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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}
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  \input{../disclaimer.tex}
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%%%%%%%%%%%%%% 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 slope of the
  tuning curve of the neural responses? 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{lifboltzmanspikes.m}.
    Call it with the following parameters:
\begin{lstlisting}
trials = 10;
tmax = 50.0;
Dnoise = 1.0;
imax = 25.0;
ithresh = 10.0;
slope=0.2;
input = 10.0;

spikes = lifboltzmanspikes( trials, input, tmax, Dnoise, imax, ithresh, slope );
    \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.

    For the two inputs use $I_1=10$ and $I_2=I_1 + 1$.


  \begin{parts}
    \part 
    First, show two raster plots for the responses to the two differrent stimuli.

    \part Measure the tuning curve of the neuron with respect to the input. That is,
    compute the mean firing rate as a function of the input
    strength. Find an appropriate range of input values.  Do this for
    different values of the \texttt{slope} parameter (values between
    0.1 and 2.0).

    \part Generate histograms of the spike counts within $W=200$\,ms of the
    responses to the two differrent stimuli $I_1$ and $I_2$. How do they depend on the slope
    of the tuning curve of the neuron?

    \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 slopes can the two stimuli be well 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 $I_2$. What is the
    probability that the stimulus was indeed $I_1$ or $I_2$,
    respectively? Find the threshold $n_{thresh}$ that
    results in the best discrimination performance.

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

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

    \part You may change the difference between the two stimuli $I_1$ and $I_2$ 
    as well as 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.

    \part For $I_1=10$ the mean firing is about $80$\,Hz. When changing the slope of the tuning curve
    this firing rate may also change. Improve your analysis by finding for each slope the input
    that results exactly in a firing rate of $80$\,Hz. Set $I_2$ on unit above $I_1$.

    \part How does the dependence of the stimulus discrimination performance on the slope change
    when you set both $I_1$ and $I_2$ such that they evoke $80$ and
    $100$\,Hz firing rate, respectively?

 \end{parts}

\end{questions}

\end{document}
