{ Y(t)>) = IV,

(2.1)

>

( .JG + 1) - MWI

1)

(2.2)

T,, - 00 < c(< 00,

Y(t))

(2.3)

for all t and c(. (5) Transition map among 2N(L+ l) state configurations the system of functional equations (1.1).

are defined

by

In combining these five characteristic features of the system of functional equations (1.1) we observe that we are now concerned with a dynamical system of the Markovian type under a formulation of finite mathematics. 3. SINGLE

NEURONIC

EQUATION

In what follows we confine ourselves to a single neuronic equation where the number of neurons in Eq. (1 .l) reduces to one. However, for the sake of brevity of notation, we rewrite our equation, without loss of generality, in the following form: II-1

1)

1) = 1 c a,x(t - k) - 0 (3.1) [ k=O which is a special case of (1) essentially with N = 1 and L = n - 1. Now let us introduce an n-dimensional vector x(t

+

6, = (60 ht-1, 6,-z, . . .YL_(n--2), &-(“-I)),

(3.2)

196

TOSIO

KITAGAWA

and hence the inner product n-1 (a, 6) = ,zO GL!f. Our concurrence

function

(3.3)

~(6,) is a real-valued

function

z(6,) = (a, 6,) - 0. The functional We may and vectors defined by

equation

(3.1) is now reduced

6 f+l = shall introduce

Mm

by (3.4)

to

= f(h).

a transformation

defined

9

(3.5) of n-dimensional

4 4-l 8

(3.6)

Ez.

L-3) t - (n -

1;

In view of the translatability property given in Sec. 2, we may and we shall confine our discussion to the transition

(3.7)

Now 6, is called to be a state at the time point t of our dynamic system, state while 6, = [6,, 6f--l, 81--2, . . ., 6t_cn-1J is called an n-dimensional configuration at the time point t of our dynamical system. The functional equation (3.1) induces a digraph (X, I) where X is the set of all 2” state configurations, that is, X = (6) with 1x1 = 2”, where the directed connection I, that is, the set of arcs (edges), is defined by the map I- : 6 -+ Lq6). (3.8) The principal three assertions:

attitude

of our approach

(a) We set up a digraph but

(b) We are interested also with transition

(X, I) induced

is summarized

by the following

by (3.8).

not only in each individual digraph given by (a), phenomena among these digraphs which are

ON NEURONIC

EQUATIONS

197

induced

by the changes of Y(6) due (a,, a1 2 . . .) a,_,, e>. (c) The dynamical behavior of equation as t + + cc is our concern, to rest (critical) state configurations, problems associated with them.

to the changes

of the vector (a, 0) =

the solution (6,) of the functional including the studies of convergence reverberation cycles, and the stability

The following examples are given in order to illustrate general problems with which we are concerned in this paper. In order to simplify an intuitive understanding of the fundamental features introduced here, we shall appeal to the notations specially suited to each individual example. EXAMPLE

3.1

Let us consider x(t + I) = I[a,x(t)

+ a,x(t

-

1) + a,x(t

- 2) - O]

(3.9)

which is a special case of (3.1) with z(a, 6) = C&)6, + a,& + a,& -Or; 0

Notation:

- 8.

ao12 I no +a1 I-n2 aij 5

ai -hi,

(Odi.jS2,

i+j)

FIG. I. Transition map associated with (3.9).

(3.10)

198

TOSIO KITAGAWA

A transition map is shown in Fig. 1, where to each route there is associated the condition under which the route is realized. The transition map includes all the combinations of possible routes which can be realized under any feasible conditions upon the set of coefficients (a,, a,, a2) and 0. For the moment we write z(6,, 6,,6,)

= a,&

+ a,&

+ a,&

There are eight state configurations (Table corresponds a function value of z(6,, a,, 6,).

0

0

0

-0

001 0 1

0

ai -

e e

1

1

al

a2

0

a2

-

+

-

fJ

(3.11)

- 0.

1) to each of which there

1 0

0

a#J -

1 1

1 0 1

da + a2 -

1

0 1 1

0

uo+ala0

+

al

i-

0 0 a2 -

0

We consider a set of the following five cases (I-V) each of which is defined by a combination of two sets of conditions upon z(6,, 6,, S,), respectively. There is a set of common conditions which each of all these cases satisfies, and also another set of conditions which one and only one of these cases satisfies, respectively. Now the common conditions are given: (1) z(6,, 6,, 1) 5 0

for any (6,, 6,)

and

(2) a2 5 0 < a, < a,. The combination (a) -8
of these two conditions

yields

- 8 < a, - 8 < a, + a, - 8,

(b) a2 - 0 < a, + u2 - 0 < a, + a, - 0 < a, + a, + u2 -

Q 5 0,

and (c) ~(a,, 4, 1) 5 ~(6,~ %, O), for any (6,, 6,). The set of specific conditions, which are mutually exclusive and characterize the set of the five Cases (I-V), is given as follows, and is illustrated in Fig. 2 (a)-(e): Case I.

-e>o.

Case II.

-e~o
CuseZZZ. Case IV. Case V.

ON NEURONIC

EQUATIONS

199

In combination with (b), for each 3-dimensional state we can complete a transition It is noted that there is of critical values defined by

the value of l[z(6,, 6,, S,)] can now be given configuration 6 = (a,, 6,, S,), and therefore route to each of the five cases (I-V). a continuous change of value of 0 and a set (a,, a,, az) such that transitions from one of

FIG. 2(a). Transitions

FIG. 2(b)-(d). Transitions

of 23 state configurations

of 2j state configurations

under (3.9) Case I.

under (3.9) Cases 11, III and IV.

200

TOSIO KITAGAWA

FIG. 2(e). Transitions

of 23 state configurations

under (3.9) Case V.

the five Cases to another does occur in the sequence 1 + II -+ III --+ IV -+ V. Case I has two reverberation cycles, while Case V is one state configuration (0, 0, 0) which is a critical (rest) configuration and to which every state configuration converges ultimately. The other Cases, II, III, and IV can be considered as intermediate transitory state configurations between Cases I and V. 4. CONCURRENCE In our functional

FUNCTIONS equation

AND

ALLIED

OPERATORS

(3.1), the value of the concurrence

function

n-l

2 an_kBk- 0 (4.1) k=O plays the decisive role. In what follows (so far we have been concerned with each fixed vector O), we may and we shall write z(6) instead of z(a, 6). Let us start with elementary observations. There are two important special cases that are pertinent to our detailed discussion. z(a, 6) =

Case A. All the values of the concurrence for all 6. Case A. All the values positive for all 6.

of the concurrence

functions

z(a, 6) are positive

functions

~(a, S) are non-

Case B. The 2”-’ values of the concurrence Function are positive, while the other 2’+’ values of the concurrence function are nonnegative. The following

propositions,

4.1 and 4.2, are immediately

apparent.

ON NEURONIC PROPOSITION

EQUATIONS

201

4.1

In Case A, denoting the corresponding for any state configuration 6,

operator

, forj=

1,2 ,...,

2’ by PA,

n-

1,

we have,

(4.2)

and

that is 8*[6(“)] = (1”).

PROPOSITION

(4.3)

4.2

In Case A, all the state con$gurations converge to the state conzguration (1, 1,. . .> 1) = (1”) which is critical (= invariant), at most after n applications of 9. Similar observations hold true for Case A. Now regarding Case B, we shall first make observations on its special case when there hold the following relations. Case B,. For any n-dimensional state . . .) 6,, 6,) with the n - 1 components we have z@-‘1, 1) and z(#“-*), 0) Case B,. For any lz-dimensional state . . .) a,, 6,) with n - 1 components we have z(P1), 1) and z(?P-‘), 0)

configuration #“) = (6,_1, 6n_-2, 6(“-l) = (6,-r, 8n_2, . . ., 6,, 6,), > 0,

(4.4)

s 0.

(4.5) configuration #“) = (6,-r, &,_2, @“-l) = 6,-r, 6n_2, . . ., 6,, 6,), 5 0

(4.6)

> 0.

(4.7)

202 PROPOSITION

TOSIO KITAGAWA 4.3

In Case B,, denoting the corresponding operator for any n-dimensional state configuration 6’“‘,

, forj

= 1,2,.

LY by YB~, we have,

. ., n -

1,

(4.8)

and

(4.9)

PROPOSITION

4.4

In Case &, denoting the corresponding operator for any n-dimensional state configuration d(“),

6n-j 6,-j_,

1

3

forj:

_Y by S’B~, we have,

= 1, 2, . . ., n -

1,

(4.10)

ON NEURONIC

203

EQUATIONS

3 that is, 2’;&6(“))

and hence

=

,

that is Y~“(S(“)) = #“). BO

(4.12)

In the consequence of (4.8), the relation (4.11) hold true evidently for the Case B. In view of these preparatory observations given just now we shall discuss the Case B furthermore in details, in order to investigate the phenomena of reverberations for Case I% A feasible set of linear inequalities and equalities among n real numbers by the concurrence-function {Q} (k = 0, 1, 2, . . .) n - 1) induced z[a, @)I specifies a certain operator 9[6(“)] to be applied to each ndimensional state configuration 6’“’ = (6,_ i, 6, -2, . . ., a,, S,), all of which constitute the n-dimensional state configuration space denoted by X, whose size IX,] = 2”. All the characteristic features of the functional equations (3.1), (3.6), and hence (3.7) are concerned with each specification off(b) introduced in (3.7) based upon (3.5) which defines their respective operator 9’. After discussing some elementary Cases A, A, Ba, and I$, it will be adequate to introduce a system of somewhat generalized notations. We introduce the following definition. DEFINITION

4.1

The following set of operators Ym, 25, {~cIJY (~E,I, {go,>, and {=%I (1 = 0, 1,2, . . .) n - 1) is introduced as a set of special cases of the operator 2 given in (3.7) each qf which is speczfied by their respective f (6) in the following way: (1) 9,

: f [@)I = 1 for all 6’“) in X,.

(2) 9,- : f [6(“)] = 0

for al/ 6(“) in X,.

204

TOSIO KITAGAWA (3) $Pcl, :f[P]

= 6,

for all ~9’) in X,.

(4) YE, :f[#“)]

= 8,

for all P)

(5) _Yp .f[P)]

= 6,6,_,

(6) L&:;f[6”“]

= 6 I1 6 _ II6 _ 2

in X,.

1 . f 6,6,

for all ~9”) in X

. * . 61

for

all IY”) innX,.

Sections 5 and 6 are devoted to a discussion of 2z0 with reference to reverberation cycles, which are basically important to the following sections of this paper. Section 7 is concerned with 2,, for general 1 2 1. In Sets. 8 and 9 we discuss 2’~, (I 2 1). It is noted that 2, = PA, 2’G = Y;i, gmO = 2?~,,, 2Pz, = %,,, 2,, = 2’~,, and dpiO = 2~~. Illustrations of these operators, given in Definition 3.1, with reference to the (n + 1)-dimensional vector (a,,, a,, a2, . . ., a,_I, e), are given in Sec. 10. 5. REVERBERATION

CYCLES

FOR

2z,,

In this section we are entirely concerned that, for any P) = (6,_ 1, &,_2, . . ., dl, i&J, 9&“-l, which is nothing For eachj,

with the operator

Bn-2, * . ., b, 43) = @,, h-1,6,-2, . . ., but the Case &, introduced in Sec. 4. 1 sj

$P&&l, =

5 II ij”_*,

(Fj_1,

Q,

such (5.1)

1, we have .

Fj_2,

62,

yz,

.

.)

m

6,,6,)

*

a)

81,

50,

6,-1,

6”-2,

3.

*)

(5.2)

Sj)

and we have z&(&l-1,

bn-2, * * -34, &J> = GLl,

h-2,

* * ., 81,

&A

(5.3)

hi-*,

. . *, 4,

&A

(5.4)

and hence =%:(d”- 1, A-2,

* * *, b,

which can simply be denoted

&I>

=

(h-1,

by 2;!#))

= S(n),

(5.5)

J@(S(“‘)

= g@),

(5.6)

and

respectively. any @),

There

holds

the semigroup

~;:s(#n)) for any pair of positive PROPOSITION

integers

property

to the effect that, for

= ~~,(‘q,@(“)),

(5.7)

s and t.

5.1

For each assigned rS(“) there exists a uniquely determined integer k (= k(#“))) such that

nonnegative

ON NEURONIC

205

EQUATIONS -

(1) Y&(6(“)) = S(n),

forh = 1, 2, . . ., k - 1,

(2) L$,@(“)) # (den)) (3) _9&$F’)

= CF,

forh = 1, 2, . . ., 2k - 1,

(4) 9&(6(n)> # 6(“)

for j = 1,2, . . ., k, and

(5) _Y~Jj(&“)) = w) (6) k is a divisor of n.

Proof. Because of the relations negative integer m such that (i)

_Y~O(~cn))= d(“),

(ii)

Y&(6(“)) # 6’“‘,

(iii)

2n = qm.

(5.6) and

(5.7), there

for h = 1, 2, . . ., m -

exists a non-

1,

(5.8)

1 In fact since l~m~22n, we canwrite 2n=qm+rwithOir
(b@)),

{~& : x = 1,2, 3, . . .} = {_fY&(6’“‘) : y = 1, 2, . . ., m> because of (5.8)-(i), lIksm--land -

there

exists

a nonnegative

integer

(5.9) k such

that

@ ($0) = g(n) a0 It is to be noted that we have, for any a@‘),

(5.10)

AT- (g(n)) = Lz-aa(6 ()n )

(5.11)

and hence that, in view of (5.7;: 9i,+j(&“))

= 9i0(y;0(6(n))

= giO(O*)

= w)),

(5.12)

forj = 1,2,. . ., k. In view of the definition of m, we have 2k = pm with a nonnegative integer p. Since k -C m, p can not be greater than 1. Hence we have m = 2k, which completes the proof of Proposition 5.1. Now we shall introduce DEFINITION

the following

definition:

5.1

A representation

of an n-dimensional

state vector

6’“’ = (6,_, 2 s n-2,. . .) 61, S,) by a catenation of a sequence of q k-dimensional state vectors #,k), 8(k) 2 ,.*., @jk) 14

206

TOSIO KITAGAWA

= 1, 2,. . .,q,

such that,forj with

. . . #O 4 ’ representation

,#n) = @‘Q) and qk = n, is called an articular articulars with length k. DEFINITION

(5.14) of ~3~“)by means

of q

5.2

An articular representation of an n-dimensional state configuration 13~“) is said to be a regular (k, m) articular representation when there exist a nonnegative integer m and a k-dimensional state contguration oCk)such that (i) (ii)

n = kg = k(2m + l), 81”) = 8s”) = . . . = a$kj)+l = . . . = &z_l

(iii)

In this case we denote $0 and bCk) is called the constituent representation. PROPOSITION

= &;+1

= 6(k),

(5.15)

SF) = Sk”) = . a s = ~5:) = . . .

= [#$jW]yj(W, articular

(5.16)

of the regular

(k, m)-articular

5.2

An n-dimensional

state

conjiguration

13~“)belongs

to a reverberation

cycle of exactly 2k Iength if and only tf there is a regular (k, m) articular representation of 8”) with the constituent articular oCk)for which there does not exist any (k,, m,) articular representation k = (2m, + l)k, with a nonnegative integer m, and a positive integer kI. In order to prove Proposition lemmas. LEMMA

5.2 it is convenient

to state the following

5.1

We have

-g& ((o’k’&k’)“,zj’k’) = o(k)(#k)$“))m CIO = (&J+j(k))$@

(5.17)

= (fVO#k))m$k) 8~~((o’k’#k’)“#k’) The proof is immediate.

= (#k)#k))m#k) .

(5.18)

ON NEURONIC

LEMMA

EQUATIONS

207

5.2

For an assigned n-dimensional state configuration a(“) let there be a sequence of state conjigurations {6(j)} (j = 0, 1,2, . . ., s - 1) where each 8(j) is a kj dimensional state configuration with C?(O)= 6”” and k, = n. Let there be a hierachicalsystem of regular (kj, 1, mj, 1) articular representations such that g[il

forj

=

($j+ll#j+ll

)

mj+l

6

lj+ll

(5.19)

3

= 0, 1,2,. . .,S - 1. Then we have, for j = 1,2, . . ., s, a(n) = pl = (~Cjl#jl)M#jl,

where Mj is recurrently

(5.20)

dejined by

Mj+l = (2mj+l + for j = 0, 1,2, . . ., s - 1, with MO = 0.

l)Mj

f

(5.21)

mj+l,

Proof. We have #“’ =

(p1p1)m1pl

$11 =

($21~ygC21

p

(~[21S[21)n12$21_

(5.22)

3

(5.23)

Hence we have

The combination $0

=

of these three formula

(5.24)

yields us

=

__ {(~~21~)~2~~21(6[21~C21)~2~C21)~~(~C218C2])~2~C21}

=

(~C2l~C21)(2m2+1)~t+m2~121

(5.25) 9

which shows that M2 = (2m2 + l)M, where MI = ml. Now we can proceed

by induction,

After these preparations

+ m,,

which completes

we proceed

(5.26) our proof,

n

to

Proof of Proposition 5.2. The sufficiency is obvious. To verify the necessity, the relation Y&(6) = (6) yields, in view of the articular representation of 6 given in (5.14), --#$‘@‘,$JG . . . 8&$ a= ~fO@+j$“’ . . . ,jW (5.27) 4 ’ which leads us to = J$k) = s&k) = . - * = q,,,

(1)

q)

(2)

@4 = Sk”’ = @’ = . . . = &k&

(3)

s$k’ = 6’;).

(5.28)

208

TOSIO KITAGAWA

Since a(/+21 = a(k) = d(k) 1 , we note that there exists an even integer 2m such that q - 1 =42m, i.e. q = 2m + 1. Now, in view of(l), (2), and (3) in (5.28), we have 6’“’ = (#k)B(k))m#Q, (5.29) as we were to prove. EXAMPLE

5.1

The reverberation cycle of exact length 2. Let us take k = 1 and a(‘) = I. 8(2mf1) = (6C1)&1))m~(1) = (1 0)“‘l yields us a - reverberation cycle of exact length 2, and ZPS,(SPm+ 1)) = B(1)(#r)#l))m = (#1)&i ))m$G = d(zm+l) = (0 l)mO and yi0(#2m+l)) = CJEo(&2m+l)) = @m+r). EXAMPLE

5.2

The reverberation cycle of exact length 4. Let us take k = 1 and the set of all the possible 2-dimensional state configurations (0 0,O 1, 1 0, 1 1}, to which conjugate relations are applied so as to restrict to 6 = 0 0, and 0 1. We note that sequential applications of _Yz, on (#2)~c2))m~cz) = (O* 12)m O2 yield us &2@m+l)l = (02 12), 02 + l(O2 12)m 1 -+ 12 (02 12)m, = (12 07,

12 = &wm+1)1,

(5.30)

&2(2m+l)l = (12 02>m l2 -+ 0(12 02)m 1 + O2 (12 02), =

#2(2m+l)l_

(5.31)

For the case JC2) = 0 1, we have $X2m+i)l = (0 l* O)m 0 1 = 0 (12 02)m 1,

(5.32)

which belongs to (5.24). In consequence, there is one and only one type of reverberation cycle with exact length 4 which is given by (5.23) and (5.24) for each positive integer m. EXAMPLE

5.3

The reverberation cycle of exact length 6. Let us take k = 3. From the set of all the possible 3-dimensional state configurations the conjugations among P) and 5, leads us to the restriction of four ones, namely, 0 0 0, 001,010,and011. (1) The case h(3) = 03. S equential (03 13)m O3 yield us

applications

of JZa, on c!V~(*~+l)l =

(03 13)m 03 + 1 (03 13)m o2 + l2 (03 13)m 0 -+ 13 (03 13)m = (13 03)m 03,

(5.33)

ON NEURONIC

EQUATIONS

209

and (13 Oqm 13 -+ 0 (13 03)m l2 -+ o2 (13 Oqm 1 + 03 (1 3 Oym = (03 13)m 03, which gives a reverberation

(5.34)

cycle of exact length 4.

(2) The case a(3) = 0’ 1. S’mce (0’ 1 1’ O)m02 I = 0’ (I3 03)m 1, this case is coincident with case (1). (3) The case h(3) = 0 1 0. Since (0 1 0 1 0 1)” 0 1 0 = (0 1)3mf’ 0, this case is reduced to a reverberation cycle discussed in Example 5.1. (4) The case d(3) = 0 l*. S’mce (0 I2 1 02)m0 l* = 0 (13 03)“’ 12, this case is coincident with case (1). EXAMPLE

5.4

The reverberation cycle of exact length 8. Let us take k = 4. Similarly as in Example 5.3, let us consider the eight 4-dimensional state configurations d(4) : (1) 0 0 0 0

(2) 0 0 0 1

(3) 0 0 1 0

(4) 0 0 1 1

(5) 0 1 0 0

(6) 0 1 0 1

(7) 0 1 1 0

(8) 0 1 1 1. (5.35)

The cases (I), (2), (4), and (8) yield a reverberation cycle of exact length of 4 induced by #4(2m+1)l = (04 14)m 04. On the other hand, (3), (5), (6), and (7) yield a reverberation cycle of exact length 4 induced by Bt4(2m+1)’ = (O* 1 0 l* 0 1)” O* 1 0. 6. TYPES AND NUMBERS OF REVERBERATION n-DIMENSIONAL STATE CONFIGURATIONS APPLICATIONS OF THE OPERATOR _?Z#, Let a positive integer numbers such that

n be decomposed

CYCLES UNDER

into the product

FOR

of prime

n = 2”p{‘ps,’ * * * pp,

where CIis a nonnegative such that

(6.1) integer and pi (i = 1, 2, . . ., Y) are prime numbers 3 5 Pl < P2 < * * * < P,,

(6.2)

with (i = 1, 2, . . .) r). Pi2 l, In view of Proposition 5.1 we state the following PROPOSITION

6.1

For each assigned a(“), there (i = 1,2, . . ., r) such that

(1)

k(#“))

= 2”p;‘py

(2)

O

S

5

(6.3) proposition.

Yi

Pi,

is a set of nonnegative

integers

{yi}

. . . pr,

(i

+

1, , . . ., r).

(6.4)

210

TOSIO KITAGAWA

Proof (6.4) comes from the relation YE= qk(@) and (5.15)-(i). Before giving an enunciation of general assertion on types and numbers of reverberation cycles for n-dimensional state configurations under applications of the operator _YzOwhen n is given by (6.1), let us discuss some special cases of n which aid in understanding a certain essential feature of the problem and an introduction of a certain procedure of difference operators which will be illustrated later in this section, It is noted that any n-dimensional state configuration in the n-dimensional state configuration space x, belongs to one and only one reverberation cycle in every case. PROPOSITION

6.2

When n = 2”, the length of any reverberation cycle is 2n, and there exists a set of 22u--(5+1) reverberation cycles which are mutually disjoint. Proof. The k = 2~. PROPOSITION

relation

n = k(2m + 1) = 2” implies

that

m = 0 and

6.3

When n = p, where p is a prime number greater than 2, there exist exactly two types of reverberation cycles: (0) One reverberation cycle of the exact length 2 induced by the state configuration (1 0)(P-1)‘2 0. (I) (2~ - 2)/2p reverberation cycles of the exact length 2~. Proof The relation n = k(2m + 1) = p yields us two solutions for (k, m), i.e. (1, (p - 1)/2) and (p, 0), each of which corresponds to (0) and (1) enunciated in the assertion of Proposition 6.3. PROPOSITION

6.4

When n = 2”p, where p is a prime number greater than 2, there exist exactly two dtflerent types of reverberation cycles: (0) (k,, m,) = (2”, (p - 1)/2) type. The number of reverberation cycles of this type is equal to 22a-(a+ I). (1) (k,, ml) = (2*p, 0) type. The number of reverberation cycles of this type is equal to (2P2’ - 22”)/2p2u = 22”-(a+1)(2(p-1)2a - 1)/p. Proof. The number of all the possible k-dimensional state configurations is equal to 2k, each of which belongs to one and only one reverberation cycle of the exact length 2k, for k = 2”. In the consequence the totai number of reverberation cycles of the exact length 2k = 2’+l is equal to 2k/2k = 2Za-(@+l), since there is no more reverberation cycle of the

ON NEURONIC

EQUATIONS

211

exact length of 2”+l. All the other reverberation length 2~2~ = 2a+1p, and hence their total number

cycles have the exact is equal to

(2YXM- 29/2=+‘p, as we were to prove. A similar

n

method

PROPOSITION

can be applied

to the following

special cases.

6.5

When n = 2~~0, where p is a prime number greater there exist the following

than 2 and p >= 2,

exact (/? + 1) types of reverberation

(1) (k,, m,,) = (2”, (pfl - 1)/2). this type is equal to 2”-(‘+ l).

The number

cycles:

of reverberation

(j) (kj, mj) = (2Olpj,(pfl-j - 1)/Z) (j = 1, 2, . . ., @. The reverberation cycles of this type is equal to (2k~ - 2kj-1)/2kj. PROPOSITION

<

number

of

6.6

When n = 2Ep,p,, 2 andpI cycles:

cycles of

p2,

where pi (i = 1,2) are prime

there exist the following

exact four

(0) (k,, m,) = (2’, (pIp2 - 1)/2). Th e number of this type is equal to 2k0/2k, = 22a-(‘f1).

(1) (k,, ml) = (2”p,, (p2 -

1)/2).

numbers greater

than

types of reverberation of reverberation

cycles

The number

of reverberation

cycles

The number

of reverberation

cycles

of this type is equal to (2kl - 2ko)/2k,. (2) (k,, m2) = (2*p,,

(pl -

of this type is equal to (2k2 -

1)/2).

2ko)/2k2.

(3) (k3, mJ) = (2=pIp2, 0). The number type is equal to (2k3 -

(2kz -

2ko) -

(2kl -

of reverberation

cycles of this

2ko) + 2ko)/2k3.

Now before giving similar assertions to the other special cases, it will be useful to introduce the following recursive set of difference operators. DEFINITION

6.1

For any prime number pi greater

than 2 and any positive integer G,

(1)

A”‘(p,)2G

2 2p’G - 2’.

(2)

AC2’(pr, ~2)2’ 2 A”‘(P~)(A”‘(P~))~‘, 7 A(1)(P2){2P’G - 2’}, 7

(3)

A’“‘(P~,

~2,

(2PZPlG

. . .) 1.42

_

2PlG)

-

(y+

pz,

_

29

7

A’“-“(~1,

7

A’“- ‘)(pl, p2, . . ., pn- 1)2pnG

. . ., P,J{A”‘~P,)~~)

-A’“-“(p,,

p2,.

. ., ~.-~)2’.

212

TOSIO KITAGAWA

COROLLARY

6.1

We have A’YPI, PZ, . . ., ~3’

= A”(P,,,

where (aI a2 * * * a,) is any permutation PROPOSITION

~az, . . ., P$~,

(6.5)

of (1, 2, . . ., n).

6.7

When n = 2”p,p,p, = Kp,p2p3, where pi (i = 1, 2, 3) are prime numbers such that 3 5 p1 < p2 < p3, there exist eight difSerent types of reverberation

cycles.

(0) (k,, m,)

= (K, (p1p2p3

-

1)/2) type. The number

of reverberation

1)/2) type. The number

of reverberation

cycles of this type is equal to 2K/2K.

(1) (k,, m,) = (p,K,

(p2p3

-

cycles of this type is equal to (2~1~ (2) (k2, m2) = (p,K,

(p1p3

-

2K)/2pIK

1)/2) type. The number

cycles of this type is equal to (2pzK - 2K)/2p2K (3) (k3, m3) = (p3K, (pIp2

= A(‘)(~,)2~/2p,K.

-

of reverberation

= A”‘(p2)2K/2p2K.

1)/2) type. The number

cycles of this type is equal to (2paK - 2K)/2p3K

of reverberation

= A(‘)(~,)2~/2p,K.

(p3 - 1)/2). The number (4) (k1,2, ml,2 ) = (p,p,K, _ ~PIK _ ~PZK cycles of this type is equal to (2 PIPS

of reverberation +

2K)/2plp2K

=

A”‘(PI 2~2)2~/2~, p2K. (5) (kI,,, m 1,3) = (pIp3K, cycles of this type is equal to (2p1p3~ - 2plK-22p2K (6) (k2,,, m2,3 ) = (p2p3K, cycles of this type is equal to (2p~P3~ -

(p2 -

1)/2).

The

+ 2K)/2p,p,K (pl -

1)/2).

2pzK - 2psK + 29/2p,p,K

number

of reverberation

= At2’(pI, ~~)2~/2p,p,K. The

number

of reverberation

= A(2)(p2,p3)2K/2p2p3K.

(7) (kI,2,3, m1,2.3 ) = (p1p2p3K, 0). The number of reverberation of this type is equal to A(3)(pI, p2, p3)2K/2p,p2p3K. PROPOSITION

cycles

6.8

When n = 2”p,p, * * * p, = KP, where pi (i = 1,2, . . ., r) are prime numbers such that 2 < p1 < p2 < . . . < p, and we put K = 2” and there exist (r + 1) classes of types of reverberation P =PlP2”‘Pr, cycles: Class

0.

(k,, m,)

= (K, (P -

1)/2).

The

number

of reverberation

cycles of this type is equal to 2K/2K. Class 1. (ki, mi) = (p,K, (Pp;l

cycles

of each type (i = 1, 2, . . ., r).

belonging

- 1)/2). The number of reverberation to this class is equal to A(‘)(pi)2”/2piK

ON NEURONIC

CZUSS 2.

EQUATIONS

213

(kil,il) mi,,) = (Pi,PilK,

reverberation

[Pp,l’~~<’

cycles of each type belonging

tki,,i,

,..., ij3

mi,,i2

1)/2]. The number

of

(i,, i, = 1, 2, . . ., I’; i, # i2).

A’2’(pi,, pi,)2K/2pi,pi2K

Class J*

-

to this class is equal to

,..., ij >

=

[

fi

PihK>(Pil,iI,...,ij -

h-l

J)!2]where 2

we have put -1

(’ >

Pit,i*....,ij = P ,bl Pi, The number is equal

of reverberation

.

cycles of each type belonging

to this class

to A(j’(p. 11)p.125* . ., Pij)2~/2-‘,~~~ pihK, (il, iz, e a-2 ij = 1, 2, a . a)7

with i, < iz < . * * < ij), where j = 3, 4, 5, . . ., r. In order to proceed to general case given by (6.1), we introduce a sequence of difference operators, which is a generalization of that given in Definition 6.1. DEFINITION

6.2

For any set of n difSerent prime numbers (pi} (pi > 2, i = 1, 2, . . ., r), any set of r nonnegative integers (vi> and positive integer G, we de$ne

(2)

for vj = 0,

for vi = 0 (3)

Ack)(;; f

;;:::e:> A(k-1)

Pl

PZ

*.

.

Pk-1 )p(p},

( 1'1 V2"'Vk_l

214

TOSIO KITAGAWA

-1 ~‘k-

1)

=

Pl

(

A(,,-1)

Pl

(

( -*(k-l)

P2

v1

V2”’ P2

Vk-1

2Pk”“G )

* * * Pk-1

v1 v2”’

for vk = 0, ’

)

’ ’ * Pk-1

Pl

We observe the following

(2prYkG - 2PkY*-‘G), for vk 2 1, 2c

. . . Vk_l

A@-1) P1 P2 ’ * vi v2”’ (

1

1

Vk-1)

P2”‘Pk-1

Pl (

COROLLARY

. . * Pk-

\‘I v2

*(k-l)

= +

P2

\Vi V2”’

’ 2PkV” - 1G

vk-1 > * Pk-1

2G

for vk 2 1, for vk = 0.

3

vk-1 >

,

corollary:

6.2

We have *(*‘(:‘f’:::~)=*‘k)(plp,...pk~,

(6.6)

~(k)(e:e::::ez>=*(k’(~,:~~~:::r~),

(6.7)

where (ccl, cc2,. . ., cxk)is any assigned permutation of (1, 2, . . ., k). b(k) Pi

P2

Pk o)=“‘k-“(~::::::e:::).

* ’ ’ Ph-1 .v_

( v1v2”

(6.8)

kl

In order to give an explicit expression notation given by the following definition. DEFINITION

of ACk) let us introduce

the

6.3

We define @

Pl

P2

’ * . Pk

VI

v2

* * * Vk

2G

E

2G

ifIl

4(Pi3 vi,

49

(6.9)

( El E2 -. * E:,> where we define $(Pi, vi, Ei) =

;‘+&“ i 3

for

Vi

2

1,

for vi = 0,

(6.10)

and Ei =

0

or 1, for

0, We observe the following

corollary.

Vi

2

1,

for vi = 0.

(6.11)

ON NEURONIC COROLLARY

215

EQUATIONS

6.3

We have

PROPOSITION

6.10

In the case n = 2”pflpr$ * - . pk where pi are prime numbers such that 3 s pi < pz < . . * -c p, andPi 2 1 (i = 1, 2, . . ., r), we have thefollowing assertions. (1) Each n-dimensional state contguration 6 belongs to one and only one reverberation cycle of exact length 2k(6) where k(6) is a divisor of n with the form k(6) = 2’tp;;‘piy:z . . . p;),

(6.17)

where n = 0, a, l 5

Yi,

5

Pi,

(j

(6.18) =

1,

2, . . .) v),

(6.19)

and (iI, iz, . . ., iY) is a subset of (1, 2, . . ., n). (2) To each assigned triplet of a nonnegative integer v, a subset (4, i,, . . ., iv) of the set (1, 2, . . ., n), with the condition 1 s i, < i2 < . * * < i, 5 n, and a set of v positive integers (yi,, yi2, . . ., yi,) such that 1 5 Yij 5 ij (j = 1, 2, . . ., v) and to each n which may be 0 or CX,there exist (6.20) d@erent reverberation cycles of the exact length 2~p~~1p~~2 - - ., p,?:. That is to say, denoting by N(I) the total number of reverberation cycles having the exact iength I we have the result that N[2k(6)] is equal to (6.20) for any k(S) given by (6.17), (6.18), and (6.19). Proof The proof of assertion (1) is immediate from Proposition 6.1. In order to prove the assertion (2), we proceed by induction with respect to v and yij (j = 1, 2, . . ., v). Special cases given in Propositions 6.2-6.9 yield us a set of starting types of reverberation cycles for application of mathematical induction.

216

TOSIO KITAGAWA

Yamaguchi the effect that

[14] gave an equivalent

result

under

another

formula

p(d)2k(J)‘d

(6.21)

d: odd

where p(d) denotes

MGbius function

in number

theory.

TABLE 2 Types and Numbers of Reverberation Cycles of n-dimensional State Configurations Under Application of zgO for n = 2-20 (1) n = 2~ (r = 1,2, 3,4) n

LRV

2a

3+1

2 4 8 16

4 8 16 32

NSC 22= 4 16 256 65536

NRC 22”-(a+

1)

1 2 16 248

TABLE 3 Types and Numbers of Reverberation Cycles of n-Dimensional State Configurations Under Application of “a, for n = 2-20 (2) n = p (prime number >= 3) n

LRV

to

NSC

NRC

3

2 6

2 6

1 1

5

2 10

2 30

1 3

7

2 14

2 126

1 9

11

2 22

2 2046

1 93

13

2 26

2 8190

1 315

17

2 34

2 131070

1 3855

19

2 38

2 524286

1 13797

ON NEURONIC EXAMPLE

217

EQUATIONS

6.1

The types and the numbers of reverberation cycles of n-dimensional state configurations under applications of _YzOfor II = 2-20. The applications of Propositions 6. I-6.10 yield a complete enumeration of all the possible types and their numbers of reverberation cycles as shown in the Tables 2-5, where the following notation is introduced : (1) LRC = I = Length of reverberation cycle. (2) NSC = N(E) = Number of n-dimensional state configurations belonging to a reverberation cycle of the specified exact length I. (3) NRC = N(I)/I = Number of reverberation cycles having the specified exact length 1. TABLE 4 Types and Numbers of Reverberation Cycles of n-Dimensional State Configurations Under Application of 2s0 (3) 2up II

LRV

NSC

NRC

6 = 2.3

4 12

4 60

4 5

10 = 2.5

4 20

4 1020

1 51

12 = 2.3

8 24

16 4080

2 170

14 = 2.7

4 28

4 16380

1 585

20 = 22.5

8 40

16 126076

2 3174

TABLE 5 Types and Numbers of Reverberation Cycles of n-Dimensional State Configurations Under Application of gz,, (4) 9, 15, 18 n

LRV

NSC

NRC

9 = 32

2 6 18

2 6 504

1 1 28

15 = 3.5

2 6 10 30

2 6 30 32730

1 1 3 1091

18 = 2.32

4 12 36

4 60 262080

1 5 7280

TOSIO KITAGAWA

218

7. 6(“)* REVERBERATION CYCLES OF n-DIMENSIONAL CONFIGURATIONS UNDER APPLICATIONS OF 9z,_ I PREPARATORY

STATE

CONSIDERATIONS

For each n-dimensional state configuration IS(“)= (&-I, Sn-*, . . .) S,, S,), we define 9%,_ (SC”))by 9- en-1(k)

= 9Z”_ (&-I, S,_z, . . ., d,-I, a,-1+1, * * *, b, &J> = @,-,:6,-l,.

* ., L+1,

L,

h-1,

* . ., 6,)

(7.1)

The repeated applications of gx,_, yield us 8”%-I(S(“)) = (6n_ l+h _ 1, Fn - l+h - 23 **.Y for each h (1 s h 5 1). It is convenient for brief descriptions notions : (1)

4 = (h-1,

h-2,

(2)

A, = (6,_1, bn_2,. . .) S”_,).

(3)

A{“) = (s,_r+c-l,

s n- z+fl- 27

* e.9

(4)

A?‘) = (S”-I+&-1,

s”-l+JL-2,

* * ., &-A

&-I,

* * *, 6h)

%-I,

to introduce

(7.2)

the following

* * .>a,-2.

4I-,),

(1

5

p

s

I).

(1

5

CL s

0.

(7.3)

(0 $ v 5 n - 1). (+4 = (Sn-l,Sn_2,. . .,S,), -(0 6 v 5 n - 1). (6) (+I = (S,_ 1, Sn_2, . . ., S,), It is obvious that A,“’ = A,, A,“’ = A,.

(5)

(1) (2)

~:._,(scn’) = (&(h& k’_” (S(“)) = (Ajh’jfI (,+izy%

(1 5 h 5 2).

(3)

Y;;;h(S(“)) = ($‘A,?i,

(1 5 h s 1).

(4)

Z’!t+h(S(n)) = (Aih’A IA Iii I (x+hjA), a” I

(7.4)

(1 s h 6 1).

c21+kjA),

(7.5)

(1 s h s 1).

In general we have in an abbreviated notation. (2)

91k! (8")) = ((AA)k I 1 (zkljA). 9;lf”cscn)) = (Ajh)(A1&;2kl+hjA),

(3)

L?$‘,f: l)z(S(“)) = @,(A,~&,+

(4)

g$;k_;l)*+h(S(n))

(1)

(1 5 h 5 1).

,,A).

= (Ajh’~,(A,~,):(zk+I)l+h,A),

(7.6)

(1 5 h s 1).

The results given in (7.5) and (7.6) are valid when the number of applications of 9’*,_, is not greater than n - 1. Now the following two cases are to be distinguished.

ON NEURONIC

EQUATIONS

219

Case 1. n = 2mZ + h - 1, (h - 2, . . ., I). We have ,;;j,:h-l(fs”)) Z!

a,-,

= (A,‘h-1$4JJV”_1). (d(n)) = (#+(A,‘&)“)

(7.7)

E 6’““.

(7.8)

Case 2. n = (2m + I)1 + h, (h = 1, 2, . . ., I). We have 2:;:“:

r)*+h- ‘(6’“‘) = ($hg! a,-, (P)

1)A,(&Q*G”_

= (A$‘;i,(A,AJ”)

@*-REVERBERATION CYCLES FOR n = 2ml+ h (h = 0, 1,2, . . ., I - 1)

THE

l).

(7.9)

55 LP*.

(7.10)

CASES

We have L??&P’)

= ~~,,_,(A(h)(AI~l)m-lAIAI), (7.11) = (A{h’(AJJ”-lAI

cn-,+ ,,A),

YkG--1(&“)*) = (d$h+k)(AIAl)mA 1 (n _ Ifk& for k - 0, 1, 2, . . ., 1 - k.

(7.12)

c!?;;j,(dcn’*) = (iI,(&i,)“-

(7.13)

and, for p = 1, 2, . . ., h -

l‘d~(,_,,+i),

1

c!?$-~~+~@(“)*)= (kt(P)i&(‘+&)m-lAr & a,-, (P)‘)

= (Ajh’A,(A,A,)“= (@(A,A,)m)

(n_,,+pjA),

(7.14)

l/l ) (7.15)

= $1.

Now we observe that ~~n+-4(~(“)*) = L?:“_,(P)*) forp

= 1, 2, . . .) 1, and in particular gJr,

In-l

(gcn,‘)

= LFz”_I(P)*)

(7.16)

that =

(p’)

=

p*

(7.17)

.

These results show that the set of n-dimensional state configurations defined by {S?&_1(6(“)*): q = 1, 2, . . ., 21) constitutes a reverberation cycle generated by #“)*, whose length is at most 21 and for which (7.16) is valid. d”)*-REVERBERATION CYCLES n = (2m + 1)1 + h (h = 1,2, . . ., I

FOR

We can proceed quite similarly have, for k = 0, 1, 2, . . ., I - h -

L&f “n-l(P*)

= St

THE

CASES

- 1) as in the previous

section.

In fact we

1,

_ (LpA&4JI,“)

= (d”(h+k)A&4JJm-1A~ ’

3 (“-l+k)A),

(7.18)

and L?;;_h1(6’“‘*) = ((&QrnAI

(n_hjA).

(7.19)

220

TOSIO KITAGAWA

Wehavefurther,forp=

1,2,3 ,...,

9;,_“;p(6’“‘*)

h-

= (njp$4,A,)VI

1 (“-h+p)A),

(7.20)

and A?;“_$(P)*)

= (Ajh’(A,A,)mA,), = (A~%l,(A,A,)“)

Now we observe that the relations (7.16) present case. In consequence, we observe that the set figurations defined by {L?~,_~#‘)*): q = reverberation cycle generated by 6(“‘*, whose which (7.16) is valid. EXAMPLE

(7.21)

* = 6(“) .

and (7.17) hold true for the of n-dimensional state con1, 2, . . ., 21) constitutes a length is at most 21 and for

7.1

reverberation cycles under the applications of YE,,_ ,. These are illustrated by the following Table 6 and Fig. 3.

6(“)*

TABLE 6 6(“)*-Reverberation

Cycles Under the Applications

Case II

Case I

*:- 1

2m 0+ 1

L-1

2n2 0

1

gw

1

s,-1(&r1L1)”

(&-16n--l)m

1 (0 1)m + 0 (1 o)m = 6(n)*

= (0 l)m --f (1 O)m = 6(n)*

=

%“_ ,(6(n)*)

1 (0 1)” + 0 (1 0)m (a) Case I FIG. 3. @‘)*-reverberation EXAMPLE

of Teg,,_,

(1 OY + (0 1)” (b) Case II cycles under the applications of Pg,_ ,.

7.2

#‘)*-reverberation

cycles under

the applications

of Yz,_?.

There are two cases, I and II, each of which has two subcases according to the value of v, Case I. II = 2ml + v = 4m + v

(v = 1,2).

Case II. n = (2m + 1)Z + v = 4m + 2 + v

(v = 1, 2).

ON NEURONTC

221

EQUATIONS TABLE

G(n)*-Reverberation

(a) Az = O* n AZ _ A2 $“S %a,_ ,(6(“)*) zpbn_ ,s(“‘* )

I

Cycles Under the Applications

of z,-,-2 Case II2

Case I,

Case I2

Case III

4m + 1 O2

4m i- 2 0*

4m + 3 O2

4m O2

12

12

12

12

A2”‘(A&)“’ 1 (02 12)“’ (12 o’)m 1

A~‘2’(A&” lZ (02 lqm 0 (1202)“’ 1

= % r.:“_,(&*)

0 (12 02)“’ (02 12)” 0

02 (12 0’)‘”

ie& _ ,(6(“)*) = gw

1 (02 12)”

12 (02 12)“’

1 (02

1”)“’ 0

Az”‘A,(A,&)” 0 l2 (02 12) o* (1202)” 1 1 02 (12 oz)m 12 (02 12)“O 0 12 (02 12)”

(A,&)‘” (02 l*)m 102 (12 oqm-l

(12 oqm 0 12 (02 12)“‘-’ 0 (02 12)“’

(b) Al = 0 1 n AZ

4m + 1 01

4m + 2 01

4m + 3 01

4m 01

A,

10

10

10

10

$o*

&(A,&)”

;i,c’l(A,A,)‘n 0 (0 12 0)” =(021qm0

1 O(Ol’0)“’ = 1(0’12)“0

FIG. 4. S(“)*-reverberation 15

1

A,“?A,(A,&” (A &)” (0 lzO)m 1 (1 O)(O 12 0)” = l*(OZlZ)mO =012(02~2)“!-10

cycles under the applications of %z,_~

222

TOSIO KITAGAWA

All the possible two-dimensional state configurations are 0 0, 0 1, 1 0, 1 1; the conjugate relation reduces to the essential two configurations OOandO 1. Hence A, = 00,

A2 = 1 1,

A,&=0011,

(7.22)

A, = 0 1,

A, = 10,

A,&

(7.23)

= 0 1 10.

Table 7 (a) and (b) shows that (7.22) and (7.23) give us the set of the same four types of 6(“)* reverberation cycles, although their A, are different. (Also see Fig. 4). EXAMPLE

7.3

#“)*-reverberation There according

cycles under the applications

are two cases I and to the value of v :

Case I.

n = 2ml-k

of YE, _ 3.

II, each of which

v = 6m + v

Case ZZ. n = (2m + 1)Z + v = 6m + 3 + v

has three

subclasses

(v = 1,2,3), (v = 1, 2, 3).

All the possible two-dimensional state configurations are 0 0 0, 0 0 1, 0 1 0, 0 1 1 and their respective conjugate configurations 1 I 1, 1 1 0, 1 0 1, 1 0 0. It is sufficient for us to consider the former four : A, = 03,

A,&

A, = 13,

= O3 13.

(7.24)

A2 = 0’ 1,

A, = 12 0,

A&& = O* 13’O;

(7.25)

A,=OlO,

A,=

A,A,

(7.26)

A, = 0 I*,

A, = 1 o*,

101,

= (0 1)”

AzA2 = 0 l3 O*.

(7.27)

As in Example 7.3, (7.24), (7.25), and (7.26) reduce to the same set of reverberation cycles. It is therefore necessary and sufficient to discuss the set of reverberation cycles corresponding to each of (7.24) and (7.26). We could prepare two tables each of which shows a detailed construction of Sx cycles for each of (7.24) and (7.26). Instead of giving such tables similar to Tables 6 and 7, we give in Fig. 5 (a)-(f) a certain description of reverberation cycles corresponding to Cases I and II, by a system of simplified notation: B” 7 (A,&)“‘,

B, = (A3A3)rn;

hB; 7 Ay)(A,a,)m,

J;

3 +,,B; 7 A$‘%,(A,A,)‘“, hBT;;

= &)(A3A3)m. 3+h@ = A:h’A,;;i,A,)“;

= ,&h’(A3ii3)“-

(7.28) (7.29) (7.30)

lck)A3,

hB);I;: = A$yA3A3)m-l(k)‘q3; 3+h~;--l

= AT).

3+hB;-1

= z?ih’. A3(A3A3)m-1(*)Ak;

(7.31)

A3(~3z3)m-l(k)~3; (7.32)

ON NEURONIC

h

forh,k=

223

EQUATIONS

(7.33)

B”k = AyyAJ3)m(“G3,

1,2,3.

b

C

e FIG. 5. (a) n = 6m+l. 6m+5. (f) n = 6m+6.

cb) n = 6mf2.

f (c) n = 6mS3.

(d) n = 6mS4.

8. TRANSITIONS OF n-DIMENSIONAL STATE CONFIGURATIONS UNDER APPLICATIONS PREPARATORY

OBSERVATIONS

define an operator state configurations by We

~&(4l-1,

&-2, =

93,

(e) n =

OF 9s,

ON zffl

defined over the set of all the n-dimensional

* * .>~,hJ

(1, h-1, (0, 6n-l,

h-2,. * *, Sz, d,), an+, . . ., d2, S,),

if 6,6,

= 0,

if 6,S, = 1.

(8.1)

224

TOSIO KITAGAWA

Our purpose is to make some fundamental observations on transition phenomena among n-dimensional state configurations under the applications of P’D,. Let us start with some preparatory observations. Observation 8.1 (1) There configurations

is a reverberation

cycle consisting

of n-dimensional

k = 1, 2, . . ., n>,

{ lk On-k;

state W)

and {Oj l”-j; j = 1, 2, . . ., f2 - 11. (8.3) (2) An n-dimensional configuration Onmk 1 Ok-’ for each k = 1,2, . . ., n is belonging to the attraction realm of the reverberation cycle defined in (8.2) and (8.3) in the sense that it is either a configuration belonging to the reverberation cycle or it attains at some configuration belonging to the reverberation cycle after finite number applications of $Pp,. Proof. (1) is immediate 0”

+

1 (y-1

_,

from the sequence 12

on-2

_+

. . . +

lk

of transpositions (y-k

_+

. . . _+

I”-

0 J

t

O”-‘It*

from 0” :

. . +

Ok I”--k

+

. . . t

02

1n-2

+

0

(8.4.1)

l”-’ + 1”

(2) is also immediate from the fact that for k = 1 and n, Onek1 Ok-l belongs to the reverberation cycle given in (l), and that, for k = 2, 3, . . ., n - 1, we have (y-k

1 ok-l

~

1 (y-k

1 ok-z.

. . +

lk-2

on-k

1 0

(8.4.2) lk

00-k

+

lk-1

On-k

1.

Observation 8.2 The configuration (0 1)” 0 in the (2171+ 1)-dimensional state configuration space attains at the configuration lZm’r belonging to the reverberation cycle defined in (1) of Observation 8.1 after 2m + 1 time applications of 9~~. The proof is immediate. The implications of these two preparatory observations are interesting in the sense that a set of state configurations consisting of one reverberation cycle under applications of the Bz, operator shows a different feature under applications of 9~~. In fact a reverberation cycle under applications of _Y,, may be said to be disintegrated into a set of state configurations consisting of transitory state configurations and those belonging of reverberation cycle(s) under the applications of 9~~.

ON NEURONIC

EQUATIONS

INTRODUCTION

OF A SET

225 OF OPERATORS

{r,?,)

In this paragraph we shall introduce a set of operators (9~~) (I = observations on 0, 1, 2, . . .) n - 1) and we shall give some introductory the mutual relationship among these operators by referring to simple examples. DEFINITION

8.1

An operator Pa, to each n-dimensional 6 = (cS~-~, a,,-2,. . ., 6,, 13,)isdefinedby y,-,(&-I,

state configuration

L2,. . 7 61, b) = (1, 6n-l, dn-3,. . .) 6,, S,),

if 6,6,-i

. . . 6, = 0

hn-*, . . ., 6,, a,),

if 6,6,-i

. . ’ 6, = 1.

= (0,6,-i, OBSERVATION

(8.5)

8.3

For each 6 = (hrt-l, &-2,

. . ., 6,, 6,) we have

9jj0(B) = 9*“(6).

(8.6)

For the set of 2” n-dimensional state configurations it is possible to consider the applications of ?Z,-, for I = 0, 1, 2, . . ., IZ - 1. We shall introduce the following definition. DEFINITION

8.2

A digraph (X,,, l-6,) is called a _YD,-digraph in the n-dimensional configuration when (1) X, is the set of all the n-dimensional 16 = (&l-1,4-2,

state

state conjigurations

. . ., b, &I)>,

and (2) T,q, is the set of all the directed arcs defined 619 L$, : 6 + LYp,(6).

(8.7)

We are interested not only in each digraph (X, l-s,) for each assigned I, but also with the transitory phenomena among these n digraphs when 1 ranges over 0, 1,2, . . ., n - 1. In fact the following simple examples are suggestive by showing a set of bridges from the graph (X,, l-g,,_,) to the graph (&

k).

EXAMPLE

8.1

The family of digraphs (X4, l-D,> (, = 0, 1, 2, 3). Each digraph (X4, r,,) (I = 3, 2, 1, 0) is shown and (c).

in Fig.

6 (a), (b),

226

TOSIO KITAGAWA

FIG. 6(a). The digraph (X,, I’&) for Tfi3 applied on Cdimensional space.

state configuration

FIG. 6(b). The digraph (X,, I’fiz) for gf12 applied on 4-dimensional space.

state configuration

FIG. 6(c). The digraph (X,, l?s,) for 4, space.

state configuration

applied on 4-dimensional

ON NEURONIC

227

EQUATIONS

A direct observation on the digraph (X4, l-p,) shows that there are a reverberation cycle and a set of state configurations each of which attains at one state configuration belonging to the reverberation cycle. By denoting state configuration in the following way,

(ij-,31, (~)-z,(pzl, (;)_(loY, (8.8)

Fig. 6 (d) yields the digraph

representation

[Fig. 6(d)].

FIG. 6(d). The digraph (X,, l?s,) for “fll applied on 4-dimensional state configuration space.

FIG. 6(e).The digraph (X,, I&) for “a0 applied on 4-dimensional

space.

state configuration

228

TOSIO KITAGAWA

In this case there are two reverberation cycles as we have shown in Sec. 6 for ~3’~~which is equivalent to 9~~. Figure 6(f) shows the clearer description of two reverberation cycles as well as their connections with the corresponding digraph (X4, l-,-J given in Fig. 6(d) by denoting an arc lost from Fig. 6(d) by a dotted arc and an added arc to Fig. 6(d) by a heavy arc in order to obtain Fig. 6 (e) and (f) from the original Fig. 6(d).

FIG. 6(f). The digraph (X,, I&) for Tpp,,applied on 4-dimensional state configuration space. yCOORDINATE

SYSTEM

SUITED

TO APPLICATION

OF “,i,

AND

“/jr

In order to have a summarized feature associated with transitions of state configurations induced by applications of _.5?~,on the n-dimensional state configuration space, it is convenient to introduce a new coordinate system suited to _.Y,-,for each fixed I, where 0 5 1 2 II - 1. DEFINITION

8.3

For each Jixed I, 1 5 1 5 n - I, and to each n-dimensional conjiguration 6 = (6,_1, c?,,-~, . . ., 6,, S,), we de$ne = r/i? l(S) * * * #?cl_ ,,(S) . F/fQ6) * . . yrb”(S),

p(6)

by giving a set of n Y&‘)(0 5 v 5 n a way that rly)(q

(8.9) 1) values taking either 1 or 0 in such

?I(‘; fj.+;l.-,‘).**6j+16j,

=

i

state

n

In

2

J'

for j =

forj=O,1,2,n-

n -

I + 1,. . ., 12-

,

1.

(8.10)

To each n-dimensional state configuration 6 there corresponds one and only one n(l)(6) expression, although different 6’s may correspond to the same one n(l)-expression. Immediately, we have the following:

ON NEURONIC

Observation

229

EQUATIONS

8.4

For each 6 in an n-dimensional either one of the following I forms

q”‘(6) =

state configuration (O<=kSl-2)

lkO’-k-’ . O&-l, i 1’-’

space, n(‘)(6) has

(8.11)

. L-t-+1,

are (n - l)-dimensional and (IZ - I + l)where &_, and kll_l+l dimensional state configurations each of which is uniquely determined by 6 respectively. It is noted that the main emphasis of Observation 8.4 is to point out the specific construction of the earlier part of q-expressions, that is, the part before the comma, and that we are not saying that to any assigned (n - I)-dimensional or (n - I + I)-dimensional state configuration (c”_i or g,,_l+l) there does exist an n-dimensional state configuration. In fact it is also obvious that there do exist a set of restrictions as to the values of &-, and 5,,--1+1. DYNAMICAL

BEHA

VI0 R INDUCED

B Y %,, _ ,

Let us denote by X$)the set of all the n-dimensional state configurations 6’“’ such that a(n) = lj-lod(n-A (J = 1,2, . . .) n - l), (8.12) > when den-j) runs through the (n - j)-dimensional state configuration space X,_j, that is to say, xl;” 7 L;)(n) : 6Cfi)= 1 j- 106(n-j), p.i) E x,_ j] (8.13) It is obvious IXl;i)l = 2”-j, (.j = 1, 2, . . ., n - 1). For the sake of completeness, let us define Xr) = [dCn) : 6(“) = i”-‘~C1), h(l) E X,], (8.14)

= ((1”~‘0) u (1”)). Now we observe PROPOSITION

the following

proposition.

8.1

(i) For each ~3~“)belonging to Xl;“, we have, ,Ir)(d(n)) = s$- l)(ij-

= ij- 10-j.

l@n-j)), 0

3

17p-“(1”-‘())

v,(n-i)(l”) (ii) Any yip- “(6) has auy ‘kze form (8.16), and (8.17), respectively.

(J = 1, 2, . . ., tl =

in-1

.o,

= l”--’ . 1.

I),

(8.15) (8.16) (8.17)

of the right-lzatzd sides of (8.15),

230

TOSIO KlTAGAWA

(iii) To each assigned n t-l) (8.16), and(8.17), we have

form given in the right-hand side of (8.15), o] = _p$,

[&n): n~-l)(&~))

= I’-‘On-j.

[h’“‘; r~-“(#“‘)

= l”_’ . q = (po},

(8.19)

[ij’“‘: n;-“(o’“‘)

= l”-’ . 11 = (I”}.

(8.20)

(8.18)

transition behaviors induced by (iv) Under the n(“-‘) expression, L?B,_, on the n-dimensional state configuration space X,, are reduced to the convergence to I”-’ . 1: 0 (r-’ .o -+ 1 o”-2 . 0 + 12 on-3 . 0 + . . . + 1.61 on-j. (8.21)

1

C

l”-’ . 1 + I”--’ . 0 +_ lo-2 0.0

DYNAMICAL

BEHAVIORS

+ . . . + 1’ on-j-1 . 0.

INDUCED

Let us begin with an observation which has the general form: y$-2’(#“‘)

BY y&,--2

on the possible expression = @n-2)

where t1c2 have the four possibilities PROPOSITION

n$‘-2)(6(“‘)

. &C2,

(8.22)

0 0,O 1, 1 0, and 1 1.

8.2

All the feasible

expressions

of (8.22) are given by the following

n + 2

farms: (1) 1”-2 * 12,

(3) o”-2 * 0 1,

(2) 1”-2 * 1 0,

(4) lnU2 . 02, and (5) 1’ O”-jP2 * O2

(j = 0, I, 2, . . ., n - 3).

(8.23)

Proof. All the n-dimensional state configurations X, = (6,) are divided into the family of n + 1 classes, where the jth class Xp) is defined by x,(j)

forj

F

[a(n)

: a(")

=

1'0

$+-l),~(n-j-l)

E

xn_j_l],

(8.24)

= 0, 1,2, . . ., n - 2 and X,(n -

1) 7 [a(“) : ~9”) = l”-l 0] = {In-r 0},

X,(n) z [#“) : ~3~“)= I”] = {I”}. Let us find the q$‘-2)(6’“‘) expression X,(j) (j = 0, 1,2. . . ., n).

to each a(“) belonging

(0) The class X,(O). We have r~-2’(O#“-” = O”--2.02, )$4)(0

l”-1

= On-2.0

for $“-1) 1.

(8.25) (8.26)

+ I”-1.

to each

(8.27) (8.28)

ON NEURONIC

EQUATIONS

231

(1) The class X,,(l). We have ~$-~)(l

0 cP-~))

= 1 One2 * O*,

for any CP-~) E Xn_*.

(8.29)

(j) The class X,,(j). We have, for any #“-i-1) E X,,_j_l 9r-2)(l.i 0 h@-j-1) 0 = 1.i On-j-z. 02 ,

(8.30)

forj = 2,3,. . .,n - 2. (n - 1) The class X,(n - 1). We have $2)(1”-1 0) zz In-2 . 1 0.

(8.31)

(n) The class X,(n). We have $/V(l”)

(8.32)

PROPOSITION

12.

= l”-2.

8.3

The transition behaviors induced by ZF,,_~ on the n-dimensional state confgure space X, are summarized under r$-‘) expressions by the following reverberation cycle with associated state conjiguration v-2 . O2 attaining to it: 1” f-y-3 .@ --) 12 (y-4.02 --t . . . _.+ 1.i (y-j-2.02 + . . . 'x

77

on-2 . 02 (p-2 . () 1 + I”-2 . 12 + I”-2 . 1 0 + I”-2 . 02 +_ This is immediate D YNAMICAL

(8.33) 02.

from the ylFe2) expression.

BEHA VIORS

Let us proceed

INDUCED

quite similarly

is given by the following

BY

z/7,,_ 3

as in the case Z;“S,_~. The specification = @(n-3). &k2&,

p/;-3)(@))

PROPOSITION

I"--3 0.

of

(8.34)

proposition:

8.4

We have rlk-3'(02

$"-2')

=

p-3.

19

=

on-3

rlf3'(02 rlp-3'(f-J

1 p-29

yp-3'(0

I"--2 0)

rp-3'((-j ul$v3'(1

0

y/$-3)(1 rl(n-3)(1J "

,,-I) p9

0

0

#-i-1))

v$n-3,(1"-2

0

=

on-3

=

f-J"-3.0

=

(y-3

=

,"-2)

#U)

@3'(1"-' q$-3'(1")

0)

03,

=

for

.02

pp--2)

_#

.()3,

.() 12,

for pa

1 O"-4.02

f 14,

In-2,

(8.37) (8.39) (8.40) (8.41)

1, .O

(8.36) (8.38)

1 0,

1 (y-4.03

(8.35)

I"-2

for #n--2)= y-2,’ for (p-2) f I"-3 0,

1,

(8.42)

=

Ij 00-j-3

=

I"-3.

1 02,

(8.43)

=

I"-3.

12 0,

(8.44)

=

I"-3.

13.

(8.45)

(2

sj

5

n

_

3),

232

TOSIO

PROPOSITION

8.5

The transition behaviors configuration

space

induced by 8,-,_ 3 on the n-dimensional

X, are summarized

under

r~jl”-~) expressions

following digraph representation consisting of one reverberation one route converging to One3 . O3 belonging to it: 1 on-4 .02 1 t on-3.

KITAGAWA

03

--f

state by the

cycle witJt

on-3 f 0 1 0

1 on-4.03

3

. . . _+

12

+

lj

(y-j-3.03

+

. . . +

O3 (8.46)

1”-3.

1

r (y-3

. 02

D YNAMICAL

1 +

on-3

. ()

BEHA VIORS

1n-3.

13

INDUCED

+

I”-3

. 12

0

t

I”-3

’ 1 02

BY 9,3,, _ 4

This method is quite similar to those given in above for the operator Yp,,_,. Now it will be sufficient to give the main result. PROPOSITION

8.6

The transition behaviors configuration following

space

induced by P’D~_~ on the n-dimensional

X,, are summarized

digraph representation

three additional

under

consisting

np-4’

expression

of one reverberation

routes each of which converges

state byTe

cycle with

to either of One4 . O4 and

o”-4 .03 I :

1 on-5 . 03 1 c on-4 . 02 1 0 12 U-6, * 03 1 c

/ J (y-4.

1

1 on-5 * 02 1 0

J

04

7 (y-4

. 03

1 +

1 (y-5

. 02

12

+

(y-4

. ()

12

()

J. 1 (y-5

. 04

1

\\ \ \ p-4.02

I 1.l on-.+4

(8.47)

12

\\\ . 04

\

ON NELJRONIC EQUATIONS DYNAMICAL

BEHAVIORS

As in the previous PROPOSITION

233

INDUCED

section,

BY P,,-~

our main result is given by the proposition.

8.7

The transition behaviors induced by 9,-,_, on the n-dimensional state conjiguration space X,, are summarized under the P$-~) expression by the following digraph representation consisting of one reverberation cycle with six additional routes which are classiJed into three classes: (1) ThejYrst class consisting of three routes converging to O”-’ ’ 05. (2) The second class consisting of two routes converging to 0nm5 . O4 1. (3) The third routes converging to OnW5. O3 12, as shown in the following jigure. ,O”-6 .05 + (y-5 .05 + lo"-6 .04, +_ on-5 .03 , 0 t i

7 \\

12 (y-7 .05

\

i2 On-7

a

04

1 +

1

on-6 . 03

1 0

+

On-5 02

02

1

I

‘\

1 13 on-8 .()5

13 (y-8.04

, +_ ,2 on-7 .03

, 0 +

, on-6 . o2

0

I on-5 . 04 , +_ , (y-6. 03

, +_ p-5.

1 o2

.OJl 0

02 12 0

t\ i ,.ip-j-5

. 05

'12

on-7 .‘o" 12 +

, on-6 02 12 0 +

o"-5.01202

1

(8.48) ,"-4 (f.05

on-5 .

03

12 c,

t

I ,"-5 $5

1 ,"-5. , 04 I ,"-5 +,2 03

cjr,-5 . 02 , 3

I In-j. ,3 02 I I"-5 !,4 (J

I ,I+5 .$5 +

on-5.0

14

o"-6o2

13co”-5~o

130

234

TOSIO KITAGAWA

9. REVERBERATION

CYCLES

UNDER

APPLICATIONS

OF 28,

The purpose of this section is to give general observations on transition phenomena induced by applications of the operator 3’s, on the ndimensional state configuration space. Some preparatory considerations are given to show the mutual relationship between the &form and $)-form which we introduced in Sec. 8. &FORM

AND

qWFORM

Any n-dimensional can be denoted by

state

configuration

gcn, = fi

()P’

6”‘) = (an-i,

6n-2, . . ., 6,&J

14i,

(9-l)

i=l

where

iil We observe,

(Pi

+

Pl

2

0,

Pi

2

l

CL

2

0,

9j

2

l

by repeated

4i) = n, (i = 2, . . .) r), (j = 1,2, . . ., r -

applications

yqp)

0

=

Bl

(9.2)

of .496,,

1

Pr

Pl PZ”‘P,+l

qrq1 qz***qr-20

= [

0 Pl + (1 - 1) 4%

...

s,

*.

where we have put q,. = (1 PROPOSITION

91

q2

(9.5)

.

When q,. 2 1, we have, q;(p))

Pr-1 PI *

qr-1

01

Pr

PI 01 ’

(9.7)

1, we have

... +

(9.6)

1) + s,..

...

PI

’

9.2

(1) When 1 5 qr 5 I -

gqr

(9.3) (9.4)

1).

0

p2 * * * pi * *. p,

41

q2

---

4i

* **

(2) When qr 2 I, we have

4r

1[

0

zzz

41

where we have put q, = (1 - 1) + s,. In combination, these two propositions

+

4r

+

PI

P2

* * * Pr-1

P2

. * .

give the following

Pr-1

1, (9.8)

proposition.

ON NEURONIC PROPOSITION

EQUATIONS

235

9.3

To each n-dimensional state configuration c’?“)given in (9.1), there corresponds a set of a nonnegative integer m = m(#“‘) < u, a positive integer s = s(6(“)0), a set of s positive integers {ui}, ui = u,[$“)] (i = 2 , s), and a set of s nonnegative integers {vi}, vi = vi[#“)] (i = 1: 2: : . .) s), such that yy#“))

[

VI

+

(1

u2

-

PHENOMENA

UNDER

+

(I

-

. . .

u3

1) v2 + (I -

Vi

TRANSITION

* . * ui ... US 1

0

=

1)

1) vj + (1 -

. . .

V,

APPLICATIONS

+

(1

1)

’

1)

-

(9.10)

OF “B,

Since we are interested in finding reverberation cycles induced by applications of LZ’B,on the n-dimensional state configuration space, we shall concentrate our observations on the set of the n-dimensional state configurations given by (9.10) at which every n-dimensional state configuration attains after m applications of _!?a, where 0 I m s n - 1. For the sake of convenience, we shall use pi and qi in the place of ui and vi respectively, and we shall be concerned with #0

=

0 P2 Pi PS q1 + (2 1) q2 + (I 1) ’ ’ * 4i + (1 1) * ’ * qs + (2 - 1) [

=

1q1+l-1 ifi

()P’

14i+(l-l)_

(9.11)

Now the yj,‘)(6’“‘) for (9.11) is given by rl”‘(#O) = I’-’ . 14’ opz+u-1) 142 oPz+u--l).

. . 14r-I

n

PROPOSITION

lPF+u-l)

14r

.

Proof. We observe 0 q1 + (Z -

0

[

l -

P 1) q2 + (Z -

4, 1

(9.11)

that

q, + (I -

...

P 1)

Pi qi + (1 =

(9.12)

9.4

Any n-dimensional state conjiguration 6”” having the form belongs to a reverberation cycle of the length 2n - (I - 1).

y4,+U--l) PI

1’

q3

+

(I

-

1)

*.

**’ PI 1) * . * q, + (I P2

1) q2 + (I -

*

1)

P3

1

(9.13) . . .

1) q3 + (1 - 1) * *. Pr-1

qr_l + (1 -

P, 1) 0

1’

236

TOSIO KITAGAWA

pJr+qr+u-

0

I)

81

41 +

Pz

(I

-

1)

q2

+

P3

(1 -

1)

q3

+

pi q2

0 =

P,

+

+

...

(E -

1)

4,

(1 -

1)

q1

+

***

1) - * -

(1 -

Pr

* . * qr +

1)

(1 -

1

(I

1)

-

q,

+

(I

1)

-

P3

*.*

Pr-1

1 *

q3+(z-1)***qr-1+(z-1) The repeated z,*

application

of (9.14) yield us

(Pi+qr+(l-l))

i:

i=l

0

X [ 41

Pz

+(Z

41

cl2

[ 1 - 1 p2 + (1 where, with pl = 0, i$l NOW, starting

(Pi

I)

p3

+

9i

[

qr

0

+

+

Cz

(I -

1)

-

1)

p2

+

p3

+

(2

41 (Z -

4r-

l>>

=

pr

+

1

(9.15)

1

(1 -

(9.16)

n.

side of (9.15), we have

q2

O [ 1 - 1 p2 + (1 =

***

1) . * *

(1 -

with the right-hand 41

2;;

+

1 >“d 1

‘*. P, 1) ” ’ 4, + (I - 1)

*** Pi 1) ” . qi + (1 -

1) q2 + (1 -

-

0

=

***

G-1

1) * * *

-

1)

p3

p3

+

+

p,

+

4r

(Z -

1) .0... .

cl2 (I -

1) 0 pr + G-1(I

1

41

[

x [ qr

0

-I- (1 -

=

q2

1) p2 + (I -

1)

0

qr-l

+

(1

.**

Pr (1 -

1)

q,

+

(I -

4r+ $2 YL?I 0 1 -

applications

1)

41

1) p2 + (Z -

1)

q2 **. p3+(Z-l).*.pr_l+(Z-1) The repeated

1

qr-2

*

of (9.18) yield

(Pi+Qi-l+(l-l))

41

1 p2 + (I -

42

1)

p3

+

(1 -

(9.17)

1

Yr-1

1) . * . p2 + (2 -

-

12

1)

-

_yP,fG-1+(1--l) Bl

x

(9.14)

P2

*.*

1) - * *

pr +

4r-

1

(I

-

qr-2

1)

0

1

(9.1s)

ON NEURONIC

=p, = [

41 +

237

EQUATIONS

0

P2

P3

1) q2 + (I -

(1 -

1)

93

+

(i -

Pr

.**

1) . . * q, + (2 -

1)

1’ (9.19)

which shows, in combination

with (9.15),

C&n-U- “@‘“‘) = d(n),

(9.20)

as we were to prove. TRANSITION COORDINATE

FORMULA SYSTEM

FOR

APPLICATIONS

OF pfi,

UNDER

#‘,

Before we give an qtr) version of Proposition 9.4, it is useful to have the following sets of transition formula for repeated applications of _%‘p, under yl(‘) coordinate system. PROPOSITION

9.5

For anyfeasible

component

<@“Iof r$)(#‘)),

where m = n - (I - 1) - q,

we have fp&‘-1

. pqq

~.&(l’-i

. ~(“#y) = 0-1 . lfy(m)oq-k

= O-1 . (yyoq-li

CJ;,(O’-’ . <‘“‘y?) = o’-1 . ok~(m)(y-k

(0 s k 5 4),

(9.21)

(0 5 k 5 41,

(9.22)

(0 5 k 5 q),

(9.23)

CJ;,(O’- 1 . tb’)Oq) = lkOl- 1 -k . Ok
. ((“)(y?) = I’-1 .

~;:-l)+k(O~-i

(0 5 k i PROPOSITION

l)),

(9.24)

1).

(9.25)

lko1-1/$4oq-Pl)-k

q - (1 -

9.6

For any feasible component fj(“) of $)(6(“)), where m = n - (I - 1) p - q, we have ~jl+f’(ll--l . ~O~9(y’qq)= 1l-l . lr(~-l)oq+J-1((m) cp 2 l>, (9.26) $ps4,+P(ll--l . pqP(y) = 0-1 . ()qy(m), (9.27) s;[+P(Oi-l

. pq)P()4)

=

1’-’

. 1P-wl$)~+(q-q(m)

(p

cJjr+P(()‘-l

. pd

=

(y-1

. ()Pl~l-(I-l)()-lpI)

(q 2 I).

10)

2

Z),

(9.28) (9.29)

It will be of some use to give the proof of Proposition 9.4 by appealing to Proposition 9.5 and hence to Proposition 9.6. It will be sufficient to have a sequence of Ys, transitions such that 16

238 1’-1

TOSIO KITAGAWA . 141

oP2+u-l)14i

fj i=Z

r-l

*

01-l

.

(

11

()Plg@

()Pi+u--l)lqi

()PP+Pl),

i=2

;

I’-’

. lPr@lr+u-l)lm

_:

I’-1.

(yr-llPr()4*+(f-1)

$

II-1

1”, 11-l

>

r-l r-2 JJ

~Pi+~l-l~~~i~P~-I+~I-l~,

r-2 /J2

. lP~-104r-I+(I--1)1PrO(lr+(l-1)14L

.

fi

lPjOlj+(~-l)

141

( j=s 5

()Pi+u-l)lqi,

ig

fi

11-l.

(

lPj()4j+CL--1)

( i=2 141

j=2

P,+u--l)lq*

‘z

>

opi+u-

O

qqi,

,

(9.30)

>

,

> *-I

+

01-1

. 04 IJ

1Pj04j+(lm1)

lPrOllr+(~-1)

( j=2

+

i--l 11-l . Oql+(L--l) jJ12 lPjOPj+(t--l)

(

whose 6”” expression

9

> 14r, >

can be given by l’-l@I

r-1 JJ

lPj+(l--l)Oqj+l

(9.31)

j=l

10. THE FAMILY OF OPERATORS AND THE LINEAR THRESHOLD FUNCTIONS ASSOCIATED WITH THE NEURONIC EQUATION OF A SINGLE NEURON This section gives some ideas on the use of the family of operators ya, _!Z5, _!Z’b,,and ye, (I = 0, 1, 2, . . ., n - 1) for the representation of the linear threshold functions associated with the neuronic equation (1.3) under its general situation. Since a more systematic consideration is scheduled to be given in a future paper [13], we are content to give a few examples for the present. EXAMPLE

IO.1

We are concerned

with the neuronic

x(t + 1) = l[a,x(t) According

to the notation

equation

+ a,x(t

-

1) - l9].

(10.1:

given in Sec. 3, we have

z(6,, 6,) = a&,

+ a,&) - 8,

(10.2

ON NEURONIC

239

EQUATIONS

which yields z(0, 0) = -8,

z(l,O)

= a, - 8,

z(0,1) = a, - 8,

z(1, 1) = a, + a, - 8.

(10.3)

We shall take into consideration all possible combinations of a,, ao, and 0. Regarding the pair a, and a,, there are eight situations such that Case 1:

0 < a, s a, < a, + a,: (0) 1 0 2

CaseII:

0~a,
CaseIII:

a0 < 0 5 a, + a,
Case IV:

a, < a, + a, < 0 5 a,: 0 2(O) I

1

Case V:

a0 + a, < a, 5 a, < 0: 2 0 1 (0)

CaseVI:

a,+~,



210(O)

CaseVII:a,
sO
12(0)0

CaseVIII:al

5 0 < a, + a,
(10.4)

I (0)20,

where 0, 1, and 2 denote ao, ai, and a, + a, respectively, the value 0. Let us consider, for instance, Case II. The value depends upon 8. In fact, we have the following.

while (0) denotes of z(6,, 6,) now

(i) 0eJ, = (-co, 0). Then z(6,, 6,) 2 ~(0, 0) = -6’ > 0 6 = (6,, 6,) in X,. Hence P’(S) reduces to dpo.

for

all

(ii) 0 E Jz = [0,ao).Then ~(0, 0) = -0 5 0, while z(6,, So) 2 ~(1, 0) = a0 - 0 > 0 for all (S,, So) # (0, 0). This implies that F,F, = 1 --f l[z(6,, 6,)] = 0 F,F,

= 0 -+ l[z&,

(10.5)

So)] = 1,

which we may and we shall write (10.6)

gfl(i,Ti) in the sense that the linear threshold side of (1.3) reduces to S?p, provided of S, and 6, respectively.

function defined in the right-hand that we use 8, and 5, in the place

(iii) 0 E J3 = [a,,al).Then z(6,, 1) 2 a, - 6'> 0 for all S, in X,, and z@,, 0) 5 a0 - 8 5 0 for all 6, in X1. That is, l[z(6,, I)] = 1 and l[z(S,, 0)] = 0, which implies that f(6) = ijo and hence is equivalent to Y,,, which we may and we shall write by _%‘pcOj. (iv) 0 E J4 = [a,,a, + al). Then ~(1, 1) = a, + a, - 8 > 0, while that z(d,, SO) 5 a1 - 8 5 0 for all (6,, 6,) # (1, 1). This implies f(6) = 6,6, and is equivalent to L?'~(I,o) in the present notation. (~1 ee.7,

= [a0 + a,, co).

which implies thatf(6)

Then z(6,, So) 5 ~(1, 1) = a, + is equivalent to _Ym.

U, -

0 5

0

240

TOSIO KITAGAWA

Similar considerations can be applied to the seven other cases. In each of the eight cases, there is a division of the whole real axis of 0 into five subintervals {.I,,} (h = 1,2,3,4, 5) in each of which we have the representation of the linear threshold functions by means of LZ~, YG, Zfl,, 9~~ (I = 0, 1) as shown in Table 8 and the auxiliary Fig. 10.1 (a), (b) and (c). We are not going to give any systematic approach to operator representations of l[z@)] for general situation. Nevertheless we have to add one more example in which simple descriptions given in Example 10.1 do not hold, but there is a need for introducing the notion of conditional operator representations of 1[2(6)] for certain cases. TABLE 8 Operator Representations

of Linear Threshold Function Given by (1.3) Subinterval

Case I. II. III. IV. v. VI. VII. VIII.

Jl

(0)102 (0)012 O(O)21 02(0)1 201(O) 210(O) 12(0)0 l(O)20

FIG. 7(a). Operator representations

J3

J4

JS

RI, 0) PO, 0)

m,o)

tm 0) m, 0) m, 0)

RI, 0)

B(l,V

in subinterval Jz of f(S) in each Cases Z - VIII.

ON NEURONIC

241

EQUATIONS

FIG. 7(b). Operator representationin

FIG. 7(c). Operator representation EXAMPLE

subinterval J3 off(S) in each of Cases I -

VIII.

in subinterval J4 off(S) in each of Cases I - VIII.

10.2

Let us consider

the neuronic

x(t + 1) = l[a,x(t) For the present

purpose

equation

+ a,x(t

-

1) + a,x(t

just mentioned,

- 2) - 01.

(10.7)

let us consider

the case when

0 < a2 c a, -c a, + a2 < a, -c a, + a2 < a, + a, < a, + a, + a,,

(10.8)

242

TOSIO KITAGAWA

which defines the subdivision of the whole Q axis into a family of nine subintervals {J,,} (h = 1, 2, . . ., 9). Let us discuss the operator representations in each of these subintervals: (i) 0 E J, = (- co, 0). Then we have 2,. (ii) for all 6,6,& Hence

0 E JZ = 6 # (0, = 0 and we have

[0, Q). Then we have ~(0, 0, 0) 2 0 and z(6,, 6,, 8,) > 0 - - 0, 0) in X,. Since (0, 0, 0) = (1, 1, l), 8,8,F, = 1, and implies z(6,, 6,, 6,) = 0 and z(6,, 6,, 6,) = 1 respectively. Y~(z,J,Q as the operator representation of l[z(6,, 6,, S,)].

(iii) f3EJ~ = [a,, al). Then we have ~(0, 0,O) 5 ~(0,0, 1) = a, - 0 5 0, while z(6,, 6,, 1) 2 z(6,, 6,, 0) 0 for all (6,, 6,) # (0,O) in X,. Hence we have _Y~(i,i). (iv) 8 E J4 = [a,, a, + q). Then we have to distinguish with the case (a) 6, = 1 and the case (b) 6, = 0. In the case (a), ~(1, 6,, 6,) > 0 for all (6,, 6,) in X,, which implies that we have 5Ym under the condition that 6, = 1. We denote this by the notation _Y0:2 as conditional representation. In the case (b), we have ~(0, I, 1) > 0 and ~(0, 6,, S,) s 0 for all (a,, S,) # (I, 1) in X,. This gives us again the conditional operator representation ~~(i,~):~ where Z denotes the condition that 6, = 0. (v) 0 E J, = [a, + az, q,). Then we have ~(1, 6,, 8,) > 0

and

~(0, a,, 6,) 5 0

for all (a,, 6,) in X,. Hence we have J?~(z). (vi) 0 E J6 = [a,, a, + u2). Similarly, as in (iv), we have the conditional operator representations Zrn:~ and Y/5(1,0):2. (vii) 8 E J7 = [a0 + a2, a, + al). Then we have the conditional operator representations _Yaj:z and P’p(r)~. But it is noted that they can be amalgamated as YD(z,~). (viii) 8 E J8 = [a, + a,, a, + a, + a2). Then we have the conditional operator representations Ya,~ and 2 D(~,o)Q. But it is noted that they can be amalgamated as Y~(z,I,o). (ix) 0 E J9 = [a0 + a, + a2, co). Then we have Ya. The results are summarized in Table 9 in which a beautiful duality is shown to exist, as we might expect. Although any systematic approach to the operator representations of linear threshold functions in (1.3) will be discussed in a future paper [13], it is adequate to discuss briefly the family of operators, which we have already discussed in the preceding sections, with regard to the parameter 8. As we have illustrated in Examples 10.1 and 10.2, the change of 0 may induce the change of operator representations. From the standpoint of the functional equation, we have to consider a function O(t) which is

ON NEURONIC

243

EQUATIONS TABLE 9

Operator representations

in each subinterval of l[z(&, &, S,)]

Operator representation

Subinterval

JS

Operator representation

Subinterval

JS

9N2)

assigned from outside to the functional equation (1.3), instead of restricting ourselves to an assigned parameter 8. That is to say, our subsequent interests are more concerned with the functional equations II-1 (10.9) x(t + 1) = 1 c a&t - k) - e(t) [ k=O 0(t) may be any assigned function. It may be periodic or nonperiodic, and it may also converge to 1 or 0. The general features of (10.9) are too complicated to give a deep general observation into the behavior of the solution. In the event that we have started from the simplest situation when d(t) is a constant 0 independent of t, and we must then discuss how the change of 0 may induce the change of operator representations. The discussions may provide preparatory considerations for dealing with the functional equation (10.9). The concurrence function n-l (10.10) z(L 1, dn-2, * ..,a,) = c aki$,-1-k - 8, k=O

1.

with a parameter

0 yields the operator M&l-,,

lk(6,-1, for any 6 (=I) = (C&-i, &-2, M,_, where we have

b-2,. h-2,

representation

_Ygorthat is,

111= 0, * * -3 ~,,O>l = 1, * *,6,

(10.11) (10.12)

. . .) 6,) E Xn_l, if and only if + a,_,

5 8 -c m,_,,

(10.13)

put

M,_,

=

max

(10.14)

I%"-') EXn__I

m,_,

3

min SC"_') EX"_,

(10.15)

244

TOSIO KITAGAWA

Now let us cGYJYJTYJTFYJYJYTJYJonsider the case when a function 0(t) satisfy the cond (10.13) for all t 2 z (an integYTYer), but O(z + 1) is greater than the values given by some state configurationsJ n-2 J ~

Uj~YTJn_1-j

3

O(l- +

l).

(10.16) TY

jtYC) YJ

Then the cTYJhange of operator representation may occur, and in fact the disintegratioYJn(s) of reverberation cycle(s) occurs just at the state configuration(s) wYJhich are occupied just at the time t = z. Transitions of thJe operator representation such as from ~?a,_ 1 to 9’~~ can be discussed iTYn connection with the change of 0(t), as we shall show in Ref. 13. TY REFERENCES TY A. AiJTJello, E. Burattini, and E. R. Caianiello, Synthesis of reverberating neural networks, Kybernetik 7, 191-195 (1970). S. Amari, Characteristics of randomly connected threshold-element networks and networks and network systems, Proc. IEEE 59, 35-47 (1971). S. Amari, Learning of patterns by self-organizing nets of threshold elements, (in Japanese) Densi Tsushin Gakukai, Document EC 71-42 (1971-11). E. R. Caianiello, Outline of a thought-processes and thinking machines, J. Theoret. Biol. 2, 204-235

(1961).

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