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New exercise for point processes

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Jan Benda
2015-10-26 23:35:23 +01:00
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pointprocesses/exercises/UT_WBMW_Black_RGB.pdf Normal file
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pointprocesses/exercises/counthist.pdf Normal file
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pointprocesses/exercises/isihist.pdf Normal file
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pointprocesses/exercises/pointprocesses01.tex
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\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}}{{\bfseries\large 27. Oktober, 2015}}
\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
jan.benda@uni-tuebingen.de}
\runningfooter{}{\thepage}{}
@@ -89,114 +89,99 @@ 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.
\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.
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
\[ P = \lambda \cdot \Delta t \]
for sufficiently small $\Delta t$.
\begin{parts}
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
Variablennamen bekommen.
\begin{solution}
\begin{lstlisting}
clear all
load poisson.mat
whos
poissonspikes = spikes;
load pifou.mat;
pifouspikes = spikes;
load lifadapt.mat;
lifadaptspikes = spikes;
clear spikes;
\end{lstlisting}
\end{solution}
\part Schreibe eine Funktion, die die Spikezeiten der ersten
\code{tmax} 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 zu plotten.
\begin{solution}
\lstinputlisting{../code/spikeraster.m}
\lstinputlisting{../code/plotspikeraster.m}
\mbox{}\\[-3ex]
\colorbox{white}{\includegraphics[width=1\textwidth]{spikeraster}}
\end{solution}
\part Write a function that generates $n$ homogeneous Poisson spike trains of a given duration $T_{max}$
with rate $\lambda$.
\begin{solution}
\lstinputlisting{hompoissonspikes.m}
\end{solution}
\part Schreibe eine Funktion, die einen einzigen Vektor mit den Interspike-Intervallen
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 Interspike-Intervallen, gegeben in Sekunden,
berechnet und dieses mit richtiger Achsenbeschriftung plottet. Die
Interspike-Intervalle sollen dabei in Millisekunden angegeben
werden. Die Funktion soll ausserdem den Mittelwert, die Standardabweichung,
und den Variationskoeffizienten der Interspike Intervalle berechnen
und diese im Plot mit angeben.
Benutze diese und die vorherige Funktion, um die Interspike-Intervall 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 Using this function, generate a few trials and display them in a raster plot.
\begin{solution}
\lstinputlisting{../code/spikeraster.m}
\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 Schreibe eine Funktion, die die Seriellen Korrelationen der
Interspike Intervalle f\"ur lags bis zu \code{maxlag} berechnet
und plottet. Die Seriellen Korrelationen $\rho_k$ f\"ur lag $k$
der Interspike Intervalle $T_i$ sind wie folgt definiert:
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i -
\langle T \rangle) \rangle}{\langle (T_i - \langle T
\rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm
var}(T_i)} = {\rm corrcoef}(T_{i+k}, T_i) \] Benutze dies Funktion,
um die Interspike Intervall 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 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?
\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 Compare the histogram with the true distribution of intervals $T$ of the Poisson process
\[ p(T) = \lambda e^{-\lambda T} \]
for various rates $\lambda$.
\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 $\Delta t$
used for generating the Poisson spikes?
\begin{solution}
The bins between the discretization have zero entries. Therefore
the other ones become higher than they should be.
\end{solution}
\part Plot the mean interspike interval, the corresponding standard deviation, and the CV
as a function of the rate $\lambda$ of the Poisson process.
Compare the ../code with the theoretical expectations for the dependence on $\lambda$.
\begin{solution}
\lstinputlisting{hompoissonisistats.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}}
\end{solution}
\part Write a function that computes serial correlations for the interspike intervals
for a range of lags.
The serial correlations $\rho_k$ at lag $k$ are defined as
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
Use this function to show that interspike intervals of Poisson spikes are independent.
\begin{solution}
\lstinputlisting{../code/isiserialcorr.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonserial100hz}}
\end{solution}
\part Write a function that generates from spike times
a histogram of spike counts in a count window of given duration $W$.
The function should also plot the Poisson distribution
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
for the rate $\lambda$ determined from the spike trains.
\begin{solution}
\lstinputlisting{../code/counthist.m}
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}}
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}}
\end{solution}
\part Write a function that computes mean count, variance of count and the corresponding Fano factor
for a range of count window durations. The function should generate tow plots: one plotting
the count variance against the mean, the other one the Fano factor as a function of the window duration.
\begin{solution}
\lstinputlisting{../code/fano.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
\end{solution}
\end{parts}
\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. Zus\"atzlich soll die Funktion
die Poisson-Verteilung
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \] mit der Rate
$\lambda$, die aus den Daten bestimmt werden kann, mit zu dem
Histogramm hineinzeichen.
\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/pointprocesses02.tex Normal file
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\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}{}
\fi
\header{{\bfseries\large \"Ubung 6\stitle}}{{\bfseries\large Statistik}}{{\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{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
\[ P = \lambda \cdot \Delta t \]
for sufficiently small $\Delta t$.
\begin{parts}
\part Write a function that generates $n$ homogeneous Poisson spike trains of a given duration $T_{max}$
with rate $\lambda$.
\begin{solution}
\lstinputlisting{hompoissonspikes.m}
\end{solution}
\part Using this function, generate a few trials and display them in a raster plot.
\begin{solution}
\lstinputlisting{../code/spikeraster.m}
\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 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?
\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 Compare the histogram with the true distribution of intervals $T$ of the Poisson process
\[ p(T) = \lambda e^{-\lambda T} \]
for various rates $\lambda$.
\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 $\Delta t$
used for generating the Poisson spikes?
\begin{solution}
The bins between the discretization have zero entries. Therefore
the other ones become higher than they should be.
\end{solution}
\part Plot the mean interspike interval, the corresponding standard deviation, and the CV
as a function of the rate $\lambda$ of the Poisson process.
Compare the ../code with the theoretical expectations for the dependence on $\lambda$.
\begin{solution}
\lstinputlisting{hompoissonisistats.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}}
\end{solution}
\part Write a function that computes serial correlations for the interspike intervals
for a range of lags.
The serial correlations $\rho_k$ at lag $k$ are defined as
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
Use this function to show that interspike intervals of Poisson spikes are independent.
\begin{solution}
\lstinputlisting{../code/isiserialcorr.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonserial100hz}}
\end{solution}
\part Write a function that generates from spike times
a histogram of spike counts in a count window of given duration $W$.
The function should also plot the Poisson distribution
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
for the rate $\lambda$ determined from the spike trains.
\begin{solution}
\lstinputlisting{../code/counthist.m}
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}}
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}}
\end{solution}
\part Write a function that computes mean count, variance of count and the corresponding Fano factor
for a range of count window durations. The function should generate tow plots: one plotting
the count variance against the mean, the other one the Fano factor as a function of the window duration.
\begin{solution}
\lstinputlisting{../code/fano.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
\end{solution}
\end{parts}
\end{questions}
\end{document}

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