\documentclass[a4paper,12pt,pdftex]{exam}

\newcommand{\ptitle}{F-I curves}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Grewe}{phone: 29 74588}%
{email: jan.grewe@uni-tuebingen.de}

\begin{document}

\input{../instructions.tex}


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\section{Quantifying the responsiveness of a neuron by its F-I curves}
The responsiveness of a neuron is often quantified using an F-I
curve. The F-I curve plots the \textbf{F}iring rate of the neuron as a
function of the stimulus \textbf{I}ntensity.

\begin{questions}
  \question In the accompanying datasets you find the
  \textit{spike\_times} of an P-unit electroreceptor of the weakly
  electric fish \textit{Apteronotus leptorhynchus} to a stimulus of a
  certain intensity, i.e. the \textit{contrast}. The spike times are
  given in milliseconds relative to the stimulus onset.
  \begin{parts}
    \part For each stimulus intensity estimate the average response
    (PSTH) and plot it. You will see that there are three parts.  (i)
    The first 200\,ms is the baseline (no stimulus) activity. (ii)
    During the next 1000\,ms the stimulus was switched on. (iii) After
    stimulus offset the neuronal activity was recorded for further
    825\,ms.

    \part Extract the neuron's activity for every 50\,ms after
    stimulus onset and for one 50\,ms slice before stimulus onset.

    For each time slice plot the resulting F-I curve by plotting the
    computed firing rates against the corresponding stimulus
    intensity, respectively the contrast.

    \part Fit a Boltzmann function to each of the F-I-curves. The
    Boltzmann function is a sigmoidal function and is defined as
    \begin{equation}
       f(x) = \frac{\alpha-\beta}{1+e^{-k(x-x_0)}}+\beta \; .
    \end{equation}
    $x$ is the stimulus intensity, $\alpha$ is the starting
    firing rate, $\beta$ the saturation firing rate, $x_0$ defines the
    position of the sigmoid, and $k$ (together with $\alpha-\beta$)
    sets the slope.

    Before you do the fitting, familiarize yourself with the four
    parameter of the Boltzmann function. What is its value for very
    large or very small stimulus intensities? How does the Boltzmann
    function change if you change either of the parameter?

    How could you get good initial estimates for the parameter?

    Do the fits and show the resulting Boltzmann functions together
    with the corresponding data.

    \part Illustrate how the F-I curves change in time also by means
    of the parameter you obtained from the fits with the Boltzmann
    function. 

    Which parameter stay the same, which ones change with time?

    Support your conclusion with appropriate statistical tests.

    \part Discuss you results with respect to encoding of different
    stimulus intensities.
  \end{parts}
\end{questions}

\end{document}
