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

\newcommand{\ptitle}{Reverse reconstruction}
\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*{Reverse reconstruction of the stimulus that evoked a neuronal response.}
To analyse the encoding properties of a neuron one often calculates the
Spike-Triggered-Average (STA). The STA is the average stimulus that
led to a spike in the neuron:
\[ STA(\tau) = \frac{1}{n} \displaystyle\sum_{i=1}^{n}{s(t_i - \tau)} \]
where $n$ is the number of spikes and $t_i$ is the time of the
$i_{th}$ spike. $\tau$ is a temporal shift relative to the spike
time. For the beginning let $\tau$ assume values in the range
$\pm50$\,ms. The STA can be estimated by cutting out snippets from the
stimulus that are centered on the respective spike time and by
subsequently averaging them. The STA can be used to reconstruct the
stimulus from the neuronal response. The reconstructed stimulus can
then be compared to the original stimulus and provides a good
impression about the features that are encoded in the neuronal
response.

\begin{questions}
  \question In the accompanying data files you find the spike
  responses of a p-type electroreceptor afferent (P-unit) and a
  pyramidal neuron recorded in the hindbrain of the weakly electric
  fish \textit{Apteronotus leptorhynchus}. The respective stimuli are
  stored in separate files. The neron is stimulated with an amplitude
  modulation of the fish's own electric field. The stored stimulus
  trace is the modulator that is applied to the field and is
  dimensionless, i.e. it has not unit. The data is sampled with
  20\,kHz temporal resolution and spike times are given in
  seconds. Start with the P-unit and, in the end, apply the same
  analyzes/functions to the responses from the pyramidal neuron.
  \begin{parts}
    \part Estimate the STA and plot it. What does it tell?
    \part Implement a function that does the reverse reconstruction and uses the STA to reconstruct the stimulus.
    \part Implement a function that estimates the reconstruction 
    error using the mean-square-error and express it relative to the
    variance of the original stimulus.
    \begin{equation}
      err = \frac{1}{N} \cdot \displaystyle\sum^{N}_{i=1}(x_i - \bar{x})^2,
    \end{equation}
    with $N$ the number of data points, $x_i$ the current value and
    $\bar{x}$, the average of all $x$.
    \part Analyze the robustness of the reconstruction: Estimate 
    the STA with less and less data and estimate the reconstruction
    error.
    \part Plot the reconstruction error as a function of the amount of data 
    used to estimate the STA and apply a statistical test to test if
    estimating the STA from more data improves the reconstruction.
    \part Repeat the above steps for the pyramidal neuron, what do you
    observe?
  \end{parts}
\end{questions}

\end{document}