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\title{Explorations of Fitness Landscapes of a Hopfield Associative Memory with Random and Evolutionary Walks}

\author{Akira Imada, Keijiro Araki\thanks{A. Imada is with the
Graduate School of Information Science, Nara Institute of Science and Technology, 
Nara, Japan. E-mail: akira-i@is.aist-nara.ac.jp .}\thanks{K. Araki is with the 
Graduate School of Information Science and Electrical Engineering, Kyusyu University,
Fukuoka, Japan. E-mail: araki@c.csce.kyusyu-u.ac.jp .}}

\pagestyle{empty}

\maketitle



\begin{abstract}
\noindent
We apply evolutionary computations to the Hopfield's neural network model of associative memory.
In the model, some of the appropriate configurations of the synaptic weights give the network a function of associative memory.
One of our goals is to obtain the distribution of these optimal configurations as the global optima in the synaptic weight space
as well as the information of local optima created together.
In other words, our aim is to know a geometry of fitness landscapes defined on weight space.
As a step toward this goal, we concentrate in this paper mainly on the local optima.
Hence, 
we use a walk by the Gaussian mutation to explore the fitness landscape, 
rather than more effective evolutionary walks, 
expecting its high probability to be trapped at the local optima.
\end{abstract}

\section{Introduction}
\noindent
Associative memory is a dynamical system which has a number of stable states with a domain of attraction around them
(Koml\'{o}s et al. 1988).
If the system starts at any {\mbox state} in the domain, it will converge to the stable state.\\
\indent
Hopfield~(1982) proposed a fully connected neural network model of associative memory
in which information is stored by being distributed among neurons.
The dynamical behaviors of the model are determined by the strengths of the synapses which are referred to as  {\it weights}.
Here, the weight from neuron $j$ to neuron $i$ is denoted as $w_{ij}$.\\
\indent
For the fully connected neural networks with $N$ neurons, 
all the possible configurations of $N^2$ weights construct the $N^2$-dimensional Euclidean space.
Hereafter, we call this a ({\it synaptic}) {\it weight space}.
Every configuration of $w_{ij}$'s is represented as a point in weight space.
Fitness values of the points are estimated according to the capability of the weight configurations
to store a set of given patterns as associative memory.\\
\indent
Gardner (1988) discussed the optimal solutions of the weight configurations in terms of their volume in weight space.
However, the number and distribution of these solutions in the space are still an open problem.
Hence, the task of our evolutionary computations is to locate 
the optima in weight space so that the information as to the number and distribution is obtained.\\
\indent
Our previous results show that as the number of patterns to be stored increases,
the task to locate one of the optimal solutions becomes difficult (Imada et al. 1997a). 
We conjecture that this phenomenon is due to the increased ruggedness of the fitness surface.
The fitness surface constructed on a configuration space is called a {\it fitness landscape}.
In our problem, a capability of a network to store patterns as associative memory
defines the fitness landscape on the synaptic weight space.
As a first step to study the geometry of the fitness landscape,
we concentrate, in this paper, on the case that fitnesses are defined only for a few patterns to be stored,
where we suspect the number of local optima is not massively large.
The landscapes are explored by a walk using the Gaussian mutation
rather than more effective evolutionary walks,
expecting it to be trapped more easily at one of the local optima.\\
\indent
Although there have been a fair amount of analyses of the fitness landscape of the Hopfield model (Amit, 1989; Bray et al. 1980; 1981),
they are all defined on vertices of the $N$-dimensional Boolean hypercube 
which represent all the possible configuration of {\it binary neuron states}.
The analysis of the landscape defined on the {\it continuous weight space} has not been explored, as far as we know.

\section{Fitness Landscape \& Evolutionary Walks}
\noindent
The concept of the fitness landscape was first introduced by Wright (1932) to study biological evolutionary processes. 
Since then, this concept has been used not only in evolutionary biology but also in chemistry, physics, computer science and so on.
In chemistry, for example, a molecule can be represented as a string of $N$ genes whose values are one of $k$ alleles, or equivalently,
a sequence of $N$ letters with each letter being chosen from an alphabet of size $k$ (Macken et al. 1991).
Twenty amino acids ($k=20$) for proteins or four nucleotides ($k=4$) for nucleic acids can be considered as examples of the alphabet.
The $k^N$ possible combinations of the letters construct a configuration space of the string.
Then, for example, the free energy of RNA folding into secondary structures (Fontana and Schuster, 1987),
or the ability of peptides to bind to a particular substrate to catalyze a specific reaction (Smith, 1970)
is assigned as a fitness value to each point in the configuration space of the strings.
We can also find another example in~physics.
The Hamiltonian energy of Ising spins defines a fitness landscape on the configuration space of $N$ spins.
In this case, each spin takes the value either 1 or $-1$ ($k=2$).
Bray et al. (1980, 1981) argued the number and distribution of meta-stable states (local optima) 
of the Hamiltonian energies.\\
\indent
To explore these fitness landscapes, we need a rule with which a point in the space moves to one of its neighbors.
Then, consecutive movements of a point to the neighbors form a {\it walk} on the landscape.
Macken et al. (1991) used {\it random point mutation}, which replaces a single letter in the string with another alphabet 
to specify neighbors of the string.
They studied the statistical properties of the landscape of 
the chemical affinity of antibody for  antigen in immune response by sampling points along ``{\it an evolutionary walk}''
in which point moves to the {\it firstly found fitter} neighbor.
Weinberger (1990) used two different {\mbox walks:} ``{\it gradient walk}'' in which walker steps to the {\it best} of its neighbors;
and ``{\it random adaptive walk}'' in which the next point is chosen {\it at random} from the set of better neighbors.
Using these two walks they investigated the Kaufman's NK landscape 
which is a model formulated in more general {\mbox form} (Kauffman and Levin, 1987).\\
\indent
The fitness landscapes described above are all constructed on {\it discrete} string space.
What we are concerned, however, is a fitness landscape defined on {\it continuous} weight space 
in which it loses sense to move to the next neighbor point.
Hence, we use a walk by the Gaussian random mutation 
which gives a small perturbation on each coordinate of a point in $N^2$-dimensional weight space.
The perturbation  on $x_i$, $i$-th coordinate, is drawn for each $i$ 
from the Gaussian distribution with mean 0 and standard deviation $\sigma_0$, i.e.,
\begin{displaymath}
x_i^{\mbox {\scriptsize {new}}} = x_i^{\mbox {\scriptsize {old}}} + {\sigma}_{0} \cdot N_{i} (1, 0).
\end{displaymath}
This forms a ``{\it random walk}'' in the continuous weight space.
Though the step size of a walk:
\begin{displaymath}
\left( \hspace{.04in} \sum_{i =  1}^{N^2} { ({\sigma}_{0} \cdot N(1, 0))^2 } \hspace{.04in} \right)^{1/2},
\end{displaymath}
are different from step to step, we can control the size on average by tuning ${\sigma}_{0}$.\\
\indent
On the other hand, if we want a ``hill-climber'', we give the mutation $\lambda $  times to the current point.
Then the point with the {\it best} fitness among these $\lambda$ mutants is selected as the next point.
If there is no higher fitness point among $\lambda$ mutants, 
the process of giving mutation $\lambda$ times and searching higher fitness point is repeated without moving.

\section {Hopfield Model \& Fitness Evaluation}
\noindent
Before proceeding further, we summarize the Hopfield model briefly, and define our fitness evaluation.\\

\noindent
The Hopfield model of associative memory consists of $N$ neurons and $N^2$ synapses.
Each neuron state is either active (+1) or quiescent ($-1$).
When an arbitrary $N$-bit length {\it bipolar pattern}, a sequence of $+1$ and $-1$,
is given to the network as an initial state,
the network changes its {\mbox state} dynamically.
Each neuron state is updated asynchronously (at most one neuron at a time)\footnote{One neuron is updated 
only once per cycle in which all the $N$ neurons must be updated. 
Hopfield chose neuron to be updated in random order, while we updated them in a pre-fixed order.
It was verified that the both cause essentially the same results.}
according to 
\begin{displaymath}
s_{i}\hspace{.02in} (t+1) = sgn \hspace{.02in} \left( \hspace{.04in} \sum_{j \ne i}^N { w_{ij} s_{j}(t)} \hspace{.04in} \right),
\end{displaymath}
where $s_{i}(t)$ is a state of $i$-th neuron at time $t$.
Provided the synaptic weights $w_{ij}$'s are determined appropriately, networks store a number of patterns as fixed point attractors.
To store $p$ bipolar patterns:
\begin{displaymath} 
{\xi}^{\nu} = ({\xi}^{\nu}_{1},\cdots,{\xi}^{\nu}_{N}), \hspace{.2in}{\nu} = 1, \cdots, p \hspace{.1in},
\end{displaymath}
where ${\xi}_{i}^{\nu}$ takes either 1 or $-1$,
Hopfield prescribed $w_{ij}$'s by the rule which was at first advocated by Hebb (1949) and later formalized by Cooper (1973) as:
\begin{displaymath}
w_{ij} = \frac{1}{N} \sum _{ \nu = 1}^{p} {{ \xi }_{i}^{ \nu }{ \xi }_{j}^{ \nu }}\hspace{.02in}(i \ne j), \hspace{.02in} w_{ii} = 0.
\end{displaymath}
Then, giving one of the stored patterns, possibly including a few errors, to the network as an initial state
will result in the stable state.\\

\noindent
Fitness Evaluation is made as follows:
When one of stored patterns ${\xi}^{\nu}$ is given to the network as an initial state,
the state of neurons varies from time to time afterwards (unless ${\xi}^{\nu}$ is stored as a fixed point).
In order for the network to function as an associative memory, these two states must be as similar as possible.
The similarity as a function of time is defined by,
\begin{displaymath}
m^{\nu}(t) = \frac{1}{N} \sum_{i=1}^{N} { {\xi}_{i}^{\nu} s^{\nu}_{i}(t) }.
\end{displaymath}
This is conventionally called an {\it overlap}.
The temporal average of $m^\mu(t)$ is calculated for each stored pattern.
Then, the fitness function is obtained by further averaging these values over all the given patterns.
That is, our fitness value $f$ is calculated as:
\begin{displaymath}
f = \frac {1}{p \cdot t_{0}}
\sum _{\nu=1}^{p}
{
\sum_{t=1}^{t_{0}}
{m^{\nu}(t)} }.
\end{displaymath}
In this paper, $t_{0}$ is set to $2N$, twice the number of neurons.
Note that the fitness~1 implies all the $p$ patterns are stored as fixed points, while fitness less than~1 includes many other cases.

\section{Experimental Results and Discussion}
\noindent
Since all the experiments in this paper were carried out  on networks with 49 ($=N$) neurons,
the combinations of 2401 (=$49 \times 49$) real-valued weights construct the 2401-dimensional weight space.
For {\it a priori} set $p$ random bipolar patterns, fitness value is assigned to each point on the space 
by the degree to how the weight configuration can store the $p$ patterns as fixed points.
Thus, different $p$ defines different fitness landscape.
The geometry of the fitness landscapes are studied with $p$ varying from 1 to 6.
Two parameters of the Gaussian walk, ${\sigma}_{0}$ and ${\lambda}$ are set to 0.002 and 200, respectively.

\subsection{Fitness Distribution}
\noindent
As a preliminary experiment,
we picked up 240,000 samples\footnote{Since usually we perform GAs with a population of 200 individuals 
and with stop criterion being 12,000 generation, the number of samples estimated fitness value is typically 240,000 in total.}
 randomly from weight space;
and fitness values of these samples are estimated
in such a way that was mentioned in the previous section with the target here being only {\it one} pre-determined random pattern.
This gives us a bird's eye view of the fitness landscape for $p=1$.
The distribution of these fitness values, shown in Fig. \ref{histogram}, was very similar to the Gaussian distribution.
This suggests that the landscape is more or less like the Kauffman's NK model (Kauffman and Levin, 1987)
in which each point is assigned random fitness value drawn from the Gaussian distribution.
Moreover, we can see that it is very difficult for the randomly chosen samples 
to be near one of {\mbox peaks} whose height is one.
Therefore, we can guess 
that the distribution of the global optima is a kind of ``{\it needles in haystack}.''
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=histogram.eps,width=5.5cm,height=5.5cm}
   \caption{\bf Fitness distribution of randomly chosen samples}
   \label{histogram}
 \end{center}
\end{figure}


\subsection{Down Hill Walk from a Peak}
\begin{table*}[hbtp]
\begin{center}
\caption{Statistics of the 100 walks to the optimum ($p=1$ and 12,000 steps at most).}
\label{table1}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
\multicolumn{1}{c}{number}
           &  \multicolumn{4}{c}{ultimate-fitness}          & \multicolumn{1}{c}{\hspace{.5in}}            
                                                               & \multicolumn{4}{c}{walk length$^\dagger$}                                                    
                                                                                                               &  \multicolumn{1}{c}{number} \\
              \cline{2-5}                                        \cline{7-10}
of patterns &  max        & min       & avg       & std      &  & max       & min       & avg       & std       & of runs \\ \hline \hline 
\multicolumn{11}{l}{(walks succeeded in locating one of the global optima)} \\
1      & 1.000       & ---       & 1.000     & ---      &  & 10959     &   18       &  347       & 143.9     & 32        \\ 
\multicolumn{11}{l}{(walks trapped at one of the local optima)} \\
1      & 0.980       & 0.853     & 0.950     & 0.00298  &  & 11450     &   10       &  905       & 251.9     & 88        \\ \hline \hline 
\end{tabular}
\end{center}
{\small $^{\dagger}$ By walk length when the walks are trapped at local optima, we means the number of steps where the last fitness change occurs.}

\hspace*{1in}\\

%\begin{table}[hbtp]
\begin{center}
\caption{Statistics of the 30 walks to the optimum ($p>1$ and 60,000 steps at most).}
\label{table2}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
\multicolumn{1}{c}{number}
           &  \multicolumn{4}{c}{ultimate-fitness$^{\ddagger}$}          & \multicolumn{1}{c}{\hspace{.5in}}            
                                                               & \multicolumn{4}{c}{walk length}                                                    
                                                                                                               &  \multicolumn{1}{c}{number} \\
              \cline{2-5}                                        \cline{7-10}
of patterns &  max        & min       & avg       & std      &  & max       & min       & avg       & std       & of runs \\ \hline \hline 
2      & 0.984       & 0.866     & 0.933     & 0.0062   &  & 51569     &   478       &  9150       & 2370     & 30        \\  
3      & 0.941       & 0.825     & 0.901     & 0.0055   &  & 58388     &  2343       & 28549       & 3393     & 30        \\  
4      & 0.941       & 0.703     & 0.877     & 0.0089   &  & 59870     &  7560       & 37392       & 2747     & 30        \\  
5      & 0.913       & 0.755     & 0.860     & 0.0073   &  & 59798     &  9767       & 37310       & 2846     & 30        \\  
6      & 0.886       & 0.650     & 0.773     & 0.0111   &  & 59481     &  9718       & 47116       & 2449     & 30        \\  \hline \hline
\end{tabular}
\end{center}
{\small $^{\dagger}$ Ultimate fitness means the fitness measured at the 60,000-th step.}\\
\end{table*}

\noindent
Then, we experiment a down hill walk from a peak.
This is simple but quite a new experiment in evolutionary computation's community.
The down-hill-walker moves to the lowest neighbor point among 
$\lambda$ mutants formed by the Gaussian mutation.
That is to say, an upside down version of the hill-climber.
In contrast to the random sampling mentioned above,
the walk is guaranteed to include at least one optimum.\\
\indent
Some peaks in weight space are already known;
a configuration determined by the Hebbian rule is an example;
or we have located other global optima using evolutionary algorithms (Imada et al. 1997b).\\
\indent
Though the primary objective of this experiment was to find an appropriate step size of the walk formed by the Gaussian mutation,
and in fact, it is from this experiment that we set the value of $\sigma_0$ to 0.002,
we can obtain further information about the peak.
Different random {\mbox walks} from a specific point are expected to start in different {\mbox directions};
and to march approximately radially from 
the starting point until it is trapped at a local/global optimum.
Hence, the observations of several random down hill walks from a peak will give us a geometry of the peak.
In Fig.~\ref{dhw}, we compare two Hebbian peaks; one is  for $p=1$ (top) and the other is for $p=6$ (bottom).
We observed 10 down-hill-walkers together with 10 random-walkers
both started from the top of the same Hebbian peak.
We showed in the figure the steepest trace of the 10 down-hill-walkers, and the flattest trace of the 10 random-walkers.
These traces seem to reflect roughly the profile of the peak they walk down.
From the observation of the down-hill-walkers,
we can see that the peak for $p=1$ has larger area around the top (basin of attraction) and much steeper side than the peak for $p=6$.
Note that the horizontal line in the case of Gaussian mutation down hill walk ($p=1$) does not mean 
the actual walk but the struggle trying to escape the basin of attraction around the top of the peak.\\

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=3000-5001-dh.eps,width=5.9cm,height=5.9cm}
   \epsfile{file=3030-5035-dh.eps,width=5.9cm,height=5.9cm}
   \caption{\bf Fitness along a down hill walk}
   \label{dhw}
 \end{center}
\end{figure}

\indent
Random-walkers, on the other hand, seem to choose paths of medium degree of slope, and we observe that 
the side of the peak is more rugged in $p=1$ than $p=6$.

\subsection{Gaussian Mutation Hill Climbing}
\noindent
In this subsection, we show the results of hill climbing walks by the Gaussian mutation,
starting at the origin at weight space.
The hill climbers explore the fitness landscape constructed for $p=1$ through $p=6$.\\

\noindent
The walk length, by which we mean the number of steps of a walk needed from its starting point to the local/global optimum that captures the walk,
is strongly related to both the density and basin size of the local optima in the landscape.
We show this following after Macken et al. (1991).\\
\noindent
Now, let us denote the probability for a point to reach optimum in exactly $k$ steps, starting at the fitness $u_0$, as $p_{k}(u_0)$.
In order for this to occur, two independent events must happen.
First, the point must reach a fitness in the range $(u, u+du)$ in $k$ steps starting at the fitness $u_0$;
Second, all the $\lambda$ mutant neighbors of the point must have the fitness lower than $u$.
Assume these probabilities are $f_k(u; u_0)du$,\hspace{.02in} ($u_0<u<1$) and $h(u)$, respectively. Then we obtain
\begin{displaymath}
p_k(u_0) = \int _{u_0}^{1} f_{k}(u; u_0) h(u)du.
\end{displaymath}
Clearly, the function $h(u)$ is roughly the {\it density of local optima in the space},
and $f_k(u; u_0)du$ should relate to the size of basin of attraction of the local optima,
though we have not known the explicit form of each function in the equation.\\

\noindent
The only case for hill-climbers to locate one of the global optima was the landscape for $p=1$
where we repeated the simulation 120 times with the stop criterion being set to 
12,000 steps unless one of the global optima is found.
The global optimum was located in 32 out of 120 runs.
The {\mbox statistics} are shown in Table~\ref{table1}.\\
\indent
In the landscape for $p>1$,
hill-climbers were all trapped at one of the local optima. We observed them by extending the stop criterion to 60,0000 steps.
Due to the tremendously long time required to calculate them,
we only repeated the simulation 30 times.
The statistics are shown in Table~\ref{table2}.\\
\indent
The fact that walkers were able to reach one of the global optima only in the landscape for $p=1$ suggests that
the {\it global} optima exist more and their basin size is larger in the landscape for $p=1$ than for $p>1$.
Then as $p$ increases, local optima become dominating the landscape.\\
\indent
The probability for a walk to escape the local optimum, however large its basin size may be,  is not zero.
It is in contrast to the mutation used in standard genetic {\mbox algorithm} with binary chromosomes where mutation flips a bit.
In fact, we observed that some walks escaped the local optima after remaining there for a long period.
As an example, we show an evolution of distance from the origin of a hill-climber in Fig. \ref{distance}. 
The walker started to move again after a fairy long period of being stagnated at a local optimum.
This suggests that the  stop criterion 60,000 steps is still not long enough.
Nevertheless, 
as is observed in the  minimum and average walk length in Table \ref{table2},
the tendency toward longer walk length to the local optima as $p$ increases
shows the increase in number and basin size of local optima.\\
\indent
Furthermore, we can observe the fitness value when a walk is terminated becomes dramatically lower as $p$ increases.
These many local optima with large basin and low hight seem to prevent hill-climbers from reaching a global optima in landscapes for $p>1$.
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=escape.eps,width=5.5cm,height=5.5cm}
   \caption{\bf Evolution of distance from the origin}
   \label{distance}
 \end{center}
\end{figure}

\noindent
Although we have used, so far, the term {\it hill} or {\it peak} for the optimal configuration of $w_{ij}$,
we must note here the following feature of our fitness landscape.
Since each $w_{ij}$ can take an arbitrary real value, there are infinite number of {\mbox equivalent} configurations 
which differ only by scaling factor.
In other words, for any scaling factor $\kappa$, $\kappa w_{ij}$ works exactly in the same way as $w_{ij}$ in updating neuron states.
Therefore, our landscape of the Hopfield network, if  defined on the weight space, 
is made up of infinitely extended {\mbox straight} {\it ridges} emanating from the origin.
The height of each ridge remains constant, but different from each other.


\section{CONCLUSION}
\noindent
We have described explorations of evolutionary walks 
on the fitness landscape of the Hopfield model of associative memory.
The landscape is defined on the {\it continuous} synaptic weight space 
by assigning to each point the ability of the network to store patterns as associative memory.
Since the space is astronomically huge and ridges are extremely narrow,
statistics of the points sampled at random or {\mbox along} a random walk cannot represent the whole space.
In this paper, we employed the walks formed by the Gaussian random mutation.
Because of its indeterministic property, which is in contrast to the standard binary mutation, the statistical analysis,
such as using {\it correlation} between two neighboring points, or  
{\it autocorrelation function} along an evolutionary walk are very resistant to be used.
Hence, as a first step to know the geometry of our landscape,
we observed down hill random walk from a peak as well as the  
Gaussian mutation hill climbing starting at the origin.\\
\indent
From the observation of these walks, we have made the following conjectures:
The {\it global} optima has the largest basin size when $p$, 
the number of patterns to be stored, is one; and the size becomes smaller as $p$ increases.
The side wall of the global ridge is steepest when $p=1$ and becomes milder as $p$ increases.
As for the number of optima, a fair amount of the global ridges exist when $p=1$, 
then local ridges become dominating the landscape as $p$ increases. 
In addition, the height of the local ridges becomes lower drastically with their basin size becoming smaller, 
which make it difficult for hill-climbers to reach the global optimum on the landscape for $p>1$.\\

\noindent
The study has not been matured at all, and many things are waiting for our explorations.
The next step will be to analytically derive the description of the walk length as a function of both the density and the basin size of local optima.


\section*{Acknowledgments}
\noindent
We thank Peter Davis at Advanced Telecommunication Research Institute (ATR)
for providing us great insight into the dynamics of the Hopfield neural networks
in the early stage of this research.




\section*{References}
\begin{description}
\setlength{\itemsep}{0pt}
\item[]Amit, D. J. (1989)
{\em Modeling Brain Function: The World of Attractor Neural Networks.}
Cambridge University Press.

\item[]Bray, A., and M. Moore (1980)
{\em Metastable states in spin glasses.}
Journal of Physics, C: Solid State Physics, 13, ppL469--L476.

\item[]Bray, A., and M. Moor (1981)
{\em Metastable States in Spin Glasses with Short-ranged Interactions.}
Journal of Physics, C: Solid State Physics, 14, pp1313--1327.

\item[]Cooper, L. N. (1973)
{\em A Possible Organization of Animal Memory and Learning.}
Proceedings of the 24th Nobel Symposium on Collective Properties of Physical Systems.
pp252--264.
%% B.Lundqvist and S.Lundqvist (eds) Academic Press, New York.

\item[]Derrida, B.,  and E. Gardner (1986)
{\em Metastable states of a Spin Glass Chain at 0 Temperature.}
Journal of Physique 47, pp959--965.

\item[]Fontana, W., and P. Schuster (1987)
{\em A Computer Model of Evolutionary Optimization.}
Biophysical Chemistry, 26, pp123--147.

\item[] Gardner, E. (1988)
{\em The Phase Space of Interactions in Neural Network Models.}
Journal of Physics, 21A, pp257-270.

\item[]Hebb, D. O. (1949)
{\it The Organization of Behavior.} Wiley.

\item[]Hopfield, J. J. (1982) 
{\em Neural Networks and Physical Systems with Emergent Collective Computational Abilities.}
Proceedings of the National Academy of Sciences, USA, 79,  pp2554-2558.

\item[]Imada, A., and K. Araki (1997a),
{\em Random Perturbations to Hebbian Synapses of Associative Memory using a Genetic Algorithm.}
Proceedings of International Work-Conference on Artificial and Natural Neural Networks.
Springer Verlag, Lecture Notes in Computer Science, No.1240, pp398--407.

\item[]Imada, A., and K. Araki (1997b)
{\em Evolution of Hopfield Model of Associative Memory by the Breeder Genetic Algorithm.}
Proceedings of the 7th International Conference on Genetic Algorithms, pp784--791.

\item[] Kauffman, S. A., and S. Levin (1987)
{\em Towards a General Theory of Adaptive Walks on Rugged Landscapes.}
Journal of Theoretical Biology, 128, pp11--45.

\item[]Koml\'{o}s, J., and R. Paturi (1988)
{\em Convergence Results in an Associative Memory Model.} 
Neural Networks 1, pp239--250.

\item[]Macken, C.,  P. Hagan, and A. Perelson (1991)
{\em Evolutionary Walks on Rugged Landscapes.}
SIAM Journal of Applied Mathematics, 51, pp799--827.

\item[]Smith, J. M. (1970)
{\em Natural Selection and the Concept of a Protein Space.}
Nature, 225. pp563.

\item[]Weinberger, E. D. (1990)
{\em Correlated and Uncorrelated Fitness Landscapes and How to Tell the Difference.}
Biological Cybernetics, 63, pp325--336.

\item[]Wright, S. (1932)
{\em The Roles of Mutation, Inbreeding, Crossbreeding and Selection in Evolution.}
Proceedings of the 6th International Congress of Genetics, pp356--366.

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