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\newif\ifpsadobeflag

\newread\psadobefile

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\catcode`\%=14


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\title{Probability Interpretation of Population Codes}

\author{Terence D. Sanger\\ 
Massachusetts Institute of Technology\\ 
Room E25-526\\ 
Cambridge, MA 02139\\ 
(617)253-5769  tds@ai.mit.edu}


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        \kern-#2}%
     \kern#1\hfil}}

\def\psfloater#1#2#3#4#5{#2.#3#4#5}

\def\psklutz#1{%
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\def\psadobeextract#1 #2 #3 #4 #5%
  \global\psadobey= #3 #6bp
  \global\psadobew= #4 #6bp
  \global\advance\psadobew by -\psadobex
  \global\psadobeh= #5 #6bp
  \global\advance\psadobeh by -\psadobey}

\def  \psadobebb #1%
    \else \psadobeflagfalse \psadobeextract#2{#3}

\def\psadobeheader#1#2{\openin\psadobefile=#2
  \global\psadobex= 72 #1bp
  \global\psadobey= 72 #1bp
  \global\psadobew= 72 #1bp
  \global\psadobeh= 72 #1bp
  \ifeof\psadobefile
    \message{Postscript file not found: #2.}\else
    \begingroup
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	  \ifeof\psadobefile
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\def\psadobescale#1#2{%
  \multiply\psadobex by #1 \divide\psadobex by #2
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  \multiply\psadobeh by #1 \divide\psadobeh by #2}

\def\psadobemagnify#1#2#3{%
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  \psfile{\psadobex}{\psadobey}{\psadobew}%
         {\psadobeh}{#1}{#3}}

\def\psadobe#1{\xpsadobe{#1}{#1}}

\def\xpsadobe#1#2{%
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  {\float=\mag \advance\float by 10000
   \psfile{\psadobex}{\psadobey}{\psadobew}%
         {\psadobeh}{\expandafter\floater\the\float}{#2}}}

\def\psadobenomag#1{%
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\def\psadobepercent#1#2{\xpsadobepercent{#1}{#2}{#2}}

\def\xpsadobepercent#1#2#3{%
  \psadobeheader{}{#2}%
  \global\divide\psadobex by 100\global\multiply\psadobex by #1%
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  \global\divide\psadobeh by 100\global\multiply\psadobeh by #1%
  {\float=\mag \multiply\float by #1\divide\float by 100
	\advance\float by 10000
	  \psfile{\psadobex}{\psadobey}{\psadobew}%
	       {\psadobeh}{\expandafter\floater\the\float}{#3}}}

\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[t]
\begin{center}

\openin\psadobefile=cod-noti.ps
\global\psadobex= 72 bp
\global\psadobey= 72 bp
\global\psadobew= 72 bp
\global\psadobeh= 72 bp
\ifeof\psadobefile
\message{Postscript file not found: cod-noti.ps.}\else
\begingroup
\catcode`\%=12
\loop
\ifeof\psadobefile
\psadobeflagfalse
\errmessage{No Bounding Box information in cod-noti.ps}\else
\read\psadobefile to \line
\expandafter\psadobebb\line %
\ifpsadobeflag\repeat
\catcode`\%=14
\endgroup\fi%
\global\divide\psadobex by 100\global\multiply\psadobex by 70%
\global\divide\psadobey by 100\global\multiply\psadobey by 70%
\global\divide\psadobew by 100\global\multiply\psadobew by 70%
\global\divide\psadobeh by 100\global\multiply\psadobeh by 70%
{\float=\mag \multiply\float by 70\divide\float by 100
\advance\float by 10000

\unhbox\psvoid%
\hbox to \psadobew{%
\kern-\psadobex%
\vbox to \psadobeh{%
\vfil%
\kern\psadobey%
\special{psfile=cod-noti.ps hscale=\expandaftert.he\float vscale=\expandaftert.he\float}%
\kern-\psadobey}%
\kern\psadobex\hfil}}
\end{center}

\label{fig-coding}
\end{figure}
}

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{\newpage
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\samepage \begin{displaymath}\sigma_i(x) = \frac{1}{s\sqrt{2\pi}}e^{-(x-m_i)^2/s^2}  \end{displaymath}
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{\newpage
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\samepage \begin{displaymath}\sigma_i(\hat{x}) = P[\hat{x}|x=m_i]  \end{displaymath}
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\hat{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline315: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{displaymath}P[x=m_i|\hat{x}] = \frac{\sigma_i(\hat{x})P[x=m_i]}{P[\hat{x}]} \end{displaymath}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\sigma_i(\hat{x})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline321: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\samepage \setbox\sizebox=\hbox{$P[x]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline323: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P[\hat{x}]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline325: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\samepage \setbox\sizebox=\hbox{$\sigma_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline327: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{equation}E[\sigma_i(x)]  =  \int \sigma_i(x) P[x]dx 
\label{eq-sigma}
\end{equation}
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{\newpage
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\stepcounter{section}
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{\newpage
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{\newpage
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{\newpage
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{\newpage
\clearpage
\samepage \begin{equation}P_y[y] = \int K(x,y) P_x[x] dx 
\label{eq-1}
\end{equation}
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K(x,y) = \delta(y-f(x)) f'(x)^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline375: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi(y) \approx
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{\newpage
\clearpage
\samepage \begin{equation}A = \int \int \phi(y)K(y,x)\tau(x)^T dy dx,  
\label{eq-A}
\end{equation}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$K(y,x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline395: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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\stepcounter{section}
{\newpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\hat{y}=f(x)+\nu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline425: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$E[\phi_j]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline429: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P[\hat{y}]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline431: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\hat{y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline433: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}I(\hat{Y};X) = H(\hat{Y}) - H(\hat{Y}|X) \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$H(\hat{Y}|X) =
H(\nu)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline439: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline441: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \begin{figure}[t]
\begin{center}

\openin\psadobefile=imageall.eps
\global\psadobex= 72 bp
\global\psadobey= 72 bp
\global\psadobew= 72 bp
\global\psadobeh= 72 bp
\ifeof\psadobefile
\message{Postscript file not found: imageall.eps.}\else
\begingroup
\catcode`\%=12
\loop
\ifeof\psadobefile
\psadobeflagfalse
\errmessage{No Bounding Box information in imageall.eps}\else
\read\psadobefile to \line
\expandafter\psadobebb\line %
\ifpsadobeflag\repeat
\catcode`\%=14
\endgroup\fi%
\global\divide\psadobex by 100\global\multiply\psadobex by 60%
\global\divide\psadobey by 100\global\multiply\psadobey by 60%
\global\divide\psadobew by 100\global\multiply\psadobew by 60%
\global\divide\psadobeh by 100\global\multiply\psadobeh by 60%
{\float=\mag \multiply\float by 60\divide\float by 100
\advance\float by 10000

\unhbox\psvoid%
\hbox to \psadobew{%
\kern-\psadobex%
\vbox to \psadobeh{%
\vfil%
\kern\psadobey%
\special{psfile=imageall.eps hscale=\expandaftert.he\float vscale=\expandaftert.he\float}%
\kern-\psadobey}%
\kern\psadobex\hfil}}
\end{center}

\label{fig-diffuse}
\end{figure}
}

{\newpage
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