\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}

Phase coupling of neuronal activity is observed in many systems. This
means that the action potentials fired by a neuron occur with a
specific phase relation to a driving periodic signal. For example,
sensory neurons in auditory systems and electrosensory systems fire in
close phase relation to the stimulus frequency.  P-type
electroreceptor afferents (P-units) of the weakly electric fish
\emph{Apteronotus leptorhynchus} are driven by the fish's
self-generated field, the electric organ discharge (EOD), and fire
action potentials phase locked to it.

In this project you quantify the strength of the coupling of P-unit
spikes to the EOD 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 $0$ for no phase locking and $1$ for perfect phase
locking.

\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 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 Plot the EOD with the evoked spikes on top.
    
    \part Illustrate the phase locking by plotting the PSTH within the
    EOD cycle.
    
    \part Implement a function that estimates the vector strength
    between the EOD and the spikes.
    
    \part Create a polar plot that shows the timing of the spikes 
    relative 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}