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\title{Chapter 6 Dynamical systems of first-order neural
networks}

\begin{abstract}  After basic concepts about finite-state
dynamical systems such as structural stability and attractors
are defined, and prior results are critically reviewed,
primitive dynamical neural networks (PDNNs) are remodelled by
incorporating spontaneous firing and distinguishing
non-neutral invariant sets from spontaneous neutral cycles. 
These PDNNs are a class of McCulloch and Pitts networks
$x(t) = \mathrm{Sgn}(Ex(t-1)-h)$ on $\{-1,1\} ^{n}$ such
that $h$ is the zero vector and all the diagonal elements of
the efficacy matrices $E$ are negative.  PDNN-definable
threshold transformations are characterized in terms of
Boolean functions, and the existence of non-neutral minimal
attractors are proved for the general dimension $n$ by
construction using [\;]-representations of Boolean
transformations.
\end{abstract}
\maketitle

\section*{6.1 Finite-state dynamical system (FSDS)}
Since a dynamical \emph{neural network} (DNN), which is the
subject of this and subsequent chapters, is a \emph{finite-state
dynamical system} (FSDS), we first describe its basic concepts in
the following by modifying the definitions of well-known concepts
about semidynamical systems on a general topological space
(Mathematical Society of Japan, 1987, pp. 487-503). 

 Let $X$ be a finite metric space with an integer-valued distance
$d$. If $S$ is a non-empty subset of $X$ then the $\epsilon
$-neighborhood of $S$,
$U_{\epsilon }S$ for a positive integer $\epsilon $ is defined by
$$
    U_{\epsilon }S = \{ x \mid  d(x,S) \leq \epsilon \}. 
$$
Let $\varphi $ be a mapping from $X\times \mathbf{Z}_{+}$ to
$X$.  For each $t \in \mathbf{Z}_{+}$ a transformation $\varphi
_{t}: X \rightarrow 
X$ is defined by $\varphi
_{t }x = \varphi (x,t)$ for every $x \in  X$.  If $\varphi
_{t}$ satisfies 
\begin{enumerate}
    \item $\varphi _{s}\circ \varphi _{t} = \varphi _{s+t}$
for all $s,t \in \mathbf{Z}_{+}$,
    \item $\varphi _{0} = I_{X}$ (the identity transformation
of $X$),
\end{enumerate}
then $\varphi $ is called a \emph{finite-state dynamical
system} (FSDS) on the \emph{state
space} $X$, whose points are called $states$.  If $F$ is a
transformation of $X$, then $F$
defines a mapping $\varphi : X\times \mathbf{Z}_{+}\rightarrow 
X$ by
$$
        \varphi (x,t) = F^{t}x, x \in  X, t \in 
\mathbf{Z}_{+}. 
$$
Then $\varphi $ is an FSDS on $X$ such that $\varphi _{t} =
F^{t}$ and called the FSDS
\emph{generated by} $F$.  Let $F_{v}$ be a transformation of
$X$ defined for each point
$v$ of an open set $U$ of $\mathbf{R}^{m}$.  If $F = F_{v}$
for some $v \in  U$, then
the FSDS generated by $F$ is called a parametrized FSDS with
its parameter space $U$.  If
$F = F_{v}$ and $F = F_{w}$ for every point $w$ of an
neighborhood of $v$ for a
parametrized FSDS generated by $F$, then the parametrized FSDS
is called
\emph{structurally stable}.

   If $\Psi$ is a set of sequences of $X$, then
$\mathrm{Im}\Psi$, the \emph{image} of $\Psi $, is $\bigcup_{V\in
\Psi }\mathbf{V}$ i.e.the union of the images of the sequences
belonging to  . A sequence $V = (v_{0},v_{1},...)$ is called
\emph{cyclic}, if there exists some $k$ such that $a_{i} = a_{j}$
for every $i$ and $j$ such that $i = j \; \mathrm{mod} \; k$ and
$a_{i} \ne a_{j}$ for every $i$ and $j$ such that $i \ne j \;
\mathrm{mod} \; k$.

     The sequence $(x,Fx,F^{2}x,...)$ is called the \emph{orbit}
starting at $x$ and denoted by $\mathrm{Orb}_{F}x$.  A cyclic
orbit is identified with an element of $\mathrm{CY}(F)$. 
That is one cyclic orbit obtained from another by shifting the
starting point is regarded as the same.  For a subset $S$ of $X$,
$\mathrm{Orb}_{F}S$ is the set of all orbits $\mathrm{Orb}_{F}x$
such that $x \in  S$.  The set $\omega _{F}x$ defined by
\begin{eqnarray*}
     \omega _{F}x = \{ y &\mid &  \mbox{For any}\;  k \in 
\mathbf{\mathbf{Z}}_+, \; \mbox{there
exists}\\
                  && \qquad \mbox{some}\;  t > k \; \mbox{such
that}\;  y = F^{t}x \}
\end{eqnarray*}
is called the \emph{limit set} of $x; \omega _{F}S$ is the
union of all limit sets $\omega
_{F}x$ such that $x \in  S$.  Clearly, for any point $x$,
$\omega _{F}x = \mathbf{C}$ for
a cycle $C \in  \mathrm{CY}(F)$.  This $C$ is called a
\emph{limit cycle} of $x$.  A subset $S$ of $X$ is called
\emph{invariant} if $FS = S$. 
Clearly $S$ is an invariant set if and only if $S =
\mathrm{Im}\Psi $ for some $\Psi \subseteq \mathrm{CY}(F)$.\\

\textbf{Definition 6.1.1}  A subset $\Phi $ of $\mathrm{CY}(F)$
is called \emph{attractive}
or an \emph{attractor} in the FSDS generated by $F$, if there
exists some $\epsilon $-neighborhood $U_{\epsilon
}(\mathrm{Im}\Phi )$ satisfying
\begin{enumerate}
        \item $F(U_{\epsilon }(\mathrm{Im}\Phi )) \subseteq 
U_{\epsilon}(\mathrm{Im}\Phi )$;
        \item $\omega _{F}(U_{\epsilon }(\mathrm{Im}\Phi )) =
\mathrm{Im}\Phi $.
\end{enumerate}

In particular, if $\Phi $ consists of one cycle, the cycle is
called attractive.
$\mathrm{CY}(F)$ is clearly an attractor.  Also, if $\Psi $
and $\Phi $ are both attractors,
then $\Psi \cup \Phi $ is also an attractor.  Therefore, an
attractor is called a \emph{minimal attractor}, if no proper
subset of it is an attractor. 
Further, an attractor $\Phi $ is called \emph{connected}, if
$\Phi  = \Psi \cup \Upsilon$, $\Psi \cap \Upsilon  = \emptyset $,
then $\mathrm{min}_{x\in \mathrm{Im}\Phi ,y\in
\mathbf{\Upsilon }}d(x,y) = 1$; otherwise, called
\emph{disconnected}.  The \emph{basin} for an attractor $\Phi
$ is the set of all points $x$ such that $F^{ k }x \in 
\mathrm{Im}\Phi $ for some $k$.  Two attractors $\Psi $ and
$\Phi $ are called \emph{separated} if $d(x,y) \geq  2$ for
any points $x$ of the basin for $\Psi $ and any point $y$ of the
basin for $\Phi $.\\

\textbf{Definition 6.1.2}  A subset $\Phi $ of $\mathrm{CY}(F)$
is called \emph{strong attractor} in the FSDS generated by $F$,
if there
exists some $\epsilon $-neighborhood $U_{\epsilon
}(\mathrm{Im}\Phi )$ satisfying
$$
     F(U_{\epsilon }(\mathrm{Im}\Phi )) =\mathrm{Im}\Phi .
$$
           
A strong attractor is clearly an attractor.


\section*{6.2 A Critical review of prior results\label{CRPR}}
In a nervous system, each neuron exhibits an impulse of one
electric state, called \emph{action potential}.  Therefore, the
state of each neuron can be distinguished by the existence and
nonexistence of an action potential.  Assuming the nervous system
consists of n neurons, we can identify each neuron with an
element of $\mathbf{N} = \{ 1,2,...,n\}$.  Then the state of the
whole
nervous system at time $t$ is expressed by a point $x(t) =
(x_{1}(t),x_{2}(t),...,x_{n}(t)) \in  \{ -1,1\} ^{n}$.  

    Let $(i,j)$ denote a synapse, where $i$ and $j$ are
integers of $\mathbf{N}$, neuron $i$ being the
\emph{postsynaptic} neuron and neuron $j$ being the
\emph{presynaptic} neuron.  Let $E$
be a real $n\times n$ matrix and $h$ be a real column
$n$-vector, where each element $E_{ij}$ expresses the
\emph{efficacy} of the synapse $(i,j)$ and $h_{i}$ expresses the
threshold value for the action potential of neuron $i$.  Then
the classical neural network model of McCulloch and Pitts (1943)
asserts that the state $x(t+1)$ is defined by the state
$x(t)$ and $E$ as follows: 
$$
\begin{array}{lll}
       Fx &=& \mathrm{Sgn}(Ex-h), \\
        x(t+1) &=& F(x(t)),
\end{array}
\eqno (6.2.1) 
$$
where
$$
(\mathrm{Sgn}(y))_{i} =  \left\{
      \begin{array}{ll}
           1 & \mbox{if $y_{i} > 0$,} \\
        -1 & \mbox{if $y_{i} \leq  0$},
       \end{array} 
            \right.
$$
Further, $\{ -1,1\} ^{n}$ is a finite metric space with the
integer-valued \emph{Hamming distance} $d_{H}(x, y)  = |\{
i\mid  x_{i} \ne  y_{i}\}|$, where $|S|$ denotes the number of
elements of the set $S$.  Therefore,
$x(0), x(1), ...$ is the orbit starting at $x(0)$, in the FSDS on
the state space $\{ -1,1\}
^{n}$ generated by the threshold transformation $F$ of $\{ -1,1\}
^{n}$.

    This discrete-time binary system is now called an
artificial neural network and is not
regarded as a representation of the activities of biological
neurons, the history of studies is
partly responsible for this claim.  However, a continuous-time
and continuous state-model is
not manageable for the description of dynamics of a large
population of neurons, so that
some abstraction to the level of (6.2.1), such as putting the
firing mechanism in a black box
and aligning action potentials, is more or less inevitable.\\

A first breakthrough was made in the early 60s by Arimoto
(1963).\\  

\textbf{Theorem 6.2.1} (Arimoto 1963)  For any $k \leq  2^{n}$,
there exists a network (6.2.1) having a $k$-cycle.\\

Later, the following Goles-Chacc's theorem  appeared
(Goles-Chacc, 1980; see also Goles
\& Olivos, 1981; Goles-Chacc et al., 1985, Proposition 2, p.
269).\\

\textbf{Theorem 6.2.2} (Goles-Chacc, 1980)  If the matrix $E$ in
(6.2.1) is symmetric, then
any limit cycle is either a loop or a 2-cycle.

\begin{proof}  Let the function $\beta : \{ -1,1\} ^{n}\times
\{ -1,1\} ^{n} \rightarrow 
\mathbf{R}$ be defined by 
$$
        \beta (x,y) = -x^{T}Ey+(x^{T}+y^{T})h, 
$$
and let $\gamma : \mathbf{Z}^{+} \rightarrow  \mathbf{R}$ be
defined
by
$$
        \gamma t = \beta (x(t),x(t-1)).
$$
Then
$$
     \gamma (t+1)-\gamma t =
-(x^{T}(t+1)-x^{T}(t-1))(Ex(t)-h),
$$
since $E$ is symmetric.  Since $\{ -1,1\} ^{n}$ is a finite
set, we can assume that $(Ex-h)_{i} \ne  0$ for every $x$ and
every $i$ in (6.2.1) by adjusting $h$.  Therefore,
$x^{T}_{i}(t+1)(Ex(t)-h)_{i} > 0$ for every $i$.  Therefore,
if $x_{i}(t+1) \ne  x_{i}(t-1)$ for some $i$, then either
$$
    x_{i}(t+1) > 0, -x_{i}(t-1) > 0, \; \mbox{and}\;
((Ex(t)-h))_{i} > 0,
$$
or
$$
   x_{i}(t+1) < 0, -x_{i}(t-1) < 0, \; \mbox{and}\;
((Ex(t)-h))_{i} < 0.
$$
Therefore, either
$$
 x(t+1) = x(t-1) \; \mbox{and}\;  \gamma (t+1)-\gamma t = 0
$$
or
$$
 x(t+1) \ne  x(t-1) \; \mbox{and}\;  \gamma (t+1)-\gamma t < 0,
$$
which proves that $x(t+1) = x(t-1)$ for any point $x(t-1)$ and
$x(t+1)$ on any limit cycle.
\end{proof}

However, the condition of the symmetry has no justification in
biological nervous systems. 
Neither Arimoto's theorem nor Goles-Chacc's theorem was
universally recognized, and
Hopfield (1982) replaced (6.2.1) with $n$ recursive equations
by which $(x(t))_{i}$ is
obtained one by one for $i = 1,...,n;$ for example,
$$
    x_{i}(t+1) =
F_{i}(x_{1}(t+1),..,x_{i-1}(t+1),x_{i}(t),..,x_{n}(t)).
$$
In this serial model, if $E$ is symmetric, $h = o$, where $o$
is the column vector whose
every coordinate is 0, and $E_{ii} = 0$ for every $i$, then any
limit cycle is a loop.  This
result, which is stronger than Goles-Chacc's, was obtained by
sacrificing the parallel
operation defined by (6.2.1) and therefore a clear departure
from the biological root.  Note
that if a state remains on a loop, then each neuron either
continues to fire at every unit time
or continues not to fire at all.  That is too extreme a case
in biological networks.  Also, from a technological point, loops
represent a very
limited amount of information.  Instea,
non-loop attractors should be the main targets.   

     Prior results concerning attractors are limited to
attractive fixed points (Amari, 1972; 
Robert, 1986; Cottrell, 1988; Blum, 1990; etc.).  First of
all, there has been some confusion
and limitation about the concept of attractors, so that no
standard definition of attractors has been established. 
For example, in spite of its subtitle, "the world of attractor
neural networks,"
Amit (1989) does not give a definition of
"attractor, and takes limit cycles for attractors (P.77).
 Kamp \& Hasler(1990, pp. 6-7) and others define a basin of
attraction
or radius of attraction for a fixed point without defining an
attractor.
Earlier, Amari (1972) called a fixed point q stable if there
exists $U_{\epsilon }q,
\epsilon \geq  1$, such that $F(U_{\epsilon }q) = q$.
More recently, Cottrell (1988) called a fixed point $q$ a
$k$-attractor
 if $Fx \in  U_{k-1}q$ for any point $x$ such that $d_{H}(q,x) =
k$.
These definitions are too limited to be established as standards
(see Example 6.3.6 in the next section).  In fact, these fixed
points are here called strong
attractors (Definition 6.1.2).
On the other hand, Robert (1986) obtained a
necessary and sufficient condition for $U_{1}q$ to satisfies (i)
and (ii)
of our Definition 6.1.1 for a fixed point $q$ (described in Kamp
\& Hasler, 1990).   

        According to Definition 6.1.1, the set of all cycles
$\mathrm{CY}(F)$ is an attractor (a
loop is a 1-cycle).  Therefore, if the transformation $F$ of
(6.2.1) is one-to-one, e.g., if $E$
is the identity matrix $I$ and $h = o$, then we obtain a
minimal attractor $\Phi $ such that
$\mathrm{Im}\Phi  = \{ -1,1\} ^{n}$.  If $E = I$ and $h_{i} =
-2$ for every $i$, $F$
has the unique loop $(11..1)$, so that there exists an $F$
having an attractive loop for every
$n$.  Further, if $E_{ij} = -1$ for every $i,j$ and $h = o$
for $n \geq  3$, then $11..1
\leftrightarrow  -1-1 ..-1$ is an attractive 2-cycle.  Example
6.3.4 (b) in the next section shows
such an attractor for $n = 2$.  Therefore, there exists an $F$
having an attractive 2-cycle
for every $n \geq  2$.   However, beyond these results and
those obtained by direct
products, it is not clear whether there exist other
attractors.  Nevertheless, the enhanced
Arimoto theorem (Theorem 5.5.2 of Chapter 5) implies the
following Arimoto-Ueda theorem.  The proof is
clear by letting $\epsilon  = n$ in Definition 6.1.1.\\  

\textbf{Theorem 6.2.3}  For any $k \leq  2^{n}$
there exists a McCulloch and
Pitts network (6.2.1) that has an attractive unique $k$-cycle.\\

The network of the enhanced Arimoto theorem has a powerful
convergence property, since
any state converges to a unique cycle regardless of the
initial state.  However, in the real
world,  it seems often rather desirable that the convergence
depends on the initial state.

    In relation to this dependence on the initial state, one
feature that has not been
incorporated in prior studies is the elementary but essential
fact that \emph{in many neurons
the postsynaptic potentials merely modify spontaneous firing
that occurs without any synaptic
input} (Kalat, 1995, p. 63).  As a result, prior studies have
failed to define what the
spontaneous or neutral activities of a single neuron or a
population of neurons are.  As a
result, we have not been able to distinguish neurons'
significant activities from their
insignificant activities.  Particularly, we have not been able
to sort out a great number of
loops or 2-cycles often appearing in the network (6.2.1).

\section*{6.3 Dynamical neural networks
(DNNs)\label{RDNN}}
In the current situations described in the last section, we
remodel and characterize dynamical
systems of neural networks that are amiable to global analysis of
population dynamics and still
retain some of the essential features of biological networks. 
The remodelled networks are not
new ones but a restricted class of the McCulloch and Pitts
model (6.2.1).  I call them \emph{primitive dynamical neural
networks}  (\emph{PDNNs}).
They are dynamic, because they are not only dynamical systems but
also each neuron
performs neutral spontaneous firing at rate 1/2 in their
prototype.  They are primitive,
because they are generated by single threshold transformations
like the McCulloch and Pitts
model and also because the threshold vectors are zeros.

Our main objective is to prove the existence of previously
unknown non-loop attractors.
Let us start with the following assumption.\\

\textbf{Assumption 6.3.1 }  The periodic firing of the action
potentials with period 2 of any neuron is neutral.\\

Assumption 6.3.1 claims that the periodic state transition $...
\rightarrow  -1 \rightarrow  1
\rightarrow  -1 \rightarrow  1 \rightarrow  -1 \rightarrow 
...$ of any neuron is neutral or
indifferent whether it is disconnected or connected with other
neurons.  In general, any 2-cycle $q \leftrightarrow  -q$ in $\{
-1,1\} ^{n}$ represents a
neutral activity.  I call it a neutral cycle and any other cycles
significant.
In this case, the generating transformation $F$ can be defined
by
$$
        E = -I, \; h = o,
$$
in (6.2.1), where $I$ is the $n\times n$ identity matrix, and
$o$ is the column $n$-vector
whose every coordinate is 0.  Here every neuron fires
spontaneously without any input from
other neurons, and any state is on a neutral cycle.  Now I make
the
following definition.\\

\textbf{Definition 6.3.2}  A \emph{primitive dynamical neural
network} (PDNN) is a
parametrized FSDS on $\{ -1,1\} ^{n}$ with parameters  $v =
(v_{12},..,v_{1n},v_{21},v_{23},..,v_{2n},..,v_{n1},..,v_{nn-1})
\in 
\mathbf{R}^{n(n-1)}$ and generated by the threshold
transformation $F$ defined by
$$
        Fx = \mathrm{Sgn}(Ex),
\eqno (6.3.1) 
$$
where $E = E_{v}$ is an $n\times n$ real matrix such that
$E_{ii} = - 1$ for every $i$
and $E_{ij} = v_{ij}$ for every other $i$ and $j$.  The $n$ is
called the \emph{dimension} of the DNN.\\

Thus in PDNNs, the prototype DNN generated by  $-I$ is modified
by further
synaptic connections. However, the threshold vector h remains o,
and the synaptic efficacy
Eii between each neuron itself remains -1.   
Here, the assumption $h = o$ is a strong constraint.  However,
if $h = o$, then $E_{ii} =
-1$  for every $i$ is equivalent to $E_{ii} < 0$ for every
$i$.  As a result, such an extreme
case as $E = I$ and $h = o$ is effectively excluded from
PDNNs.
    In a supposed biological system from which the current
model is abstracted, each neuron
is self-oscillatory.  Therefore, a network in which a neuron
is not self-oscillatory but
performs "self-sustained" firing (central pattern generation)
by means of post synaptic
rebound and inhibitory input from surrounding neurons is not
covered here (see Coombes \&
Doole, 1996).  The time period between $t = i$ and $t = i+1$
is assumed here to be the
sum of the time for the action potential and the absolute
refractory period.  Then the relative
refractory period and spontaneous release of inhibitory
chemical transmitters contribute to the
negativity of diagonal elements.

     If a neuron is spontaneously firing in this model, then
the firing rate per unit time is
$1/2$.  On the other hand, the maximum firing rate is 1 and
the minimum firing rate 0.  
Therefore, the firing rate of any neuron cannot exceed 2 times
the spontaneous firing rate. 
Alternatively, we can construct a DNN of spontaneous firing
rate $1/3$ or less.  For
example, the periodic state transition, $\cdot \cdot \cdot
\rightarrow  -1 \rightarrow  -1
\rightarrow  1 \rightarrow  -1 \rightarrow  -1 \rightarrow  1
\rightarrow \cdot \cdot \cdot $,
is neutral in a DNN of spontaneous firing rate $1/3$.   A
state $x(t+1)$ depends on $x(t)$
and $x(t-1)$ in this DNN.  Therefore, a DNN of spontaneous
firing rate $1/3$ or less can
incorporate \emph{temporal summation} in its postsynaptic
potential, while the postsynaptic
potential $(Ex(t))_{i}$ of a PDNN of spontaneous firing rate
$1/2$ is only \emph{spatial
summation}.  Therefore, such an alternative model far better
reflects a real nervous system. 
Further, any real nervous system is not autonomous.  The
dynamics of the system depends
on information that changes at every unit time and that is
input from the outside of the
system, from neurons of other nervous systems and/or from
external stimulus.  Still further,
the rigid synchronization (alignment) of firing for all
neurons is unrealistic.  However,
mathematical analysis will be extremely difficult in such DNN
models that possibly simulate
real nervous systems.  Therefore, as a first step, we have to
limit our subject to an extremely
simplified model with spontaneous firing rate $1/2$ and does
not claim any immediate
application to real nervous systems except a preparation for
the analysis of such DNN
models.  

    If transformations $F$ and $G$ of $\{ -1,1\} ^{n}$ are
orthogonally similar, then the
DNNs generated by $F$ and $G$ are called \emph{orthogonally
similar}.  A transformation
that generates a PDNN is not necessarily self-dual.  However,
we have the following proposition.\\

\textbf{Proposition 6.3.3}  The PDNN defined by Definition 6.3.2
is
\emph{structurally stable}, if
and only if $F$ is self-dual. 

\begin{proof}  Assume that the PDNN defined by Definition 6.3.2
is structurally stable.  Then
$(F-)x = F(-x) = \mathrm{Sgn}(E(-x)) =
\mathrm{Sgn}(-(Ex))$ for every $x$.  Also
there is no $i$ such that $(Ex)_{i} = 0$ for some $x$;
therefore $\mathrm{Sgn}(-(Ex)) = -
\mathrm{Sgn}(Ex) = -Fx$, so that $F- = -F$, that is $F$
is self-dual.  Conversely,
assume $F$ is self-dual.  Then there is no $i$ such that
$(Ex)_{i} = 0$ for some $x$. 
Therefore there exists a neighborhood $U_{\epsilon }v$ such
that if $w \in  U_{\epsilon }v$,
then $\mathrm{Sgn}(E_{w}x) = \mathrm{Sgn}(Ex)$ for every $x$. 
Therefore the PDNN is
structurally stable.
\end{proof}

The set of all significant cycles in $\mathrm{GRAPH}(F)$ is
denoted by $\mathrm{SCY}(F)$.  We are concerned with an attractor
that is
a subset of $\mathrm{SCY}(F)$, since any neural activities that
are
significant and less sensitive to disturbance such as failures in
synchronization are described
with such a significant attractor. Further, a desirable
attractor would be minimal and
connected.  The following examples illustrate a variety in the
present PDNN model.\\

\textbf{Example 6.3.4}  Complete picture for dimension 2. 
Orthogonally non-similar transformations are described below.
$$
E =
 \begin{bmatrix}
   -1 & \epsilon \\
   \delta & -1
 \end{bmatrix},
$$
\begin{enumerate}
\item  $\epsilon ,\delta  < -1$:
$$
    11 \leftrightarrow  -1-1
\eqno \mbox{(Neutral cycle),}
$$
$$
    1-1\partial ,  -11\partial 
\eqno\mbox{(Non-attractive loops).}
$$
$\mathrm{SCY}(F) = \{ (1-1), (-11)\}$ is not attractive.
\item $\epsilon  < -1, -1 < \delta  < 1$:
$$
1-1 \rightarrow  11 \leftrightarrow  -1-1 \leftarrow  -11
\eqno \mbox{ (Attractive neutral cycle).}
$$
\item $\epsilon  < -1, 1 < \delta $:
$$
11 \rightarrow  -11 \rightarrow  -1-1 \rightarrow  1-1
\rightarrow  11
\eqno\mbox{ (Attractive 4-cycle).}  
$$
\item $-1 < \epsilon , \delta  < 1$:
$$
11 \leftrightarrow  -1-1, \; 1-1 \leftrightarrow  -11
\eqno\mbox{ (Neutral cycles).}
$$
\item $\epsilon  = -1$ or 1, or $\delta  = -1$ or 1: 
Structurally unstable. 
\end{enumerate}

\textbf{Example 6.3.5}
$$
E =
 \begin{bmatrix}
   -1 & 2 & 2 & 2 \\
   2 & -1 & -2 & 2 \\
   2 & -2 & -1 & 2 \\
   2 & 2 & 2 & -1 
 \end{bmatrix},
$$
$$
\begin{array}{llllll}
    1-111 & \rightarrow  & 1111\partial ,& -1 1-1-1&
\rightarrow & -1-1-1-1\partial  \\
   & \nearrow  &       &         & \nearrow &\\ 
    11-11 &   &       & -1-1 1-1&  &  
\end{array}
\eqno
\mbox{(Non-attractive loops),}
$$
$$
    1-11-1 \leftrightarrow  -1-111, \;  -11-11
\leftrightarrow  11-1-1
\eqno
\mbox{(Non-attractive 2-cycles),}
$$
$$
\begin{array}{lllllll}
    111-1& \rightarrow & 1-1-11& \leftrightarrow & -111-1&
\leftarrow &  -1-1-11\\
         & \nearrow &       &  &    & \nwarrow &\\
    -1111&  &       &  &       & &    1-1-1-1
\end{array}
\eqno
\mbox{(Neutral 2-cycle).}
$$
$\mathrm{SCY}(F) = \{ (1111), (-1-1-1-1), (1-11-1,-1-111),
(-11-11, 11-1-1)\}$ is not attractive.\\

\textbf{Example 6.3.6}
$$
E =
 \begin{bmatrix}
   -1 & 2 & 2 & 2 \\
   4 & -1 & 2 & 2 \\
   2 & 2 & -1 & 2 \\
   2 & 2 & 2 & -1 
\end{bmatrix},
$$
$$
\begin{array}{lllll}
 1-1-1-1& \rightarrow & -11-1-1&  &\\
        &  &        & \searrow     &\\
        &  & -1-11-1& \rightarrow & -1-1-1-1\partial\\ 
        &  &        & \nearrow &\\
        &  & -1-1-11&  &
\end{array}
\eqno
\mbox{(Attractive loop),}
$$
$$
\begin{array}{lllll}
   -1111& \rightarrow & 1-111& &\\ 
        &  &      & \searrow &\\  
        &  & 11-11& \rightarrow & 1111\partial\\ 
        &  &      & \nearrow &\\ 
        &  & 111-1&  &     
\end{array}
\eqno
\mbox{(Attractive loop),}
$$

$$
    11-1-1 \leftrightarrow  -1-111, 1-11-1 \leftrightarrow 
-11-11, 1-1-11 \leftrightarrow  -
111-1
\eqno \mbox{(Neutral cycles).}
$$
$(-1-1-1-1)$ and $(1111)$ are neither stable in Amari (1972)
nor a $k$-attractor in Cottrell (1988).\\

\textbf{Example 6.3.7}
$$
E =
 \begin{bmatrix}
   -1 & -2 & 2 & 2 \\
   2 & -1 & -2 & 2 \\
   2 & 2 & -1 & 2 \\
   2 & -2 & 2 & -1 
\end{bmatrix},
$$
$$
\begin{array}{lllllll}
  -1111& \rightarrow &  1-11-1&  & 1-111&  &       \\
       &  &        & \searrow &      & \searrow &       \\  
       &  & -1-11-1& \rightarrow & 1-1-11& \rightarrow &
1111\partial  \\ 
       &  &        & \nearrow &       &  &       \\ 
  111-1& \rightarrow & -1-111&   &       &  &      
\end{array}
\eqno
\mbox{(Non-attractive loop),}
$$
$$
\begin{array}{lllllll}
1-1-1-1& \rightarrow & -11-11&  & -11-1-1&  &\\
        &  &      & \searrow &       & \searrow &\\  
        &  & 11-11& \rightarrow & -111-1& \rightarrow &
-1-1-1-1\partial  \\ 
        &  &      & \nearrow &       &  &\\ 
 -1-1-11& \rightarrow & 11-1-1&   &       &  &      
\end{array}
\eqno
\mbox{(Non-attractive loop),}
$$
$\mathrm{SCY}(F) = \{ (1111), (-1-1-1-1)\}$ is a disconnected
minimal attractor.

\section*{6.4 PDNN-definable transformations\label{DDT}}
In this section we characterize the PDNN-definable
transformations for spontaneous firing rate 1/2 defined by:\\

\textbf{Definition 6.4.1}  If a transformation $F$ of $\{ -1,1\}
^{n}$ can be defined by $Fx =
\mathrm{Sgn}(Ex)$ for an $n\times n$ real matrix $E$ such that
$E_{ii} = -1$, then $F$ is
called DNN-definable.\\

\textbf{Theorem 6.4.2}  A self-dual threshold transformation $F$
of $\{ -1,1\}$ is PDNN-
definable if and only if $\mathrm{Var}(i^{-}F) \leq 
\mathrm{Var}(F)$ for every $i$. 

\begin{proof}  Let $F$ be a self-dual threshold transformation
of $\{ -1,1\} ^{n}$.  Then by
Proposition 4.3.1 of Chapter 4, there exists an $n\times n$ real
matrix $W$ such that $Fx =
\mathrm{Sgn}(Wx)$, and there is no point $x$ such that
$(Wx)_{i} = 0$ for some $i$.  If
$W_{ii} = 0$ for some $i$, then replacing $W_{ii}$ with some
$\epsilon _{ii}$ such that
$|\epsilon _{ii}|$ is sufficiently small does not
change $\mathrm{Sgn}(Wx)$. 
Therefore, we obtain a matrix $W\prime $ such that $W\prime
_{ii} \ne  0$ for every $i$
and $\mathrm{Sgn}(W\prime x) = \mathrm{Sgn}(Wx)$.  Next,
divide the ith row of
$W\prime $ by $|W\prime _{ii}|$ to obtain
$W\prime\prime $.  Suppose
$W\prime\prime _{ii} = 1$ for some $i$, and
$\mathrm{Sgn}((W\prime\prime x)_{i}) =
\mathrm{Sgn}((W\prime\prime i^{-}x)_{i})$ for every $x$, then
replacing $W\prime\prime
_{ii} = 1$ with $-1$ does not change
$\mathrm{Sgn}((W\prime\prime x)_{i})$ for every
$x$.

    Suppose $F$ is not PDNN-definable.  Then we obtain a matrix
$W$ such that $Fx =
\mathrm{Sgn}(Wx)$, $W_{ii} = 1$ for some $i$, and $(Wq)_{i} >
0$ and $(W(i^{-
}q))_{i} < 0$ for some $q$.  Therefore, $(W(-i^{-}q))_{i} >
0$.  Thus $d_{H}((q,-i^{-}q)
= n-1$, and both $q$ and $-i^{-}q$ are on the same side of the
hyper plane of
$\mathbf{R}^{n}$ defined by $(Wx)_{i} = 0$.  Further, since
$W_{ii} = 1$, we have
$q_{i} = 1$ and so $(-i^{-}q)_{i} = 1$.  Therefore, for any $r
\in  \{ -1,1\} ^{n}$ such
that $r_{1} = 1$, at least one of $r$ and  $-(i^{-}r)$ must be
on the same side of the hyper
plane as $q$.  Therefore the set $\{ x \mid  x_{i} = 1, and
(Fx)_{i} = 1, x \in  (-
1,1)^{n}\}$ contains at least $2^{n-2}+1$.  Therefore,
$\mathrm{Var}(i^{-}F) >
\mathrm{Var}(F)$.

    Suppose $F$ is PDNN-definable.  Then, by the above result,
there exists a real matrix $E$
such that $Fx = \mathrm{Sgn}(Ex)$ for every $x$ and $E_{ii} =
-1$ for every $i$.  Let
$E_{i}$ be the $i$th row-vector of $E$ and $E\prime _{i}$ be
the row-vector obtained by
replacing $-1$ with 1 for the $i$th coordinate of $E_{i}$. 
Then
\begin{eqnarray*}
   & & |\{ x \mid  x_{i} = 1, (E_{i},x) < 0, x \in  \{
-1,1\} ^{n}\}| \\
     &\geq & |\{ x \mid  x_{i} = 1, (E\prime _{i},x) < 0,
x \in  \{ -1,1\} ^{n}\}| \\
     &=& |\{ x \mid  x_{i} = 1, (-E_{i},x) < 0,  x \in  \{
-1,1\} ^{n}\}|,
\end{eqnarray*}
where $(\cdot , \cdot )$ denotes the inner product.  Therefore
$\mathrm{Var}(i^{-}F) \leq 
\mathrm{Var}(F)$ for every $I$.
\end{proof}

\textbf{Corollary 6.4.3}  Let $F$ be a self-dual threshold
transformation of $\{ -1,1\} ^{n}$. 
Then $F$ is orthogonally equivalent to a PDNN-definable
threshold transformation of $\{ -
1,1\} ^{n}$.
 
\begin{proof}  Let $F$ be a self-dual threshold transformation
of $\{ -1,1\} ^{n}$.  Let $J
= \{ i \mid  i \in  \mathbf{N}, \mathrm{Var}(i^{-}F) >
\mathrm{Var}(F)\}$.  If $G = J^{-}F$,
then $\mathrm{Var}(i^{-}G) \leq  \mathrm{Var}(G)$ for every
$i$.  Therefore, $G$ is
PDNN-definable.
\end{proof}

\textbf{Corollary 6.4.4}  If $F$ is a self-dual threshold
transformation such that $(Fx)_{i} =
x_{i}$  for every $x$ for some $i$, then $F$ is not
PDNN-definable.\\

\textbf{Corollary 6.4.5}  Let $T$ be an orthogonal
transformation of $\{ -1,1\} ^{n}$.  Then
$T$ is PDNN-definable if and only if for every $i$ there exists
some $x$ depending on $i$
such that $(Tx)_{i} \ne  x_{i}$.

\begin{proof}  Let $D$ be the matrix representing $T$, i.e.,
$Tx = Dx$ for every $x$.  If
$(Tx)_{i} = x_{i}$ for every $x$ for some $i$, that is,
$D_{ii} = 1$, then
$\mathrm{Var}(i^{-}T) > \mathrm{Var}(T)$, so that $T$ is not
PDNN-definable.  Let
$D_{ii} = 0$ or $-1$ for every $i$.  If $D_{ii} = 0$ for some
$i$, then replacing $D_{ii}$ with some $\epsilon _{ii} < 0$  such
that $|\epsilon _{ii}|$ is sufficiently small does not change
$\mathrm{Sgn}(Dx)$. 
Therefore we obtain a matrix $D\prime $ such that $D\prime _{ii}
= -1$ for every $i$ and $\mathrm{Sgn}(D\prime x)
= Dx$, so that $T$ is PDNN-definable.
\end{proof}


\textbf{Proposition 6.4.6} If $F$ is PDNN-definable, and $G$ is
orthogonally similar to $F$,
then $G$ is also PDNN-definable.

\begin{proof}  Let $F$ be PDNN-definable and $G$ be a threshold
transformation similar to
$F$.  By definition, $Fx = \mathrm{Sgn}(Ex)$, $E_{ii} = -1$
for every $i$, and there
exists an orthogonal transformation $T$ of $\{ -1,1\} ^{n}$
such that $G = T^{-1}FT$. 
Let $D$ be the matrix representing $T$.  Then $Gx =
T^{-1}(FTx)= D^{-
1}\mathrm{Sgn}(EDx) = \mathrm{Sgn}(D^{-1}EDx)$.  It is clear
that $(D^{-1}ED)_{ii}
= -1$ for every $i$.  Therefore, $G$ is PDNN-definable.
\end{proof}    

If a transformation $H$ of $\{ -1,1\} ^{n}$ is the direct
product of transformations $F$ of
$\{ -1,1\} ^{m}$ and $G$ of $\{ -1,1\} ^{n-m}$, i.e.,
$$
H\begin{bmatrix}x\\y\end{bmatrix}
 =\begin{bmatrix}Fx\\Gy\end{bmatrix}
$$
then clearly $H$ is PDNN-definable if and only if both $F$ and
$G$ are PDNN-definable.  In
this case, we call the PDNN generated by $H$ the \emph{direct
product} of the PDNN
generated by $F$ and the PDNN generated by $G$.

We now switch from transformations of $\{ -1,1\} ^{n}$  to
corresponding transformations
of $\mathbf{Q}^{n}$. First, the following proposition immediately
follows from Theorem 6.4.2.\\  

\textbf{Proposition 6.4.7}  A self-dual transformation
$[f_{1},...,f_{n}]$ of
$\mathbf{Q}^{n}$ is PDNN-definable, if and only if $|f_{i}| \geq 
2^{n-2}$ for every $i$. $\langle f\rangle$ is PDNN-definable iff
$|f| \geq  2^{n-2}$.
$\langle\langle f\rangle\rangle$ is PDNN-definable iff $|f| \geq 
2^{n-2}$.\\

\textbf{Corollary 6.4.8} If F is a minimal threshold
transformation,
 then $\bar{\neg}F$ is PDNN-definable.\\

\section*{6.5  Attractive loops}

The following proposition indicates basic properties of
attractive loops of the present PDNNs. 
In particular, Proposition 6.5.1 (a) implies that there is no
connected attractor consisting of
more than two loops.\\

\textbf{Proposition 6.5.1}  Let $F$ be a PDNN-definable self-dual
transformation of
$\mathbf{Q}^{n}$.  (a)  If $(q)$ and $(r)$ are different loops
of $F$, then $d_{H}(q, r)
\geq  2$.  (b)  If $Fx = q$ for every $x$ such that
$d_{H}(x,q) = 1$, then $(q)$ is an
attractive loop.

\begin{proof}  (a)  Without loss of generality let $q = l$ and
$r = 1^{-}l$, where $l =
1\cdot \cdot \cdot 1$, and $(q)$ and $(r)$ be loops of a
PDNN-definable transformation $F$
of $\mathbf{Q}^{n}$.  If $F = [f_{1}, f_{2},...,f_{n}]$, then
$1^{-}F = [p_{1}\cdot \neg
f_{1},f_{2},...,f_{n}]$ by Proposition 2.4.1 of Chapter 2. 
Therefore $q \in  x_{1}\cdot \neg
f_{1}$ and $\bar{\neg}r = 1^{-}o \in  p_{1}\cdot \neg f_{1}$,
since
$(q)$ and $(r)$ are loops
of $F$.   Therefore, as the proof of Theorem 6.4.2, $p_{1}\cdot
\neg f_{1}$ contains at least
$2^{n-2}+1$ points.  Therefore $\mathrm{Var}(i^{-}F) >
\mathrm{Var}(F)$, which
contradicts the fact that $F$ is PDNN-definable.
(b).  Let $F$ be a PDNN-definable transformation of 
$\{ -1,1\}^{n}$ defined by $Fx = \mathrm{Sgn}(Ex)$ for every
$x \in  \{ -1,1\}^{n}$. 
Without loss of generality let $q = l$, and let $F(i^{-}q) =
q$ for every $i$.  Then
$E(i^{-}q) > o$ for $i = 1,2,..,n$.  Therefore, adding these
$n$ inequalities, we obtain 
$(n-2)E(q) > o$, so that $Fq = q$.
\end{proof}
 
 A PDNN is called symmetric if the matrix $E$ is symmetric in
Definition 6.3.2.  From
Goles-Chacc's theorem (Theorem 6.2.1) it follows.\\

\textbf{Theorem 6.5.2} $\mathrm{SCY}(F)$ of any symmetric PDNN
generated by $F$ is either empty or consists of loops or/and
2-cycles.\\

The above theorem means that in symmetric PDNNs, any neuron
ultimately either performs a
spontaneous firing or continues to fire at the maximum rate 1
or does not fire at all.\\

    Now, in order to find an attractor in symmetric PDNNs,
consider the simple PDNN generated by $F = \mathrm{Sgn}(Ex)$,
where
$$
E =
\begin{bmatrix}
-1 & \epsilon & \cdot & \cdot & \cdot & \cdot & \epsilon \\
\epsilon & -1 & \epsilon &\cdot & \cdot & \cdot & \epsilon \\ 
\epsilon & \epsilon & -1 & \epsilon & \cdot & \cdot & \epsilon\\ 
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\epsilon & \cdot & \cdot & \cdot & \cdot & \epsilon & -1
\end{bmatrix},
\eqno (6.5.1)  
$$
            
Let the transformation of $\mathbf{Q}^{n}$ corresponding to $F$
be $G =
[g_{1},...,g_{n}]$.  Let $d(x)$ be the density of $x$, i.e. $d(x)
= |\{ i \mid x_{i} = 1\}|$.  Then
$(Fx)_{i} = \mathrm{Sgn}((2d(x)-n-1)\epsilon  - 1)$, if $x_{i}
= 1$.  Therefore, $x \in 
g_{i}$ if and only if $x_{i} = 1$ and $(2d(x)-n-1)\epsilon  -
1 < 0$. 

    First let $\epsilon  > 0$.  Then, $x \in  g_{i}$ iff
($x_{i} = 1$, $d(x) \leq  n$,  and
$\epsilon  < 1/(n-1))$, or ($x_{i} = 1$, $d(x) \leq  n-1$,
$\epsilon  < 1/(n-3))$ or
($x_{i} = 1$, $d(x) \leq  n-2$, and $\epsilon  < 1/(n-5)$), or
so forth.  Therefore, if $0 < \epsilon  < 1/(n-1)$ then $G =
\langle p_{1}\rangle$ and this PDNN is
generated by $\bar{\neg}$.  If $1/(n-1) < \epsilon  < 1/(n-3)$,
then
$$
G = \langle p_{1}\cdot S_{1}\{\neg p_{2},..,\neg p_{n}\}\rangle ,
\eqno (6.5.2)  
$$
and
$(l)$ and $(o)$ are loops.  If $1/(n-3) < \epsilon  < 1/(n-5)$
then
$$
G = \langle p_{1}\cdot S_{2}\{\neg p_{2},..,\neg p_{n}\}\rangle ,
\eqno (6.5.3)  
$$
and $(l)$ and $(o)$ are attractors, $GU_{1}l = l$ and
$GU_{1}o = o$, and $U_{1}l$ and $U_{1}o$ are
respectively the basins for the
attractors.  If $1/(n-5) < \epsilon  < 1/(n-7)$ then
$$
G = \langle p_{1}\cdot S_{3}\{\neg p_{2},..,\neg p_{n}\}\rangle ,
\eqno (6.5.4)  
$$
and $U_{2}l$ and $U_{2}o$ are the basins for the attractors, and
so forth. This sequence ends, when $1/(n-(n-1)) < \epsilon$ for
even $n$, where
$$
G = \langle p_{1}\cdot S_{n/2}\{\neg p_2,..,\neg p_n\}\rangle ,
\eqno (6.5.5)  
$$
and $U_{n/2-1}l$ and $U_{n/2-1}o$ are the basins for the
attractors $(l)$ and $(o)$.  The sequence ends, when $1/(n-(n-2))
= 1/2 < \epsilon$  for odd $n$, where
$$
G = \langle p_1\cdot S_{(n-1)/2}\{\neg p_2,..,\neg p_n\}\rangle ,
\eqno (6.5.6)
$$
and $U_{(n-1)/2-1}l$ and $U_{(n-1)/2-1}o$ are the basins for the
attractors $(l)$ and $(o)$. 

Further, points outside the basins for the
attractors are on neutral cycles.  In particular, if $d(x) = n/2$
or $d(x) = (n+1)/2$, then $Gx = \bar{\neg}x$, so that any point
with density $n/2$ or $(n+1)/2$ is always on a neutral cycle.

    Next, let $\epsilon  < 0$.  Then $Gl = \bar{\neg}l$.  If $Gx
\ne  \bar{\neg}x$, $Gx \ne  l$,
and $Gx \ne  \bar{\neg}l$, then $Gx = x$, so that
$(2d(x)-n+1)\epsilon  + 1 < 0$ and
$(2d(x)-n-1)\epsilon  - 1 > 0$.  From the former inequality
follows $2d(x) \geq  n$, and from the latter follows $2d(x) \leq 
n$, so that $d(x) = n/2$. 
Therefore, if $\omega _{G}x = \mathbf{V}$ for some $V \in
\mathrm{SCY}(G)$, then
$d(x) = n/2$, so that $\mathrm{SCY}(G)$ is not attractive.  From
the above arguments, we have obtained:\\

\textbf{Theorem 6.5.3}  If $n \geq  4$ and $\epsilon  > 1/(n-3)$
in the symmetric PDNN
generated by $F$ defined by (6.5.1), then $\mathrm{SCY}(G)$
consists of two separate
attractive loops $(l)$ and $(o)$.  If $\epsilon  < 1/(n-3)$,
then $\mathrm{SCY}(G)$ is either empty or not attractive.\\

Note that all PDNNs and their attractors have their isometrically
similar
counterparts.  The following corollaries are obtained by
construction
with direct products.\\

\textbf{Corollary 6.5.4}  There exists a symmetric PDNN generated
by $F$ of dimension $n
= rm$ for every $r \geq  4$ and every positive integer $m$
such that $\mathrm{SCY}(F)$
consists of $2^{m}$ loops, each being attractive.\\

\textbf{Corollary 6.5.5}  If $n \geq  5$,  then there exists a
symmetric PDNN of dimension $n$ that
has a significant attractive 2-cycle.\\


As illustrated in Example 6.3.5, even if a symmetric PDNN
generated by $F$ is irreducible,
i.e., not a direct product, $\mathrm{SCY}(F)$ may contain a
significant 2-cycle.  While
each neuron fires at either maximum or minimum rate when state
on a loop is attained,
some neurons fires at maximum or minimum rate and the other
neurons perform neutral
firing when a non-neutral 2-cycle is attained.  Therefore, as
long as non-neutral neurons are
concerned, there is no difference between a non-neutral
2-cycle and a loop in smaller
dimension.

\section*{6.6  Attractive cycles}
PDNNs which are circular and symmetric at the same time were
completely described by
Theorem 6.5.3 in the last section.  In this section we show the
existence of non-loop attractors in circular PDNNs. First, the
following proposition is a basis for constructing a PDNN with
having an attractor.\\

\textbf{Proposition 6.6.1}  Let $F$ be a self-dual
transformation of $\mathbf{Q}^{n}$,
$\Phi $ be an attractor in the FSDS generated by $F$, and
$\Phi $ be self-dual, i.e. $\bar{\neg}\mathrm{Im}\Phi  =
\mathrm{Im}\Phi $.  Then
$\Psi  =
\mathrm{CY}(\bar{\neg}F|\mathrm{Im}\Phi )$, where
$|\mathrm{Im}\Phi$ denotes
the restriction to $\mathrm{Im}\Phi$, is an attractor in the
FSDS on $\mathbf{Q}^{n}$ generated by $\bar{\neg}F$.

\begin{proof}  Since $\Phi $ is attractive, there exists a
neighborhood $U_{\epsilon
}(\mathrm{Im}\Phi )$ such that $F(U_{\epsilon
}(\mathrm{Im}\Phi )) \subseteq 
U_{\epsilon }(\mathrm{Im}\Phi )$, and $F^{k}(U_{\epsilon
}(\mathrm{Im}\Phi )) =
\mathrm{Im}\Phi $ for some positive integer $k$.  Therefore,
$(\bar{\neg}F)(U_{\epsilon
}(\mathrm{Im}\Phi )) \subseteq  \bar{\neg}(U_{\epsilon
}(\mathrm{Im}\Phi )) =
U_{\epsilon }(\mathrm{Im}\Phi )$.  Since $F$ is self-dual,
$(\bar{\neg}F)^{k} = \bar{\neg}^{k}F^{k}$,  Therefore,
$(\bar{\neg}F)^{k}(U_{\epsilon
}(\mathrm{Im}\Phi )) = \bar{\neg}^{k}F^{k}(U_{\epsilon
}(\mathrm{Im}\Phi )) =
\bar{\neg}^{k}\mathrm{Im}\Phi  =
\mathrm{Im}\Phi $.  Hence, $\Psi $ is an attractor in the
FSDS on $\mathbf{Q}^{n}$
generated by $\bar{\neg}F$, since $\mathrm{Im}\Psi  =
\mathrm{Im}\Phi $.
\end{proof}

According to Proposition 6.5.1, if $F$ is a minimal threshold
transformation, then $\bar{\neg}F$ is PDNN-definable.  Therefore,
if we have a minimal threshold transformation having an
attractor, then we can obtain a PDNN having an attractor by
Proposition 6.6.1.The 4-cycle attractors and in the following
Theorem 6.6.2 and two-3-cycle attractors in
Theorem 6.6.4 are respectively constructed by modifying Examples
4.4.2 and 4.4.3 of Chapter 4, which are one-to-one
transformations.   
In the following, $\mathrm{Orb}_{\langle \bar{\neg},
\rho\rangle}D$ for a subset $D$ of $\mathbf{Q}^{n}$ is denoted by
$[D]$.\\

\textbf{Theorem 6.6.2}  There exists a circular PDNN generated by
$F$ of dimension $4m$
for any $m \geq  1$ such that $\mathrm{SCY}(F)$ is a connected
minimal attractor
consisting of one 4-cycle.

\begin{proof}  Let $F = \langle f \rangle $ be the circular
self-dual transformation of
$\mathbf{Q}^{4m}$ defined by
$$
        f = p_{1}\cdot p_{2}\cdot \neg p_{4}\cdot p_{6}\cdot
\neg p_{8}\cdot \cdot \cdot
p_{4m-2}\cdot \neg p_{4m}.
$$
Clearly $f$ is a threshold function, so that $F$ is a
threshold transformation.  Let $a =
11001100\cdot \cdot \cdot 1100$ and $A = \mathrm{Orb}_{\rho }a$. 
Then $A = (a,\rho
a,\rho ^{2}a,\rho ^{3}a) \in  \mathrm{CY}(F)$, since $Fa = \rho
a$.

First, we show that $F(\mathrm{Car} F) \subseteq \mathbf{A}$. 
Let $x \in f$.  Clearly,
$x_{4k+2} = 1$ and $x_{4k+4} = 0$ for every $k$.  If $x_{4k+1} =
1$, then $\rho^{-4k}x \in f$; if $x_{4k+1} = 0$, then
$\rho^{-4k}x \notin
\bar{\neg}f$, since $x_{4k} =
0$.  Therefore, $(Fx)_{4k+1} = 0$.  We have $x_{4k+2} = 1$, so
that $(\rho^{-(4k+1)}x)_{1} = 1$, but $x_{1} = 1$, so that
$(\rho^{-(4k+1)}x)_{4(m-k-1)+4} = 1$. 
Therefore, $\rho^{-(4k+1)}x \notin f$, so that $(Fx)_{4k+2} = 1$.
If $x_{4k+3} = 1$,
then $\rho^{-(4k+2)}x \notin f$, since $(\rho^{-(4k+2)}x)_{4m} =
1$; if $x_{4k+3} =
0$, then $\rho^{-(4k+2)}x \in \bar{\neg}f$.  Therefore,
$(Fx)_{4k+3} = 1$. We have
$x_{4k+4} = 0$, so that $(\rho^{-(4k+3)}x)_1 = 0$, but $x_1 = 1$,
so that $(\rho^{-(4k+3)}x)_{4(m-k-1)+2} = 1$.  Therefore,
$\rho^{-(4k+3)}x \notin \bar{\neg}f$, so that $(Fx)_{4k+2} = 0$.
  Therefore, $Fx = \rho a$. Therefore, $Ff \subseteq \mathbf{A}$,
 so that $F(\mathrm{Car}F) \subseteq \mathbf{A}$,
since $\mathrm{Car}F = [f]$ and $[\mathbf{A}] = \mathbf{A}$.

     Next, $U_1\mathbf{A} \subseteq \mathrm{Car}F$.  In fact, if
$x = 1^{-}a$, then $x \in \rho^{4}f$.
  If $x = 2^{-}a$, then $x \in  \bar{\neg}\rho f$. If $x =
3^{-}a$,
 then $x \in  f$.  If $x = 4^{-}a$, then $x \in \rho^{3}f$.
   Therefore, because of the circularity and
self-duality of $F$, $U_1\mathbf{A} \subseteq \mathrm{Car}F$.

    The above arguments have shown that $A$ is an attractor
and the only non-loop cycle in the FSDS generated by $F$.  By
Proposition 6.5.1, $\bar{\neg}F$ is PDNN-definable.
  By Proposition 6.6.1, $\mathrm{CY}(\bar{\neg}F|\mathbf{A})$ is
also an attractor in the PDNN. Clearly it
consists
of one 4-cycle.  It is also
$\mathrm{SCY}(\bar{\neg}F)$ and a connected minimal attractor.
\end{proof}

It is clear from the above proof, a flow graph of $F$ in Theorem
6.6.2
is
$$
   [f] \rightarrow [\neg 1\cdot 2\cdot 3\neg 4\cdot \neg 5\cdot
6\cdot 7\neg 8\cdot \cdot \cdot
\neg (4m-3)\cdot (4m-2)\cdot (4m-1)\cdot \neg 4m]\partial .
$$

\textbf{Example 6.6.3} $\mathrm{GRAPH}( \bar{\neg}F)$ in Theorem
6.6.2 for $m = 1$. 
$$
 0000 \leftrightarrow 1111, \qquad 0101 \leftrightarrow 1010,
$$
$$
\begin{array}{lllllll}
        0010& & & &   &   &   1110 \\
          & \searrow & &    &   & \swarrow &         \\
          0111& \rightarrow &  1100& \rightarrow & 1001&
\leftarrow & 0100 \\
          &   &   \uparrow  &   & \downarrow &  &      \\
       0001& \rightarrow & 0110& \leftarrow & 0011& \leftarrow
& 1101  \\
           & \nearrow &    &    &   & \nwarrow &    \\
       1011&  &    &   &    &  & 1000 
\end{array}
$$

\textbf{Theorem 6.6.4}  There exists a circular PDNN generated by
$F$ of dimension $3m$
for any $m \geq  2$ such that $\mathrm{SCY}(F)$ is a connected
minimal attractor
consisting of two 3-cycles.

\begin{proof}  Let $F = \langle f \rangle $ be the circular
self-dual transformation of
$\mathbf{Q}^{n}$ defined by
$$
  f = p_{1}\cdot p_{2}\cdot p_{5}\cdot p_{8}\cdot p_{11}\cdot
\cdot \cdot p_{3m-1}
S_{m-1}\{ \neg p_{3},\neg p_{4},\neg p_{6},\neg p_{7},..,\neg
p_{3m-3},\neg p_{3m-2},\neg p_{3m}\}. 
$$
Clearly $f$ is a threshold function, so that $F$ is a
threshold transformation.  Let $a =
110110\cdot \cdot \cdot 110$ and $A = \mathrm{Orb}_{\rho
^{-1}\bar{\neg}}a$.  Then $A =
(a,\rho ^{-1}\bar{\neg}a,..,(\rho ^{-1}\bar{\neg})^{5}a) \in 
\mathrm{CY}(F)$, since $Fa = \rho
^{-1}\bar{\neg}a$.  We will show $F(U_{1}\mathbf{A}) \subseteq 
\mathbf{A}$.  In fact,
$F(1^{-}a) = \rho ^{-1}\bar{\neg}a$, $F(2^{-}a) = a$, and
$F(3^{-}a) = \rho ^{-1}\bar{\neg}a$. 
Therefore, because of the circularity and self-duality of $F$,
$F(U_{1}\mathbf{A})
\subseteq  \mathbf{A}$. 

    It is shown in the following that $Fx \in  \mathbf{A}$ or
$F^{2}x \in  \mathbf{A}$ for every $x \in  f$.\\
(Case 1) $x_{3}\cdot x_{6}\cdot \cdot \cdot x_{3(m-1)}\cdot
x_{3m} = 1$:  Since at least $m-1$ coordinates of $x$ are 0,
$x_{3k-2} = 0$ for every $2 \leq  k \leq  m$, so that $x
= 1^{-}\rho a$.  Therefore, $Fx = \bar{\neg}a \in  \mathbf{A}$.\\ 
(Case 2) $x_{3}\cdot x_{6}\cdot \cdot \cdot x_{3(m-1)}\cdot
x_{3m} = 0$:  Then $Fx = 01x_{3}01x_{6}01\cdot \cdot \cdot
x_{3(m-1)}01x_{3m}$.  Clearly $(F^{2}x)_{3k-2} = 0$ for every
$k$.  Since $x_{3k} = 0$ for at least one $k$,
$(F^{2}x)_{3k-1} = 1$ for every $1 \leq  k \leq  m$.  If
$(Fx)_{3k} = x_{3k} = 1$ for some $1 \leq  k \leq  m$ then
$Fx \notin \rho ^{3k-1}f$, so that $(F^{2}x)_{3k} = 1$.  If
$(Fx)_{3k} = x_{3k} = 0$ for some $1 \leq  k \leq  m$, then
$\bar{\neg}(Fx) \in \rho ^{3k-1}f$, so that $(F^{2}x)_{3k}
= 1$.  Therefore, $F^{2}x = \rho a \in  \mathbf{A}$.  

    The above arguments have shown that $A$ is an attractor
and a unique non-loop cycle in
the FSDS generated by $F$.  By Proposition 6.5.1 $\bar{\neg}F$ is
PDNN-definable.  By Proposition 6.6.1,
$\mathrm{CY}(\bar{\neg}F|\mathbf{A})$ is also
an attractor in the PDNN.  Clearly it consists of two 3-cycle.
It is also $\mathrm{SCY}(\bar{\neg}F)$ and a connected
minimal attractor.
\end{proof}

\end{document}