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%---------------------------------IEEEtran.sty---------------------
%
% Style file `IEEEtran.sty' created by Gerry Murray, June 1992, gmurray@ieeepub.org
% The sample file (supplied) is based on this `IEEEtran.sty'.
% The purpose of this style file is spelled out in the title.
% If, for some reason you're unable to see the title, then here it is again:
%
% This Is A Sample File Using The `IEEEtran.sty',\\ To Help You Estimate
% Your Page Count And Facilitate Input-Processing Of Your Compuscript
% 
% You`ll notice that it is very, very simple and there is an
% important reason behind this. 
% As an author, you should primarily concern yourself with
% the CONTENT of the compuscript, leaving the FORMAT to the IEEE.
% We have sophisticated software, in house and different to LATEX,
% which we use to typeset transaction papers. 
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% Have Fun.

\endinput

-----------------------------Sample-----------------------------

% Style file `IEEEtran.sty' created by Gerry Murray, June 1992,
% gmurray@ieeepub.org
%
% The sample file (supplied) is based on this `IEEEtran.sty'.
% The purpose of this style file is spelled out in the title.
% If, for some reason you're unable to see the title, then here it is
% again:
%
% This Is A Sample File Using The `IEEEtran.sty',\\ To Help You Estimate
% Your Page Count And Facilitate Input-Processing Of Your Compuscript
% 
% You`ll notice that it is very, very simple and there is an
% important reason behind this. 
% As an author, you should primarily concern yourself with
% the CONTENT of the compuscript, leaving the FORMAT to the IEEE.
% We have sophisticated software, in house and different to LATEX,
% which we use to typeset transaction papers. 
% The single most important reason for making this style file 
% available was due to the fact that many authors (using TEX
% and LATEX) were supplying their compuscripts using style files 
% that were SINGLE COLUMN PAGE WIDE. It was obvious from the coding
% that many had spent time inserting commands that made the 
% displayed equations `pretty', spreading them right across the page.
% But of course, IEEE transactions are DOUBLE COLUMN, with the 
% column width being 21 pica. Thus, we ended up deleting these
% typesetting `niceties' and breaking the page wide displayed
% equations so that they would FIT inside the style-required 21pc. 
% By using this style file (and by all means feel free to modify it)
% you'll find that the displayed equations, algorithms, nomenclature
% lists, etc. that you supply to us (on disk) will undergo minimal
% change.
% Obviously, if you have used macros to facilitate the 
% keyboarding of your compuscript, PLEASE supply them on the disk.
% And last but not least, it's wonderful to receive a `.bib' file
% containing the hundreds of references in your database file,
% but what we REALLY need is the `.bbl' file, which is the
% reference list that will be seen when your paper is published.
%
% ***************** E-MAILING YOUR FILE TO US ******************
% Great! ...just ask the editor (responsible for the transaction)
% to give you an e-mail address (if you don't already know it).
% **************************************************************



\documentstyle[IEEEtran]{article}    % Specifies the document style.

%
\hyphenation{compu-script }
\begin{document}
\bibliography{plain}
\nocite{*}

\title{This Is A Sample File Using The `IEEEtran.sty',\\ To Help You
Estimate Your Page Count And Facilitate Input-Processing Of Your
Compuscript} 
\author{ERB, Woody, Pheff, Bont, Tranman, IP, Dalton, Christine and OOZ}


\maketitle               
\begin{abstracttext}
The theoretical analysis and derivation of artificial neural systems
consist 
essentially 
of manipulating symbolic mathematical objects according to certain 
mathematical and biological knowledge. A simple observation has been
made 
that this work can be done more efficiently with computer assistance
by using and extending methods and systems of symbolic computation. 
In this paper, after presenting the mathematical characteristics of
neural 
systems and a brief review on Liapunov stability theory, we present some
features and 
capabilities of existing systems and our extension
for manipulating objects occurring in the analysis of neural systems. 
Then, some strategies and a toolkit developed in MACSYMA for computer
aided 
analysis and derivation are described. A concrete example is given to 
demonstrate the derivation of a hybrid neural system, i.e.\ a system
which in its 
learning rule combines elements of supervised and unsupervised learning. 
The future work and directions on this topic are indicated.
\end{abstracttext}
\begin{keywords}
CA system, computer aided analysis and derivation,  
Liapunov function, neural system, symbolic computation.
\end{keywords}




\auffil{This would be where the author affiliation would be, together
with the IEEE log Number, like so.\\ 
The research of D.\ Wang is supported by a grant from SIEMENS AG Munich.
He is with the Research Institute for Symbolic Computation,
Johannes Kepler University, A-4040 Linz, Austria. 

$^{~}~~$B.\ Sch\"urmann is with the Corporate Research and Development,
ZFE IS INF 2, Siemens AG, 8000 M\"unchen 83, Germany.}
\bigskip
\noindent    
\centerline{\sc I. Introduction}

\smallskip
\noindent
SINCE the early 1940s a large number of artificial neural 
systems have been 
proposed by neural scientists. The dynamical behavior of these systems
may 
be mathematically described by sets of coupled equations like
differential 
equations for formal neurons with graded response. The investigation of
essential features of neural systems such as stability and 
adaptation depends strongly upon the state of the mathematical theory to
be 
applied and on a concrete and efficient analysis of dynamical equations.
Unlike abstract theoretical research in which the mathematical objects 
adopted are frequently assumed to be of certain canonical form, 
the neurodynamics is usually complicated due to various biological facts
which should be taken account of to a degree as large as possible. 
Consequently, this makes the analysis and derivation very complex, 
sometimes to an extent which is beyond human capacity, and 
the traditional methods and tools of mathematics are not always
sufficient. 
It is therefore proposed in [19] to use and extend
the methods and software systems of symbolic 
computation for handling, analyzing and constructing neurodynamics and
its 
related objects. The present paper is the continuation of our work in
this 
direction. The attempt is to demonstrate how symbolic computation can be 
applied to aid the analysis and derivation of neural systems. 

In contrast to the approximative character of numerical calculations,
symbolic 
computation treats objects with semantics 
like functions, formulae and programs. A variety of software systems for 
performing symbolic computation have been developed for research and
applications in natural and technical sciences. However, the existing
systems cannot be directly used for the analysis and
derivation of neural systems as the operations on the occurring objects,
particularly those involving an unspecified number of arguments like
indefinite summations,
have not yet been taken into account. To achieve our goal, some 
rules for differentiating and integrating indefinite summations with
respect 
to indexed variables were proposed [20].
A toolkit has been designed and implemented in MACSYMA for manipulating 
these objects occurring in the analysis and derivation of neural systems 
[21]. 

In the next section, we introduce the general method 
and techniques for the stability analysis of 
artificial neural systems. The role of symbolic computation for
representing 
and manipulating the objects concerning neural systems is discussed in 
Section III. In Section IV we present some strategies for using 
computer algebra (CA)
systems and their extension to analyse the stability of neural 
systems and to derive novel stable systems. A brief description of a
toolkit 
developed in MACSYMA is also provided.
A concrete example is given in Section V to illustrate the derivation
of a hybrid model by our toolkit. Section VI contains a discussion on
future 
developments. The paper is closed with a brief summary.
\bigskip
\noindent    
\centerline{\sc II. Stability Analysis of Neural Systems}

\smallskip
\noindent
Consider artificial neural systems which are described by coupled
systems of 
differential equations of the form 
$$\stackrel{.}{x}=F(x, w, K)\eqno(1)$$
and
$$\stackrel{.}{w}=G(x, w, K)\eqno(2)$$

\medskip
\noindent
where $x=(x_1(t), ..., x_n(t))$ is the activation state vector, 
$w=(w_{ij}(t))$ is the weight matrix of dimension $n\times n$, $n$ is
the 
number of nodes and $K$ is an external time-independent pattern vector. 
Such systems of
differential equations which describe the neural model will occasionally 
be named {\em neurodynamics}.

Once a neural model is proposed, its main features are 
represented by its dynamic behavior. The adaptability of the latter 
is often suggested by experimental results in neurophysiological
research. 
>From a mathematical and theoretical point of view, the essential
question 
of the dynamic behavior as discussed in [7] is as follows. 

\medskip
\noindent
{\bf Question 1:}
{\em Under what conditions does the neural system (1)-(2) always
approach 
an equilibrium point in response to an arbitrary, but sustained, input
pattern?}

\medskip
In a satisfactory analysis of this question, the behavior of the neural
system in response to arbitrary initial data, an arbitrary sustained
input 
pattern, and an arbitrary choice of system parameters should be
provided. 
Also an account of how many equilibrium points exist and of how they are 
approached through time is desirable. 

In addition, while an individual equilibrium point of (1)-(2) is
specified, 
many questions concerning its local behavior could arise. For example,
we may ask 

\medskip
\noindent
{\bf  Question 2:} 
{\em Is a given equilibrium point stable, and if so,
what is its domain of attraction and how to characterize the
trajectories 
of the system near the equilibrium point?}

\medskip
The mathematical analysis of the first question is called a global 
stability analysis,
whereas that of the second is called a local stability analysis. 
Since the 1960s, a variety of contributions (e.g., [1, 5, 6, 10]) 
to the global and local analysis of
neural systems have been made. The major mathematical method used for
this
analysis is Liapunov's second method. 

Let $\Delta$ be an arbitrary set in $R^{3n}$ and
$L=L(x, w, K)$
be a $C^1$ function on $\Delta$ to $R$. We say that $L$ is a 
{\em global Liapunov function\/} over $\Delta$ for (1)-(2) if 
 the differential of $L$ along trajectories of the dynamic system,
$$\stackrel{.}{L}\equiv \frac{dL}{dt}=
\sum_{i=1}^n\frac{\partial L}{\partial x_i}\cdot
\stackrel{.}{x}_i+\sum_{i=1}^n\sum_{j=1}^n\frac{\partial L}{\partial
w_{ij}}\cdot
\stackrel{.}{w}_{ij},\eqno(3)$$
does not change sign on $\Delta$. Define
$$E=\{(x,w)\in\bar{\Delta}|~\stackrel{.}{L}=0\},$$
where $\bar{\Delta}$ is the closure of $\Delta$. Let $M$ be the largest 
invariant set in $E$ (for definition of invariant set, see [9]).
Then the central part of the Liapunov method is
the following fundamental\medskip
\noindent
{\bf Stability Theorem 1$^{\rm [9]}$:} 
{\em If $L$ is a global Liapunov function on $\Delta$ for (1)-(2), 
then each solution $(x(t), w(t))$ of (1)-(2) that remains in $\Delta$ 
for all $t>0$ approaches $M\cup\{\infty\}$ as $t\rightarrow\infty$. 
If $M$ is bounded,
then either $(x(t), w(t))\rightarrow M$ or $(x(t),w(t))\rightarrow
\infty$ as
$t\rightarrow \infty$.} 

\medskip
This is an extended Liapunov stability theorem, referred as to LaSalle 
invariant principle, 
and contains the usual Liapunov-like theorems on stability and
instability of
autonomous systems. 

For the local stability analysis, 
we let $(x^0, w^0)$ be an equilibrium point of (1)-(2) and $\delta$ be a
small neighborhood of $(x^0, w^0)$ in $R^{3n}$. A $C^1$ function
$L=L(x, w, K)$ defined on $\delta$ to $R$ with
$L(x^0, w^0, K)=0$ is called a {\em local Liapunov function\/} if it is
positive definite (i.e., positive whenever $(x, w)\neq (x^0, w^0)$), 
and with a differential $dL/dt$ along the trajectories 
being continuous and of fixed negative sign in $\delta$.
The following is the classic stability theorem due basically to
Liapunov.

\medskip
\noindent
{\bf Stability Theorem 2:} 
{\em If there is a Liapunov function $L$ over $\delta$ for (1)-(2), 
then the equilibrium point $(x^0, w^0)$ is stable. If $dL/dt$ is
moreover 
negative definite, then $(x^0, w^0)$ is asymptotically stable.}

\medskip
The above two theorems have served as a theoretical basis for stability
analysis of neural systems. So far a considerable
 amount of work has been done for 
various specialized models. Significant contributions were made in [1,
5], for 
example, to the local stability analysis 
while [6] contains most of the early global stability results. 
In [3],
a global Liapunov function is constructed for content-addressable memory
networks with symmetric and constant weights which includes a number of
popular
models as specialized cases. This Liapunov function was generalized in
[17]
by using instead of symmetric detailed balance weights. In [13],
a careful local stability analysis for the Hopfield model was made.

As both the global and local stability theorems consist of the
determination 
of corresponding Liapunov functions, but the theorems themselves 
are merely existential in character and give no handle on the exact 
construction, the discovery of specific Liapunov functions 
depends strongly upon specific mathematical knowledge,
various technical derivations and extensive experiments. 
When the Liapunov functions are constructed,
important information about the trajectories of systems will be 
provided without any knowledge of the trajectories themselves.
Therefore,
for a practical stability analysis the central problem remains to 
find explicitly the associated Liapunov function $L$ for a given
dynamical system. To design stable and adaptive artificial neural
systems on the other hand, one has to seek such dynamics 
which in the ideal case both reflect the biological 
features (at least on a rudimentary level) and associate a Liapunov
function. 
Thus, the main task in this 
context is the exact determination of such a pair of mathematical
objects. 

It is clear that in addition to the application 
of a mathematical theory, the analysis and derivation of neurodynamics 
consist of a certain amount of experiments and manipulations on the
objects 
which are considered to be possible candidates as Liapunov functions 
and adaptive 
neurodynamics. As a matter of fact, these objects have become more and
more 
complicated due to diverse biological facts as we mentioned earlier. 
A natural question now is how to efficiently and accurately deal with
these 
complex mathematical objects. To answer this question,
 the approach of automatic analysis and derivation by computer is
desirable.
As proposed in [19], methods
and software systems of symbolic computation can be demonstrated to well
suit 
this need. 

It is noted that in this work, our emphasis is mainly on the 
manipulation of symbolic mathematical objects, without much concern for 
theoretical proof about stability and the biological performance of
neural 
systems.\bigskip
\noindent    
\centerline{\sc III. The Role of Symbolic Computation}

\smallskip
\noindent
Symbolic computation treats objects with semantics such as logic
formulae,
geometric objects, algebraic functions and computer programs. It
contains
computer algebra, computational geometry, automated reasoning and
automatic
programming as subareas. The development of symbolic computation,
including
algorithm design and system implementation together with applications,
has
received more and more attention due to its important role in dealing
with large and complicated symbolic objects in various technical
sciences.
In this section, we present a few fundamental features of symbolic 
computation for the analysis and 
derivation of neural systems.

\bigskip
\noindent
{\em A. Representation of Objects}

\smallskip
To treat symbolic objects, the first task is their representation, 
particularly, in a software system. Within the existing systems
such as CA systems, graphical kernel systems and theorem
provers designed for different purposes, most symbolic objects
including mathematical numbers, symbols and functions, geometric bodies,
logic formulae and programs can be represented.

The objects occurring in the analysis and derivation of neural systems
are essentially mathematical objects such as elementary functions, 
unspecified functions, indexed variables and functions, indefinite 
summations and matrices. Most of these objects can be represented in
existing CA systems. However, these systems are
not capable to represent objects like the Kronecker
symbol, piece-wise functions and indefinite summations with non-interval
domains. Such capabilities could be improved for special purposes
without
major difficulties.

\bigskip
\noindent
{\em B. Operations on Objects}

\smallskip
The main task for the analysis and derivation of neural systems is the
manipulation of the occurring objects. One major manipulation is 
formal operations, including arithmetic operation,
functional operation, matrix operation,
differential and integral operation and limit operation. CA 
systems provide a user-friendly interface for non-specialized users to
perform
most of these operations while the number of arguments is given.
Nevertheless, the objects occurring in the analysis and derivation of
neural systems are often related to an indefinite number of arguments 
or dimension. The existing CA systems 
have their emphasis on traditional algebraic objects and are either 
unable to perform operations on these objects, or provide wrong 
computations. For example, expressions of the form
\[\sum_{i=1}^nw_{ij}S_i(x_i)\]
appear in most neural systems. To analyze stability and derive learning
rules, we need frequently to calculate the partial derivative and
integral 
of the above expression with respect to an indexed variable $x_k$ or
$w_{kl}$. Unfortunately, this cannot be done with the available CA 
systems.
Thus, the operations among objects with an indefinite number of
arguments 
have to be extended and modified. One of the authors investigated the 
possibility 
for such extension and suggested some rules for differentiating and
integrating indefinite summations with respect to indexed variables
which 
we have to deal with for the analysis and 
derivation of some neural systems. These rules, together with others, 
have been included in a toolkit to be described in Subsection IV.D
for manipulating indefinite summations.  

\bigskip
\noindent
{\em C. Simplification of Objects}

\smallskip
While an object is given or obtained from a sequence of operations which 
might be very complicated, it is important to know whether this object
can be replaced by an equivalent but simpler one and how to find such
an object automatically. This is the problem of simplification, which 
belongs partially to the subarea of automated deduction. As an example,
let us look at the expression
\[1-\frac{1}{n}\sum_{i=1}^n[\cos^2i\theta+\sin(i+1)\theta
\sin(i-1)\theta].\]
By a careful analysis, one finds that this expression is the
same as the much simpler one
\[\sin^2\theta.\]
The current CA systems have integrated simplification mechanisms and 
some of them are able to simplify the above expression.
Simplifications can be carried out automatically or with human
interaction. 
Some special-purpose systems have implemented more power for automatic 
simplification. 

As the manipulation of objects with indefinite number of arguments has
not 
been taken into account of by the popular CA systems, 
simplifications of some of such objects cannot be done within these
systems.
In our toolkit developed in MACSYMA, we have include several rules to 
simplify objects with indefinite summations as will be seen from the 
example given in Section V.


\bigskip
\noindent
{\em D. Data Transformation}

\smallskip
Symbolic computation systems are highly interactive with powerful
devices for input and output of objects. One can expect the input,
output and terminal display of the objects to be very text-like. Some
tools have been developed for transforming the input/output from
the form of one system to that of another, and to \TeX or other
word-processor form. So one is able to directly include the 
objects produced by the symbolic computation systems into his/her
text.

As one purpose, symbolic computation may be the preprocessing for
numeric computation, especially in the case of neural research. On the
one hand, most CA systems have capabilities for doing
numeric computation; therefore, the symbolic objects can be used for
further numeric experiments within the same system. 
On the other hand, it is also possible to
encode the objects processed by CA systems into the
numeric (like C, Fortran, Pascal etc.) programs. In addition, 
the CA systems may also be used to generate large numeric 
computing programs and codes. This is somewhat in the category
of automatic programming and will be be discussed briefly in the last 
section. 

\bigskip
\noindent    
\centerline{\sc IV. Computer Aided Analysis and Derivation}

\smallskip
\noindent
In the previous section, we have discussed four features and
capabilities 
of methods and systems of symbolic computation for the analysis and
derivation of neural systems. In this section we present some strategies
for using CA systems and their extension
to aid the analysis and derivation of neural systems.  

\bigskip
\noindent
{\em A. Computation Checking}

\smallskip
While a stability analysis of a neural system is made or a possibly 
stable system is derived by humans, CA systems can be
used to check the computation and derivation step by step. This is
necessary particularly when the objects become too complex. 
Symbolic computation
guarantees the correctness and rapidity of the performance. Intuitively, 
this seems
to be an easy and trivial task. But the real situation is actually not
as
simple as expected. The reason is that two computations on the same
objects
can yield two equivalent but totally different expressions as different
simplification techniques and rules may be used. One cannot easily
determine whether one expression is the same as the other. Therefore,
for computation checking, the equivalence proof of different expressions
beyond computation must be considered, which requires very powerful
deduction mechanisms.

Some operations and simplification rules concerning expressions with 
indefinite sums have been designed and implemented 
for the toolkit in MACSYMA (see Subsection D), and the computation and
derivation of a number of neural systems have been checked, where some 
mistakes in the literature were discovered (see an example in [21]).

\bigskip
\noindent
{\em B. Constructing Liapunov Functions}

\smallskip
As presented in Section II, a stable and adaptive artificial neural
system
is characterized by the existence of a Liapunov function.
A sufficient condition for a function $L$ to be a Liapunov function is
$$\frac{\partial L}{\partial
x_i}=-[A_i(x)]^2\stackrel{.}{x}_i,~~~~\eqno(4)$$
$$\frac{\partial L}{\partial w_{ij}}=-[B_{ij}(w)]^2\stackrel{.}{w}_{ij}.
\eqno(5)$$
If the dynamics (1)-(2) of the neural system is known 
from some experimental neurophysiological facts,
the stability analysis of neurodynamics is mainly to construct
a Liapunov function $L$.
For this purpose, one useful strategy is to insert (1) and
(2) into (4) and (5), and integrate both with respect to $x_i$ 
and $w_{ij}$ respectively. In this case, different $A_i$ and $B_{ij}$
need
to be examined. 

The other strategy for obtaining a Liapunov function is based on two
steps. 
First, specialize the equations to dimensions 1 or 2 for instance,   
and then construct their Liapunov functions. Secondly, try to extend the 
Liapunov functions so obtained to the general case of indefinite
dimension 
$n$ and verify 
whether the extended ones are still Liapunov functions. 

In both cases, necessary experiments are involved and CA 
systems become then helpful tools. In MACSYMA together with our toolkit, 
we are by now able to perform almost all operations including
differentiation 
and integration of objects needed for practical experiments. 
This provides many more possibilities for constructing suitable
Liapunov functions.

\bigskip
\noindent
{\em C. Deriving Neurodynamics}

\smallskip
A Liapunov function is not necessarily the energy of the 
system. However, the term {\em energy function}, {\em objective
function\/} 
or {\em cost function}, is often used instead of Liapunov function in 
the neural networks community. 
If a problem-specific objective function $L$ is given,
the other direction is to derive the neurodynamics, in particular,
the learning rule. This is frequently achieved by minimizing the
function 
$L$ in weight space, where the position of the minimum is located at the 
weights' values at the end of the learning phase.
In this case, one may take various possible energy functions and 
use $(4)$ and $(5)$. 

Often the dynamics for the activation states $x$ is specified first. The
objective function need then be chosen in order to derive an adaptive
dynamic 
equation for the weights, namely, the learning rule. We have made some
experiments in this direction and found that this procedure is quite
feasible. 
The example given in Section V constitutes such a case.

Again, while a possible neurodynamics is derived, it may be very
complicated. 
Much work would remain for simplifying such dynamics either by
transforming 
it to a simpler form or by introducing some constraints. Even though the 
dynamics may be mathematically and biologically suitable, one may still
need to 
consider various modifications and simplifications from computational
aspects.
It is convenient to do all of them by use of a symbolic computation
tool.

\bigskip
\noindent
{\em D. A Toolkit in MACSYMA}

\smallskip
To test the analytic derivation of neural systems on the computer as 
discussed in the previous sections, we have developed a toolkit within
the
CA system MACSYMA. This toolkit together
with its base system MACSYMA has served well for our investigation in
this 
approach. The current version of the toolkit contains three parts: a
database 
including a number of neural systems well known from literature, a
library of 
functions for manipulating indefinite summations and a few subsidiary
functions 
for help, literature inquiry and reading neural systems from the
database.

The database serves for studying and comparing various neural systems
such as
the Cohen and Grossberg, the Hopfield, the standard backpropagation, 
the recurrent backpropagation 
models etc. These models can simply be called by the toolkit if needed. 
The MACSYMA system itself contains very extensive functions for
algebraic, 
differential, integral and matrix operation, equation 
solving and algebraic simplification on objects with a definite number
of 
arguments. It provides also the indefinite sum expression with interval
domain
and a few functions for treating this object. 
Nevertheless, these functions are currently limited and give often
incorrect
computations due to MACSYMA's design philosophy. For the analysis and 
derivation of neural systems, the main manipulation is among objects
with 
indefinite summations. Therefore, the MACSYMA built-in functions for
these 
objects have to be modified and some new functions for manipulating
these 
objects must be designed and implemented. To this end, we have
established a 
library of such modified and new functions. They are DF, DFINTOSUM, INT,

REWRITEDF, SUMCOMB, SUMCONT, SUMCONV, SUMDIST, SUMP, SUMSIMP, SUMSIMPD 
and SUMSIMPM. 

The functions WHO, WHERE and CALL are used respectively for listing all 
neural systems included in the database, literature details of each
system, 
and reading systems from the database to MACSYMA session. 
The description of all functions can be displayed by the help function
HELP. 
This makes the toolkit easy to be used. Of course, the toolkit is still 
undergoing changes and will
be extended from time to time by adding more functions and completing
the 
database. A detailed description of the current version of the toolkit
is 
given in [21]. Using the implemented functions, we are now able to 
deal with the majority of neural systems known from the literature.     

\bigskip
\noindent    
\centerline{\sc V. A Concrete Example}

\smallskip
\noindent
In the main part of this section, we present an example to illustrate
the 
derivation of a hybrid neural model by using our toolkit. 
The Liapunov function of the hybrid model consists of an adaptive
Hopfield term
and a quadratic error function. Hopfield type models are very successful
in 
restoring incomplete and/or noise-corrupted patterns whereas models
which minimize
the quadratic error between the actual and the desired output are
suitable for input 
data classification. The hybrid model discussed below combines the
virtues of both
approaches. This example is extracted from recent results in [18] which
we 
briefly describe first. 

\bigskip
\noindent
{\em A. The Neural Model}

\smallskip
The dynamics for the activation states $z_k$ of nodes $k$ considered 
is described by a set of coupled
differential equations of the form
$$\stackrel{.}{z}_k=f_k(z_k)[-g_k(z_k, K_k)+\sum_{s=1}^nw_{sk}S_s(z_s)],
~~k=1, ..., n. \eqno(6)$$
We assume the activation states $z_k$ of the 
neurons to change much more rapidly 
with time than the weights $w_{ij}$. Then, at the fixed point
$z^{\infty}$, 
the equation (6) becomes
$$g_k(z_k^{\infty}, K_k)=\sum_{s=1}^nw_{sk}S_s(z_s^{\infty}).\eqno(7)$$
The existence of fixed points is guaranteed, if the neural system is 
dynamically stable. This is the case if either $w_{ij}=w_{ji}$
(symmetry,
cf.\ [8]), or $\mu_jw_{ij}=\mu_iw_{ji}$ (detailed balance,
cf.\ [17]), or $\sum_{i,j}w_{ij}^2<1/[\max|S_s'(z_s)|]^2$
(finite norm, cf.\ [2]). Henceforth, we assume either one of the first
two 
conditions to hold.

The main goal below is to derive a hybrid learning rule by
proposing the following objective function
\[L\!=\!\frac{1}{4}\!\sum_{s=1}^n\!\sum_{k=1}^n\frac{\mu_k\lambda_{sk}w_{sk}^2}{\eta_{sk}}
-\frac{1}{2}\!\sum_{s=1}^n\!\sum_{k=1}^n\mu_kw_{sk}S_s(z_s^{\infty})S_k(z_k^{\infty})\]
\[+\sum_{k=1}^n\mu_k\int_0^{z_k^{\infty}}g_k(\xi_k, 
K_k)\frac{dS_k(\xi_k)}{d\xi_k}d\xi_k~~~~~~~~~~\]
$$+\frac{1}{2}\sum_{k=1}^n\alpha_k[S_k(K_k)-S_k(z_k^{\infty})]^2,
~~~~~~~~~~~~~~\eqno(8)$$ 
where
\[\eta_{sk}=\mu_{max(s, k)}/\mu_k.\] 
This objective function is a combination of a Hopfield-type energy
function
(first 3 terms) and an error function as used in multi-layer
perceptrons.
Accordingly, taking the gradient of $L$ in weight space yields a
learning
rule which contains Hebbian as well as backpropagation terms. Whether
such a 
hybrid formulation may permit more efficient learning than either the
Hebb rule or the delta rule alone, has yet to be checked for specific 
applications. In any case, (8) leads to a generalization of the standard
learning algorithms and thus may be useful for efficient neural
programming.

The following calculations are carried out in MACSYMA 4.103 with our
toolkit.
Following the MACSYMA input lines, we use the text form instead of the
display of computer output for space limitation.
All these expressions are exactly a copy of the original forms. 

\bigskip
\noindent
{\em B. Derivation of a Hybrid Learning Rule}

\smallskip
We calculate now the differential of $L$ along trajectories of the 
neural system which need be derived. It is assumed that each
$z_r^{\infty}$ is
dependent on the weight $w_{ij}$.

Calculating first the partial derivative of $L$ with respect to 
$z_r^{\infty}$ and simplifying it by SUMSIMP, we have

\medskip
\noindent
{\tt (C5) SUMSIMP(DF(LL, Z[R]));\footnote{Since there is no 
difference between small letters and capital letters in MACSYMA, 
we use a double letter instead of the normal capital letter. 
In MACSYMA, the input lines are automatically indexed by labels
of the form {\tt (Ci)} where {\tt i} is an integer, incremented with
each new line and the subscripts are entered by using a pair 
of square brackets (cf.\ [12]). 
Here we simply use z for $z^{\infty}$, and the same 
below.}} 
\[\frac{\partial L}{\partial z_r^{\infty}}
=
-\frac{1}{2}S_r'\{\sum_{s=1}^nS_s(z_s^{\infty})(\mu_sw_{rs}+w_{sr}\mu_r)
~~~~~~~~~~\]
\[~~~~~~~~~~-2\mu_rg_r(z_r^{\infty},K_r)-2\alpha_r[S_r(z_r^{\infty})-S_r(K_r)]\},\]
provided that $1\leq r\leq n$.                          
Assume now that $w_{ji}$ is dependent on $w_{ij}$ (cf.\ [8, 17]). Then

\medskip
\noindent
{\tt (C6) SUMSIMP(DF(LL,W[I,J]))+SUMSIMP(DF(LL,

~~W[J,I]))*'DIFF(W[J,I],W[I,J]);} 
\[\tilde{L}=\frac{\partial L}{\partial w_{ij}}+
\frac{\partial L}{\partial w_{ji}}\frac{dw_{ji}}{dw_{ij}}~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~\]
\[=-\frac{\mu_i[S_i(z_i^{\infty})\eta_{ji}S_j(z_j^{\infty})-\lambda_{ji}w_{ji}]
}{2\eta_{ji}}\frac{dw_{ji}}{dw_{ij}}\]
\[-\frac{\mu_j[\eta_{ij}S_i(z_i^{\infty})S_j(z_j^{\infty})-\lambda_{ij}w_{ij}]
}{2\eta_{ij}},~~\]
provided that $1\leq i, j\leq n$.
Hence

\medskip
\noindent
{\tt (C7) SUM(D5*'DIFF(Z[R],W[I,J]),R,1,N)+D6;
\[\frac{dL}{dw_{ij}}=\sum_{r=1}^n\frac{\partial L}{\partial
z_r^{\infty}}
\frac{dz_r^{\infty}}{dw_{ij}}+\tilde{L}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\]                             

\[=-\frac{1}{2}\sum_{r=1}^nS_r'\frac{dz_r^{\infty}}{dw_{ij}}
\{\sum_{s=1}^nS_s(z_s^{\infty})(\mu_sw_{rs}+w_{sr}\mu_r)\]
\[~~-2\mu_rg_r(z_r^{\infty},K_r)
-2\alpha_r[S_r(z_r^{\infty})-S_r(K_r)]\}\]
\[-\frac{\mu_i[(S_i(z_i^{\infty})\eta_{ji}S_j(z_j^{\infty})-\lambda_{ji}w_{ji}]
}{2\eta_{ji}}\frac{dw_{ji}}{dw_{ij}}~~~~\]
\[-\frac{\mu_j[\eta_{ij}S_i(z_i^{\infty})S_j(z_j^{\infty})-\lambda_{ij}w_{ij}]
}{2\eta_{ij}}.~~~~~~~~~\]
Substituting $g_r(z_r^{\infty},K_r)$ by using the fixed point equation
(7) and 
simplifying then the resulting expression, we obtain

\medskip
\noindent
{\tt (C8) FACTORSUM(SUMCONT(SUBST('SUM(W[R,S]*SS}

[S](Z[S]),S,1,N),G[R](Z[R],KK[R]),D7)));} 
\[\frac{dL}{dw_{ij}}=
\{\eta_{ij}\eta_{ji}\sum_{r=1}^nS_r'\frac{dz_r^{\infty}}{dw_{ij}}
\{\sum_{s=1}^nS_s^{\infty}(z_s^{\infty})
[(2\mu_r-\mu_s)w_{rs}\]
\[-w_{sr}\mu_r]+2\alpha_rS_r(z_r^{\infty})-2\alpha_rS_r(K_r)\}~~~~\]
\[~~~~-\mu_i\eta_{ij}S_i(z_i^{\infty})
\eta_{ji}S_j(z_j^{\infty})\frac{dw_{ji}}{dw_{ij}}
+\mu_i\eta_{ij}\lambda_{ji}w_{ji}\frac{dw_{ji}}{dw_{ij}}\]
\[~~~~~~-\eta_{ij}S_i(z_i^{\infty})\mu_j\eta_{ji}S_j(z_j^{\infty})
+\lambda_{ij}w_{ij}\mu_j\eta_{ji}\}/(2\eta_{ij}\eta_{ji}).\]

OK let's finish up with the references list and the biography section.




\begin{yourbibliography}{99}
\bibitem{}This is a reference that might appear in a compuscript and
since it is in electronic form, no rekeying is required ...only
editorial `manipulation'. 
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.
\bibitem{}This is another reference that might appear in a compuscript
and since it is in electronic form, no rekeying is required ...only
editorial `manipulation'.

\end{yourbibliography}





\begin{yourbiography}{Gerry Murray}
...will process your compuscript, adding `value' to your file using a
sophisticated text editor. Jeremy Barth will standardize the dimensions
you've used, run a spell check, apply SGML tags to structure your
document and validate all coding. The editor will `on-screen edit' the
file, `size' your artwork and supply you with a laser proof by mail, or
by fax, or even email you a postscript version of the file which you can
view on your workstation (using OpenWindows, NeXT, or an appropriate
postscript viewer). When the editor is finished incorporating your
amendments, Tom Bontrager, Mark Pheffer, Dalton Patterson or Chaucer
Tran will receive the file and apply their abundant knowledge of
typesetting to produce a document of the highest possible quality. A
postscript file will be generated, output at 2032dpi on high quality RC
paper for final examination by the editor. Christine will incorporate
the artwork, the resulting camera-ready-copy will then be shot, the film
supplied to the printer and your paper printed in the transaction for
all to admire.  
\end{yourbiography}

\end{document}






----------------------------------How To------------------------------


The ***REAL*** way to use `IEEEtran.sty' is as follows:

Use \documentstyle{article} or your favourite style file, pulling in
all your wonderful symbolics ***as you are writing your paper*** and as you and
the reviewer(s) are putting your wits together to culminate in a masterpiece!

When you are happy with the ***ultimate unequivocal final version*** .....

***THEN***...

1) Change the \documentstyle to use `IEEEtran.sty'

2) Change the symbolics so that the file actually contains the numbers
   i.e. blah blah \cite{fred} and so on and so on
   should be changed to:
   blah blah [3] and so on and so on 
   ....one author (who used the style file) did a smart thing 
       AFTER he had decided upon a final version...
   He put his \cites and other symbolics on a line on their own and 
   commented them out (from the formatting) by putting a % sign before each
   symbolic. Then, on the next line he just inserted the copy-matching 
   numerical, like this:

      Well, according the Fred Bloggs
      %\cite{FredBloggs}
      [24]
      the value of $\alpha$ should be even greater than what we think it
      should be.

   Thus, HE knows he put in the correct (copy-matching) numerical and WE
   can send him back an author-proof that correctly matches his submission.
 
3) The above also applies to the Figure cites (and any `auto-numbering' feature,
   standard or synonymous with your system).

4) Change the \section'ing to small caps and \sub/\subsubsection'ing to italics. 

In essence, by you actually putting in the CORRECT COPY MATCHING numericals
we don't have to jump through hoops due to incomplete files being sent to us
(the wrong `.bib', `.bbl', the wrong versions of figures etc).  Also, and 
more importantly, the #'s that are on your hard-copy (and in the reviewer's
hands) will be the same ones that you receive in your author proof. 
We cannot guarantee this if symbolics are used, you see!

5) Figure captions can be part of the text (in between paragraphs) like this:
   
       And in Fig. 3 we see that the value of $\alpha$ increases
     exponentially.

     Fig. 3\quad This is the caption for figure 3 showing some $\alpha$.

       And after the caption we continue on with the next paragraph, like this.

And since we've been doing disks for quite some time now --- many, many
submissions are incomplete and out of date --- much grief!!!

---Not so if the FILE contains EVERYTHING AS IS ON THE H/C !!!

Then you should be able to get a fairly accurate printout of what your
paper (author proof) will look like.

If some of the math splashes over the 21pica width then you should break it
as you see fit (and mathematically correct too!).

I know, I know, seems like an awful lot of work ....but not really...after all
you want your paper published quickly, right?
Also, we want to process your paper quickly and efficiently, and assure you
that what you send in WILL actually be sent back to you sans mistakes 
(cites etc.) In addition, if the math is broken --- it's even better for all.

In essence, `IEEEtran.sty' is not really for formatting (paper printout)
- but is REALLY for our INPUT PROCESSING which is chalk and cheese, frankly.

I hope this gives you some insight.

Gerry




