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

\newcommand{\ptitle}{f-I curves}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
{email: jan.benda@uni-tuebingen.de}

\begin{document}

\input{../instructions.tex}


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\section{Quantifying the responsiveness of a neuron using the f-I
  curve}
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.

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{questions}
  \question Estimate the $f$-$I$-curve for the onset and the steady
  state response.
  \begin{parts}
    \part Estimate for each stimulus intensity 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 in 50\,ms time windows before
    stimulus onset (baseline activity), immediately after stimulus
    onset (onset response), and 50\,ms before stimulus offset (steady
    state response).

    Plot the resulting $f$-$I$ curves by plotting the three computed
    firing rates against the corresponding stimulus intensities
    (contrasts).

  \end{parts}

  \question Fit a Boltzmann function to each of the $$-$I$-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.
  
  \begin{parts}
    \part Before you do the fitting, familiarize yourself with the
    four parameters 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 the parameters?
    
    \part Can you get good initial estimates for the parameters?

    \part Do the fits and show the resulting Boltzmann functions
    together with the corresponding data.
    
    \part Illustrate how the fit to the $f$-$I$ curves changes during
    the fitting process. You can plot the parameters as a function of
    fit iterations.  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}
