\documentclass[addpoints,11pt]{exam}
\usepackage{url}
\usepackage{color}
\usepackage{hyperref}

\pagestyle{headandfoot}
\runningheadrule
\firstpageheadrule
\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}
\runningfooter{}{}{}
\pointsinmargin
\bracketedpoints

%\printanswers
%\shadedsolutions

%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{listings}
\lstset{
 basicstyle=\ttfamily,
 numbers=left,
 showstringspaces=false,
 language=Matlab,
 breaklines=true,
 breakautoindent=true,
 columns=flexible,
 frame=single,
% captionpos=t,
 xleftmargin=2em,
 xrightmargin=1em,
% aboveskip=11pt,
 %title=\lstname,
% title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
 }


\begin{document}
%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
\sffamily
% \begin{flushright}
% \gradetable[h][questions]
% \end{flushright}

\begin{center}
  \input{../disclaimer.tex}
\end{center}

%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{questions}
  \question You are recording the activity of a neuron in response to
  two different stimuli $I_1$ and $I_2$ (think of them, for example,
  of two light intensities with different intensities $I_1$ and $I_2$
  and the activity of a ganglion cell in the retina). Within an
  observation time of duration $W$ the neuron responds stochastically
  with $n$ spikes.

  How well can an upstream neuron discriminate the two
  stimuli based on the spike counts $n$? How does this depend on the
  duration $W$ of the observation time? How is this related to the fano factor
  (the ratio between the variance and the mean of the spike counts)?

  The neuron is implemented in the file \texttt{lifadaptspikes.m}.
  Call it with the following parameters:
    \begin{lstlisting}
trials = 10;
tmax = 50.0;
input = 65.0; 
Dnoise = 0.1;
adapttau = 0.2;
adaptincr = 0.5;

spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr );
    \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.

    Think of calling the \texttt{lifadaptspikes()} 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{lifadaptspikes()} 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}, \texttt{adapttau}, and
    \texttt{adaptincr} parameter.

    For the two inputs $I_1$ and $I_2$ use
    \begin{lstlisting}
input = 65.0; % I_1
input = 75.0; % I_2
    \end{lstlisting}

  \begin{parts}
    \part 
    Show two raster plots for the responses to the two different stimuli.

    \part Generate histograms of the spike counts within $W$ of the
    responses to the two different stimuli. How do they depend on the
    observation time $W$ (use values between 1\,ms and 1\,s)?

    \part Think about a measure based on the spike count histograms
    that quantifies how well the two stimuli can be distinguished
    based on the spike counts. Plot the dependence of this measure as
    a function of the observation time $W$.

    For which observation times can the two stimuli perfectly discriminated?

    \underline{Hint:} A possible readout is to set a threshold
    $n_{thresh}$ for the observed spike count.  Any response smaller
    than the threshold assumes that the stimulus was $I_1$, any
    response larger than the threshold assumes that the stimulus was
    $I_2$. For a given $W$ find the threshold $n_{thresh}$ that
    results in the best discrimination performance.

    \part Also plot the Fano factor as a function of $W$. How is it
    related to the discriminability?

    \uplevel{If you still have time you can continue with the
      following question:}

    \part You may change the two stimuli $I_1$ and $I_2$ and the
    intrinsic noise of the neuron via \texttt{Dnoise} (change it in
    factors of ten, higher values will make the responses more
    variable) and repeat your analysis.

 \end{parts}

\end{questions}

\end{document}
