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{\newpage
\clearpage
\samepage % latex2html id marker 305
\parbox{5in}{\bf
Recent evidence of population coding in motor cortex has
led some researchers to claim that certain
variables such as hand direction or force may be coded within a
Cartesian coordinate system with respect to extrapersonal space.  These claims
are based on the ability  to predict the rectangular coordinates of hand
movement direction using a ``population vector'' computed from
multiple cells' firing rates.  I show
here that such a population vector can always be found given a very
general set of assumptions.  
Therefore the existence of a population vector 
constitutes only weak support for the explicit use of a particular coordinate
representation by motor cortex.
}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 35
d_i(M) \approx b_i + k_i \cos (\theta_i - \theta_M )
\label{eq-theta}
\end{equation}
}

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\samepage \begin{equation}M \approx P = \sum_{i=1}^N C_i d_i 
\label{eq-pop} 
\end{equation}
}

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{\newpage
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\samepage \begin{equation}d_i(M) \approx b_i + b_{ix} m_x + b_{iy} m_y + b_{iz} m_z
\label{eq-linear}
\end{equation}
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\end{equation}
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\samepage \begin{equation}d_i(M) \approx b_i + k_i C_i \cdot M
\label{eq-lin2}
\end{equation}
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\samepage \begin{equation}% latex2html id marker 77
d(\theta_M) = \sum_{k=0}^4 \alpha_k \cos(k\theta_M + \phi_k)
\label{eq-fourier}
\end{equation}
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{\newpage
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\samepage \begin{eqnarray*}% latex2html id marker 81
E[d(\theta_M)] & = & E\left[\sum_{k=0}^4 \alpha_k \cos(k\theta_M +
						\phi_k)\right] \\ 
	& = & \alpha_0 \\ 
    E[d(\theta_M)x] 
	& = & E\left[\sum_{k=0}^4 \alpha_k \cos(k\theta_M +
						\phi_k)\cos(\theta_M)\right] \\ 
	& = & E[\alpha_1 \cos(\theta_M + \phi_1)\cos(\theta_M)] \\ 
	& = & \left( \frac{\alpha_1}{2}\right) \cos(\phi_1) \\ 
    E[d(\theta_M)y] 
	& = & E\left[\sum_{k=0}^4 \alpha_k \cos(k\theta_M +
						\phi_k)\sin(\theta_M)\right] \\ 
	& = & E[\alpha_1 \cos(\theta_M + \phi_1)\sin(\theta_M)]\\ 
	& = & \left( \frac{\alpha_1}{2}\right) \sin(\phi_1)
\end{eqnarray*}
}

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\end{equation}
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{\newpage
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\samepage \begin{equation}D \approx QM 
\label{eq-qm}
\end{equation}
}

{\newpage
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\samepage \begin{equation}M \approx HD \approx HQM 
\label{eq-output} 
\end{equation}
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