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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 the adaptation time-constant.}
Stimulating a neuron with a constant stimulus for an extended period of time
often leads to a strong initial response that relaxes over time. This
process is called adaptation and is ubiquitous. Your task here is to
estimate the time-constant of the firing-rate adaptation in P-unit
electroreceptors of the weakly electric fish \textit{Apteronotus
  leptorhynchus}.

\begin{questions}
  \question In the accompanying datasets you find the
  \textit{spike\_times} of an P-unit electroreceptor to a stimulus of
  a certain intensity, i.e. the \textit{contrast} which is also stored
  in the file. The contrast of the stimulus is a measure relative to
  the amplitude of fish's field, it has no unit. The data is sampled
  with 20\,kHz sampling frequency and spike times are given in
  milliseconds relative to the stimulus onset.
  \begin{parts}
    \part Estimate for each stimulus intensity the PSTH and plot
    it. You will see that there are three parts.  (i) The first
    200\,ms is the baseline (no stimulus) activity. (ii) During the
    next 1000\,ms the stimulus was switched on. (iii) After stimulus
    offset the neuronal activity was recorded for further 825\,ms.
    \part Estimate the adaptation time-constant for both the stimulus
    on- and offset. To do this fit an exponential function to the
    data. For the decay use:
    \begin{equation}
       f_{A,\tau,y_0}(t) = y_0 + A \cdot e^{-\frac{t}{\tau}},
    \end{equation}
    where $y_0$ the offset, $A$ the amplitude, $t$ the time, $\tau$
    the time-constant.
    For the increasing phases use an exponential of the form:
    \begin{equation}
       f_{A,\tau, y_0}(t) = y_0 + A \cdot \left(1 - e^{-\frac{t}{\tau}}\right ),
    \end{equation}
    \part Plot the best fits into the data.
    \part Plot the estimated time-constants as a function of stimulus intensity.
  \end{parts}
\end{questions}

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