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

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

\begin{document}

\input{../instructions.tex}

%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Quantifying the coupling of action potentials to the EOD.}
Phase coupling of neuronal activity is observed in several
system. This means that the action potentials fired by a neuron occur
with specific phase relation to the driving periodic signal. For example sensory
neurons in the auditory system and the electrosensory system fire in
close phase relation to the stimulus frequncy.  P-type electroreceptor
afferents (P-units) of the weakly electric fish \emph{Apteronotus
  leptorhynchus} are driven by the fish's self-generated field, the
EOD and fire action potentials phase locked to it. In this project you
have to quantify the strength of this coulpling using the
\textbf{vector strength}:
\begin{equation}
 VS = \sqrt{\left(\frac{1}{n}\sum_{i=1}^{n}\cos
 \alpha_i\right)^2 + \left(\frac{1}{n}\sum_{i = 1}^{n} \sin \alpha_i
 \right)^2},
\end{equation}
with $n$ the number of spikes and $\alpha_i$ the timing of the each
spike expressed as the phase relative to the EOD. The vector strength
varies between $0$ and $1$ for no phase locking to perfect phase
locking, respectively.

\begin{questions}
  \question In the accompanying datasets you find recordings of the
  ``baseline'' activity of P-unit electroreceptors (in the absence of
  an external stimulus) of different weakly electric fish of the
  species \textit{Apteronotus leptorhynchus}.  The files further
  contain respective recordings of the \textit{eod}, i.e. the fish's
  electric field. The data is sampled with 20\,kHz and the spike times
  are given in seconds.
  \begin{parts}
    \part Illustrate the phase locking by plotting the PSTH within the EOD cycle.
    \part Implement a function that estimates the vector strength
    between the \textit{EOD} and the spikes.
    \part Create a polar plot that shows the timing of the spikes 
    relatve to the EOD.
    \part Apply an appropriate statistical test to check whether locking is statistically significant.
    \part Analyze the baseline responses of each fish and extract measures as were introduced in chapter 10 of the script. Plot the results
    appropriately.
    \part Does the vector strength correlate with the EOD frequency or the reponse variability (CV)?
  \end{parts}
\end{questions}

\end{document}