<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt,pdftex]{exam}

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

\begin{document}

\input{../instructions.tex}


%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\section{Estimating the adaptation time-constant}
Stimulating a neuron with a constant stimulus for an extended period of time
often leads to a strong initial response that relaxes over time. This
process is called adaptation. Your task here is to
estimate the time-constant of the firing-rate adaptation in P-unit
electroreceptors of the weakly electric fish \textit{Apteronotus
  leptorhynchus}.

\begin{questions}
  \question In the accompanying datasets you find the
  \textit{spike\_times} of an P-unit electroreceptor to a stimulus of
  a certain intensity, i.e. the \textit{contrast} which is also stored
  in the file. The contrast of the stimulus is a measure relative to
  the amplitude of fish's field and is given in percent. The data is sampled
  with 20\,kHz sampling frequency and spike times are given in
  milliseconds (not seconds!) relative to the stimulus onset.
  \begin{parts}
    \part Estimate for each stimulus intensity the PSTH. 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. Find an appropriate bin-width
    for the PSTH.

    \part Estimate the adaptation time-constant for both the stimulus
    on- and offset. To do this fit an exponential function
    $f_{A,\tau,y_0}(t)$ to appropriate regions of the data:
    \begin{equation}
       f_{A,\tau,y_0}(t) = A \cdot e^{-\frac{t}{\tau}} + y_0,
    \end{equation}
    where $t$ is time, $A$ the (positive or negative) amplitude of the
    exponential decay, $\tau$ the adaptation time-constant, and $y_0$
    an offset.

    Before you do the fitting, familiarize yourself with the three
    parameter of the exponential function. What is the value of
    $f_{A,\tau,y_0}(t)$ at $t=0$? What is the value for large times? How does
    $f_{A,\tau,y_0}(t)$ change if you change either of the parameter?

    Which of the parameter could you directly estimate from the data
    (without fitting)?

    How could you get good estimates for the other parameter?

    Do the fit and show the resulting exponential function together
    with the data.

    \part Do the estimated time-constants depend on stimulus intensity?

    Use an appropriate statistical test to support your observation.
  \end{parts}
\end{questions}

\end{document}
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