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\firstpageheader{Scientific Computing}{Project Assignment}{11/02/2014
  -- 11/05/2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
\firstpagefooter{}{}{{\bf Supervisor:} Jan Benda}
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%\printanswers
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\begin{document}
%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{center}
  \input{../disclaimer.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{questions}
  \question You are recording the activity of a neuron in response to
  constant stimuli of intensity $I$ (think of that, for example,
  as a current $I$ injected via a patch-electrode into the neuron).

  Measure the tuning curve (also called the intensity-response curve) of the
  neuron.  That is, what is the mean firing rate of the neuron's response
  as a function of the input $I$. How does this depend on the level of
  the intrinsic noise of the neuron?

  The neuron is implemented in the file \texttt{lifspikes.m}.  Call it
  with the following parameters:
    \begin{lstlisting}
trials = 10;
tmax = 50.0;
input = 10.0;  % the input I
Dnoise = 1.0;  % noise strength

spikes = lifspikes( trials, input, tmax, Dnoise );
    \end{lstlisting}
    The returned \texttt{spikes} is a cell array with \texttt{trials} elements, each being a vector
    of spike times (in seconds) computed for a duration of \texttt{tmax} seconds.
    The input is set via the \texttt{input} variable, the noise strength via \texttt{Dnoise}.

    Think of calling the \texttt{lifspikes()} function as a
    simple way of doing an electrophysiological experiment. You are
    presenting a stimulus with a constant intensity $I$ that you set. The
    neuron responds to this stimulus, and you record this
    response. After detecting the timepoints of the spikes in your
    recordings you get what the \texttt{lifspikes()} function
    returns. The advantage over real data is, that you have the
    possibility to simply modify the properties of the neuron via the
    \texttt{Dnoise} parameter.

  \begin{parts}
    \part First set the noise \texttt{Dnoise=0} (no noise). Compute
    and plot the mean firing rate (number of spikes within the
    recording time \texttt{tmax} divided by \texttt{tmax} and averaged
    over trials) as a function of the input for inputs ranging from 0
    to 20.

    \part Do the same for various noise strength \texttt{Dnoise}. Use $D_{noise} = 1e-3$,
    1e-2, and 1e-1. How does the intrinsic noise influence the response curve?

    \part Show some interspike interval histograms for some
    interesting values of the input and the noise strength.

    \part How does the coefficient of variation $CV_{isi}$ (standard
    deviation divided by mean) of the interspike intervalls depend on
    the input and the noise level?
    

 \end{parts}

\end{questions}

\end{document}
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