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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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\begin{document}
%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{center}
  \input{../disclaimer.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{questions}
  \question An important property of sensory systems is their ability
  to discriminate similar stimuli. For example, discrimination of two
  colors, light intensities, pitch of two tones, sound intensities, etc.
  Here we look at the level of a single neuron. What does it mean that
  two similar stimuli can be discriminated given the spike train
  responses that have been evoked by the two stimuli?

  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 different light intensities, $I_1$ and $I_2$, and the spiking
  activity of a ganglion cell in the retina). The neuron responds to a
  stimulus with a number of spikes. You (an upstream neuron) can count
  the number of spikes of this response within an observation time of
  duration $T$. For perfect discrimination, the number of spikes
  evoked by the stronger stimulus within $T$ is always larger than for
  the smaller stimulus. The situation is more complicated, because the
  number of spikes evoked by one stimulus is not fixed but varies,
  such that the number of spikes evoked by the stronger stimulus could
  happen to be lower than the number of spikes evoked by the smaller
  stimulus.

  The central questions of this project are:
  \begin{itemize}
  \item How can an upstream neuron discriminate two stimuli based
    on the spike counts $n$?
  \item How does this depend on the duration $T$ of the observation
    time?
  \end{itemize}
  
  The neuron is implemented in the file \texttt{lifspikes.m}.
  Call it like this:
  \begin{lstlisting}
trials = 10;
tmax = 50.0;
input = 15.0; 
spikes = lifspikes(trials, input, tmax);
  \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 intensity of the
  stimulus is given by \texttt{input}.

  Think of calling the \texttt{lifspikes()} function as a simple way
  of doing an electrophysiological experiment. You are presenting a
  stimulus with an intensity $I$ that you set. The neuron responds to
  this stimulus, and you record this response. After detecting the
  time points of the spikes in your recordings you get what the
  \texttt{lifspikes()} function returns.

  For the two inputs $I_1$ and $I_2$ to be discriminated use
  \begin{lstlisting}
input = 14.0; % I_1
input = 15.0; % I_2
  \end{lstlisting}

  \begin{parts}
    \part 
    Show two raster plots for the responses to the two different
    stimuli.  Use an appropriate time window and an appropriate
    number of trials for the spike raster.

    Just by looking at the raster plots, can you discriminate the two
    stimuli? That is, do you see differences between the two
    responses?

    \part Generate properly normalized histograms of the spike counts
    within $T$ (use $T=100$\,ms) of the spike responses to the two
    different stimuli. Do the two histograms overlap? What does this
    mean for the discriminability of the two stimuli?

    How do the histograms of the spike counts depend on the
    observation time $T$? Plot them for four different values of $T$
    (use values of 10\,ms, 100\,ms, 300\,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 $T$.

    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 $T$ find the threshold $n_{thresh}$ that
    results in the best discrimination performance. How can you
    quantify ``best discrimination'' performance?

 \end{parts}

\end{questions}

\end{document}
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