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\begin{document}

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\title{
  \vspace{0.3cm}
  \begin{minipage}[c]{0.12\textwidth}
    \includegraphics[width=\textwidth]{tum-lbl.eps}
  \end{minipage}
  \hfill
  \begin{minipage}[c]{0.7\textwidth}
    \begin{center}
    \bf  
    Minimal Model of Prey Localization \\ through the Lateral-Line
    System
    \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[c]{0.12\textwidth}
    \includegraphics[width=\textwidth]{tum-lbl.eps}
  \end{minipage}
}

\author{
  \vspace*{-0.5cm} \\
  Jan-Moritz P. Franosch$^1$, 
  Andreas Elepfandt$^2$, and J. Leo van Hemmen$^1$ \\
  $^1$ Physik Department, TU M\"{u}nchen,
  85747 Garching bei M\"{u}nchen,
  Germany \\
  $^2$ Institut f\"{u}r Biologie, Humboldt Universit\"{a}t,
  Invalidenstra{\ss}e 43, 10115 Berlin, Germany
}

\date{}


\maketitle

\thispagestyle{empty}

\vspace*{-0.7cm}
\textbf{ The clawed frog \emph{Xenopus} is a predator catching prey at
  night by detecting water movements.  We present a general method, a
  `minimal model' based on a minimum-variance estimator, to explain
  prey detection through the frog's lateral-line organs.  Waveform
  reconstruction allows \emph{Xenopus} to determine both direction and
  character of the prey and even to distinguish two simultaneous wave
  sources.}

\begin{figure}[H]
  \includegraphics[width=0.49\columnwidth]{figure1.eps}
  \begin{minipage}[b]{0.49\columnwidth}
    \caption[]{ \small \bf 
      The clawed frog's lateral-line organs can be seen as
      white ``stitches''.
    }
  \end{minipage}
  \label{fig:Xenopus}
\end{figure}

In the present case water can be taken as a linear system
\citep{Lamb:1932} where the deflection $y_i$ of cupula $i$ is linear
in the stimulus $x^{\mathbf{p}}$ at position $\mathbf{p}$ on the water
surface,
\begin{equation}
  y_i(t) = (h_i^{\mathbf{p}} \star x^{\mathbf{p}})(t) = \int_{-\infty}^\infty 
  h_i^{\mathbf{p}} (\tau) \, x^{\mathbf{p}}(t-\tau) \,\mathrm{d}\tau \,.
  \label{eq:M}
\end{equation}
Here $h_i^{\mathbf{p}}$ is the so-called impulse response at cupula
$i$. An approximation of the transfer function is given by
\begin{equation}
  \label{eq:H}
  H(\omega) = \sqrt{\frac{r_0}{r}} \, D_{\Delta\varphi} \,
  \exp\left[\frac{4 \nu k^3}{\omega} (r_0-r) + i k(r_0-r) \right] \,.
\end{equation}
We minimize the expectation value of the least-squares error
\begin{equation}
  \label{eq:E}
  ||x^{\mathbf{p}}-\hat{x}^{\mathbf{p}}||^{2} = 
  \int_{0}^{T_{I}} [x^{\mathbf{p}}(t) - 
  \hat{x}^{\mathbf{p}}(t)]^2 \,\mathrm{d}t \,.
\end{equation}
The solution minimizing the error in \eqref{eq:E} can be shown to be
\begin{equation}
  \label{eq:x}
  \fcolorbox{black}{yellow}{
  $\displaystyle
  \hat{x}^{\mathbf{p}} = \sum_j s_j^{\mathbf{p}} \star y_j \,, \quad   
  S_j^{\mathbf{p}}(\omega) = 
  \frac{H_j^{\mathbf{p}*}(\omega)}{\sum_i |H_i^{\mathbf{p}}(\omega)|^2
    + \sigma^2}
  $}
\end{equation}
The functions $S_j^{\mathbf{p}}$ are the Fourier transforms of the
\emph{reverse} transfer functions $s_j^{\mathbf{p}}$.

\begin{figure}[h]
  \includegraphics[width=0.7\columnwidth]{figure2.eps}
  \begin{minipage}[b]{0.29\columnwidth}
    \caption[]{
       \small \bf \\ Connections of a neu\-ron, sensitive for direction
        $\varphi$, to the lateral-line organs.
    }
  \end{minipage}
  \label{fig:neuron}
\end{figure}

\textcolor{myred}{Membrane potential}
$\textcolor{myred}{V^{\mathbf{p}}} \approx \hat{x}^{\mathbf{p}}$ of
the spike-response \cite{Gerstner:NNII-1994} neuron 
\begin{equation}
  V^{\mathbf{p}}(t) = 
  \sum_{i,k,f}
  J_{ik}^{\mathbf{p}} \varepsilon(t-t_i^f-\Delta_{ik}^{\mathbf{p}}) +
  \sum_{i,k,f'} J_{ik}^{\mathbf{p}'} \varepsilon(t-t_i^{f'}-\Delta_{ik}^{\mathbf{p}'})
\end{equation}
where the $t_i^f$ are the firing times of the nerve from lateral-line
organ $i$ and $\Delta_{ik}^{\mathbf{p}}$ is the delay time of synapse
$k$ with synaptic strength $J_{ik}^{\mathbf{p}}$.

\newpage
\vspace*{-1.2cm}
\begin{figure}[H]
  \begin{center}
  \includegraphics*[width=0.37\textwidth]{figure3.eps}
    \vspace*{-2.5ex}
    \caption[]{ \small \bf
      Neurons \textcolor{mygreen}{responding} strongest ($\varphi
      \approx 0$) tell \emph{Xenopus} the direction of the wave
      source.  The \textcolor{myred}{membrane potential} of these
      neurons gives \emph{Xenopus} an approximation to the actual wave
      form and allows the animal to distinguish different kinds of
      prey.  }
    \label{fig:map_neuron}
  \end{center}
  \begin{center}
    \includegraphics[width=0.37\textwidth]{figure4.eps}
    \vspace*{-2.5ex}
    \caption[]{ \small \bf
      \textbf{Top:} \emph{Xenopus'} experimental response angle
      \cite{Claas:JCP-1996} versus stimulus angle. Left: intact
      \emph{Xenopus}. Right: lateral-line organs at the right-hand
      side have been deactivated.  \textbf{Bottom:} Response of our
      neuronal model. }
    \label{fig:dots_neuron}
  \end{center}
  \begin{center}
  \includegraphics*[width=0.37\textwidth]{figure5.eps} 
    \vspace*{-2.5ex}
    \caption[]{ \small \bf
      `Map' like that of \figref{fig:map_neuron} for 
      \emph{two} wave sources, positioned at $\varphi=-45^{\circ}$ and
      $45^{\circ}$. With the help of its
      \textcolor{myred}{evaluations}, \emph{Xenopus} could easily
      distinguish position and waveform of the sources, as in
      experiment \cite{Elepfandt:NL-1986}.}
    \label{fig:map2}
  \end{center}
\end{figure}

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\bibitem{Lamb:1932}
Lamb, H. (1932) \emph{Hydrodynamics} (Cambridge University Press) 6th edn.
  \S\S226ff, 246, 331, 332.
  
\bibitem{Gerstner:NNII-1994} Gerstner, W. \& van Hemmen, J.~L. (1994)
  in \emph{Models of Neural Networks II}, eds. Domany, E., van Hemmen,
  J.~L. \& Schulten, K. (Springer, New York) chap.~1 pp. 39--47.

\bibitem{Claas:JCP-1996}
Claas, B. \& M\"{u}nz, H. (1996) \emph{J. Comp. Physiol.} \textbf{178},
  253--268.

\bibitem{Elepfandt:NL-1986}
Elepfandt, A. (1986) \emph{Neurosci. Lett. (Suppl.)} \textbf{26}, 380.

\end{thebibliography}



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