59 lines
2.7 KiB
TeX
59 lines
2.7 KiB
TeX
\documentclass[a4paper,12pt,pdftex]{exam}
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\newcommand{\ptitle}{Reverse reconstruction}
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\input{../header.tex}
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\firstpagefooter{Supervisor: Jan Grewe}{phone: 29 74588}%
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{email: jan.grewe@uni-tuebingen.de}
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\begin{document}
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\input{../instructions.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Reverse reconstruction of the stimulus that evoked a neuronal response.}
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To analyse encoding properties of a neuron one often calculates the
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Spike-Triggered-Average (STA). The STA is the average stimulus that
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led to a spike in the neuron:
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\[ STA(\tau) = \frac{1}{n} \displaystyle\sum_{i=1}^{n}{s(t_i - \tau)} \]
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where $n$ is the number of spikes and $t_i$ is the time of the
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$i_{th}$ spike. $\tau$ is a temporal shift relative to the spike
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time. For the beginning let $\tau$ assume values in the range
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$\pm50$\,ms. It can be estimated by cutting out snippets form the
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stimulus centered on the respective spike time and averaging them. The
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Spike-Triggered-Average can be used to reconstruct the stimulus from
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the neuronal response. The reconstructed stimulus can then be compared
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to the original stimulus.
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\begin{questions}
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\question In the accompanying data files you find the spike responses of
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a p-type electroreceptor afferent (P-unit) and a pyramidal neuron
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recorded in the hindbrain of the weakly electric fish
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\textit{Apteronotus leptorhynchus}. The respective stimuli are
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stored in separate files. The data is sampled with 20\,kHz temporal
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resolution and spike times are given in seconds. Start with the
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P-unit and, in the end, apply the same analyzes/functions to the
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responses from the pyramidal neuron.
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\begin{parts}
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\part Estimate the STA and plot it. What does it tell?
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\part Implement a function that does the reverse reconstruction and uses the STA to reconstruct the stimulus.
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\part Implement a function that estimates the reconstruction
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error using the mean-square-error and express it relative to the
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variance of the original stimulus.
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\begin{equation}
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err = \frac{1}{N} \cdot \displaystyle\sum^{N}_{i=1}(x_i - \bar{x})^2,
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\end{equation}
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with $N$ the number of data points, $x_i$ the current value and
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$\bar{x}$, the average of all $x$.
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\part Analyze the robustness of the reconstruction: Estimate
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the STA with less and less data and estimate the reconstruction
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error.
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\part Plot the reconstruction error as a function of the amount of data
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used to estimate the STA and apply a statistical test to test if
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estimating the STA from more data improves the reconstruction.
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\part Repeat the above steps for the pyramidal neuron, what do you
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observe?
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\end{parts}
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\end{questions}
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\end{document}
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