146 lines
4.9 KiB
TeX
146 lines
4.9 KiB
TeX
\documentclass[addpoints,11pt]{exam}
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\usepackage{url}
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\usepackage{color}
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\usepackage{hyperref}
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\pagestyle{headandfoot}
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\runningheadrule
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\firstpageheadrule
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\firstpageheader{Scientific Computing}{Project Assignment}{11/02/2014
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-- 11/05/2014}
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%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\firstpagefooter{}{}{{\bf Supervisor:} Jan Benda}
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\runningfooter{}{}{}
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\pointsinmargin
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\bracketedpoints
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%\printanswers
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%\shadedsolutions
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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basicstyle=\ttfamily,
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numbers=left,
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showstringspaces=false,
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language=Matlab,
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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% captionpos=t,
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xleftmargin=2em,
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xrightmargin=1em,
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% aboveskip=11pt,
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%title=\lstname,
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% title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
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}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
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\sffamily
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% \begin{flushright}
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% \gradetable[h][questions]
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% \end{flushright}
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\begin{center}
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\input{../disclaimer.tex}
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\end{center}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{questions}
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\question You are recording the activity of two neurons in response
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to a constant stimulus $I$ (think of it, for example, of a sound
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wave with intensity $I$ and the activity of an auditory neuron).
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For different inputs $I$ the interspike interval ($T$) distribution looks
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quite different. You want to compare these distributions to
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the following three standard distributions.
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The first is the exponential distribution of a Poisson spike train:
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\begin{equation}
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\label{exppdf}
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p_{exp}(T) = \lambda e^{-\lambda T}
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\end{equation}
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where $\lambda$ is the mean firing rate of the response.
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The second distribution is the inverse Gaussian:
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\begin{equation}
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\label{invgauss}
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p_\mathrm{ig}(T) = \frac{1}{\sqrt{4 \pi D T^{3}}} \exp \left[ - \frac{(T - \mu)^{2} }{4 D T \mu^{2}} \right]
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\end{equation}
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where $\mu$ is the mean interspike interval and
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$D=\textrm{var}(T)/(2\mu^3)$
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is the so called diffusion coefficient.
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The third one was derived for neurons driven with colored noise:
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\begin{equation}\label{pcn}
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p_\mathrm{cn}(T)=\frac{1}{2\tau\sqrt{4\pi\epsilon\gamma_1^3}}\exp\left[-\frac{(T-\mu)^2}{4\epsilon\tau^2\gamma_1}\right]\left\{\frac{[(\mu-T)\gamma_2+2\gamma_1\tau]^2}{2\gamma_1\tau^2}-\epsilon(\gamma_2^2-2\gamma_1e^{-T/\tau})\right\}
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\end{equation}
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with $\gamma_1(T)=T/\tau+e^{-T/\tau}-1$, $\gamma_2(T)=1-e^{-T/\tau}$
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and correlation time of the colored noise $\tau$.
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Eq.~(\ref{pcn}) thus has the three parameter $\mu$, $\epsilon>0$, and $\tau$.
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The squared coefficient of variation (standard deviation of the
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interspike intervals divided by their mean) of the density
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eq.~(\ref{pcn}) is given by
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\begin{equation}
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\label{cvpcn}
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C_V^2=\frac{2}{\delta}\left[\epsilon\left(1-\frac{1-e^{-\delta}}{\delta}\right)+\epsilon^2\left(e^{-\delta}+\frac{(1-e^{-\delta})(1-2e^{-\delta})}{\delta}\right)\right]
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\end{equation}
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with $\delta=\mu/\tau$.
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The two neurons are implemented in the files \texttt{pifouspikes.m}
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and \texttt{lifouspikes.m}. Call them with the following
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parameters:
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\begin{lstlisting}
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trials = 10;
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tmax = 50.0;
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input = 10.0; % the input I
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Dnoise = 1.0; % noise strength
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outau = 1.0; % correlation time of the noise in seconds
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spikespif = pifouspikes( trials, input, tmax, Dnoise, outau );
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spikeslif = lifouspikes( trials, input, tmax, Dnoise, outau );
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\end{lstlisting}
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The returned \texttt{spikespif} and \texttt{spikeslif} are cell
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arrays with \texttt{trials} elements, each being a vector of spike
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times (in seconds) computed for a duration of \texttt{tmax}
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seconds. The input is set via the \texttt{input} variable.
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\begin{parts}
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\part For both model neurons find the inputs $I_i$ required to
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make them fire with a mean rate of 10, 20, 50, and 100\,Hz.
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\part Compute interspike interval distributions of the two model
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neurons for these inputs $I_i$.
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\part Compare the interspike interval distributions with the exponential
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distribution eq.~(\ref{exppdf}) and the inverse Gaussian
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eq.~(\ref{invgauss}) by measuring their parameters from the
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interspike intervals. How well do they describe the real
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distributions for the different conditions?
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\part Also fit eq.~(\ref{pcn}) to the data using maximum (log)-likelihood.
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How well does this function describe the data?
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Compare the fitted value for $\tau$ with the one used for the
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model (\texttt{outau}).
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\uplevel{If you still have time you can continue with the following question:}
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\part Compare the measured coefficient of variation with eq.~(\ref{cvpcn}).
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\part Repeat your analysis for different values of the intrinsic
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noise strengh of the neurons \texttt{Dnoise}. Increase or decrease
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it in factors of ten.
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\end{parts}
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\end{questions}
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\end{document}
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