[pointprocesses] cleaned up m and pdf files
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TEXFILES=$(wildcard pointprocesses-?.tex)
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TEXFILES=$(wildcard pointprocesses-?.tex) psth.tex
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include ../../exercises.mk
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Binary file not shown.
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% generate spike times:
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rate = 20.0;
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spikes = hompoissonspikes( 10, rate, 50.0 );
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% isi histogram:
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isivec = isis( spikes );
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isihist( isivec );
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hold on
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% theoretical density:
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xmax = 5.0/rate;
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x = 0:0.0001:xmax;
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y = rate*exp(-rate*x);
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plot( 1000.0*x, y, 'r', 'LineWidth', 3 );
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% plot details:
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title( sprintf( 'Poisson spike trains, rate=%g Hz, nisi=%d', rate, length( isivec ) ) )
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xlim( [ 0.0 1000.0*xmax ] )
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ylim( [ 0.0 1.1*rate ] )
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legend( 'data', 'poisson' )
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hold off
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rates = 1:1:100;
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avisi = [];
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sdisi = [];
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cvisi = [];
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for rate = rates
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spikes = hompoissonspikes( 10, rate, 100.0 );
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isivec = isis( spikes );
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av = mean( isivec );
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sd = std( isivec );
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cv = sd/av;
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avisi = [ avisi av ];
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sdisi = [ sdisi sd ];
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cvisi = [ cvisi cv ];
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end
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f = figure;
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subplot( 1, 3, 1 );
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scatter( rates, 1000.0*avisi, 'b', 'filled' );
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hold on;
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plot( rates, 1000.0./rates, 'r' );
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hold off;
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xlabel( 'Rate \lambda [Hz]' );
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ylim( [ 0 1000 ] );
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title( 'Mean ISI [ms]' );
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legend( 'simulation', 'theory 1/\lambda' );
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subplot( 1, 3, 2 );
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scatter( rates, 1000.0*sdisi, 'b', 'filled' );
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hold on;
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plot( rates, 1000.0./rates, 'r' );
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hold off;
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xlabel( 'Rate \lambda [Hz]' );
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ylim( [ 0 1000 ] )
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title( 'Standard deviation ISI [ms]' );
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legend( 'simulation', 'theory 1/\lambda' );
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subplot( 1, 3, 3 );
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scatter( rates, cvisi, 'b', 'filled' );
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hold on;
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plot( rates, ones( size( rates ) ), 'r' );
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hold off;
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xlabel( 'Rate \lambda [Hz]' );
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ylim( [ 0 2 ] )
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title( 'CV' );
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legend( 'simulation', 'theory' );
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@ -1,19 +0,0 @@
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function spikes = hompoissonspikes( trials, rate, tmax )
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% Generate spike times of a homogeneous poisson process
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% trials: number of trials that should be generated
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% rate: the rate of the Poisson process in Hertz
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% tmax: the duration of each trial in seconds
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% returns a cell array of vectors of spike times
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dt = 3.33e-5;
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p = rate*dt;
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if p > 0.2
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p = 0.2
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dt = p/rate;
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end
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x = rand( trials, ceil(tmax/dt) );
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spikes = cell( trials, 1 );
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for k=1:trials
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spikes{k} = find( x(k,:) < p ) * dt;
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end
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end
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\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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@ -1,12 +1,12 @@
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\documentclass[12pt,a4paper,pdftex]{exam}
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\newcommand{\exercisetopic}{Point Processes}
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\newcommand{\exercisenum}{X2}
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\newcommand{\exercisetopic}{Integrate-and-fire models}
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\newcommand{\exercisenum}{13}
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\newcommand{\exercisedate}{January 19th, 2021}
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\input{../../exercisesheader}
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\firstpagefooter{Prof. Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}
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\firstpagefooter{Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\input{../../exercisestitle}
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \qt{Homogener Poisson Prozess}
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Wir wollen den homogenen Poisson Prozess benutzen um Spikes zu generieren,
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mit denen wir die Analysfunktionen des vorherigen \"Ubungszettel \"uberpr\"ufen k\"onnen.
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Ein homogener Poisson Prozess mit der Rate $\lambda$ (gemessen in Hertz) ist ein Punktprozess,
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bei dem die Wahrschienlichkeit eines Ereignisses unabh\"angig von der Zeit $t$ und
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unabh\"angig von vorherigen Ereignissen ist.
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Die Wahrscheinlichkeit $P$ eines Ereignisses innerhalb eines Bins der Breite $\Delta t$ ist
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\[ P = \lambda \cdot \Delta t \]
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f\"ur gen\"ugend kleine $\Delta t$.
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\begin{parts}
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\part Schreibe eine Funktion die $n$ homogene Poisson Spiketrains
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einer gegebenen Dauer $T_{max}$ mit Rate $\lambda$ erzeugt.
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\begin{solution}
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\lstinputlisting{hompoissonspikes.m}
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\end{solution}
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\part Benutze diese Funktion um einige Trials von Spikes zu erzeugen
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und plotte diese als Spikeraster.
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\begin{solution}
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\begin{lstlisting}
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spikes = hompoissonspikes( 10, 100.0, 0.5 );
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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=0.7\textwidth]{poissonraster100hz}}
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\end{solution}
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\part Berechne Histogramme aus den Interspikeintervallen von $n$
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Poisson Spiketrains mit der Rate $\lambda=100$\,Hz. Ver\"andere
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\"uber die Dauer $T_{max}$ der Spiketrains und die Anzahl $n$ der
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Trials die Anzahl der Intervalle und ver\"andere auch die Binbreite
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des Histograms (fang mit 1\,ms an). Wieviele Interspikeintervalle
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werden ben\"otigt, um ein ``sch\"ones'' Histogramm zu erhalten? Wie
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lange m\"usste man also von dem Neuron ableiten?
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\begin{solution}
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About 5000 intervals for 25 bins. This corresponds to a $5000 /
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100\,\hertz = 50\,\second$ recording of a neuron firing with
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100\,\hertz.
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\end{solution}
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\part Vergleiche Interspike-Intervall Histogramme von Poisson-Spikes
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verschiedener Raten $\lambda$ mit der theoretisch zu erwartenden Verteilung
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der Intervalle $T$ des Poisson Prozesses
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\[ p(T) = \lambda e^{-\lambda T} \; .\]
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\begin{solution}
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\lstinputlisting{hompoissonisih.m}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}}
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\end{solution}
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\part \extra Was pasiert mit den Histogrammen, wenn die Binbreite
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der Histogramme kleiner als das bei der Erzeugung der Poisson
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Spiketrains verwendete $\Delta t$ ist?
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\begin{solution}
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Die Bins zwischen der durch $\Delta t$ vorgegebenen
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Diskretisierung haben den Wert 0. Dadurch werden aber die anderen
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durch die Normierung h\"oher als sie sein sollten.
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\end{solution}
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\part Plotte den Mittelwert der Interspikeintervalle, die
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dazugeh\"orige Standardabweichung und den Variationskoeffizienten
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als Funktion der Rate $\lambda$ des Poisson Prozesses. Vergleiche
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die Ergebnisse mit den theoretischen Erwartungen (siehe Vorlesungsskript).
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\begin{solution}
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\lstinputlisting{hompoissonisistats.m}
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\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}}
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\end{solution}
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\part Plotte die seriellen Korrelationen von Poisson-Spiketrains und
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erkl\"are kurz das Ergebniss.
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\begin{solution}
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\mbox{}\\[-2ex]\hspace*{2cm}
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\colorbox{white}{\includegraphics[width=0.8\textwidth]{poissonserial100hz}}\\
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Alle Korrelationen zwischen Interspikeintervallen sind Null, da
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beim Poisson Prozess das Auftreten jedes Spikes unabh\"angig von
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den vorherigen Spikes ist.
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\end{solution}
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\part Vergleiche Histogramme von Spikecounts gez\"ahlt in Fenstern
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der Breite $W$ mit der Poisson Verteilung
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\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \; , \]
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wobei die Rate $\lambda$ aus den Daten bestimmt werden
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soll. Hinweis: es gibt eine \code{matlab} Funktion, die die
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Fakult\"at $n!$ berechnet.
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\begin{solution}
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%\lstinputlisting{counthist.m}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}}
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\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}}
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\end{solution}
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\part Schreibe eine Funktion, die die mittlere Anzahl, die Varianz
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und den Fano-Faktor der Anzahl der Spikes in einem Fenster der
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Breite $W$ bestimmt. Benutze die Funktion, um diese Parameter f\"ur
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verschiedene Fensterbreiten $W$ zu bestimmen. Zwei Plots sollen aus
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den Ergebnissen angefertigt werden: (i) Varianz gegen Mittelwert der counts.
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(ii) Fano Faktor als Funktion der Fensterbreite.
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\begin{solution}
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\lstinputlisting{fanoplot.m}
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\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
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\end{solution}
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\end{parts}
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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{inversegauss.m}
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\lstinputlisting{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 $D$.
|
||||
\begin{solution}
|
||||
\begin{lstlisting}
|
||||
...
|
||||
% annotation:
|
||||
misi = mean( isis );
|
||||
sdisi = std( isis );
|
||||
disi = sdisi^2.0/2.0/misi^3;
|
||||
text( 0.6, 0.7, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' )
|
||||
text( 0.6, 0.6, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' )
|
||||
text( 0.6, 0.5, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' )
|
||||
text( 0.6, 0.4, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' )
|
||||
...
|
||||
\end{lstlisting}
|
||||
\end{solution}
|
||||
|
||||
\part Compare intersike interval histograms obtained from the LIF and PIF models with the inverse Gaussian.
|
||||
\begin{solution}
|
||||
\lstinputlisting{lifisih.m}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih01}}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih10}}\\
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih08}}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih16}}
|
||||
\end{solution}
|
||||
|
||||
\part Plot the firing rate (inverse mean interspike interval),
|
||||
mean interspike interval, the corresponding standard deviation,
|
||||
CV, and diffusion coefficient as a function of the input to the LIF
|
||||
and the PIF with noise strength set to 0.01.
|
||||
\begin{solution}
|
||||
\lstinputlisting{lifisistats.m}
|
||||
Leaky integrate-and-fire:\\
|
||||
\colorbox{white}{\includegraphics[width=0.8\textwidth]{lifisistats}}\\
|
||||
Perfect integrate-and-fire:\\
|
||||
\colorbox{white}{\includegraphics[width=0.8\textwidth]{pifisistats}}
|
||||
\end{solution}
|
||||
|
||||
\part Plot the firing rate as a function of input of the LIF and the PIF for various values
|
||||
of the noise strength.
|
||||
\begin{solution}
|
||||
\lstinputlisting{lifficurves.m}
|
||||
Leaky integrate-and-fire:\\
|
||||
\colorbox{white}{\includegraphics[width=0.7\textwidth]{lifficurves}}\\
|
||||
Perfect integrate-and-fire:\\
|
||||
\colorbox{white}{\includegraphics[width=0.7\textwidth]{pifficurves}}
|
||||
\end{solution}
|
||||
|
||||
\part Use the functions for computing serial correlations, count statistics and fano factors
|
||||
to further explore the statistics of the integrate-and-fire models!
|
||||
|
||||
\end{parts}
|
||||
|
||||
\end{questions}
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -1,163 +0,0 @@
|
||||
\documentclass[12pt,a4paper,pdftex]{exam}
|
||||
|
||||
\newcommand{\exercisetopic}{Point Processes}
|
||||
\newcommand{\exercisenum}{X3}
|
||||
\newcommand{\exercisedate}{January 19th, 2021}
|
||||
|
||||
\input{../../exercisesheader}
|
||||
|
||||
\firstpagefooter{Prof. Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\input{../../exercisestitle}
|
||||
|
||||
\begin{questions}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\question \qt{Statistik von Spiketrains 2}
|
||||
In Ilias findet ihr die Dateien \code{poisson.mat},
|
||||
\code{pifou.mat}, und \code{lifadapt.mat}. Jede dieser Dateien
|
||||
enth\"alt mehrere Trials von Spiketrains von einer bestimmten Art
|
||||
von Neuron. Die Spikezeiten sind in Sekunden gemessen.
|
||||
|
||||
Mit den folgenden Aufgaben wollen wir die Statistik der Spiketrains
|
||||
der drei Neurone miteinander vergleichen.
|
||||
|
||||
Bereits im letzten \"Ubungszettel erstellte Funktionen d\"urfen (sollen!)
|
||||
wiederverwendet werden.
|
||||
\begin{parts}
|
||||
\part Lade die Spiketrains aus den drei Dateien. Stelle sie in Rasterplots dar.
|
||||
|
||||
\part Plotte die Interspike-Intervall Verteilungen.
|
||||
|
||||
Annotiere die Plots mit dem Mittelwert, der
|
||||
Standardabweichung, und dem Variationskoeffizienten der
|
||||
Interspikeintervalle, sowie der mittleren Feuerrate.
|
||||
|
||||
\part Vergleiche die ISI-Histogramme mit der ISI Verteilung eines Poisson Prozesses
|
||||
der Rate $\lambda$:
|
||||
\[ p(T) = \lambda e^{-\lambda T} \; .\]
|
||||
|
||||
\part Erstelle Return-Maps f\"ur die drei Spiketrains, also jedes
|
||||
Interspike-Intervall $T_{i+1}$ gegen das vorherige Intervall $T_i$
|
||||
geplottet.
|
||||
|
||||
\part Schreibe eine Funktion, die die seriellen Korrelationen der
|
||||
Interspikeintervalle f\"ur Lags bis zu \code{maxlag} berechnet und
|
||||
plottet. Die Seriellen Korrelationen $\rho_k$ f\"ur Lag $k$ der
|
||||
Interspikeintervalle $T_i$ sind die Korrelationskoeffizienten
|
||||
zwischen den Interspikeintervallen $T_i$ und den um das Lag $k$
|
||||
verschobenen Intervallen $T_{i+k}$:
|
||||
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i -
|
||||
\langle T \rangle) \rangle}{\langle (T_i - \langle T
|
||||
\rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm
|
||||
var}(T_i)} = {\rm corr}(T_{i+k}, T_i) \]
|
||||
|
||||
Benutze diese Funktion, um die Interspikeintervall-Korrelationen
|
||||
der drei Neurone zu vergleichen.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/isiserialcorr.m}
|
||||
\lstinputlisting{../code/plotserialcorr.m}
|
||||
\colorbox{white}{\includegraphics[width=1\textwidth]{serialcorr}}
|
||||
\end{solution}
|
||||
|
||||
\end{parts}
|
||||
|
||||
\continue
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\question \qt{Homogener Poisson Prozess}
|
||||
Wir wollen den homogenen Poisson Prozess benutzen um Spikes zu
|
||||
generieren, mit denen wir die Analysfunktionen des vorherigen
|
||||
Aufgaben \"uberpr\"ufen k\"onnen.
|
||||
|
||||
Ein homogener Poisson Prozess mit der Rate $\lambda$ (gemessen in
|
||||
Hertz) ist ein Punktprozess, bei dem die Wahrscheinlichkeit eines
|
||||
Ereignisses unabh\"angig von der Zeit $t$ und unabh\"angig von
|
||||
vorherigen Ereignissen ist. Wenn wir die Zeitachse in kleine Bins
|
||||
der Breite $\Delta t$ einteilen, dann ist
|
||||
\[ P = \lambda \cdot \Delta t \]
|
||||
die Wahrscheinlichkeit innerhalb eines Bins ein Ereignis (``spike'')
|
||||
zu erhalten. $\Delta t$ muss daf\"ur klein genug sein, so dass $P<0.1$.
|
||||
\begin{parts}
|
||||
|
||||
\part Schreibe eine Funktion die $n$ homogene Poisson Spiketrains
|
||||
einer gegebenen Dauer $T_{max}$ mit Rate $\lambda$ erzeugt.
|
||||
|
||||
Falls das nicht gelingt, benutze f\"ur die folgenden Aufgaben
|
||||
soweit m\"oglich spikes aus der Datei \code{poisson.mat}.
|
||||
\begin{solution}
|
||||
\lstinputlisting{hompoissonspikes.m}
|
||||
\end{solution}
|
||||
|
||||
\part Benutze diese Funktion um einige Trials von Spikes zu erzeugen
|
||||
und plotte diese als Spikeraster.
|
||||
\begin{solution}
|
||||
\begin{lstlisting}
|
||||
spikes = hompoissonspikes( 10, 100.0, 0.5 );
|
||||
spikeraster( spikes )
|
||||
\end{lstlisting}
|
||||
\mbox{}\\[-3ex]
|
||||
\colorbox{white}{\includegraphics[width=0.7\textwidth]{poissonraster100hz}}
|
||||
\end{solution}
|
||||
|
||||
\part Berechne Histogramme aus den Interspikeintervallen von $n$
|
||||
Poisson Spiketrains mit der Rate $\lambda=100$\,Hz. Wie viele bins
|
||||
werden f\"ur ein ``sch\"ones'' ISI-Histogramm ungef\"ahr ben\"otigt?
|
||||
Ver\"andere \"uber die Dauer $T_{max}$ der Spiketrains und die
|
||||
Anzahl $n$ der Trials die Anzahl der Intervalle. Wieviele
|
||||
Interspikeintervalle werden ben\"otigt, um ein ``sch\"ones''
|
||||
Histogramm zu erhalten? Wie lange m\"usste man also von dem Neuron
|
||||
ableiten?
|
||||
\begin{solution}
|
||||
About 5000 intervals for 25 bins. This corresponds to a $5000 /
|
||||
100\,\hertz = 50\,\second$ recording of a neuron firing with
|
||||
100\,\hertz.
|
||||
\end{solution}
|
||||
|
||||
\part Vergleiche Interspike-Intervall Histogramme von Poisson-Spikes
|
||||
verschiedener Raten $\lambda$ mit der theoretisch zu erwartenden Verteilung
|
||||
der Intervalle $T$ des Poisson Prozesses
|
||||
\[ p(T) = \lambda e^{-\lambda T} \; .\]
|
||||
Achte darauf, dass die Bins des Histograms nicht kleiner als $\Delta t$ sind!
|
||||
\begin{solution}
|
||||
\lstinputlisting{hompoissonisih.m}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}}
|
||||
\end{solution}
|
||||
|
||||
\part \extra Was passiert mit den Histogrammen, wenn die Binbreite
|
||||
der Histogramme kleiner als das bei der Erzeugung der Poisson
|
||||
Spiketrains verwendete $\Delta t$ ist?
|
||||
\begin{solution}
|
||||
Die Bins zwischen der durch $\Delta t$ vorgegebenen
|
||||
Diskretisierung haben den Wert 0. Dadurch werden aber die anderen
|
||||
durch die Normierung h\"oher als sie sein sollten.
|
||||
\end{solution}
|
||||
|
||||
\part Plotte den Mittelwert der Interspikeintervalle, die
|
||||
dazugeh\"orige Standardabweichung und den Variationskoeffizienten
|
||||
als Funktion der Rate $\lambda$ des Poisson Prozesses. Vergleiche
|
||||
die Ergebnisse mit den theoretischen Erwartungen (siehe Vorlesungsskript).
|
||||
\begin{solution}
|
||||
\lstinputlisting{hompoissonisistats.m}
|
||||
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}}
|
||||
\end{solution}
|
||||
|
||||
\part Plotte die seriellen Korrelationen von Poisson-Spiketrains und
|
||||
erkl\"are kurz das Ergebniss.
|
||||
\begin{solution}
|
||||
\mbox{}\\[-2ex]\hspace*{2cm}
|
||||
\colorbox{white}{\includegraphics[width=0.8\textwidth]{poissonserial100hz}}\\
|
||||
Alle Korrelationen zwischen Interspikeintervallen sind Null, da
|
||||
beim Poisson Prozess das Auftreten jedes Spikes unabh\"angig von
|
||||
den vorherigen Spikes ist.
|
||||
\end{solution}
|
||||
|
||||
\end{parts}
|
||||
|
||||
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
@ -1,198 +0,0 @@
|
||||
\documentclass[12pt,a4paper,pdftex]{exam}
|
||||
|
||||
\usepackage[german]{babel}
|
||||
\usepackage{pslatex}
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
\usepackage{xcolor}
|
||||
\usepackage{graphicx}
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
|
||||
\pagestyle{headandfoot}
|
||||
\ifprintanswers
|
||||
\newcommand{\stitle}{L\"osungen}
|
||||
\else
|
||||
\newcommand{\stitle}{\"Ubung}
|
||||
\fi
|
||||
\header{{\bfseries\large \stitle}}{{\bfseries\large Punktprozesse}}{{\bfseries\large 27. Oktober, 2015}}
|
||||
\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
|
||||
jan.benda@uni-tuebingen.de}
|
||||
\runningfooter{}{\thepage}{}
|
||||
|
||||
\setlength{\baselineskip}{15pt}
|
||||
\setlength{\parindent}{0.0cm}
|
||||
\setlength{\parskip}{0.3cm}
|
||||
\renewcommand{\baselinestretch}{1.15}
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
language=Matlab,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
numberstyle=\tiny,
|
||||
title=\lstname,
|
||||
showstringspaces=false,
|
||||
commentstyle=\itshape\color{darkgray},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\continue}{\ifprintanswers%
|
||||
\else
|
||||
\vfill\hspace*{\fill}$\rightarrow$\newpage%
|
||||
\fi}
|
||||
\newcommand{\continuepage}{\ifprintanswers%
|
||||
\newpage
|
||||
\else
|
||||
\vfill\hspace*{\fill}$\rightarrow$\newpage%
|
||||
\fi}
|
||||
\newcommand{\newsolutionpage}{\ifprintanswers%
|
||||
\newpage%
|
||||
\else
|
||||
\fi}
|
||||
|
||||
%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\qt}[1]{\textbf{#1}\\}
|
||||
\newcommand{\pref}[1]{(\ref{#1})}
|
||||
\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\input{instructions}
|
||||
|
||||
|
||||
\begin{questions}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\question \qt{Statistik von Spiketrains}
|
||||
In Ilias findet ihr die Dateien \code{poisson.mat},
|
||||
\code{pifou.mat}, und \code{lifadapt.mat}. Jede dieser Dateien
|
||||
enth\"alt mehrere Trials von Spiketrains von einer bestimmten Art
|
||||
von Neuron. Die Spikezeiten sind in Sekunden gemessen.
|
||||
|
||||
Mit den folgenden Aufgaben wollen wir die Statistik der Spiketrains
|
||||
der drei Neurone miteinander vergleichen.
|
||||
\begin{parts}
|
||||
\part Lade die Spiketrains aus den drei Dateien. Achte darauf,
|
||||
dass sie verschiedene Variablen\-namen bekommen.
|
||||
\begin{solution}
|
||||
\begin{lstlisting}
|
||||
clear all
|
||||
% not so good:
|
||||
load poisson.mat
|
||||
whos
|
||||
poissonspikes = spikes;
|
||||
load pifou.mat;
|
||||
pifouspikes = spikes;
|
||||
load lifadapt.mat;
|
||||
lifadaptspikes = spikes;
|
||||
clear spikes;
|
||||
% better:
|
||||
clear all
|
||||
x = load( 'poisson.mat' );
|
||||
poissonspikes = x.spikes;
|
||||
x = load( pifou.mat' );
|
||||
pifouspikes = x.spikes;
|
||||
x = load( 'lifadapt.mat' );
|
||||
lifadaptspikes = x.spikes;
|
||||
\end{lstlisting}
|
||||
\end{solution}
|
||||
|
||||
\part Schreibe eine Funktion, die die Spikezeiten der ersten
|
||||
$t_{max}$ Sekunden in einem Rasterplot visualisiert. In jeder
|
||||
Zeile des Rasterplots wird ein Spiketrain dargestellt. Jeder
|
||||
einzelne Spike wird als senkrechte Linie zu der Zeit des
|
||||
Auftretens des Spikes geplottet. Benutze die Funktion, um die
|
||||
Spikeraster der ersten 1\,s der drei Neurone nebeneinander zu plotten.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/spikeraster.m}
|
||||
\lstinputlisting{../code/plotspikeraster.m}
|
||||
\mbox{}\\[-3ex]
|
||||
\colorbox{white}{\includegraphics[width=1\textwidth]{spikeraster}}
|
||||
\end{solution}
|
||||
|
||||
\part Schreibe eine Funktion, die einen einzigen Vektor mit den
|
||||
Interspikeintervallen aller Trials von Spikezeiten zur\"uckgibt.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/isis.m}
|
||||
\end{solution}
|
||||
|
||||
\part Schreibe eine Funktion, die ein normiertes Histogramm aus
|
||||
einem Vektor von Interspikeintervallen, gegeben in Sekunden,
|
||||
berechnet und dieses mit richtiger Achsenbeschriftung plottet.
|
||||
Die Interspikeintervalle sollen dabei in Millisekunden angegeben
|
||||
werden. Die Funktion soll zus\"atzlich den Mittelwert, die
|
||||
Standardabweichung, und den Variationskoeffizienten der
|
||||
Interspikeintervalle berechnen und diese im Plot mit angeben.
|
||||
|
||||
Benutze die vorherige und diese Funktion, um die
|
||||
Interspikeintervall Verteilung der drei Neurone zu vergleichen.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/isihist.m}
|
||||
\lstinputlisting{../code/plotisih.m}
|
||||
\mbox{}\\[-3ex]
|
||||
\colorbox{white}{\includegraphics[width=1\textwidth]{isihist}}
|
||||
\end{solution}
|
||||
|
||||
\part Schreibe eine Funktion, die die seriellen Korrelationen der
|
||||
Interspikeintervalle f\"ur Lags bis zu \code{maxlag} berechnet und
|
||||
plottet. Die Seriellen Korrelationen $\rho_k$ f\"ur Lag $k$ der
|
||||
Interspikeintervalle $T_i$ sind die Korrelationskoeffizienten
|
||||
zwischen den Interspikeintervallen $T_i$ und den um das Lag $k$
|
||||
verschobenen Intervallen $T_{i+k}$:
|
||||
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i -
|
||||
\langle T \rangle) \rangle}{\langle (T_i - \langle T
|
||||
\rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm
|
||||
var}(T_i)} = {\rm corr}(T_{i+k}, T_i) \]
|
||||
|
||||
Benutze diese Funktion, um die Interspikeintervall-Korrelationen
|
||||
der drei Neurone zu vergleichen.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/isiserialcorr.m}
|
||||
\lstinputlisting{../code/plotserialcorr.m}
|
||||
\colorbox{white}{\includegraphics[width=1\textwidth]{serialcorr}}
|
||||
\end{solution}
|
||||
|
||||
\part Schreibe eine Funktion, die aus Spikezeiten
|
||||
Histogramme aus der Anzahl von Spikes, die in Fenstern gegebener L\"ange $W$
|
||||
gez\"ahlt werden, erzeugt und plottet.
|
||||
|
||||
Wende diese Funktion auf die drei
|
||||
Datens\"atze an. Probiere verschiedene Fenstergr\"o{\ss}en $W$ aus.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/counthist.m}
|
||||
\lstinputlisting{../code/plotcounthist.m}
|
||||
\colorbox{white}{\includegraphics[width=1\textwidth]{counthist}}
|
||||
\end{solution}
|
||||
|
||||
|
||||
\end{parts}
|
||||
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
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pointprocesses/exercises/poissonisihist.pdf
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pointprocesses/exercises/poissonisihist.pdf
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@ -1,42 +1,20 @@
|
||||
\documentclass[12pt,a4paper,pdftex]{exam}
|
||||
|
||||
\usepackage[german]{babel}
|
||||
\usepackage{natbib}
|
||||
\usepackage{graphicx}
|
||||
\usepackage[small]{caption}
|
||||
\usepackage{sidecap}
|
||||
\usepackage{pslatex}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\setlength{\marginparwidth}{2cm}
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
\newcommand{\exercisetopic}{Time-dependent firing rates}
|
||||
\newcommand{\exercisenum}{12}
|
||||
\newcommand{\exercisedate}{January 19th, 2021}
|
||||
|
||||
%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
|
||||
\pagestyle{headandfoot} \header{{\bfseries\large Exercise
|
||||
}}{{\bfseries\large Time-dependent firing rate}}{{\bfseries\large January, 14, 2020}}
|
||||
\firstpagefooter{Dr. Jan Grewe}{Phone: 29 74588}{Email:
|
||||
jan.grewe@uni-tuebingen.de} \runningfooter{}{\thepage}{}
|
||||
\input{../../exercisesheader}
|
||||
|
||||
\setlength{\baselineskip}{15pt}
|
||||
\setlength{\parindent}{0.0cm}
|
||||
\setlength{\parskip}{0.3cm}
|
||||
\renewcommand{\baselinestretch}{1.15}
|
||||
\firstpagefooter{Dr. Jan Grewe}{}{jan.grewe@uni-tuebingen.de}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
\renewcommand{\solutiontitle}{\noindent\textbf{Solution}\par\noindent}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\vspace*{-6.5ex}
|
||||
\begin{center}
|
||||
\textbf{\Large Introduction to scientific computing.}\\[1ex]
|
||||
{\large Jan Grewe, Jan Benda}\\[-3ex]
|
||||
Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
|
||||
\end{center}
|
||||
\input{../../exercisestitle}
|
||||
|
||||
\begin{questions}
|
||||
|
||||
\question Plot the time-dependent firing rate of a neuron. Calculate
|
||||
the firing rate from the \emph{instantaneous firing rate} (based on the
|
||||
interspike interval). Use the \code{lifoustim.mat}. The dataset
|
||||
@ -45,14 +23,14 @@
|
||||
duration of each trial is 30 seconds.
|
||||
|
||||
\begin{parts}
|
||||
\part{} Write a function that takes three arguments: the spike
|
||||
\part Write a function that takes three arguments: the spike
|
||||
times of a single trial, the trial duration and the temporal
|
||||
resolution. The function should return two variables: the time axis and the
|
||||
time-dependent firing rate.
|
||||
\part{} Write a script that applies the above function to estimate
|
||||
\part Write a script that applies the above function to estimate
|
||||
the time-dependent firing rate of each trial. Plot the firing rates of the individual responses
|
||||
and the average response as a function of time into the same graph.
|
||||
\part{} Extend your program that it saves the figure with the width and height of 8.5\,cm using a fontsize of 10\,pt for labels.
|
||||
\part Extend your program that it saves the figure with the width and height of 8.5\,cm using a fontsize of 10\,pt for labels.
|
||||
See Chapter 3 in the script, or browse the Matlab help for more
|
||||
information. Store the figure in pdf format.
|
||||
\end{parts}
|
||||
@ -65,10 +43,10 @@
|
||||
|
||||
\question{} Some trials are different than the others.
|
||||
\begin{parts}
|
||||
\part{} Use a rasterplot to identify them. In which sense
|
||||
\part Use a rasterplot to identify them. In which sense
|
||||
are they different? Save the rasterplot in pdf
|
||||
format. Use the same figure specifications as above and make sure it is properly labeled.
|
||||
\part{} Identify those trials in which the spike count
|
||||
\part Identify those trials in which the spike count
|
||||
deviates more than $2\sigma$ (twice the standard deviation) from the average.
|
||||
\end{parts}
|
||||
\end{questions}
|
||||
|
||||
Reference in New Issue
Block a user