\documentstyle[12pt,eclepsf]{article}
\textheight 260mm
\topmargin 5mm
\textwidth 180mm
\oddsidemargin -7mm

\pagestyle{myheadings}
\markright{(ACS \& CISM -- 2004 at Elk, Poland)}
\thispagestyle{empty}

\begin{document}
\begin{Large}


%% --- (title) ---
\begin{center}
{\LARGE{\bf How a Peak on a Completely-flatland-elsewhere}}\\
\hspace*{1in}\\
{\LARGE{\bf can be Searched for?}}
\hspace*{1in}\\
\hspace*{1in}\\
{\large{\bf A Fitness Landscape of Associative Memory by Spiking Neurons}}
\hspace*{1in}\\
\hspace*{1in}\\
\hspace*{1in}\\ 
Akira Imada  \\
\hspace*{1in}\\
Brest State Technical University \\ 
Belarus
\end{center}

%% --- (to be more specific) ---
\clearpage
\begin{center}
{\bf 
We came accross \\
\hspace*{1in}\\
an extremely difficult \\
PEAK \\
to be searched for \\
\hspace*{1in}\\
in a \\
FITNESS LANDSCAPE \\
\hspace*{1in}\\
explored by an \\
EVOLUTIONARY COMPUTATION
}
\end{center}

%% --- (Figure) ---
\clearpage
\noindent
{\bf $\Box$ A reason for the title --- A tiny-flat-island in a huge-lake}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=a-needle.eps,width=18cm,height=8cm}
 \end{center}
\end{figure}

%% --------------------
\clearpage
\begin{center}
{\bf 
To realize an \\
\hspace*{1in}\\
ASSOCIATIVE MEMORY\\
\hspace*{1in}\\
we use \\
\hspace*{1in}\\
SPIKING NEURONS \\
\hspace*{1in}\\
whose learning is by\\
\hspace*{1in}\\
EVOLUTIONARY COMPUTATIONS.
}
\end{center}

%%------------------------------------------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ Contents}\\
{\bf
\begin{itemize}
\item[1.] What is Associative Memory?
   \begin{itemize}
   \item[$\cdot$] Traditional Model by Hopfield.
   \item[]\hspace*{1.8in}vs.
   \item[$\cdot$] A Model using Spiking Neurons. \\
   \end{itemize}
\item[2.] Can we evolve Spiking Neurons?
   \begin{itemize}
   \item[$\cdot$] We study it by observing Fitness Landscape. \\
   \end{itemize}
\clearpage
\hfill (cont'd)
\item[3.] Results of our Experiments.
   \begin{itemize}
   \item[$\cdot$] A downhill walk from Hebb's peak!
   \item[$\cdot$] How difficult to find other peaks!\\
   \end{itemize}
\item[4.] A proposition of Test-function.\\
\item[5.] Summary \& Concluding Remarks.
\end{itemize}
}

%%%%%%%%%%
\clearpage
\noindent
$\Box${\bf Associative Memory:}
{\bf
\begin{itemize}
\item[$\bullet$] stores patterns
   \begin{itemize}
   \item in a distributed way (among neurons),
   \end{itemize}
\item[$\bullet$] recalls patterns
   \begin{itemize}
   \item from noisy and/or partial input. \\ 
   \item[] \hspace*{1.5in}$\Downarrow$ i.e.
   \end{itemize}
\item[$\bullet$] gives us
   \begin{itemize}
   \item perfect recollection from imperfect information
   \end{itemize}
\end{itemize}
}

%%%%%%%%%%  Realization  %%%%%%%%%%
\clearpage
\noindent
$\Box${\bf The Associative Memory can be realized with:}\\
\begin{itemize}
{\bf
\item[$\bullet$] Fully-connected Neural Network Model (Hopfield type)
\item[$\bullet$] Artificial Immune System Model
\item[$\bullet$] Spiking Neurons 
   \begin{itemize}
      \item[](To be more biologically plausible)
   \end{itemize}
\item[$\bullet$] etc.}
\end{itemize}

%%%%%%%%%%  What is Associative Memory  %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=what-is-am.eps,width=18cm,height=10cm}
 \end{center}
\end{figure}

%%%%%%%%%%  Category of trajectory  %%%%%%%%%%
\clearpage
%%\noindent
%%$\Box$ {\bf Category of Trajectory}\\

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=trajectory.eps,width=18cm,height=10cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
%%\noindent
$\Box$ {\bf A Schematic Diagram of the Hopfield Model}\\
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=fig-nn-2.eps,width=12cm,height=6cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\hspace*{1in}\\
\noindent
{\bf $\Box$ State Transition of the Hopfield Model}
\begin{center}
\hspace*{1in}\\
\begin{Huge}
\begin{displaymath}
s_i(t+1)= sgn \Biggl(\sum_j^N w_{ij}s_j(t)\Biggr)
\end{displaymath}
\end{Huge}
\end{center}

\clearpage
\hspace*{1in}\\
\noindent
{\bf $\Box$ Hebbian Weights}\\

\begin{center}
\begin{Huge}
\begin{displaymath}
w_{ij}=\frac{1}{N}\sum_{\mu=1}^{p}\xi_i^\nu \xi_j^\nu, 
       \hspace{.2in} (i \ne j),
       \hspace{.2in} w_{ij}=0.
\end{displaymath}
\end{Huge}
\end{center}

\hspace*{1in}\\
\hspace*{1in}
(To store $p$ patterns: 
$\mbox{\bf x}^{\mu} = (\xi_1^\mu, \xi_2^\mu, \cdots \xi_N^\mu)$ 
\hspace*{.15in} $\mu = 1, 2, \cdots, p$)

\hspace*{1in}\\
\hspace*{1in}\\
\hspace*{1in}\\

\clearpage
\hspace*{1in}\\
\noindent
{\bf $\Box$ Hopfield Model has a Crucial Drawback.}\\

\begin{itemize}
\item $N$ neurons store only $2N$ patterns at most (Gardner).
\begin{displaymath}
p<2N.
\end{displaymath}
\end{itemize}

% --------------------
\clearpage
\noindent
{\bf $\Box$ From the Hopfield Model to Spiking Neuron Model}\\
\begin{center}
{\bf
Influence from other neurons \\
via\\
ELECTRIC CURRENTS \\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
changes  \\
MEMBRANE VOLTAGE \\
of the neuron. \\
\clearpage
\hfill (cont'd)\\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
If it exceeds a threshold \\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
the neuron emits \\ 
a SPIKE.
}
\end{center}

%%-----------------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ Pyramidal Cells \& Interneurons}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=wilson.eps,width=18cm,height=7cm}
 \end{center}
\end{figure}

\clearpage
\begin{large}
\begin{center}
{\bf PYRAMIDAL CELL}\\
\hspace*{1in}\\
$\Vert$ \\
(positive current) \\
$\Downarrow$ \\
\hspace*{1in}\\
{\bf PYRAMIDAL CELL}\\
\hspace*{1in}\\
$\Uparrow$ \\
(negative current)
\hspace*{1in}\\
$\Vert$ \\
\hspace*{1in}\\
{\bf INTERNEURON}\\
\end{center}
\end{large}

%------------------------------------------------------------
\clearpage 
\noindent
{\bf $\Box$ Response of a Single Neuron to an External Stimulus:}

\begin{itemize}
\item {\bf Spike rate vs stimuli (Naka-Rushton function)}
\end{itemize}

\begin{displaymath}
S(P) = \left\{
\begin{array}{ll}
MP^n/(\sigma^n+P^n) & \hspace{.2in}\mbox{if} \hspace{.1in} P \ge 0 \\
0 & \hspace{.2in}\mbox{if} \hspace{.1in} P <0 \\
\end{array}
\right.
\end{displaymath}

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=naka-rushton.eps,width=15cm,height=5cm}
 \end{center}
\end{figure}

\clearpage
\hfill (cont'd)\\
\begin{itemize}
\item {\bf Spike rate under time-variant stimulus:}
\end{itemize}

\begin{Huge}
\begin{displaymath}
\frac{dr(t)}{dt} = \frac{1}{\tau}(-r(t) + S(P))
\end{displaymath}
\end{Huge}

%------------------------------------------------------------
\clearpage 
\noindent
{\bf $\Box$ Response of Multiple Neurons}

\begin{itemize}
\item {\bf Stimulus to a Pyramidal Cell}\\
\end{itemize}

\indent
$P_i$ in the previous Eq.
\hspace*{1in}\\
\begin{displaymath}
S(P_i) = M P_i^n/(\sigma^n+P_i^n)
\end{displaymath}
\hspace*{1in}\\
\indent
is now expressed as\\

\begin{Huge}
\begin{displaymath}
P_i = (\sum_{j=1}^{N} w_{ij} \cdot R_j - g \cdot G)_{+}^2
\end{displaymath}
\end{Huge}


\clearpage
\hfill (cont'd)
\begin{itemize}
\item {\bf Spiking Ratio of a Pyramidal Cell}\\
\end{itemize}

And \\
\begin{displaymath}
\frac{dr}{dt} = \frac{1}{\tau}(-r + M P^n/(\sigma^n+P^n)) \\
\end{displaymath}

becomes\\
%% \begin{Huge}
\begin{displaymath}
\tau_R \frac{dR_i}{dt} = -R_i +
\frac{100 \displaystyle{(\sum_{j=1}^{N} w_{ij} R_j - 0.1 G)_{+}^2}}
     {100 + \displaystyle{(\sum_{j=1}^{N} w_{ij} R_j - 0.1 G)_{+}^2}}
\end{displaymath}
%% \end{Huge}

\clearpage
\hfill (cont'd)
\begin{itemize}
\item {\bf Spiking Ratio of Interneuron}
\end{itemize}

\begin{displaymath}
\tau_G \frac{dG}{dt} = -G - 0.07 \sum_{j=1}^{N} R_i
\end{displaymath}

%------------------------------------------------------------ (NEW)
\clearpage 
\noindent
{\bf $\Box$ How to encode patterns?}\\


For the pattern


\begin{displaymath}
(\xi_1, \xi_2, \cdots, \xi_N)
\end{displaymath}


we encode as

\begin{Huge}
\begin{displaymath}
\xi_i = \left\{
\begin{array}{ll}
1 & \hspace{.2in}\mbox{if} \hspace{.1in} R_i \ge M/2 \\
0 & \hspace{.2in}\mbox{if} \hspace{.1in} R_i < M/2 \\
\end{array}
\right.
\end{displaymath}
\end{Huge}

\hspace*{4in}$\Rightarrow$Rate Coding




%%------------------------------------------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ GA Implementation:} \\
\begin{large}
\begin{itemize}
\item[1.] Represent a series of $w_{ij}$ as a {\it population} of strings ({\it chromosome}).
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=chromosome-1.eps,width=15cm,height=1cm}
 \end{center}
\end{figure}
\item[2.] Evaluate {\it fitness} by ``How good is each individual?''
\item[3.] Generate an initial {\it population} at random.
\clearpage
\hfill (cont'd)
\item[4.] Evolve them with
   \begin{itemize}
   \item {\it Selection}
   \item {\it Crossover}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=crossover.eps,width=15cm,height=1.5cm}
 \end{center}
\end{figure}
%% \clearpage
%% \hfill (cont'd)
   \item {\it Mutation}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=mutation.eps,width=15cm,height=0.8cm}
 \end{center}
\end{figure}
   \end{itemize}
\item[5.] Better Solutions from {\it generation} to {\it generation.}
\end{itemize}
\end{large}
%------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ A Conceptual Illustration of Fitness Landscape}\\

\begin{figure}[h]
 \begin{center}
   \epsfile{file=fitness-ls-2d.eps,width=11.25cm,height=6cm}
 \end{center}
\end{figure}

%%----------------------------------------------------------------------------
\clearpage 
\noindent
{\bf $\Box$ One specific solution were already known --- Hebbian peak.}
\hspace*{1in}\\
\begin{center}
``When fire then wire'' principle\\
\hspace*{1in}\\
$\Downarrow$\\
\end{center}

\begin{displaymath}
w_{ij}= sgn(R_i - M/2) \cdot sgn(R_j - M/2)
\end{displaymath}


%%------------------------------------------------------------------------------------------------
\clearpage
$\bullet$ {\bf A Downhill Walk by Flipping zero to one:}
\begin{figure}[h]
 \begin{center}
   \epsfile{file=zero-to-one.eps,width=18cm,height=8cm}
 \end{center}
\end{figure}

%%----------------------------------------------------------------------------(NEW)
\clearpage
\noindent
{\bf $\Box$ Search for an island}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=island-and-needle.eps,width=14cm,height=8cm}
 \end{center}
\end{figure}

%%----------------------------------------------------------------------------(NEW)
\clearpage
\noindent
{\bf $\Box$ Hinton \& Nowlan's Search for A-needle-in-haystack}\\
\begin{itemize}
\item A-needle $\Rightarrow$ Only one configuration of 20 bits of binary string.
\item Haystack $\Rightarrow$ $2^{20}-1$ search points.
   \begin{itemize}
   \item[$\cdot$] say, $(11111111110000000000)$ is assigned fitness one, \\
                  while others are fitness zero.
   \end{itemize}

%%----------------------------------------------------------------------------(NEW)
\clearpage
\noindent
{\bf $\Box$ Black box to detect a needle}
\begin{figure}[hbtp]
 \begin{center}  
   \epsfile{file=needle-b-box.eps,width=10.5cm,height=7.5cm}
 \end{center}
\end{figure}

%%----------------------------------------------------------------------------
\clearpage
\hfill (cont'd)
\item {\bf lifetime learning of each individual (Baldwin Effect).}
   \begin{itemize} 
   \item[$\cdot$] about 25\% are ``1'',  25\% are ``0'', and the rest of the  50\% are ``?''~. \\
   \item[$\cdot$] They are evaluated with all the ``?''~position being assigned ``1'' or ``0''
                  at random $\Rightarrow$ {\it learning}\\
   \item[$\cdot$] Each individual repeats the learning up to 1000 times \\
   \item[$\cdot$] If it reaches the point of fitness one at the $n$-th trial, \\
         then the {\it degree to which learning succeeded} is calculated as
\begin{displaymath}
1+19\cdot (1000-n)/1000.
\end{displaymath}
\end{itemize}
\end{itemize}

%%----------------------------------------------------------------------------(NEW)
\clearpage
\noindent
{\bf $\Box$ Search for a needle}

\begin{figure}[h]
 \begin{center}
   \epsfile{file=hinton-nowlan.eps,width=15cm,height=8cm}
 \end{center}
\end{figure}

%%------------------------------------------------------------------------------------------------
\clearpage
\begin{large}
\begin{itemize}
\setlength{\itemsep}{0pt}
\item[] {\bf Test Function --- A Tiny Flat Island in a Huge Lake:}
   \begin{itemize}
   {\it
   \item[] Assuming a $n$-dimensional hypercube all of whose coordinate $x_i$ are
      \begin{displaymath}
      -1 \le x_i \le 1  \hspace{.1in} (i=1, \cdots, n),
      \end{displaymath}
   find an algorithm to locate a point in the region $A$
      \begin{displaymath}
      A: \{(x_1, x_2, \cdots , x_n) \hspace{.1in} \vert \hspace{.1in} 
      0 \le x_i \le a \hspace*{.1in}  (i=1,\cdots, n) \},
      \end{displaymath}
   where all the points in $A$ are assigned fitness = 1 and elsewhere 0. 
}
   \end{itemize}
\end{itemize}
\end{large}

\begin{figure}[h]
 \begin{center}
   \epsfile{file=all-positive-area.eps,width=3cm,height=3cm}
 \end{center}
\end{figure}

%%----------------------------------------------------------------------------(NEW)
\clearpage
\noindent
{\bf $\Box$ How can AIS search for a needle, if any?}\\

\noindent
Function-optimization, pattern-classification, anomaly-detection, or ...?
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=rastrigin-2.eps,width=4.0cm,height=3.0cm}
   \epsfile{file=two-gaussian-domain-2.eps,width=4.0cm,height=3.0cm}
   \epsfile{file=origin-vs-els.eps,width=4.0cm,height=3.0cm}
   \epsfile{file=a-needle-0.eps,width=5.0cm,height=3.0cm}
 \end{center}
\end{figure}
\begin{itemize}
\item Cronal Selection?
\item Negative Selection?
\item $\cdots$
\item With what and how?
\end{itemize}


%%------------------------------------------------------------------------------------------------
\clearpage
\begin{large}
\noindent
{\bf $\Box$ Conclusion}
\begin{itemize}
\item[$\bullet$] Can {\it Artificial Immune System} search for a needle? \\
\item[$\bullet$] If not, what method would be? \\
\hspace*{1in}\\
\hspace*{2in}--- A Call for Challanges!
\end{itemize}
\end{large}



\end{Large}
\end{document}