\documentclass[addpoints,10pt]{exam}
\usepackage{url}
\usepackage{color}
\usepackage{hyperref}

\pagestyle{headandfoot}
\runningheadrule
\firstpageheadrule
\firstpageheader{Scientific Computing}{Project Assignment}{11/05/2014
  -- 11/06/2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
\firstpagefooter{}{}{}
\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=10pt,
 %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 sound waves with different intensities $I_1$ and
  $I_2$). Within an observation time of duration $W$ the neuron
  responds stochastically with $n_i$ spikes.

  How well can an upstream neuron discriminate the two
  stimuli based on the spike counts $n_i$? 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)?

  \begin{parts}
    \part 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.

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

    Show two raster plots for the responses to the two differrent stimuli.

    \part Generate histograms of the spike counts within $W$ of the
    responses to the two differrent 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?

    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 $I_2$. What is the
    probability that the stimulus was indeed $I_1$ or $I_2$,
    respectively? 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}
