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\title{Associative Memory by Evolving Spiking Neurons}
\author{Akira Imada\\
        Departments of Computers and Computer Systems\\
        Brest State Technical University\\
        Moskowskaja 267, Brest, 224017 Republic of Belarus\\
        e-mail: ai@brpi.unibel.by}
\date{}

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\begin{abstract}
\noindent
We simulate an associative memory model by spiking neural network in which we
use an evolutionary computation to determine synaptic weights.
\end{abstract}

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%%We simulate an associative memory model by spiking neural network in which we
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\section*{Introduction}
\noindent

Associative memory is a memory in which we can store information
and recall it from its partial and/or imperfect stimuli. A new 
information is stored as an attractor and if the stimuli is
within the basin size of the attractor, it leads to the attractor. 
This resembles human memory in a sense.
We can recognize a friend despite her/his face will never be 
the same when we first met. 
Hopfield (1992) proposed a fully connected neural network model of 
associative memory in which information is stored distributedly 
among neurons as attractors. The model had been well studied
for more than a decade since then, and we now know it's not so 
practical partly due to its storage capacity. 
The dynamics of neurons' state still attract interests of e.g. 
some physicist and exists some unknown issue though.

\section*{Associative Memory by Evolving Spiking Neurons}
In this paper, we study spiking neurons to realize the associative memory.
Spiking neural networks was originally proposed by ... as a more 
biologically plausible model. ... proposed associative memory model
using spiking neurons where learning scheme was given analytically.

The dynamics of a network of $N$ spiking neurons are described as
\begin{displaymath}
\dot{V_i}(t)= f(V_i(t), W_{i1}(t),W_{i2}(t),\cdots W_{in}(t))+ I_i(t), \hspace{.1in}
(i = 1, 2, \cdots, N)
\end{displaymath}
where $V_i(t)$ is the membrane potential of neuron $i$, $W_{ij}$'s are
$n$ variables necessary for spiking behavior each of which is also expressed as  
$\dot{W}_{ij}(t) = g_j(V_i(t), W_{i1}(t),W_{i2}(t),\cdots W_{in}(t))+ I_i(t), \hspace{.1in}
(j = 1, 2, \cdots, n)$,
and $I_i(t)$ is an electric current to the neuron $i$ from the other neurons including
ones via plastic synapses $J_{ij}$ which is an efficacy of the synapse from neuron $j$ to $i$.
The function $f(\cdot)$, $g(\cdot)$ and number of parameters $n$ depend on Model we employ,
such as Hodgkin-Huxley, FitzHugh-Nagumo, etc. (See Yoshioka (in press) more in detail.)

Pattern to be stored is defined as a firing timing within a prespecified duration $T$ 
under the assumption that every neuron fires only once during the duration $T$. 
That is, $\mu$-th pattern is defined as:
\begin{displaymath}
s_j^{\mu} \in [0,T), \hspace{.1in} (i = 1, 2, \cdots, N)
\end{displaymath}

As Edleman (1987) once proposed Neuronal Evolution, we assume evolution
exist as a neuron's level in our brain, not a specie's level. 
Here, we propose a method to determine $W{ij}$ by means of {\it evolution}

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\section*{Reference}
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\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[]Yoshioka, M (in press)
{\em The Spike-timing-dependent Learning Rule to Encode Spatiotemporal
Patterns in a Network of Spiking Neurons.}
Physical Review.

\item[]Edelman, G.\ M.\ (1987)
{\em Neural Darwinism: The Theory of Neuronal Group Selection}
Basic.

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