diff --git a/pointprocesses/exercises/pointprocesses01.tex b/pointprocesses/exercises/pointprocesses01.tex index 93fb077..e537be7 100644 --- a/pointprocesses/exercises/pointprocesses01.tex +++ b/pointprocesses/exercises/pointprocesses01.tex @@ -103,6 +103,7 @@ jan.benda@uni-tuebingen.de} \begin{solution} \begin{lstlisting} clear all + % not so good: load poisson.mat whos poissonspikes = spikes; @@ -111,6 +112,14 @@ jan.benda@uni-tuebingen.de} 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} diff --git a/pointprocesses/exercises/pointprocesses02.tex b/pointprocesses/exercises/pointprocesses02.tex index 5728c21..0e849e3 100644 --- a/pointprocesses/exercises/pointprocesses02.tex +++ b/pointprocesses/exercises/pointprocesses02.tex @@ -11,11 +11,11 @@ \usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry} \pagestyle{headandfoot} \ifprintanswers -\newcommand{\stitle}{: L\"osungen} +\newcommand{\stitle}{L\"osungen} \else -\newcommand{\stitle}{} +\newcommand{\stitle}{\"Ubung} \fi -\header{{\bfseries\large \"Ubung 6\stitle}}{{\bfseries\large Statistik}}{{\bfseries\large 27. Oktober, 2015}} +\header{{\bfseries\large \stitle}}{{\bfseries\large Punktprozesse 2}}{{\bfseries\large 27. Oktober, 2015}} \firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email: jan.benda@uni-tuebingen.de} \runningfooter{}{\thepage}{} @@ -89,26 +89,27 @@ jan.benda@uni-tuebingen.de} \begin{questions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\question \qt{Homogeneous Poisson process} -We use the Poisson process to generate spike trains on which we can test and imrpove some -standard analysis functions. - -A homogeneous Poisson process of rate $\lambda$ (measured in Hertz) is a point process -where the probability of an event is independent of time $t$ and independent of previous events. -The probability $P$ of an event within a bin of width $\Delta t$ is +\question \qt{Homogener Poisson Prozess} +Wir wollen den homogenen Poisson Prozess benutzen um Spikes zu generieren, +mit denen wir die Analysfunktionen des vorherigen \"Ubungszettel \"uberpr\"ufen k\"onnen. + +Ein homogener Poisson Prozess mit der Rate $\lambda$ (measured in Hertz) ist ein Punktprozess, +bei dem die Wahrschienlichkeit eines Ereignisses unabh\"angig von der Zeit $t$ und +unabh\"angig von vorherigen Ereignissen ist. +Die Wahrscheinlichkeit $P$ eines Ereignisses innerhalb eines Bins der Breite $\Delta t$ ist \[ P = \lambda \cdot \Delta t \] -for sufficiently small $\Delta t$. +f\"ur gen\"ugend kleine $\Delta t$. \begin{parts} - \part Write a function that generates $n$ homogeneous Poisson spike trains of a given duration $T_{max}$ - with rate $\lambda$. + \part Schreibe eine Funktion die $n$ homogene Poisson Spiketrains + einer gegebenen Dauer $T_{max}$ mit rate $\lambda$ erzeugt. \begin{solution} \lstinputlisting{hompoissonspikes.m} \end{solution} - \part Using this function, generate a few trials and display them in a raster plot. + \part Benutze diese Funktion um einige Trials von Spikes zu erzeugen + und plotte diese als Spikeraster. \begin{solution} - \lstinputlisting{../code/spikeraster.m} \begin{lstlisting} spikes = hompoissonspikes( 10, 100.0, 0.5 ); spikeraster( spikes ) @@ -117,41 +118,31 @@ for sufficiently small $\Delta t$. \colorbox{white}{\includegraphics[width=0.7\textwidth]{poissonraster100hz}} \end{solution} - \part Write a function that extracts a single vector of interspike intervals - from the spike times returned by the first function. - \begin{solution} - \lstinputlisting{../code/isis.m} - \end{solution} - - \part Write a function that plots the interspike-interval histogram - from a vector of interspike intervals. The function should also - compute the mean, the standard deviation, and the CV of the intervals - and display the values in the plot. - \begin{solution} - \lstinputlisting{../code/isihist.m} - \end{solution} - - \part Compute histograms for Poisson spike trains with rate - $\lambda=100$\,Hz. Play around with $T_{max}$ and $n$ and the bin width - (start with 1\,ms) of the histogram. - How many - interspike intervals do you approximately need to get a ``nice'' - histogram? How long do you need to record from the neuron? + \part Berechne Histogramme aus den Interspikeintervallen von $n$ + Poisson Spiketrains mit der Rate $\lambda=100$\,Hz. Ver\"andere + \"uber die Dauer $T_{max}$ der Spiketrains und die Anzahl $n$ der + Trials die Anzahl der Intervalle und ver\"andere auch die Binbreite + des Histograms (fange mit 1\,ms an). 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. + 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 Compare the histogram with the true distribution of intervals $T$ of the Poisson process + \part Vergleiche das Histogramm mit der zu erwartenden Verteilung + der Intervalle $T$ des Poisson Prozesses \[ p(T) = \lambda e^{-\lambda T} \] - for various rates $\lambda$. + mit rate $\lambda$. \begin{solution} \lstinputlisting{hompoissonisih.m} \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}} \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}} \end{solution} - \part What happens if you make the bin width of the histogram smaller than $\Delta t$ + \part \extra Was pasiert mit den Histogrammen, wenn die Binbreite der Histogramme kleiner + als das bei der Erzeugung der $\Delta t$ der used for generating the Poisson spikes? \begin{solution} The bins between the discretization have zero entries. Therefore