81 lines
3.0 KiB
TeX
81 lines
3.0 KiB
TeX
\documentclass[a4paper,12pt,pdftex]{exam}
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\newcommand{\ptitle}{f-I curves}
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\input{../header.tex}
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\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
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{email: jan.benda@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{Quantifying the responsiveness of a neuron using the f-I
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curve}
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The responsiveness of a neuron is often quantified using an $f$-$I$
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curve. The $f$-$I$ curve plots the \textbf{f}iring rate of the neuron
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as a function of the stimulus \textbf{I}ntensity.
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In the accompanying datasets you find the \textit{spike\_times} of an
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P-unit electroreceptor of the weakly electric fish \textit{Apteronotus
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leptorhynchus} to a stimulus of a certain intensity, i.e. the
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\textit{contrast}. The spike times are given in milliseconds relative
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to the stimulus onset.
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\begin{questions}
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\question Estimate the $f$-$I$-curve for the onset and the steady
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state response.
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\begin{parts}
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\part Estimate for each stimulus intensity the average response
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(PSTH) and plot it. You will see that there are three parts: (i)
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The first 200\,ms is the baseline (no stimulus) activity. (ii)
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During the next 1000\,ms the stimulus was switched on. (iii) After
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stimulus offset the neuronal activity was recorded for further
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825\,ms.
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\part Extract the neuron's activity in 50\,ms time windows before
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stimulus onset (baseline activity), immediately after stimulus
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onset (onset response), and 50\,ms before stimulus offset (steady
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state response).
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Plot the resulting $f$-$I$ curves by plotting the three computed
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firing rates against the corresponding stimulus intensities
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(contrasts).
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\end{parts}
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\question Fit a Boltzmann function to each of the $$-$I$-curves. The
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Boltzmann function is a sigmoidal function and is defined as
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\begin{equation}
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f(x) = \frac{\alpha-\beta}{1+e^{-k(x-x_0)}}+\beta \; .
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\end{equation}
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$x$ is the stimulus intensity, $\alpha$ is the starting firing rate,
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$\beta$ the saturation firing rate, $x_0$ defines the position of
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the sigmoid, and $k$ (together with $\alpha-\beta$) sets the slope.
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\begin{parts}
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\part Before you do the fitting, familiarize yourself with the
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four parameters of the Boltzmann function. What is its value for
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very large or very small stimulus intensities? How does the
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Boltzmann function change if you change the parameters?
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\part Can you get good initial estimates for the parameters?
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\part Do the fits and show the resulting Boltzmann functions
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together with the corresponding data.
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\part Illustrate how the fit to the $f$-$I$ curves changes during
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the fitting process. You can plot the parameters as a function of
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fit iterations. Which parameter stay the same, which ones change
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with time?
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Support your conclusion with appropriate statistical tests.
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\part Discuss you results with respect to encoding of different
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stimulus intensities.
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\end{parts}
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\end{questions}
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\end{document}
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