143 lines
5.3 KiB
TeX
143 lines
5.3 KiB
TeX
\documentclass[addpoints,10pt]{exam}
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\usepackage{url}
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\usepackage{color}
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\usepackage{hyperref}
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\usepackage{graphicx}
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\pagestyle{headandfoot}
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\runningheadrule
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\firstpageheadrule
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\firstpageheader{Scientific Computing}{Integrate-and-fire models}{Oct 28, 2014}
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%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\firstpagefooter{}{}{}
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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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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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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=10pt,
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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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\sffamily
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{questions}
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\question \textbf{Statistics of integrate-and-fire neurons}
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For the following use different variants of the leaky integrate-and-fire models provided in \texttt{lifspikes.m},
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\texttt{lifouspikes.m}, and \texttt{lifadaptspikes.m} do generate some spike train data.
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Use the functions you wrote for the Poisson process to analyze the statistics of the spike trains.
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\begin{parts}
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\part Generate a few trials of the two models for two different inputs
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that result in qualitatively different spike trains and display
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them in a raster plot. Decide for a noise strength (good values to try are 0.001, 0.01, 0.1, 1).
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\begin{solution}
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\begin{lstlisting}
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spikes = pifspikes( 10, 1.0, 0.5, 0.01 );
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%spikes = pifspikes( 10, 10.0, 0.5, 0.01 );
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%spikes = lifspikes( 10, 11.0, 0.5, 0.001 );
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%spikes = lifspikes( 10, 15.0, 0.5, 0.001 );
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spikeraster( spikes )
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\end{lstlisting}
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\mbox{}\\[-3ex]
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{pifraster02}}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{pifraster10}}\\
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{lifraster10}}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{lifraster15}}
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\end{solution}
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\part The inverse Gaussian describes the interspike interval distribution of a PIF driven with white noise:
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\[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \]
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where $\langle T \rangle$ is the mean interspike interval and
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\[ D = \frac{\langle(T - \langle T \rangle)^2\rangle}{2 \langle T \rangle^3} \]
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is the diffusion coefficient (variance of the interspike intervals
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$T$ divided by two times the mean cubed). Show in two plots how
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this distribution depends on $\langle T \rangle$ and $D$.
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\begin{solution}
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\lstinputlisting{simulations/inversegauss.m}
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\lstinputlisting{simulations/inversegaussplot.m}
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\colorbox{white}{\includegraphics[width=0.98\textwidth]{inversegauss}}
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\end{solution}
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\part Extent your function plotting an interspike interval histogram
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to also report the diffusion coefficient $D$.
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\begin{solution}
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\begin{lstlisting}
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...
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% annotation:
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misi = mean( isis );
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sdisi = std( isis );
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disi = sdisi^2.0/2.0/misi^3;
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text( 0.6, 0.7, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' )
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text( 0.6, 0.6, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' )
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text( 0.6, 0.5, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' )
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text( 0.6, 0.4, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' )
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...
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\end{lstlisting}
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\end{solution}
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\part Compare intersike interval histograms obtained from the LIF and PIF models with the inverse Gaussian.
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\begin{solution}
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\lstinputlisting{simulations/lifisih.m}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih01}}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih10}}\\
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih08}}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih16}}
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\end{solution}
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\part Plot the firing rate (inverse mean interspike interval),
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mean interspike interval, the corresponding standard deviation,
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CV, and diffusion coefficient as a function of the input to the LIF
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and the PIF with noise strength set to 0.01.
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\begin{solution}
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\lstinputlisting{simulations/lifisistats.m}
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Leaky integrate-and-fire:\\
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\colorbox{white}{\includegraphics[width=0.8\textwidth]{lifisistats}}\\
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Perfect integrate-and-fire:\\
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\colorbox{white}{\includegraphics[width=0.8\textwidth]{pifisistats}}
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\end{solution}
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\part Plot the firing rate as a function of input of the LIF and the PIF for various values
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of the noise strength.
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\begin{solution}
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\lstinputlisting{simulations/lifficurves.m}
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Leaky integrate-and-fire:\\
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\colorbox{white}{\includegraphics[width=0.7\textwidth]{lifficurves}}\\
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Perfect integrate-and-fire:\\
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\colorbox{white}{\includegraphics[width=0.7\textwidth]{pifficurves}}
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\end{solution}
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\part Use the functions for computing serial correlations, count statistics and fano factors
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to further explore the statistics of the integrate-and-fire models!
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\end{parts}
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\end{questions}
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\end{document}
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