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\firstpageheader{Scientific Computing}{Project Assignment}{WS 2016/17}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
\firstpagefooter{}{}{{\bf Supervisor:} Jan Grewe}
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\section*{Estimating cellular properties of different cell types.}
You will analyse data from intracellular \texit{in vitro} recordings
of pyramidal neurons from two different maps of the electrosensory
lateral line lobe (ELL) of the weakly electric fish
\textit{Apteronotus leptorhynchus}. The resistance and capacitance of
the membrane are typically estimated by injecting hyperpolarizing
current pulses into the cell. From the respective responses we can
calculate the membrane resistance by applying Ohm's law ($U = R \cdot
I$). To estimate the membrane capacitance we need to fit an
exponential function $y = a \cdot e^{(b \cdot x)}$to the response to
get the membrane time-constant $\tau$. With the knowledge of $R$ and
$\tau$ we can estimate the capacitance $C$ from the simple relation
$\tau = R \cdot C$.

\begin{questions}
  \question{} The accompanying dataset (input\_resistance.zip)
  contains datasets from cells originating from two different parts of
  the ELL, the medial segment (MS) and the centro-medial segment
  (CMS). Each mat-file contains four variables. (i) \textit{V} the
  average membrane potential of 20 repeated current injections, (ii)
  \textit{V\_std} the across-trial standard deviation of the
  responses, (iii) \textit{t} a vector representing the recording
  time (in ms), and (iv) \textit{I} a vector containing the time-course of the
  injected current.
  
  \begin{parts}
    \part{} Create plots of the raw data. Plot the average response as
    a function of time. This plot should also show the across-trial
    variability. Also plot the time-course of the injected
    current. \\[0.5ex]
    \part{} Estimate the imput resistances of each cell.\\[0.5ex]
    \part{} Fit an exponential to the initial few milliseconds of the
    current-on response. Use a gradient-descent approach to do
    this.\\[0.5ex]
    \part{} Estimate the membrane capacitance of each cell. Compare
    $R$, $I$ and $\tau$ between cells of the two segments.\\[0.5ex]
    \part{} Optional: use a double exponential and see, if the fit gets better.
  \end{parts}
\end{questions}

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