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\firstpageheader{Scientific Computing}{Project Assignment}{11/05/2014
  -- 11/06/2014}
%\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}
\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)?

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}

  \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}
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