\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 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 time course of the
    trial-averaged 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 (mean over trials and standard
    deviation) 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 the onset and steady-state
  $f$-$I$-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.
  
  \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 Use a statistical test to evaluate which of the onset and
    steady-state responses differ significantly from the baseline
    activity.
    
    \part Discuss you results with respect to encoding of different
    stimulus intensities.
  \end{parts}
\end{questions}

\end{document}
