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\title{
               \LARGE{THRESHOLD TRANSFORMATIONS\\
                              AND\\
             DYNAMICAL SYSTEMS OF NEURAL NETWORKS}\\
                \large{A COMBINATORIAL APPROACH}\\
               \large{4th Edition (V 4.0)} 
}         

\author{
                           Takao UEDA
}
\maketitle\section*{Preface}
Neural networks have been a hot field in recent years.  One of
the reasons is that they are now used for practical decision
making through computer software and hardware.  However, unlike
widely-used methods such as statistical decision and linear
programming, validity of applying neural networks is uncertain,
since limitations in application, methods of evaluating errors,
and the like are not established.

    Information processing on a neural network is mathematically
summarized as \emph{threshold transformations}.  Because of its
non-linearity, threshold transformations are hard to analyze. 
However, clarifying mathematical properties of threshold
transformations is essential for valid applications.

    Various models of neural networks have been proposed. 
However, according to Amit, they fall basically into two types,
\emph{feed forward networks} and \emph{attractor neural
networks}.  In essence, a simple network of the first type is a
threshold transformation from the state space  $X = \{-1,1\}
^{n}$ into the state space $Y = \{ -1,1\} ^{m}$, and a simple
network of the second type is a threshold transformation into $X
= \{ -1,1\} ^{n}$ to itself, that is, a \emph{threshold
transformation} of $X$.  The first type is mainly concerned with
the function itself regarding the state spaces $X$ and $Y$ as
different.  A network of the second type is concerned with the
\emph{dynamical system} generated by the transformation.  Kamp
and Hasler's book, which contains most prior main results on the
second type, calls it a \emph{recursive neural network}.  I call
it a \emph{dynamical neural network} (DNN).  The DNNs more
closely approximate central nervous systems, and mathematically
more interesting.  But, what is an attractor in a neural network? 
Strangely enough, you can not find its definition in a book of
Amit or Kamp and Hasler.  Amari and others defined some kinds of
attractors called \emph{stability}, but they are too limited to
be established as
standards.  Therefore, I had to start with the groundwork of
defining basic concepts in the finite-state dynamical system
(Chapter 6.1).

    I first came into contact with the classical neural network
model of McCulloch and Pitts in the early 1970s, but did not find
any mathematically significant results, so that I was not much
interested in that.  Then in the mid 70s, I was informed of
Arimoto's theorem (published in 1963) by M. Sato at the applied
mathematics section of a semiannual meeting of the Mathematical
Society of Japan.  When I visited Arimoto at his office in Osaka
University to get a copy of his paper containing the result, he
expressed a negative opinion about the neural network model
because of its excessive simplification of reality and rather
discouraged me to go into that field.  But I had different
criteria, and soon later, got the concept of the variation of a
Boolean transformation and \emph{minimal Boolean
transformations}.  The \emph{variation} of a transformation of
$\{ -1,1\} ^{n}$ is the total number of coordinates of $\{ -1,1\}
^{n}$ changed by the transformation.  A transformation $F$ is
\emph{minimal} if the variation of $F$ is equal or less than the
variation of any transformation $G$ such that $G = TF$ for some
orthogonal transformation $T$.  In particular, the only minimal
orthogonal transformation is the identity $I$.  However, I could
not utilize the concepts for the study of threshold
transformations at that time.

    Without much progress, I interrupted the study of
mathematical science, partly because I was disappointed with my
ability in mathematics and partly because I did not expect I
could ever touch reality through mathematical science.  I left
Japan in 1981 to come to the United States, and studied
psychology, philosophy, and literature in most of the 80s.   

    By the time I restarted the study of threshold
transformations in the late 80s, my notion of reality had
changed.  The environment concerning neural networks had also
changed owing to the spreading of their practical application to
signal processing and various decision making and also owing to
the much publicized work of Hopfield.  Further, I could easily
simulate a model and implement an algorithm, since I had owned a
personal computer for word processing.  Soon, I succeeded in
obtaining several simplest one-to-one threshold  transformations
for the purpose of extracting their non-linear properties, by
introducing [\;]-representations of self-dual transformations
and focusing on the class of minimal circular transformations
(Chapter 4.4).  The [\;]-representations are effectively used
throughout this seminar note.

    The concepts of minimal and circular Boolean transformations
also led me to attack and solve an outstanding hard problem of
constructing a kind of Gray code for necklaces.  The results are
both theoretically and practically significant in the area of
combinatorial Gray code, but outside the scope of this seminar
note.

    For years I had been unable to utilize my results on one-to-
one threshold transformations for the study of neural networks. 
Then just after the New Year's Day of 1995, the idea of
incorporating \emph{spontaneous firing} suddenly came to my mind
with its relation to maximal or minimal threshold
transformations, when I was browsing in Amit's book, Modeling
Brain Function, at a book store.  For me, it was l'oeuf de Colomb
(Columbus's egg).  Amit's book was defective, because it did not
distinguish attractors from limit cycles, but it discussed
maximum, minimum, and average firing rates.   And I recalled
spontaneous firing, which I had almost forgotten but had learned
in an undergraduate course at the University of California at
Santa Cruz that used the first edition of Kalat's text book,
\emph{Biological Psychology}.  

    In my primitive DNNs, the prototype DNN, in which each
neuron is disconnected from each other and performs spontaneous
firing at rate 1/2, is represented by a transformation $-I$,
which is the \emph{maximal} orthogonal transformation.  Thus the
concept of minimal threshold transformations is related to a
primitive  DNN model that incorporates spontaneous firing. In
fact, if $F$ is a minimal threshold transformation, then $-F$
generates a primitive DNN, and an attractor of $F$ is easily
converted to an attractor of $-F$. 

    However, my earlier and primary objective was to clarify non-
linear properties of threshold transformations.  That is why I
concentrated on one-to-one threshold transformations in the first
place.  The readers will find that a main difference
between a linear one-to-one transformation, that is, an
$orthogonal transformation$, of $\{ -1,1\} ^{n}$ and a non-linear
(i.e. non-orthogonal) one-to-one threshold transformation of $\{
-1,1\} ^{n}$ is the selectiveness in the latter.  For
example, in a conventional digital computer, its CPU performs, in
one machine cycle, parallel processing of a basic unary operation
such as one-bit right rotation or complementation of any data
word, which is a member of the set of $n$-bit binary strings
$\mathbf{Q}^{n} = \{0,1\} ^{n}$.  These operations are
\emph{isometries}, which are equivalent to orthogonal
transformations of $\{ -1,1\} ^{n}$.  On the other hand, a CPU
capable of non-linear threshold transformations can perform, in
one machine cycle, a selective rotation or complementation in
that, if $n$ is even, a data word is rotated only when one
arbitrarily predetermined $n$-string or one of those obtained by
rotating it is loaded into the register.  Similarly, a data word
is complemented only when it is one arbitrarily predetermined
$n$-string.

    Studying one-to-one threshold transformations is also good
preparation for neural networks, since not only common methods
can be applied to the two subjects, but also a DNN having an
attractor is often constructed by modifying a one-to-one
threshold transformation.  In fact, an enhanced Arimoto theorem
in Chapter 5.5, and most attractors in Chapter 6 were obtained in
this way. 

    Meanwhile, in the process of seeking for one-to-one threshold
transformations, I sometimes found it harder to prove a given
transformation to be one-to-one than to be a threshold
transformation.  Then I discovered that most one-to-one minimal
threshold transformations are \emph{reflective}.  Further, proof
that a complex threshold transformation is one-to-one is
comparatively simplified by proving it to be reflective.  By a
reflective transformation I mean a one-to-one transformation such
that its inverse is \emph{orthogonally similar} to itself.  This
kind of transformation had already appeared as a binary-reflected
Gray code.  It also includes isometries of $\mathbf{Q}^{n}$, as I
proved in Chapter 3.2.  At present, I have not found any one-to-
one threshold transformation that is incompressible, minimal, and
non-reflective.   By compression, some of the minimal threshold
transformations can be further reduced to simpler ones (Chapter
5.2).

    The readers may be somewhat confused by the use of both $\{ -
1,1\} ^{n}$ and $\mathbf{Q}^{n}$ throughout the seminar note for
the
domain of threshold transformations.  However, this use is by no
means accidental.  Threshold transformations are connected with
two different mathematical structures: the real $n$-space
$\mathbf{R}^{n}$ and the Boolean algebra.  Such concepts as
\emph{linear separability} and orthogonal transformations belong
to the first.  The use of $\{ -1,1\} ^{n}$ is comfortable in this
context.  On the other hand, any Boolean transformations
$\mathbf{Q}^{n}$ can be expressed in compact form using the
\emph{Boolean operations} $\vee$ (OR) and $\cdot$ (AND) in the
\emph{Boolean algebra} $\mathbf{Q}$, so that the use of
$\mathbf{Q}^{n}$ allows us to treat threshold transformations as
a special class of general Boolean transformations.  Moreover,
the Boolean operations $\vee$ and $\cdot$ are basic non-linear
threshold functions from $\mathbf{Q}^{2}$ to $\mathbf{Q}$, so
that threshold transformations can be effectively dealt with in
terms of Boolean operations.  Therefore, we use both $\{ -1,1\}
^{n}$ and $\mathbf{Q}^{n}$ depending on demands.

    The shortcoming that the firing rate of any neuron can not
exceed 2 times the spontaneous firing rate in the primitive DNN
 model on the state space $X = \{ -1,1\} ^{n}$ described in
Chapter 6 is
due to a greater problem that any state $x(t+1)$ depends only on
$x(t)$ for a given \emph{efficacy matrix} $E$ and time $t$. 
Therefore the \emph{postsynaptic potential} $(Ex(t))_{i}$ is
\emph{spatial summation}.  However, the postsynaptic potential
should also include \emph{temporal summation} to simulate real
nervous systems.  Therefore, a primitive DNN model with temporal
summation
and spontaneous firing rate 1/3 is described in Chapter 7.  That
DNN is no longer a dynamical system on $X =\{-1,1\} ^{n}$, and
generated by a threshold function from $X\times X$ to $X$.  The
results are a variety of more interesting attractors, although
exact analysis becomes more tedious.

    The fundamental limitation of the DNNs described in Chapters
6 and 7 is that they are autonomous, that is, the stable periodic
firing patterns that are represented by attractors are completely
determined by the efficacy matrices of synaptic connections and
the initial states. However,  the dynamics of any biological
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.  In
autonomous models, if a minimal attractor consists of more than
one cycle, then there are some fluidity of shifting from one
pattern to another caused by noise, even with a change in firing
rate in some cases.  Therefore, my next step was to formulate 
non-autonomous DNNs, expand the concepts of attractors, and prove
their existence.

    New concepts are necessary with non-autonomous DNNs.  One is
bi-dependence, and another is invariance with the timing of
input.  The bi-dependence means that asymptotic properties of a
DNN are dictated neither by the initial state nor the input, but
dependent on both of them.  The invariance with the timing of
input means that assuming a periodic input sequence, asymptotic
properties are independent of any shift in arrival of the input
sequence.  In my definition, attractors in non-autonomous DNNs
must  satisfy this condition.  In the final Chapter 8, we will
see how autonomous DNNs described in Chapter 6 are modified by
input, so that a non-attractive limit cycle becomes attractive, a
non-unique attractor becomes unique, and an attractor consisting
of more than one cycle becomes an attractive cycle, but still the
convergence to the attractor depends on the initial state.

    It seems that when McCulloch and Pitts originally created a
neural network model, they did not intend to use it for practical
decision making but for the analysis of real neural activities. 
My motives for studying neural networks were first mathematical
and then biological.  The incorporation of spontaneous firing in
my primitive DNN model, a feature distinct from prior models,
served my
these two motives.  However, in the immediate future, the present
results may be used in engineering rather than in biology,
such as for a new computer architecture.  The main reason, I
think, is not that the DNN is "unrealistic", but that currently
available experimental data are either too microcosmic or too
macrocosmic for biological applications of our results.

      A summary of each chapter is described in the
\emph{abstract} at the beginning of the chapter, so that readers
who want to know main results may first refer to the
\emph{abstracts}.  In addition to basic concepts and
notations in the field of discrete mathematics, most of which are
defined in Chapter 1,  I had to introduce various new definitions
and notations.  Symbols and Notations attached in the final pages
will serve the readers for easy reference. \\


Takao Ueda\\     

New York

---Written for the first edition, October 1999\\

I have updated Chapter 6 with newly found attractors, since I
compiled the first edition for the first time.  The volume has
grown too much to be contained in one chapter.  Also,  writing
each time the conversion from $F$ to $-F$ with the attractor
wastes pages and makes less readable, so that, in this second
edition, I have housed attractors of minimal transformations
without the conversion in a separate Chapter 7.\\

---Added for the second edition, February 2001\\


I rewrote Chapter 7, 8, 9 on the common ground of extensive
representations of Boolean
transformations.  Consequently, as Boolean [ ]-representations
are basic tools for the first half part, the integer-valued
extensive representations are now basic tools for analysis of
attraction in the second half part.  I also added Chapter 10 to
illustrate tangible applications.\\


---Added for the third edition, August 2002\\

I revised Chapters 3.4, 4.4, 4.5, 5.2, and 7.1-7.5 to
incorporate a class of one-to-one threshold transformations
having single cycles and derived transformations having
attractors.   Particularly I reorganized Chapter 7 with the new
results.\\

     ---Added for the fourth edition, May 2005.
\pagebreak
\section*{Contents}
\begin{tabular}{ll}
&\textbf{Preface} \\
&\textbf{Chapter 1.  Introduction} \\
1.1 &  Basic notations \\
1.2 & Permutations \\
1.3 & P\'{o}lya actions\\
1.4 & Boolean functions\\
&\textbf{Chapter 2  Representations of basic Boolean
transformations}\\
2.1 & Boolean isometries\\
2.2 & Minimal and maximal transformations\\
2.3 & Self-dual transformations\\
2.4 & [ ]-representations\\
2.5 &  Circular and skew-circular transformations\\
2.6 &  Flow graphs\\
&\textbf{Chapter 3  Reflective transformations}\\ 
3.1 &  Reflective transformations\\
3.2 &  Boolean isometries are reflective\\
3.3 & Reflectiveness for [\;]-representations\\
3.4 & Circular reflective transformations\\
3.5 & Reflectiveness of orbit connections\\
& \textbf{Chapter 4  Threshold functions and transformations}\\
4.1 &  Basic properties of threshold functions\\
4.2 &  Combinatorial properties\\
4.3 & Basic properties of transformations\\
4.4 &  Circular one-to-one transformations \\
4.5 &  Skew-circular one-to-one transformations\\
&\textbf{Chapter 5  Modification of threshold transformations}\\
5.1 & Orbit connection\\
5.2 & Compression and expansion\\
5.3 & Incompressible transformations\\
5.4 &  Compressible transformations\\
5.5 &  An enhanced Arimoto theorem\\
& \textbf{Chapter 6  Dynamical systems of first-order neural
networks}\\
6.1 &  Finite-state dynamical system (FSDS)\\
6.2 &  A critical review of prior results\\
6.3 &  Dynamical neural networks (DNN)\\
6.4 &  PDNN-definable transformations\\
6.5 &  Attractive loops\\
6.6 &  Attractive cycles\\
&\textbf{Chapter 7  Calculus of attraction}\\
7.1 & Extended representations and neighborhood functions\\
7.2 & Multiple attractors\\
7.3 & Multi-cycle attractors\\
7.4 & Single-cycle attractors I\\
7.5 & Single-cycle attractors II\\
7.6 & Attractors derived from polynomials\\
7.7 & Orbit modification\\
7.8 & A temporary review
\end{tabular}

\pagebreak
\begin{tabular}{ll}
&\textbf{Chapter 8  Attractors in second-order neural networks}\\
8.1 & PDNNs of spontaneous firing rate 1/3\\
8.2 & PDNN-definable transformations\\
8.3 & Attractive loops\\
8.4 & Attractive 4-cycles \\
8.5 & Attractive cycles for $n \equiv  2$ mod 6\\
8.6 & Attractive cycles for $n \equiv  0$ or 4 mod 6\\
8.7 &  Attractive cycles for odd n\\
8.8 &  A temporary review\\
& \textbf{Chapter 9 Attractors in Non-autonomous Neural
Networks}\\
9.1 &  Introduction\\
9.2 &  Definition of attractors\\
9.3 & Boolean and extended representations\\
9.4 & Attractive loops\\
9.5 & An attractiveness condition\\
9.6 & Attractors in skew-circular PDNNs\\
9.7 & Attractors in circular PDNNs\\
& \textbf{Chapter 10. Case study: Neural integrators}\\
10.1 & Introduction\\
10.2 & Autonomous DNNs having multiple tonic attractors\\
10.3 & Non-autonomous DNNs having burst input\\
10.4 & Determination of burst patterns\\
10.5 & Circular threshold constructions\\
10.6 & Functions of burst generators\\
10.7 & Generator-integrator feedback system\\
&\textbf{References}\\
&\textbf{Index}\\
&\textbf{Symbols and Notations}\end{tabular}

\end{document}

