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\title{Playpen: Toward an Architecture for Modeling the Development of
Spatial Cognition\thanks{
This report has been prepared in the form of an HTML document, which is
available on the World-Wide Web at
{\tt ftp://ftp.cs.indiana.edu/gasser/Playpen/TR1/index.html}.
It makes extensive use of color figures, animated applets, and hyperlinks.
There is a Postscript version of the report available by anonymous ftp
to {\tt ftp.cs.indiana.edu} at {\tt /pub/gasser/Playpen/TR1.ps}.
This includes the color figures but not the applets and hyperlinks.
Therefore we highly recommend the version on the Web.
}}
\author{$^{1,2}$Michael Gasser and $^{1}$Eliana Colunga-Leal\\ 
$^{1}$Computer Science Department\\ 
$^{2}$Linguistics Department\\ 
$^{1}$Cognitive Science Program\\ 
Indiana University\\ 
Bloomington, IN 47405, USA\\ 
{\tt \{gasser,ecolunga\}@cs.indiana.edu}}


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\stepcounter{subsubsection}
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{\newpage
\clearpage
\samepage \begin{equation}\Delta a_{i}^{t} = h_{i}^{t} (a^{max}_{i} - (a_{i}^{t-1} -
	 D_{i} a_{i}^{t-1}))
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}h_{i} = 
\sum_{j=1}^{n}
a_{j} \cdot w_{ij} \cdot \Phi_{ij} (\varphi_{i} - \varphi_{j})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta \varphi_{i} =
\frac{\pi}{\sum_{j=1}^{n}}
\sum_{j=1}^{n} a_{j} \cdot w_{ij} \cdot
\Phi_{ij}^{\prime} (\varphi_{i} - \varphi_{j}),
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure*}% latex2html id marker 1176
[htb]\centerline{\textsc{Demo \thedemo }: \textbf{Object Units} (only accessible in the WWW version of the report)}\addtocounter{demo}{1}\label{demo:Object Units}\end{figure*}
}

{\newpage
\clearpage
\samepage % latex2html id marker 1947
\thedemo
}

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{\newpage
\clearpage
\samepage \begin{figure*}% latex2html id marker 1208
[htb]\centerline{\textsc{Demo \thedemo }: \textbf{Relation Units} (only accessible in the WWW version of the report)}\addtocounter{demo}{1}\label{demo:Relation Units}\end{figure*}
}

{\newpage
\clearpage
\samepage % latex2html id marker 1960
\thedemo
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}h_{i}^{t} = 
\sum_{j=1}^{n} \sum_{k=0}^{m}
a_{j}^{t-k} \cdot w_{ij}^{k} \cdot \Phi_{ij}^{k} (\varphi_{i}^{t} -
\varphi_{j}^{t-k})
\label{delay}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure*}% latex2html id marker 1224
[htb]\centerline{\textsc{Demo \thedemo }: \textbf{Delay Connections} (only accessible in the WWW version of the report)}\addtocounter{demo}{1}\label{demo:Delay Connections}\end{figure*}
}

{\newpage
\clearpage
\samepage % latex2html id marker 1965
\thedemo
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}\Delta w_{ij} \propto
\breve{a}_{i}^{+} \cdot \breve{a}_{j}^{+} \cdot
\Phi_{ij}(\breve{\varphi}_{i}^{+} - \breve{\varphi}_{j}^{+}) -
\breve{a}_{i}^{-} \cdot \breve{a}_{j}^{-} \cdot
\Phi_{ij}(\breve{\varphi}_{i}^{-} - \breve{\varphi}_{j}^{-}),
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure*}% latex2html id marker 1232
[htb]\centerline{\textsc{Demo \thedemo }: \textbf{Learning} (only accessible in the WWW version of the report)}\addtocounter{demo}{1}\label{demo:Learning}\end{figure*}
}

{\newpage
\clearpage
\samepage % latex2html id marker 1976
\thedemo
}

\stepcounter{section}
\appendix
\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}F = E + S
\label{energy}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}S = \sum_{i=1}^{n} \int_{rest_{i}}^{a_{i}} f_{i}^{-1} (a) da
\label{penalty}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}E = -\frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n} a_{i} \cdot w_{ij}
   \cdot a_{j} \cdot \Phi_{ij}(\varphi_{i} - \varphi_{j})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Phi_{i \rightarrow j} = \Phi_{j \rightarrow i}
\label{cpl_symmetry}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Phi_{i \rightarrow j}' = - \Phi_{j \rightarrow i}'
\label{cplprime_antisymmetry}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f_{i}^{-1} (\breve{a}_{i}) = \breve{h}_{i} =
\sum_{j=1}^{n} \breve{a}_{j} w_{ij} \breve{\Phi}_{j \rightarrow i}
\label{equilibrium_input}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta \breve{\varphi}_{i} =
\frac{\pi}{\sum_{j=1}^{n} w_{ij}}
\sum_{j=1}^{n} \breve{a}_{j} w_{ij} \breve{\Phi}_{j \rightarrow i}' = 0
\label{equilibrium_phase}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}J = \breve{F}^{(+)} - \breve{F}^{(-)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\breve{E} = -\frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n}
\breve{a}_{i} w_{ij} \breve{a}_{j} \breve{\Phi}_{j \rightarrow i}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\breve{E} =
-\frac{1}{2} \left(
2\breve{a}_{i}w_{ij}\breve{a}_{j}\breve{\Phi}_{j \rightarrow i}
+ \sum_{k=1}^{n}
\sum_{\stackrel{l=1}{\scriptscriptstyle k,l\neq i,j;k,l\neq j,i}}^{n}
 \breve{a}_{k} w_{kl} \breve{a}_{l} \breve{\Phi}_{l \rightarrow k}
  \right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\frac{\partial\breve{E}}{\partial w_{ij}} & = &
-\frac{1}{2}
\left[
 2\breve{a}_{i}\breve{a}_{j} 
\breve{\Phi}_{j \rightarrow i} \nonumber + 
2 w_{ij} \breve{a}_{i} \breve{\Phi}_{j \rightarrow i}
\frac{\partial \breve{a}_{j}}{\partial w_{ij}} +
2 w_{ij} \breve{a}_{j}
\breve{\Phi}_{j \rightarrow i}
\frac{\partial \breve{a}_{i}}{\partial w_{ij}} +
\right. \\ 
& & \breve{a}_{i} w_{ij} \breve{a}_{j}
\left(
\breve{\Phi}_{j \rightarrow i}'
 \left( \frac{\partial\breve{\varphi}_{i}}{\partial w_{ij}}
- \frac{\partial\breve{\varphi}_{j}}{\partial w_{ij}}\right) +
\breve{\Phi}_{i \rightarrow j}'
\left( \frac{\partial\breve{\varphi}_{j}}{\partial w_{ij}}
- \frac{\partial\breve{\varphi}_{i}}{\partial w_{ij}}\right)
\right) \nonumber + \\ 
& & 
\left. \sum_{k=1}^{n} \sum_{\stackrel{l=1}
{\scriptscriptstyle k,l\neq i,j;k,l\neq j,i}}^{n} w_{kl} 
\left(
\breve{a}_{k} \breve{\Phi}_{l \rightarrow k}
\frac{\partial \breve{a}_{l}}{\partial w_{ij}} +
\breve{a}_{l} \breve{\Phi}_{l \rightarrow k}
\frac{\partial \breve{a}_{k}}{\partial w_{ij}} +
\breve{a}_{k} \breve{a}_{l} \breve{\Phi}_{l \rightarrow k}'
\left( \frac{\partial\breve{\varphi}_{k}}{\partial w_{ij}}
- \frac{\partial\breve{\varphi}_{l}}{\partial w_{ij}}\right) \right)
\right]
\label{energy_hairy}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}& \breve{a}_{i} w_{ij} \breve{a}_{j}
\left(
\breve{\Phi}_{j \rightarrow i}'
\left( \frac{\partial\breve{\varphi}_{i}}{\partial w_{ij}}
- \frac{\partial\breve{\varphi}_{j}}{\partial w_{ij}} \right) +
\breve{\Phi}_{i \rightarrow j}'
\left( \frac{\partial\breve{\varphi}_{j}}{\partial w_{ij}}
- \frac{\partial\breve{\varphi}_{i}}{\partial w_{ij}} \right)
\right) \\ 
& = 2 w_{ij} \breve{a}_{i} \breve{a}_{j}
\breve{\Phi}_{j \rightarrow i}'
\frac{\partial\breve{\varphi_{i}}}{\partial w_{ij}} +
2 w_{ij} \breve{a_{i}} \breve{a_{j}}
\breve{\Phi}_{i \rightarrow j}'
\frac{\partial\breve{\varphi_{j}}}{\partial w_{ij}} \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\sum_{k=1}^{n} \sum_{\stackrel{l=1}
{\scriptscriptstyle k,l\neq i,j;k,l\neq j,i}}^{n} w_{kl} 
\left( \breve{\Phi}_{l \rightarrow k}'
\left( \frac{\partial\breve{\varphi_{k}}}{\partial w_{ij}}
- \frac{\partial\breve{\varphi_{l}}}{\partial w_{ij}}\right) \right) =
2 \sum_{k=1}^{n} \sum_{\stackrel{l=1}
{\scriptscriptstyle k,l\neq i,j;k,l\neq j,i}}^{n}
w_{kl} \breve{a_{k}} \breve{a_{l}}
\breve{\Phi}_{l \rightarrow k}'
\frac{\partial\breve{\varphi_{k}}}{\partial w_{ij}}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial\breve{E}}{\partial w_{ij}} =
-\frac{1}{2} \left(
2\breve{a}_{i}\breve{a}_{j}\breve{\Phi}_{j \rightarrow i} +
2 \sum_{k=1}^{n}\frac{\partial\breve{a}_{k}}{\partial w_{ij}}
\sum_{l=1}^{n}w_{kl}\breve{a_{l}}
\breve{\Phi}_{l \rightarrow k} +
2 \sum_{k=1}^{n} \breve{a_{k}}
\frac{\partial\breve{\varphi_{k}}}{\partial w_{ij}}
\sum_{l=1}^{n}w_{kl}\breve{a_{l}}
\breve{\Phi}_{l \rightarrow k}'
\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial \breve{E}}{\partial w_{ij}} =
-\breve{a}_{i}\breve{a}_{j}\breve{\Phi}_{j \rightarrow i} -
 \sum_{k=1}^{n} \breve{h}_{k} \left(\frac{\partial
 \breve{a}_{k}}{\partial w_{ij}}\right) -
\sum_{k=1}^{n} \Delta\breve{\varphi_{k}} \breve{a}_{k}
\frac{\partial\breve{\varphi}_{k}}{\partial w_{ij}}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial \breve{E}}{\partial w_{ij}} =
-\breve{a}_{i}\breve{a}_{j}\breve{\Phi}_{j \rightarrow i} -
 \sum_{k=1}^{n} \breve{h}_{k} \left(\frac{\partial
 \breve{a}_{k}}{\partial w_{ij}}\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial \breve{S}}{\partial w_{ij}} =
\sum_{k=1}^{n}f_{k}^{-1}(\breve{a}_{k})
\frac{\partial \breve{a}}{\partial w_{ij}}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial \breve{F}}{\partial w_{ij}} =
-\breve{a}_{i}\breve{a}_{j}\breve{\Phi}_{j \rightarrow i}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial \breve{J}}{\partial w_{ij}}
  \propto \breve{a}_{i}^{(+)} \breve{a}_{j}^{(+)}
\breve{\Phi}_{j \rightarrow i}^{(+)} -
\breve{a}_{i}^{(-)} \breve{a}_{j}^{(-)}
\breve{\Phi}_{j \rightarrow i}^{(-)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta w_{ij} \propto \breve{a}_{i}^{(+)} \breve{a}_{j}^{(+)}
\breve{\Phi}_{j \rightarrow i}^{(+)} -
\breve{a}_{i}^{(-)} \breve{a}_{j}^{(-)}
\breve{\Phi}_{j \rightarrow i}^{(-)}
\end{equation}
}

\stepcounter{section}

\end{document}
