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%% \title{A Study of Synaptic Plasticity of an 
%% Associative Memory Model by Spiking Neurons}


\title{A Mesa Rather than a Peak?
--- A fitness landscape on weight space
of an application using spiking neurons under rate coding.}


%% Can climbers clime the mesa?
%% Mesa like a needle in haystack.

\author{Akira Imada \\
\hspace*{0.1in}\\
Brest State Technical University\\
           Moskowskaja 267, 224017 Brest, Republic of Belarus\\
           akira@bsty.by}
\date{}

\begin{document}
\maketitle
\pagestyle{empty}
\thispagestyle{empty}

\begin{abstract}
\noindent
{\it We simulate an associative memory model using spiking neurons.
Those models in which we specify interaction among McCulloch-Pitts neurons by
a Hebbian-like learning algorithm, for example, already exist.
The Hopfield model is one of these examples. 
Though we have still many unknown issues in the Hopfield model of associative memory,
we have some inevitable drawbacks as well, such as small storage capacity. 
To overcome these drawbacks, and more importantly, to be more biologically 
plausible, we explore the model using spiking neurons.}
\end{abstract}

\hspace*{1in}\\
\noindent
{\bf keyword:} {\it spiking neurons, associative memory, synaptic weight space,
evolutionary hill climbing, downhill walk}

\section{Introduction}
\noindent
Associative memory is a memory system in which we can store information
and recall it later from its partial and/or imperfect stimuli. 
An information is stored as a number of stable states with a domain of attraction
around each of the stable states.
If the system starts with any stimulus within the domain 
it will converge to the attractor following a trajectory, hopefully a short one. 
This models human memory in the sense that
we can recognize our friend's face even without meeting for a long time, or
we can recall a song immediately after listening to a very beginning part of the song.
Hopfield~\cite{hopf} proposed a fully connected neural network model of 
associative memory in which a set of patterns is stored distributedly 
among neurons as attractors. Since then the model had been fairly well studied
for more than a decade, and we now know it is not so practical,
partly due to its small storage capacity. 
Though we still have some interesting unknown issues in the model
such as the number and distribution of the weight solutions which give a network
a function of associative memory
and these issues still attract interests of some physicists,
we study another model using spiking neurons instead of McCulloch-Pitts neurons \cite{mccu}
. The goal is to overcome issues
like small storage capacity and, more importantly,  
to look for more biologically plausible models of human memory.

\section{Spiking Neuron Model}
\noindent
Some regions in our brain such as {\it neocortex} or {\it hippocampus} are said to be 
made up of two categories of neurons, that is, {\it pyramidal cells} and {\it interneurons}.
Typically, the pyramidal cells communicate with each other via {\it excitatory synapses} 
(positive influences), while interneurons send signals to pyramidal cells via
{\it inhibitory synapses} (negative influences). \\

\noindent
As  Wilson~\cite{wils} wrote in his book, Marr~\cite{marr} was one of the first to propose 
this hippocampal model involving both recurrent excitation via Hebbian \cite{hebb}
synapses and inhibition.

\subsection{A description of the model by differential equations.}
Probably one of the most fundamental differential equations is 
\begin{equation}\label{eq:1}
\frac{dr}{dt} = -\frac{1}{\tau}r
\end{equation}
where $r$ is assumed to be a function of time and $\tau$ is referred 
to as {\it time constant}. \\

\noindent
In this paper the variable of our interest is 
the number of spikes emitted from one neuron in unit time as a function of time. 
We call it {\it spiking ratio}  and denote it $r(t)$.
In his book, Wilson~\cite{wils} wrote that
a single neuron which emits spike train 
when it receives an external stimulus $P(t)$ could be modeled simply by adding $S(P)$ 
to the right hand side of Eq.~(\ref{eq:1}) without describing explicitly shape and timing 
of each individual spike. That is,
\begin{equation}
\frac{dr(t)}{dt} = \frac{1}{\tau}(-r(t) + S(P))
\end{equation}
As stimulus $S(P)$,  Wilson~\cite{wils} proposed to employ, among many alternatives, 
Naka-Rushton~\cite{naka} function:

\begin{equation}\label{eq:1}
S(P) = \left\{
\begin{array}{ll}
%%\frac{MP^N}{\sigma^N+P^N} & \hspace{.2in}\mbox{if} \hspace{.1in} P \le 0 \\
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{equation}
where $m$ and $\sigma$ are called {\it saturation} and {\it semi-saturation constant},
respectively, and $n$ is a integer parameter for its graph to fit a phenomena.\\

\noindent
We now assume $N$ pyramidal cells and implicit number of interneurons. 
We simulate these pyramidal cells with spiking neurons which interact with each other
using electric current via plastic synapses.
Pyramidal cells are also interacted by interneurons by global inhibition.
To be more specific, stimuli to one pyramidal cell are given from 
all the other pyramidal cells as well as 
interneuron cells whose number is reduced to only one in this paper,
via synaptic strength. See Fig.~1 bellow.


\begin{figure}[hbtp] 
 \begin{center}
   \epsfile{file=wilson.eps,width=8cm,height=3cm}
\label{fig:1}
\caption{Schematic diagram with three, instead of $N$, pyramidal cells (P) and one global 
interneuron cell (G). Connections with arrow indicate excitatory synapses and with closed circle
inhibition.}
 \end{center}
\end{figure}

\noindent
The synaptic strength from pyramidal cell $j$ to $i$ is denoted as $w_{ij}$ and 
all the inhibitory synapses from interneuron is assumed to have a value $g$.
Then stimulus to the $i$-th pyramidal cell $P_i$ is described as
\begin{displaymath}
P_i = (\sum_{j=1}^{N} w_{ij} \cdot R_j - g \cdot G)_{+}^2
\end{displaymath}
where $(\cdot)_{+}$ means that we use the value if and only if inside the parentheses is 
positive and zero otherwise.
Following Wilson~\cite{wils} we experiment with
$\sigma = 10$, $M = 100$, and $n=2$ in Eq.~(\ref{eq:1}).\\

\noindent
Thus, our equation of spiking ratio of the $i$-th pyramidal cell $R_i$ and 
the one of the interneuron $G$ is given as

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

\begin{equation}\label{eq:2}
\tau_G \frac{dG}{dt} = -G - 0.07 \sum_{j=1}^{N} R_i
\end{equation}

\noindent
where, both $\tau_R$ and $\tau_G$ are to be set to 10.
Also note that $w_{ii}$ $(i = 1, \cdots N)$ should be set to all zero.

\subsection{To encode patterns.}
We have a fair amount of ways to encode $N$-bit patterns using $N$ spiking neurons.
(1) {\it Firing-rate} of a neuron within certain time window could express one bit of 
a continuous real number, or binary number according to whether the rate exceeds
a threshold or not. (2) {\it Time-to-first-spike} could express one bit of continuous real number.
In this case, we neglect informations about spike trains after the first one.
(3) {\it Spike-timing} also could encode real number on the condition that emergence of spike 
of every neuron is periodic and only one spike in one period. 
Or, in this scheme binary bit could also be encoded according to
presence/absence of spike in the period. In this paper, we exploit the first scheme. 

\subsection{To store \& recall patterns.}
In what he calls {\it CA3 network} in his book, Wilson~\cite{wils}  employed 256 pyramidal cells 
so that these cells represent a pattern constructed by $16\times16$ array of pixels. 
The network also incorporates one interneuron cell to provide pyramidal 
cells feedback inhibition.
The task of the network is to recognize 4 given patterns from its noisy input.
Each of the 4 patterns is represented by 32 active cells plus other 224 quiet cells.
Network has learned to recognize these 4 patterns by modifying the synapses according to

\begin{equation}
w_{ij} = k \cdot \mbox{sgn}(R_i - 0.5M) \cdot \mbox{sgn}(R_j-0.5M)
\end{equation}
if the previous value of $w_{ij} = 0$, otherwise 
$w_{ij} = 1$ is not modified.
Here $k$ is set to 0.016 and note that $\mbox{sgn}(x) = 1$ if $x>0$ and 0 otherwise.
A noisy input of a pattern is constructed by randomly picking up
about one-third of the active cells of the selected pattern
with adding them other 20 quiet cells, also chosen at random,
after turning them active.\\

\noindent
Then one of these 4 patterns is given to the network, that is, network starts 
the dynamics with the pattern as the initial configuration of its neurons' state.
Network updates the state according to Eq.~(\ref{eq:2}).
The dynamics is observed during a total of 100ms (assuming step of $dt$ of $dr/dt$ 1ms), 
while the noisy input remains ON for the 1st 20 ms.

\section{Experiment}
In his book, Wilson \cite{wils} assigns values of $w_{ij}$ employing the Hebbian algorithm,
and succeeded in storing 4 patterns and recalling each of them from each of their
noisy input. We tested Wilson's experiment with the same set of patterns and 
succeeded in obtaining the similar results when we also apply the Hebbian algorithm to store
the patterns.

\noindent
The synaptic weights plasticity is very robust. If we modify lots of weights, the network 
still recall the pattern correctly. That is to say, the top region of the Hebbian
peak in weight space is broad. However if explore outside edge of the top region of 
the peak, all neurons become immediately quiescent or fire at their maximum rate.

\subsection{Are Hebbian weights unique?}
Then the question is ``Are Hebbian weights unique solution?''
Or, ``Are Hebbian weights
biologically plausible?'' In order to tackle this issue, we search for other 
configuration of weight values. For the purpose we employ an evolutionary computation.

\subsection{Searching for other solutions.}
The number of synapses among pyramidal cells is $N^2$, including the one to itself.
We represent one candidate solution of a weight configuration by one dimensional
binary string of length $N^2 = 256 \times 256 = 65,536$ called chromosome.
Fitness, the degree to which how good each corresponding phenotype performs, 
is evaluated as
the ratio of how many pixels out of 256 are
correctly recovered when it's noisy input is given as the initial state, 
using Wilson's example of 4 patterns.
Result, however, is unpredictably horrible. It is as if a  search for
only one needle in a large haystack. What we observed is a series of fitness zero 
even after observing thousands of randomly created chromosomes. 

\subsection{Downhill walk from the top of Hebbian peak.}
In order to know what is going on, we learn 
what does the fitness landscape on the weight space look like
in which the hight of the land is almost everywhere zero except for already known
Hebbian peak. \\

\noindent 
Downhill walk is in two different ways. The one is by flipping only the bit that is one.
The other is opposite, that is, by flipping the bit one to zero. 
Both are by flipping bit by bit.  

\section {Results \& Discussion}
\noindent
The results are shown in Fig.~2 and Fig.~3.

\begin{figure}[h]
 \begin{center}
   \epsfile{file=one-to-zero.eps,width=8cm,height=6.2cm}
\caption{A downhill walk from top of the Hebbian peak with flipping only a gene
whose allele is one.}
 \end{center}
\end{figure}
\begin{figure}[h]
 \begin{center}
   \epsfile{file=zero-to-one.eps,width=8cm,height=6.2cm}
\caption{A downhill walk from top of the Hebbian peak with flipping only a gene
whose allele is zero.}
 \end{center}
\end{figure}

\noindent
As shown in both figures, the top region of the peak is not extremely narrow.
Of course the ratio of area of top region of the peak is a kind of infinitesimal
if we compare it to the whole area of search space. However, this is not
so special case. Generally, the solution to a problem does not have a top region
but just a point. What is problematic here is a very steep slope of side wall 
of the peak. It is terribly difficult to find a-little-better-solutions-than-others.\\

\noindent
The results are very different from the similar experiment in Hopfield model.
As our previous report the solutions are not unique at all and it is quite
easy to find one of the other solutions. See Imada~\cite{imad}, for example.

\section{Conclusion}
We have described a neural network model of associative memory using spiking neurons.
We explore the Wilson's model \cite{wils} in which 
each neuron emits spike train and 
when spiking ratio exceeds a threshold the corresponding pixel is ON and otherwise OFF.
Wilson \cite{wils} specified the strength (weight) of communication among neurons via synapses
by Hebbian principle \cite{hebb}. 
It was ascertained here too, however, we doubt this is not the 
only solution. In order to other possible configurations of weight values, 
we have  explored, in this paper, the search space by evolutionary computations, 
but the result was 
worse than a-needle-in-hay-stack. We study why and conclude this is due to the shape
of the peak in the fitness landscape. That is to say, side walls of the peak 
that we already know the location (Hebbian peak) is too steep to be climbed. \\

\noindent
As a next step, we now are simulating a hill-climbing who tries to climb the peak
from just the outskirt of the peak. We also plan to experiment of an evolution
in which each individual employs his/her learning during lifetime,
like the famous experiment by Hinton and Nowlan \cite{hint}, so that we expect
that we will find a tiny needle from hay stack.



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Proceedings of the National Academy of Sciences, USA 79,  pp.2554-2558.

\bibitem{hebb}D. O. (1949)
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\bibitem{mccu}W. S. McCulloch and W. Pitts (1943) 
{\em A Logical Calculus of Ideas Immanent in Nervous Activity.}
Bulletin of Mathematical Biophysics, 5, pp.115--113.

\bibitem{wils}H. R. Wilson (1999)
{\em Spikes, Decisions, and Actions: The Dynamical Foundations of Neuroscience.}
Oxford University Press.

\bibitem{marr}D. Marr (1971)
{\em A Theory for Archicortex.}
Philosophical Transactions of the Royal Society of London, B262, pp.23--81,

\bibitem{naka}K. I. Naka and W. A. Rushton (1966)
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\bibitem{imad}A. Imada (2002)
{\em A Fate of Two Evolutionary Walkers after the Departure from the Origin.}
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\bibitem{hint}G. E. Hinton and S.J. Nowlan (1987)
{\em How Learning can Guide Evolution.}
Complex Systems, 1, pp.495--502. 

\end{thebibliography}

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