Merge branch 'master' of raven.am28.uni-tuebingen.de:scientificComputing
This commit is contained in:
commit
729b8f281b
29
designpattern/lecture/Makefile
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29
designpattern/lecture/Makefile
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@ -0,0 +1,29 @@
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||||
BASENAME=designpattern
|
||||
|
||||
PYFILES=$(wildcard *.py)
|
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PYPDFFILES=$(PYFILES:.py=.pdf)
|
||||
|
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all : pdf
|
||||
|
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# script:
|
||||
pdf : $(BASENAME)-chapter.pdf
|
||||
|
||||
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex $(PYPDFFILES)
|
||||
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
|
||||
|
||||
$(PYPDFFILES) : %.pdf : %.py
|
||||
python $<
|
||||
|
||||
clean :
|
||||
rm -f *~
|
||||
rm -f $(BASENAME).aux $(BASENAME).log
|
||||
rm -f $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out
|
||||
rm -f $(PYPDFFILES) $(GPTTEXFILES)
|
||||
|
||||
cleanall : clean
|
||||
rm -f $(BASENAME)-chapter.pdf
|
||||
|
||||
watchpdf :
|
||||
while true; do ! make -q pdf && make pdf; sleep 0.5; done
|
||||
|
||||
|
||||
225
designpattern/lecture/designpattern-chapter.tex
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225
designpattern/lecture/designpattern-chapter.tex
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\documentclass[12pt]{report}
|
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|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Grewe \& Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
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\date{WS 15/16}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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||||
% \newcommand{\tr}[2]{#1} % en
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% \usepackage[english]{babel}
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\newcommand{\tr}[2]{#2} % de
|
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\usepackage[german]{babel}
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|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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|
||||
|
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%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
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\usepackage{xcolor}
|
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\pagecolor{white}
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||||
|
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\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
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\put(0,4){\line(1,0){170}}%
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\multiput(0,2)(10,0){18}{\line(0,1){4}}%
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\multiput(0,3)(1,0){170}{\line(0,1){2}}%
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\put(0,0){\makebox(0,0){{\tiny 0}}}%
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\put(10,0){\makebox(0,0){{\tiny 1}}}%
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\put(20,0){\makebox(0,0){{\tiny 2}}}%
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\put(30,0){\makebox(0,0){{\tiny 3}}}%
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\put(40,0){\makebox(0,0){{\tiny 4}}}%
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\put(50,0){\makebox(0,0){{\tiny 5}}}%
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\put(60,0){\makebox(0,0){{\tiny 6}}}%
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\put(70,0){\makebox(0,0){{\tiny 7}}}%
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\put(80,0){\makebox(0,0){{\tiny 8}}}%
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\put(90,0){\makebox(0,0){{\tiny 9}}}%
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\put(100,0){\makebox(0,0){{\tiny 10}}}%
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\put(110,0){\makebox(0,0){{\tiny 11}}}%
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\put(120,0){\makebox(0,0){{\tiny 12}}}%
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\put(130,0){\makebox(0,0){{\tiny 13}}}%
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\put(140,0){\makebox(0,0){{\tiny 14}}}%
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\put(150,0){\makebox(0,0){{\tiny 15}}}%
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\put(160,0){\makebox(0,0){{\tiny 16}}}%
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\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
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\end{picture}\par}
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|
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% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
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||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
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%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
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% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
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\setcounter{bottomnumber}{0}
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\setcounter{totalnumber}{2}
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|
||||
% float placement fractions:
|
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\renewcommand{\textfraction}{0.2}
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\renewcommand{\topfraction}{0.8}
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\renewcommand{\bottomfraction}{0.0}
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||||
\renewcommand{\floatpagefraction}{0.5}
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||||
|
||||
% spacing for floats:
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||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
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\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
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\setlength{\intextsep}{12pt plus 2pt minus 2pt}
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||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
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||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
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||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
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||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
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\newcommand{\figsb}{Figs.}
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\newcommand{\figrefb}[1]{\figb~\fref{#1}}
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||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{darkgray},
|
||||
keywordstyle=\color{blue},
|
||||
stringstyle=\color{green},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\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}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\include{designpattern}
|
||||
|
||||
\end{document}
|
||||
166
designpattern/lecture/designpattern.tex
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166
designpattern/lecture/designpattern.tex
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@ -0,0 +1,166 @@
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\chapter{Design Pattern}
|
||||
|
||||
Beim Programmieren sind sich viel Codes in ihrer Grundstruktur sehr
|
||||
\"ahnlich. Viele Konstrukte kommen in den verschiedensten Kontexten
|
||||
immer wieder in \"ahnlicher Weise vor. In diesem Kapitel stellen wir
|
||||
einige dieser ``Design pattern'' zusammen.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Plotten einer mathematischen Funktion}
|
||||
Eine mathematische Funktion ordnet einem beliebigen $x$-Wert einen
|
||||
$y$-Wert zu. Um eine solche Funktion zeichnen zu k\"onnen, m\"ussen
|
||||
wir uns eine Wertetabelle aus vielen $x$-Werten und den
|
||||
dazugeh\"origen Funktionswerten $y=f(x)$ erstellen.
|
||||
|
||||
Wir erstellen uns dazu einen Vektor mit geeigneten $x$-Werten, die von
|
||||
dem kleinsten bis zu dem gr\"o{\ss}ten $x$-Wert laufen, den wir
|
||||
plotten wollen. Die Schrittweite f\"ur die $x$-Werte w\"ahlen wir
|
||||
klein genug, um eine sch\"one glatte Kurve zu bekommen. F\"ur jeden
|
||||
Wert $x_i$ dieses Vektors berechnen wir den entsprechenden
|
||||
Funktionswert und erzeugen damit einen Vektor mit den $y$-Werten. Die
|
||||
Werte des $y$-Vektors k\"onnen dann gegen die Werte des $x$-Vektors
|
||||
geplottet werden.
|
||||
|
||||
Folgende Programme berechnen und plotten die Funktion $f(x)=e^{-x^2}$:
|
||||
\begin{lstlisting}
|
||||
xmin = -1.0;
|
||||
xmax = 2.0;
|
||||
dx = 0.01; % Schrittweite
|
||||
x = xmin:dx:xmax; % Vektor mit x-Werten
|
||||
y = exp(-x.*x); % keine for Schleife! '.*' fuer elementweises multiplizieren
|
||||
plot(x, y);
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = -1:0.01:2; % Vektor mit x-Werten
|
||||
y = exp(-x.*x); % keine for Schleife! '.*' fuer elementweises multiplizieren
|
||||
plot(x, y);
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = -1:0.01:2; % Vektor mit x-Werten
|
||||
plot(x, exp(-x.*x));
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Skalieren und Verschieben nicht nur von Zufallszahlen}
|
||||
Zufallsgeneratoren geben oft nur Zufallszahlen mit festen Mittelwerten
|
||||
und Standardabweichungen (auch Skalierungen) zur\"uck. Multiplikation
|
||||
mit einem Faktor skaliert die Standardabweichung und Addition einer Zahl
|
||||
verschiebt den Mittelwert.
|
||||
|
||||
\begin{lstlisting}
|
||||
% 100 random numbers draw from a Gaussian distribution with mean 0 and standard deviation 1.
|
||||
x = randn(100, 1);
|
||||
|
||||
% 100 random numbers drawn from a Gaussian distribution with mean 4.8 and standard deviation 2.3.
|
||||
mu = 4.8;
|
||||
sigma = 2.3;
|
||||
y = randn(100, 1)*sigma + mu;
|
||||
\end{lstlisting}
|
||||
|
||||
Das ist manchmal auch sinnvoll f\"ur \code{zeros} oder \code{ones}:
|
||||
\begin{lstlisting}
|
||||
x = -1:0.01:2; % Vektor mit x-Werten
|
||||
plot(x, exp(-x.*x));
|
||||
% Plotte f\"ur die gleichen x-Werte eine Linie mit y=0.8:
|
||||
plot(x, zeros(size(x))+0.8);
|
||||
% ... Linie mit y=0.5:
|
||||
plot(x, ones(size(x))*0.5);
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{for Schleifen \"uber Vektoren}
|
||||
Manchmal m\"ochte man doch mit einer for-Schleife \"uber einen Vektor iterieren:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
for i=1:length(x)
|
||||
% Benutze den Wert des Vektors x an der Stelle des Indexes i:
|
||||
do_something( x(i) );
|
||||
end
|
||||
\end{lstlisting}
|
||||
|
||||
Wenn in der Schleife das Ergebnis in einen Vektor gespeichert werden soll,
|
||||
sollten wir uns vor der Schleife schon einen Vektor f\"ur die Ergebnisse
|
||||
erstellen:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = zeros(size(x)); % Platz fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Schreibe den Rueckgabewert der Funktion get_something an die i-te
|
||||
% Stelle von y:
|
||||
y(i) = get_something( x(i) );
|
||||
end
|
||||
% jetzt koennen wir den Ergebnisvektor weiter bearbeiten:
|
||||
mean(y)
|
||||
\end{lstlisting}
|
||||
|
||||
Die Berechnungen in der Schleife k\"onnen statt einer Zahl auch einen Vektor
|
||||
zur\"uckgeben. Wenn die L\"ange diese Vektors bekannt ist, dann kann vorher
|
||||
eine entsprechend gro{\ss}e Matrix angelegt werden:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = zeros(length(x),10); % Platz fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Schreibe den Rueckgabewert der Funktion get_something - jetzt ein
|
||||
% Vektor mit 10 Elementen - in die i-te
|
||||
% Zeile von y:
|
||||
y(i,:) = get_something( x(i) );
|
||||
end
|
||||
% jetzt koennen wir die Ergebnismatrix weiter bearbeiten:
|
||||
mean(y, 1)
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
Alternativ k\"onnen die in der Schleife erzeugten Vektoren zu einem
|
||||
einzigen, durchgehenden Vektor zusammengestellt werden:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = []; % Leerer Vektor fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Die Funktion get_something gibt uns einen Vektor zurueck:
|
||||
z = get_something( x(i) );
|
||||
% dessen Inhalt h\"angen wir an unseren Ergebnissvektor an:
|
||||
y = [y z(:)];
|
||||
end
|
||||
% jetzt koennen wir dem Ergebnisvektor weiter bearbeiten:
|
||||
mean(y)
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Normierung von Histogrammen}
|
||||
Meistens sollten Histogramme normiert werden, damit sie vergleichbar
|
||||
mit anderen Histogrammen oder mit theoretischen
|
||||
Wahrscheinlichkeitsverteilungen werden.
|
||||
|
||||
Die \code{histogram} Funktion macht das mit den entsprechenden Parametern automatisch:
|
||||
\begin{lstlisting}
|
||||
x = randn(100, 1); % irgendwelche reellwertige Daten
|
||||
histogram(x, 'Normalization', 'pdf');
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = randi(6, 100, 1); % irgendwelche integer Daten
|
||||
histogram(x, 'Normalization', 'probability');
|
||||
\end{lstlisting}
|
||||
|
||||
So geht es aber auch:
|
||||
\begin{lstlisting}
|
||||
x = randn(100, 1); % irgendwelche reellwertige Daten
|
||||
[h, b] = hist(x); % Histogram berechnen
|
||||
h = h/sum(h)/(b(2)-b(1)); % normieren zu einer Wahrscheinlichkeitsdichte
|
||||
bar(b, h); % und plotten.
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = randi(6, 100, 1); % irgendwelche integer Daten
|
||||
[h, b] = hist(x); % Histogram berechnen
|
||||
h = h/sum(h); % normieren zu Wahrscheinlichkeiten
|
||||
bar(b, h); % und plotten.
|
||||
\end{lstlisting}
|
||||
@ -1,4 +1,4 @@
|
||||
function [ counts, bins ] = counthist( spikes, w )
|
||||
function [counts, bins] = counthist(spikes, w)
|
||||
% computes count histogram and compare them with Poisson distribution
|
||||
% spikes: a cell array of vectors of spike times
|
||||
% w: observation window duration for computing the counts
|
||||
@ -18,18 +18,17 @@ function [ counts, bins ] = counthist( spikes, w )
|
||||
end
|
||||
% histogram of spike counts:
|
||||
maxn = max( n );
|
||||
[counts, bins ] = hist( n, 0:1:maxn+1 );
|
||||
[counts, bins ] = hist( n, 0:1:maxn+10 );
|
||||
counts = counts / sum( counts );
|
||||
if nargout == 0
|
||||
bar( bins, counts );
|
||||
hold on;
|
||||
% Poisson distribution:
|
||||
rate = mean( r );
|
||||
x = 0:1:20;
|
||||
x = 0:1:maxn+10;
|
||||
l = rate*w;
|
||||
y = l.^x.*exp(-l)./factorial(x);
|
||||
plot( x, y, 'r', 'LineWidth', 3 );
|
||||
xlim( [ 0 20 ] );
|
||||
hold off;
|
||||
xlabel( 'counts k' );
|
||||
ylabel( 'P(k)' );
|
||||
|
||||
@ -24,9 +24,9 @@ function isihist( isis, binwidth )
|
||||
misi = mean( isis );
|
||||
sdisi = std( isis );
|
||||
disi = sdisi^2.0/2.0/misi^3;
|
||||
text( 0.5, 0.6, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' )
|
||||
text( 0.5, 0.5, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' )
|
||||
text( 0.5, 0.4, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' )
|
||||
text( 0.95, 0.8, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized', 'HorizontalAlignment', 'right' )
|
||||
text( 0.95, 0.7, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized', 'HorizontalAlignment', 'right' )
|
||||
text( 0.95, 0.6, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized', 'HorizontalAlignment', 'right' )
|
||||
%text( 0.5, 0.3, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' )
|
||||
end
|
||||
|
||||
|
||||
BIN
pointprocesses/code/lifadapt.mat
Normal file
BIN
pointprocesses/code/lifadapt.mat
Normal file
Binary file not shown.
@ -3,8 +3,8 @@ function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr
|
||||
% with an adaptation current
|
||||
% trials: the number of trials to be generated
|
||||
% input: the stimulus either as a single value or as a vector
|
||||
% tmaxdt: in case of a single value stimulus the duration of a trial
|
||||
% in case of a vector as a stimulus the time step
|
||||
% tmaxdt: in case of a single value stimulus: the duration of a trial
|
||||
% in case of a vector as a stimulus: the time step
|
||||
% D: the strength of additive white noise
|
||||
% tauadapt: adaptation time constant
|
||||
% adaptincr: adaptation strength
|
||||
|
||||
BIN
pointprocesses/code/lifoustim.mat
Normal file
BIN
pointprocesses/code/lifoustim.mat
Normal file
Binary file not shown.
@ -2,8 +2,8 @@ function spikes = lifspikes( trials, input, tmaxdt, D )
|
||||
% Generate spike times of a leaky integrate-and-fire neuron
|
||||
% trials: the number of trials to be generated
|
||||
% input: the stimulus either as a single value or as a vector
|
||||
% tmaxdt: in case of a single value stimulus the duration of a trial
|
||||
% in case of a vector as a stimulus the time step
|
||||
% tmaxdt: in case of a single value stimulus: the duration of a trial
|
||||
% in case of a vector as a stimulus: the time step
|
||||
% D: the strength of additive white noise
|
||||
|
||||
tau = 0.01;
|
||||
|
||||
20
pointprocesses/code/lifspikesoustim.m
Normal file
20
pointprocesses/code/lifspikesoustim.m
Normal file
@ -0,0 +1,20 @@
|
||||
function spikes = lifspikesoustim(trials, tmax, D, Iou, Dou, tauou )
|
||||
% Generate spike times of a leaky integrate-and-fire neuron with frozen
|
||||
% Ohrnstein-Uhlenbeck stimulus
|
||||
% trials: the number of trials to be generated
|
||||
% tmax: the duration of a trial
|
||||
% D: the strength of additive white noise
|
||||
% Iou: the mean input
|
||||
% Dou: noise strength of the frozen OU noise
|
||||
% tauou: time constant of the OU noise
|
||||
|
||||
dt = 1e-4;
|
||||
input = zeros(round(tmax/dt), 1);
|
||||
n = 0.0;
|
||||
noise = sqrt(2.0*Dou)*randn(length(input), 1)/sqrt(dt);
|
||||
for i=1:length(noise)
|
||||
n = n + ( - n + noise(i))*dt/tauou;
|
||||
input(i) = Iou + n;
|
||||
end
|
||||
spikes = lifspikes(trials, input, dt, D );
|
||||
end
|
||||
BIN
pointprocesses/code/pifou.mat
Normal file
BIN
pointprocesses/code/pifou.mat
Normal file
Binary file not shown.
@ -1,9 +1,10 @@
|
||||
function spikes = pifouspikes( trials, input, tmaxdt, D, outau )
|
||||
% Generate spike times of a perfect integrate-and-fire neuron
|
||||
% with Ornstein-Uhlenbeck noise (colored noise).
|
||||
% trials: the number of trials to be generated
|
||||
% input: the stimulus either as a single value or as a vector
|
||||
% tmaxdt: in case of a single value stimulus the duration of a trial
|
||||
% in case of a vector as a stimulus the time step
|
||||
% tmaxdt: in case of a single value stimulus: the duration of a trial
|
||||
% in case of a vector as a stimulus: the time step
|
||||
% D: the strength of additive white noise
|
||||
% outau: time constant of the colored noise
|
||||
|
||||
|
||||
24
pointprocesses/code/plotcounthist.m
Normal file
24
pointprocesses/code/plotcounthist.m
Normal file
@ -0,0 +1,24 @@
|
||||
w = 0.1;
|
||||
cmax = 8;
|
||||
pmax = 0.5;
|
||||
subplot(1, 3, 1);
|
||||
counthist(poissonspikes, w);
|
||||
xlim([0 cmax])
|
||||
set(gca, 'XTick', 0:2:cmax)
|
||||
ylim([0 pmax])
|
||||
title('Poisson');
|
||||
|
||||
subplot(1, 3, 2);
|
||||
counthist(pifouspikes, w);
|
||||
xlim([0 cmax])
|
||||
set(gca, 'XTick', 0:2:cmax)
|
||||
ylim([0 pmax])
|
||||
title('PIF OU');
|
||||
|
||||
subplot(1, 3, 3);
|
||||
counthist(lifadaptspikes, w);
|
||||
xlim([0 cmax])
|
||||
set(gca, 'XTick', 0:2:cmax)
|
||||
ylim([0 pmax])
|
||||
title('LIF adapt');
|
||||
savefigpdf(gcf, 'counthist.pdf', 20, 7);
|
||||
19
pointprocesses/code/plotisih.m
Normal file
19
pointprocesses/code/plotisih.m
Normal file
@ -0,0 +1,19 @@
|
||||
maxisi = 300.0;
|
||||
subplot(1, 3, 1);
|
||||
poissonisis = isis(poissonspikes);
|
||||
isihist(poissonisis, 0.001);
|
||||
xlim([0, maxisi])
|
||||
title('Poisson');
|
||||
|
||||
subplot(1, 3, 2);
|
||||
pifouisis = isis(pifouspikes);
|
||||
isihist(pifouisis, 0.001);
|
||||
xlim([0, maxisi])
|
||||
title('PIF OU');
|
||||
|
||||
subplot(1, 3, 3);
|
||||
lifadaptisis = isis(lifadaptspikes);
|
||||
isihist(lifadaptisis, 0.001);
|
||||
xlim([0, maxisi])
|
||||
title('LIF adapt');
|
||||
savefigpdf(gcf, 'isihist.pdf', 20, 7);
|
||||
17
pointprocesses/code/plotserialcorr.m
Normal file
17
pointprocesses/code/plotserialcorr.m
Normal file
@ -0,0 +1,17 @@
|
||||
maxlag = 10;
|
||||
rrange = [-0.5, 1.05];
|
||||
subplot(1, 3, 1);
|
||||
isiserialcorr(poissonisis, maxlag);
|
||||
ylim(rrange)
|
||||
title('Poisson');
|
||||
|
||||
subplot(1, 3, 2);
|
||||
isiserialcorr(pifouisis, maxlag);
|
||||
ylim(rrange)
|
||||
title('PIF OU');
|
||||
|
||||
subplot(1, 3, 3);
|
||||
isiserialcorr(lifadaptisis, maxlag);
|
||||
ylim(rrange)
|
||||
title('LIF adapt');
|
||||
savefigpdf(gcf, 'serialcorr.pdf', 20, 7);
|
||||
13
pointprocesses/code/plotspikeraster.m
Normal file
13
pointprocesses/code/plotspikeraster.m
Normal file
@ -0,0 +1,13 @@
|
||||
subplot(1, 3, 1);
|
||||
spikeraster(poissonspikes, 1.0);
|
||||
title('Poisson');
|
||||
|
||||
subplot(1, 3, 2);
|
||||
spikeraster(pifouspikes, 1.0);
|
||||
title('PIF OU');
|
||||
|
||||
subplot(1, 3, 3);
|
||||
spikeraster(lifadaptspikes, 1.0);
|
||||
title('LIF adapt');
|
||||
|
||||
savefigpdf(gcf, 'spikeraster.pdf', 15, 5);
|
||||
BIN
pointprocesses/code/poisson.mat
Normal file
BIN
pointprocesses/code/poisson.mat
Normal file
Binary file not shown.
@ -12,9 +12,9 @@ function spikes = poissonspikes( trials, rate, tmax )
|
||||
p = 0.1
|
||||
dt = p/rate;
|
||||
end
|
||||
spikes = cell( trials, 1 );
|
||||
spikes = cell(trials, 1);
|
||||
for k=1:trials
|
||||
x = rand( 1, round(tmax/dt) ); % uniform random numbers for each bin
|
||||
spikes{k} = find( x < p ) * dt;
|
||||
x = rand(round(tmax/dt), 1); % uniform random numbers for each bin
|
||||
spikes{k} = find(x < p) * dt;
|
||||
end
|
||||
end
|
||||
|
||||
14
pointprocesses/code/psth.m
Normal file
14
pointprocesses/code/psth.m
Normal file
@ -0,0 +1,14 @@
|
||||
function p = psth(spikes, dt, tmax)
|
||||
% plots a PSTH of the spikes with binwidth dt
|
||||
t = 0.0:dt:tmax+dt;
|
||||
p = zeros(1, length(t));
|
||||
for k=1:length(spikes)
|
||||
times = spikes{k};
|
||||
[h, b] = hist(times, t);
|
||||
p = p + h;
|
||||
end
|
||||
p = p/length(spikes)/dt;
|
||||
t(end) = [];
|
||||
p(end) = [];
|
||||
plot(t, p);
|
||||
end
|
||||
@ -1,15 +1,24 @@
|
||||
function spikeraster( spikes )
|
||||
function spikeraster(spikes, tmax)
|
||||
% Display a spike raster of the spike times given in spikes.
|
||||
% spikes: a cell array of vectors of spike times
|
||||
% tmax: plot spike raster upto tmax seconds
|
||||
|
||||
ntrials = length(spikes);
|
||||
for k = 1:ntrials
|
||||
times = 1000.0*spikes{k}; % conversion to ms
|
||||
times = spikes{k};
|
||||
times = times(times<tmax);
|
||||
if tmax < 1.5
|
||||
times = 1000.0*times; % conversion to ms
|
||||
end
|
||||
for i = 1:length( times )
|
||||
line([times(i) times(i)],[k-0.4 k+0.4], 'Color', 'k' );
|
||||
end
|
||||
end
|
||||
xlabel( 'Time [ms]' );
|
||||
if tmax < 1.5
|
||||
xlabel( 'Time [ms]' );
|
||||
else
|
||||
xlabel( 'Time [s]' );
|
||||
end
|
||||
ylabel( 'Trials');
|
||||
ylim( [ 0.3 ntrials+0.7 ] )
|
||||
|
||||
|
||||
12
pointprocesses/code/spikerate.m
Normal file
12
pointprocesses/code/spikerate.m
Normal file
@ -0,0 +1,12 @@
|
||||
function r = spikerate(spikes, duration)
|
||||
% returns the average spike rate of the spikes
|
||||
% for the first duration seconds
|
||||
% spikes: a cell array of vectors of spike times
|
||||
|
||||
rates = zeros(length(spikes),1);
|
||||
for k = 1:length(spikes)
|
||||
times = spikes{k};
|
||||
rates(k) = sum(times<duration)/duration;
|
||||
end
|
||||
r = mean(rates);
|
||||
end
|
||||
BIN
pointprocesses/exercises/UT_WBMW_Black_RGB.pdf
Normal file
BIN
pointprocesses/exercises/UT_WBMW_Black_RGB.pdf
Normal file
Binary file not shown.
BIN
pointprocesses/exercises/counthist.pdf
Normal file
BIN
pointprocesses/exercises/counthist.pdf
Normal file
Binary file not shown.
@ -14,6 +14,6 @@ function spikes = hompoissonspikes( trials, rate, tmax )
|
||||
x = rand( trials, ceil(tmax/dt) );
|
||||
spikes = cell( trials, 1 );
|
||||
for k=1:trials
|
||||
spikes{k} = find( x(k,:) >= 1.0-p ) * dt;
|
||||
spikes{k} = find( x(k,:) < p ) * dt;
|
||||
end
|
||||
end
|
||||
|
||||
BIN
pointprocesses/exercises/isihist.pdf
Normal file
BIN
pointprocesses/exercises/isihist.pdf
Normal file
Binary file not shown.
@ -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}}{{\bfseries\large 27. Oktober, 2015}}
|
||||
\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
|
||||
jan.benda@uni-tuebingen.de}
|
||||
\runningfooter{}{\thepage}{}
|
||||
@ -89,114 +89,110 @@ 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
|
||||
\[ P = \lambda \cdot \Delta t \]
|
||||
for sufficiently small $\Delta t$.
|
||||
\begin{parts}
|
||||
\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 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
|
||||
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 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
|
||||
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 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 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}
|
||||
|
||||
\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}
|
||||
199
pointprocesses/exercises/pointprocesses02.tex
Normal file
199
pointprocesses/exercises/pointprocesses02.tex
Normal file
@ -0,0 +1,199 @@
|
||||
\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 2}}{{\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{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 $\lambda$ (gemessen 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
|
||||
\[ P = \lambda \cdot \Delta t \]
|
||||
f\"ur gen\"ugend kleine $\Delta t$.
|
||||
\begin{parts}
|
||||
|
||||
\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 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. 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 (fang 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.
|
||||
\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
|
||||
\[ p(T) = \lambda e^{-\lambda T} \; .\]
|
||||
\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 pasiert 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}
|
||||
|
||||
\part Vergleiche Histogramme von Spikecounts gez\"ahlt in Fenstern
|
||||
der Breite $W$ mit der Poisson Verteilung
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \; , \]
|
||||
wobei die Rate $\lambda$ aus den Daten bestimmt werden
|
||||
soll. Hinweis: es gibt eine \code{matlab} Funktion, die die
|
||||
Fakult\"at $n!$ berechnet.
|
||||
\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 Schreibe eine Funktion, die die mittlere Anzahl, die Varianz
|
||||
und den Fano-Faktor der Anzahl der Spikes in einem Fenster der
|
||||
Breite $W$ bestimmt. Benutze die Funktion, um diese Parameter f\"ur
|
||||
verschiedene Fensterbreiten $W$ zu bestimmen. Zwei Plots sollen aus
|
||||
den Ergebnissen angefertigt werden: (i) Varianz gegen Mittelwert der counts.
|
||||
(ii) Fano Faktor als Funktion der Fensterbreite.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/fano.m}
|
||||
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
|
||||
\end{solution}
|
||||
|
||||
\end{parts}
|
||||
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
BIN
pointprocesses/exercises/serialcorr.pdf
Normal file
BIN
pointprocesses/exercises/serialcorr.pdf
Normal file
Binary file not shown.
BIN
pointprocesses/exercises/spikeraster.pdf
Normal file
BIN
pointprocesses/exercises/spikeraster.pdf
Normal file
Binary file not shown.
@ -224,48 +224,3 @@
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{\tr{Homogeneous Poisson process}{Homogener Poisson Prozess}}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{poissonraster100hz}
|
||||
\caption{\label{hompoissonfig}Rasterplot von Poisson-Spikes.}
|
||||
\end{figure}
|
||||
|
||||
The probability $p(t)\delta t$ of an event occuring at time $t$
|
||||
is independent of $t$ and independent of any previous event
|
||||
(independent of event history).
|
||||
|
||||
The probability $P$ for an event occuring within a time bin of width $\Delta t$
|
||||
is
|
||||
\[ P=\lambda \cdot \Delta t \]
|
||||
for a Poisson process with rate $\lambda$.
|
||||
|
||||
\subsection{Statistics of homogeneous Poisson process}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
|
||||
\caption{\label{hompoissonisihfig}Interspike interval histograms of poisson spike train.}
|
||||
\end{figure}
|
||||
|
||||
\begin{itemize}
|
||||
\item Exponential distribution of intervals $T$: $p(T) = \lambda e^{-\lambda T}$
|
||||
\item Mean interval $\mu_{ISI} = \frac{1}{\lambda}$
|
||||
\item Variance of intervals $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$
|
||||
\item Coefficient of variation $CV_{ISI} = 1$
|
||||
\item Serial correlation $\rho_k =0$ for $k>0$ (renewal process!)
|
||||
\item Fano factor $F=1$
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Count statistics of Poisson process}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
|
||||
\caption{\label{hompoissoncountfig}Count statistics of poisson spike train.}
|
||||
\end{figure}
|
||||
|
||||
Poisson distribution:
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
|
||||
@ -56,29 +56,38 @@ erzeugt. Zum Beispiel:
|
||||
\item Diffusions Koeffizient $D_{ISI} = \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Interval return maps}
|
||||
Scatter plot von aufeinander folgenden Intervallen $(T_{i+k}, T_i)$ getrennt durch das ``lag'' $k$.
|
||||
\subsection{Korrelationen der Intervalle}
|
||||
In ``return maps'' werden die um das ``Lag'' $k$ verz\"ogerten
|
||||
Intervalle $T_{i+k}$ gegen die Intervalle $T_i$ geplottet. Dies macht
|
||||
m\"ogliche Abh\"angigkeiten von aufeinanderfolgenden Intervallen
|
||||
sichtbar.
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{returnmapexamples}
|
||||
\includegraphics[width=1\textwidth]{serialcorrexamples}
|
||||
\caption{\label{returnmapfig}Interspike-Intervall return maps and serial correlations.}
|
||||
\caption{\label{returnmapfig}Interspike-Intervall return maps und
|
||||
serielle Korrelationen zwischen aufeinander folgenden Intervallen
|
||||
im Abstand des Lags $k$.}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Serielle Korrelationen der Intervalle}
|
||||
Korrelationskoeffizient zwischen aufeinander folgenden Intervallen getrennt durch ``lag'' $k$:
|
||||
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
|
||||
$\rho_0=1$ (Korrelation jedes Intervalls mit sich selber).
|
||||
Solche Ab\"angigkeiten werden durch die serielle Korrelation der
|
||||
Intervalle quantifiziert. Das ist der Korrelationskoeffizient
|
||||
zwischen aufeinander folgenden Intervallen getrennt durch ``Lag'' $k$:
|
||||
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)}
|
||||
= {\rm corr}(T_{i+k}, T_i) \]
|
||||
\"Ublicherweise wird die Korrelation $\rho_k$ gegen den Lag $k$
|
||||
aufgetragen (\figref{returnmapfig}). $\rho_0=1$ (Korrelation jedes
|
||||
Intervalls mit sich selber).
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Z\"ahlstatistik}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthist100hz10ms}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
|
||||
\caption{\label{countstatsfig}Count Statistik.}
|
||||
\end{figure}
|
||||
% \begin{figure}[t]
|
||||
% \includegraphics[width=0.48\textwidth]{poissoncounthist100hz10ms}\hfill
|
||||
% \includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
|
||||
% \caption{\label{countstatsfig}Count Statistik.}
|
||||
% \end{figure}
|
||||
|
||||
Statistik der Anzahl der Ereignisse $N_i$ innerhalb von Beobachtungsfenstern $i$ der Breite $W$.
|
||||
\begin{itemize}
|
||||
@ -87,21 +96,69 @@ Statistik der Anzahl der Ereignisse $N_i$ innerhalb von Beobachtungsfenstern $i$
|
||||
\item Varianz der Anzahl: $\sigma_N^2 = \langle (N - \langle N \rangle)^2 \rangle$.
|
||||
\item Fano Faktor (Varianz geteilt durch Mittelwert): $F = \frac{\sigma_N^2}{\mu_N}$.
|
||||
\end{itemize}
|
||||
|
||||
Insbesondere ist die mittlere Rate der Ereignisse $r$ (``Spikes pro Zeit'', Feuerrate) gemessen in Hertz
|
||||
\[ r = \frac{\langle N \rangle}{W} \; . \]
|
||||
|
||||
% \begin{figure}[t]
|
||||
% \begin{minipage}[t]{0.49\textwidth}
|
||||
% Poisson process $\lambda=100$\,Hz:\\
|
||||
% \includegraphics[width=1\textwidth]{poissonfano100hz}
|
||||
% \end{minipage}
|
||||
% \hfill
|
||||
% \begin{minipage}[t]{0.49\textwidth}
|
||||
% LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
|
||||
% \includegraphics[width=1\textwidth]{lifadaptfano10-100ms}
|
||||
% \end{minipage}
|
||||
% \caption{\label{fanofig}Fano factor.}
|
||||
% \end{figure}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Homogener Poisson Prozess}
|
||||
F\"ur kontinuierliche Me{\ss}gr\"o{\ss}en ist die Normalverteilung
|
||||
u.a. wegem dem Zentralen Grenzwertsatz die Standardverteilung. Eine
|
||||
\"ahnliche Rolle spilet bei Punktprozessen der ``Poisson Prozess''.
|
||||
|
||||
Beim homogenen Poisson Prozess treten Ereignisse mit einer festen Rate
|
||||
$\lambda=\text{const.}$ auf und sind unabh\"angig von der Zeit $t$ und
|
||||
unabh\"angig von den Zeitpunkten fr\"uherer Ereignisse. Die
|
||||
Wahrscheinlichkeit zu irgendeiner Zeit ein Ereigniss in einem kleinen
|
||||
Zeitfenster der Breite $\Delta t$ zu bekommen ist
|
||||
\[ P = \lambda \cdot \Delta t \; . \]
|
||||
|
||||
\begin{figure}[t]
|
||||
\begin{minipage}[t]{0.49\textwidth}
|
||||
Poisson process $\lambda=100$\,Hz:\\
|
||||
\includegraphics[width=1\textwidth]{poissonfano100hz}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}[t]{0.49\textwidth}
|
||||
LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
|
||||
\includegraphics[width=1\textwidth]{lifadaptfano10-100ms}
|
||||
\end{minipage}
|
||||
\caption{\label{fanofig}Fano factor.}
|
||||
\includegraphics[width=1\textwidth]{poissonraster100hz}
|
||||
\caption{\label{hompoissonfig}Rasterplot von Spikes eine homogenen
|
||||
Poisson Prozesse mit $\lambda=100$\,Hz.}
|
||||
\end{figure}
|
||||
|
||||
Beim inhomogenen Poisson Prozess h\"angt die Rate $\lambda$ von der
|
||||
Zeit ab: $\lambda = \lambda(t)$.
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
|
||||
\caption{\label{hompoissonisihfig}Interspikeintervallverteilungen
|
||||
zweier Poissonprozesse.}
|
||||
\end{figure}
|
||||
|
||||
Der homogne Poissonprozess hat folgende Eigenschaften:
|
||||
\begin{itemize}
|
||||
\item Die Intervalle $T$ sind exponentiell verteilt: $p(T) = \lambda e^{-\lambda T}$ .
|
||||
\item Das mittlere Intervall ist $\mu_{ISI} = \frac{1}{\lambda}$ .
|
||||
\item Die Varianz der Intervalle ist $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$ .
|
||||
\item Der Variationskoeffizient ist also immer $CV_{ISI} = 1$ .
|
||||
\item Die seriellen Korrelationen $\rho_k =0$ for $k>0$, da das
|
||||
Auftreten der Ereignisse unabh\"angig von der Vorgeschichte ist. Ein
|
||||
solcher Prozess wird auch Erneuerungsprozess genannt (``renewal
|
||||
process'').
|
||||
\item Die Anzahl der Ereignisse $k$ innerhalb eines Fensters der L\"ange W ist Poissonverteilt:
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
|
||||
\item Der Fano Faktor ist immer $F=1$ .
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
|
||||
\caption{\label{hompoissoncountfig}Z\"ahlstatistik von Poisson Spikes.}
|
||||
\end{figure}
|
||||
|
||||
198
programming/code/firing_rates.py
Normal file
198
programming/code/firing_rates.py
Normal file
@ -0,0 +1,198 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import scipy.io as spio
|
||||
import scipy as sp
|
||||
import seaborn as sb
|
||||
from IPython import embed
|
||||
sb.set_context("paper")
|
||||
|
||||
|
||||
def set_axis_fontsize(axis, label_size, tick_label_size=None, legend_size=None):
|
||||
"""
|
||||
Sets axis, tick label and legend font sizes to the desired size.
|
||||
|
||||
:param axis: the axes object
|
||||
:param label_size: the size of the axis label
|
||||
:param tick_label_size: the size of the tick labels. If None, lable_size is used
|
||||
:param legend_size: the size of the font used in the legend.If None, the label_size is used.
|
||||
|
||||
"""
|
||||
if not tick_label_size:
|
||||
tick_label_size = label_size
|
||||
if not legend_size:
|
||||
legend_size = label_size
|
||||
axis.xaxis.get_label().set_fontsize(label_size)
|
||||
axis.yaxis.get_label().set_fontsize(label_size)
|
||||
for tick in axis.xaxis.get_major_ticks() + axis.yaxis.get_major_ticks():
|
||||
tick.label.set_fontsize(tick_label_size)
|
||||
|
||||
l = axis.get_legend()
|
||||
if l:
|
||||
for t in l.get_texts():
|
||||
t.set_fontsize(legend_size)
|
||||
|
||||
|
||||
def get_instantaneous_rate(times, max_t=30., dt=1e-4):
|
||||
time = np.arange(0., max_t, dt)
|
||||
indices = np.asarray(times / dt, dtype=int)
|
||||
intervals = np.diff(np.hstack(([0], times)))
|
||||
inst_rate = np.zeros(time.shape)
|
||||
|
||||
for i, index in enumerate(indices[1:]):
|
||||
inst_rate[indices[i-1]:indices[i]] = 1/intervals[i]
|
||||
return time, inst_rate
|
||||
|
||||
|
||||
def plot_isi_rate(spike_times, max_t=30, dt=1e-4):
|
||||
times = np.squeeze(spike_times[0][0])[:50000]
|
||||
time, rate = get_instantaneous_rate(times, max_t=50000*dt)
|
||||
|
||||
rates = np.zeros((len(rate), len(spike_times)))
|
||||
for i in range(len(spike_times)):
|
||||
_, rates[:, i] = get_instantaneous_rate(np.squeeze(spike_times[i][0])[:50000],
|
||||
max_t=50000*dt)
|
||||
avg_rate = np.mean(rates, axis=1)
|
||||
rate_std = np.std(rates, axis=1)
|
||||
|
||||
fig = plt.figure()
|
||||
ax1 = fig.add_subplot(311)
|
||||
ax2 = fig.add_subplot(312)
|
||||
ax3 = fig.add_subplot(313)
|
||||
|
||||
ax1.vlines(times[times < (50000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
|
||||
ax1.set_ylabel("skpikes", fontsize=12)
|
||||
set_axis_fontsize(ax1, 12)
|
||||
|
||||
ax2.plot(time, rate, label="instantaneous rate, trial 1")
|
||||
ax2.set_ylabel("firing rate [Hz]", fontsize=12)
|
||||
ax2.legend(fontsize=12)
|
||||
set_axis_fontsize(ax2, 12)
|
||||
|
||||
ax3.fill_between(time, avg_rate+rate_std, avg_rate-rate_std, color="dodgerblue",
|
||||
alpha=0.5, label="standard deviation")
|
||||
ax3.plot(time, avg_rate, label="average rate")
|
||||
ax3.set_xlabel("times [s]", fontsize=12)
|
||||
ax3.set_ylabel("firing rate [Hz]", fontsize=12)
|
||||
ax3.legend(fontsize=12)
|
||||
ax3.set_ylim([0, 450])
|
||||
set_axis_fontsize(ax3, 12)
|
||||
|
||||
fig.set_size_inches(15, 10)
|
||||
fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95)
|
||||
fig.set_facecolor("white")
|
||||
fig.savefig("../lectures/images/instantaneous_rate.png")
|
||||
plt.close()
|
||||
|
||||
|
||||
def get_binned_rate(times, bin_width=0.05, max_t=30., dt=1e-4):
|
||||
time = np.arange(0., max_t, dt)
|
||||
bins = np.arange(0., max_t, bin_width)
|
||||
bin_indices = bins / dt
|
||||
hist, _ = sp.histogram(times, bins)
|
||||
rate = np.zeros(time.shape)
|
||||
|
||||
for i, b in enumerate(bin_indices[1:]):
|
||||
rate[bin_indices[i-1]:b] = hist[i-1]/bin_width
|
||||
return time, rate
|
||||
|
||||
|
||||
def plot_bin_rate(spike_times, bin_width, max_t=30, dt=1e-4):
|
||||
times = np.squeeze(spike_times[0][0])
|
||||
time, rate = get_binned_rate(times)
|
||||
rates = np.zeros((len(rate), len(spike_times)))
|
||||
for i in range(len(spike_times)):
|
||||
_, rates[:, i] = get_binned_rate(np.squeeze(spike_times[i][0]))
|
||||
avg_rate = np.mean(rates, axis=1)
|
||||
rate_std = np.std(rates, axis=1)
|
||||
|
||||
fig = plt.figure()
|
||||
ax1 = fig.add_subplot(311)
|
||||
ax2 = fig.add_subplot(312)
|
||||
ax3 = fig.add_subplot(313)
|
||||
|
||||
ax1.vlines(times[times < (100000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
|
||||
ax1.set_ylabel("skpikes", fontsize=12)
|
||||
ax1.set_xlim([0, 5])
|
||||
set_axis_fontsize(ax1, 12)
|
||||
|
||||
ax2.plot(time, rate, label="binned rate, trial 1")
|
||||
ax2.set_ylabel("firing rate [Hz]", fontsize=12)
|
||||
ax2.legend(fontsize=12)
|
||||
ax2.set_xlim([0, 5])
|
||||
set_axis_fontsize(ax2, 12)
|
||||
|
||||
ax3.fill_between(time, avg_rate+rate_std, avg_rate-rate_std, color="dodgerblue",
|
||||
alpha=0.5, label="standard deviation")
|
||||
ax3.plot(time, avg_rate, label="average rate")
|
||||
ax3.set_xlabel("times [s]", fontsize=12)
|
||||
ax3.set_ylabel("firing rate [Hz]", fontsize=12)
|
||||
ax3.legend(fontsize=12)
|
||||
ax3.set_xlim([0, 5])
|
||||
ax3.set_ylim([0, 450])
|
||||
set_axis_fontsize(ax3, 12)
|
||||
|
||||
fig.set_size_inches(15, 10)
|
||||
fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95)
|
||||
fig.set_facecolor("white")
|
||||
fig.savefig("../lectures/images/binned_rate.png")
|
||||
plt.close()
|
||||
|
||||
|
||||
def get_convolved_rate(times, sigma, max_t=30., dt=1.e-4):
|
||||
time = np.arange(0., max_t, dt)
|
||||
kernel = sp.stats.norm.pdf(np.arange(-8*sigma, 8*sigma, dt),loc=0,scale=sigma)
|
||||
indices = np.asarray(times/dt, dtype=int)
|
||||
rate = np.zeros(time.shape)
|
||||
rate[indices] = 1.;
|
||||
conv_rate = np.convolve(rate, kernel, mode="same")
|
||||
return time, conv_rate
|
||||
|
||||
|
||||
def plot_conv_rate(spike_times, sigma=0.05, max_t=30, dt=1e-4):
|
||||
times = np.squeeze(spike_times[0][0])
|
||||
time, rate = get_convolved_rate(times, sigma)
|
||||
|
||||
rates = np.zeros((len(rate), len(spike_times)))
|
||||
for i in range(len(spike_times)):
|
||||
_, rates[:, i] = get_convolved_rate(np.squeeze(spike_times[i][0]), sigma)
|
||||
avg_rate = np.mean(rates, axis=1)
|
||||
rate_std = np.std(rates, axis=1)
|
||||
|
||||
fig = plt.figure()
|
||||
ax1 = fig.add_subplot(311)
|
||||
ax2 = fig.add_subplot(312)
|
||||
ax3 = fig.add_subplot(313)
|
||||
|
||||
ax1.vlines(times[times < (100000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
|
||||
ax1.set_ylabel("skpikes", fontsize=12)
|
||||
ax1.set_xlim([0, 5])
|
||||
set_axis_fontsize(ax1, 12)
|
||||
|
||||
ax2.plot(time, rate, label="convolved rate, trial 1")
|
||||
ax2.set_ylabel("firing rate [Hz]", fontsize=12)
|
||||
ax2.legend(fontsize=12)
|
||||
ax2.set_xlim([0, 5])
|
||||
set_axis_fontsize(ax2, 12)
|
||||
|
||||
ax3.fill_between(time, avg_rate+rate_std, avg_rate-rate_std, color="dodgerblue",
|
||||
alpha=0.5, label="standard deviation")
|
||||
ax3.plot(time, avg_rate, label="average rate")
|
||||
ax3.set_xlabel("times [s]", fontsize=12)
|
||||
ax3.set_ylabel("firing rate [Hz]", fontsize=12)
|
||||
ax3.legend(fontsize=12)
|
||||
ax3.set_xlim([0, 5])
|
||||
ax3.set_ylim([0, 450])
|
||||
set_axis_fontsize(ax3, 12)
|
||||
|
||||
fig.set_size_inches(15, 10)
|
||||
fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95)
|
||||
fig.set_facecolor("white")
|
||||
fig.savefig("../lectures/images/convolved_rate.png")
|
||||
plt.close()
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
spike_times = spio.loadmat('lifoustim.mat')["spikes"]
|
||||
plot_isi_rate(spike_times)
|
||||
plot_bin_rate(spike_times, 0.05)
|
||||
plot_conv_rate(spike_times, 0.025)
|
||||
78
programming/exercises/plotting_psth.tex
Normal file
78
programming/exercises/plotting_psth.tex
Normal file
@ -0,0 +1,78 @@
|
||||
\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}
|
||||
|
||||
%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
|
||||
\pagestyle{headandfoot} \header{{\bfseries\large \"Ubung
|
||||
}}{{\bfseries\large Peri Stimulus Time Histogram}}{{\bfseries\large 28. Oktober, 2015}}
|
||||
\firstpagefooter{Dr. Jan Grewe}{Phone: 29 74588}{Email:
|
||||
jan.grewe@uni-tuebingen.de} \runningfooter{}{\thepage}{}
|
||||
|
||||
\setlength{\baselineskip}{15pt}
|
||||
\setlength{\parindent}{0.0cm}
|
||||
\setlength{\parskip}{0.3cm}
|
||||
\renewcommand{\baselinestretch}{1.15}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
\renewcommand{\solutiontitle}{\noindent\textbf{L\"osung:}\par\noindent}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\vspace*{-6.5ex}
|
||||
\begin{center}
|
||||
\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[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}
|
||||
|
||||
\begin{questions}
|
||||
\question Graphische Darstellung der zeitabh\"angigen Antwort eines
|
||||
Neurons. PSTH auf Basis der instantanen Feuerrate. Verwendet den Datensatz \code{}
|
||||
\begin{parts}
|
||||
\part Schreibt eine Funktion, die einen Vektor mit Spikezeiten,
|
||||
die Dauer des Trials, und die zeitliche Aufl\"osung entgegennimmt
|
||||
und die Zeitachse sowie die Feuerrate zur\"uckgiebt.
|
||||
\part Benutzt diese Funktion in einem Skript und stellt das PSTH
|
||||
eines einzelnen Trials sowie den Mittelwert \"uber alle Trials
|
||||
dar.
|
||||
\part Erweitert das Programm so, dass die Abbildung den Standards
|
||||
z.B. vom \textit{Journal of Neuroscience} entspricht
|
||||
(Schriftgr\"o{\ss}e, Abbildungsgr\"o{\ss}e).
|
||||
\part Die Abbildung sollte als pdf gespeichert werden.
|
||||
\end{parts}
|
||||
|
||||
\question Wie zuvor nur unter Verwendung der Binning Methode.
|
||||
|
||||
\question Wie zuvor nur unter Verwendung der Faltungsmethode.
|
||||
|
||||
\question Entscheidet euch f\"ur eine Varainte und erweitert das
|
||||
entsprechende Skript, sodass auch die Interspikeintervallverteilung
|
||||
und die Verteilung der Spikecounts dargestellt werden. Die
|
||||
Abbildungen sollten nat\"urlich ``publikationsreif'' sein und
|
||||
gespeichert werden.
|
||||
|
||||
\question Einige trials sind anders als die \"Ubrigen. Benutzt den
|
||||
Rasterplot um sie zu finden. Speichert alle Abbildungen in
|
||||
``publikationsreifer'' Form.
|
||||
\begin{parts}
|
||||
\part Benutzt den Rasterplot um sie zu finden.
|
||||
\part Plottet die Verteilung der Spike counts.
|
||||
\part Filtert all die trials heraus, deren spike count mehr als
|
||||
$2\sigma$ vom Mittelwert abweicht.
|
||||
\part Plottet das PSTH vor und nach dem Filtern.
|
||||
\end{parts}
|
||||
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
BIN
programming/lectures/images/binned_rate.png
Normal file
BIN
programming/lectures/images/binned_rate.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 58 KiB |
BIN
programming/lectures/images/convolved_rate.png
Normal file
BIN
programming/lectures/images/convolved_rate.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 80 KiB |
BIN
programming/lectures/images/instantaneous_rate.png
Normal file
BIN
programming/lectures/images/instantaneous_rate.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 81 KiB |
@ -176,14 +176,14 @@
|
||||
}
|
||||
\only <4> {
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.4\columnwidth]{images/badbarright}
|
||||
\includegraphics[width=0.3\columnwidth]{images/badbarright}
|
||||
\end{figure}
|
||||
\vspace{0.5cm}
|
||||
\url{https://en.wikipedia.org/wiki/Misleading_graph}
|
||||
}
|
||||
\only <5> {
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.4\columnwidth]{images/badbarleft}
|
||||
\includegraphics[width=0.3\columnwidth]{images/badbarleft}
|
||||
\end{figure}
|
||||
\vspace{0.5cm}
|
||||
\url{https://en.wikipedia.org/wiki/Misleading_graph}
|
||||
@ -352,19 +352,18 @@ saveas(fig, 'spike_detection.pdf', 'pdf')
|
||||
\item Deutliche Unterscheidbarkeit von Kurven.
|
||||
\item Keine suggestive Darstellung.
|
||||
\item Ausgewogenheit von Linienst\"arken Schrift- und Plotgr\"o{\ss}e.
|
||||
\item Vermeidung von Suggestiven Darstellungen.
|
||||
\item Fehlerbalken, wenn sie angebracht sind.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[plain]
|
||||
\huge{2. Spiketrain Analyse I}
|
||||
\huge{2. Spiketrain Analyse}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse I}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Rasterplot}
|
||||
\begin{figure}
|
||||
\centering
|
||||
@ -374,92 +373,221 @@ saveas(fig, 'spike_detection.pdf', 'pdf')
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse I}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Rasterplot}
|
||||
\"Ubung:
|
||||
\begin{enumerate}
|
||||
\item Ladet die Datei: electrorecptor\_spike\_times.mat aus dem
|
||||
\item Ladet die Datei: \code{lifoustim.mat} aus dem
|
||||
Ilias Ordner.
|
||||
\item Der Datensatz enth\"alt die Zeiten von Aktionspotentialen.
|
||||
\item Schaut euch den Inhalt und skizziert wie das Problem gel\"ost
|
||||
werden k\"onnte.
|
||||
\item Erzeugt einen Rasterplot.
|
||||
\item Erzeugt einen sch\"onen Rasterplot der Zellantworten, speichert ihn.
|
||||
\item Welche Information liefert er, welche Information ist schwer
|
||||
abzulesen?
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH}
|
||||
|
||||
Darstellung der Feurreate eines Neuron als Funktion der Zeit. Es
|
||||
gibt verschiedene Methoden dieses \textbf{P}eri \textbf{S}timulus
|
||||
\textbf{T}ime \textbf{H}istogram zu erstellen.
|
||||
\begin{enumerate}
|
||||
\item Auf Basis der \textit{instantanen} Feuerrate.
|
||||
\item Auf Basis des Zeithistogramms.
|
||||
\item Durch Faltung der Zellantwort mit einem Gauss Kern.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse I}
|
||||
\framesubtitle{Feuerrate \"uber
|
||||
die Zeit}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Instantane Feuerrate ---}
|
||||
\Large{Berechnung des PSTHs durch die Instantane Feurerrate:}
|
||||
\normalsize
|
||||
\begin{enumerate}
|
||||
\item Die Feuerrate kann aus dem Abstand zwischen zwei
|
||||
aufeinanderfolgenden Aktionspotentialen
|
||||
(\textbf{Interspikeinterval}) berechnet werden. \pause
|
||||
\item Auch als \textbf{Instantane Feuerrate} bezeichnet.
|
||||
\item Die \textbf{Instantane Feuerrate} wird aus dem Kehrwert des
|
||||
Interspikeintervals berechnet.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Instantane Feuerrate ---}
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=0.5\columnwidth]{images/isi}
|
||||
\includegraphics[width=0.9\columnwidth]{images/instantaneous_rate}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse I}
|
||||
\framesubtitle{Feuerrate \"uber
|
||||
die Zeit}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Instantane Feuerrate ---}
|
||||
Berechnung des PSTHs durch die Instantane Feurerrate:
|
||||
\textbf{Vorteile:}
|
||||
|
||||
\begin{enumerate}
|
||||
\item Z.B. das \textbf{P}eri- \textbf{S}timulus - \textbf{T}ime -
|
||||
\textbf{H}istogram, \textbf{PSTH}
|
||||
\item Wird berechnet indem die Zeit in ``bins'' geteilt wird die
|
||||
Anzahl Spikes pro bin gezaehlt werden.
|
||||
\item Die Anzahl wird dann in eine Feuerrate umgerechnet.
|
||||
\end{enumerate}\pause
|
||||
\item Sehr einfach zu Berechnen.
|
||||
\item Macht keine Annahmen \"uber ein Zeitraster, oder die Zeitskala
|
||||
der neuronalen Verarbeitung.
|
||||
\end{enumerate}
|
||||
|
||||
\textbf{Nachteile:}
|
||||
\begin{enumerate}
|
||||
\item Die Feuerrate ist nie null, auch wenn f\"ur lange Zeit kein
|
||||
Aktionspotential auftritt.
|
||||
\item Verh\"alt sich im Fourrier Raum nicht sehr sch\"on.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Binning Methode ---}
|
||||
\Large{Binning Methode:}
|
||||
\normalsize
|
||||
\begin{enumerate}
|
||||
\item Die Zeitachse wird in gleich gro{\ss}e Abschnitte ``bins''
|
||||
unterteilt.
|
||||
\item F\"ur jedes ``bin'' wird die Anzahl vorkommender
|
||||
Aktionspotentiale gez\"ahlt.
|
||||
\item Der Spike-count pro bin muss nun noch in die Rate umgerechnet
|
||||
werden.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Binning Methode ---}
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=0.5\columnwidth]{images/binning}
|
||||
\includegraphics[width=0.9\columnwidth]{images/binned_rate}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Binning Methode ---}
|
||||
|
||||
Berechnung des PSTHs durch die Binning Methode:
|
||||
|
||||
\textbf{Vorteile:}
|
||||
\begin{enumerate}
|
||||
\item Sehr einfach zu Berechnen.
|
||||
\item Zeigt nur da Aktivit\"at an, wo auch Aktionspotentiale
|
||||
generiert wurden.
|
||||
\end{enumerate}
|
||||
|
||||
\textbf{Nachteile:}
|
||||
\begin{enumerate}
|
||||
\item Mach Annahmen \"uber die relevante Zeitskala neuronaler
|
||||
Verarbeitung.
|
||||
\item Die Zeitachse wird diskretisiert.
|
||||
\item Verh\"alt sich im Fourrier Raum nicht sehr sch\"on.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Faltungsmethode ---}
|
||||
\Large{Faltung mit einem Gauss Kern:}
|
||||
\normalsize
|
||||
\[r_{est}(t) = \int_{-\infty}^{\infty}d\tau \omega(\tau)\rho(t-\tau) \]
|
||||
|
||||
wobei $\omega(\tau)$ der Gauss Kern und $\rho(t)$ die Antwortfunktion ist.
|
||||
|
||||
Gl\"ucklicherweise m\"ussen wir das nicht selbst implementieren...
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Spiketrain Analyse I}
|
||||
\framesubtitle{Feuerrate \"uber die Zeit}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Zeitabh\"angige Feuerrate, PSTH --- Faltungsmethode ---}
|
||||
\large Algortihmus:
|
||||
\normalsize
|
||||
\begin{enumerate}
|
||||
\item Z.B. das \textbf{P}eri- \textbf{S}timulus - \textbf{T}ime -
|
||||
\textbf{H}istogram, \textbf{PSTH}
|
||||
\item Wird berechnet indem man die Aktivit\"at bin\"ar ausdr\"uckt.
|
||||
\item Jede 1 wird dann duch einen ``Kern'' ersetzt.
|
||||
\item Der Vorgang heisst Faltung (\verb+conv+).
|
||||
\end{enumerate}\pause
|
||||
\item Die neuronalen Antworten werden ``bin\"ar'' ausgedr\"uckt.
|
||||
\item Ein Filterkern wird berechnet, der das Integral 1 hat.
|
||||
\item Mithilfe der Faltung (\code{conv} Funktion) wird jede 1 durch
|
||||
den ``Kern'' ersetzt.\\
|
||||
\code{conv(x, kern, 'mode', 'same')}
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Feurerrate als Funktion der Zeit --- Faltungsmethode ---}
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=0.5\columnwidth]{images/conv}
|
||||
\includegraphics[width=0.9\columnwidth]{images/convolved_rate}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame} [fragile]
|
||||
\frametitle{Spiketrain Analyse I}
|
||||
\framesubtitle{\"Ubung}
|
||||
\begin{enumerate}
|
||||
\item Berechnet die Feuerrate eines Neurons mit einer der drei Methoden.
|
||||
\item Die Abbildung soll f\"ur eine einspaltige Abbildung im
|
||||
\textit{Journal of Neuroscience} geeignet sein
|
||||
(\url{http://www.jneurosci.org/site/misc/ifa_illustrations.xhtml}).
|
||||
\item Erzeugt/ver\"andert/erweitert das Programm zum plotten so, dass
|
||||
die Abbildung automatisch erstellt und gespeichert wird.
|
||||
\item Speichert die Abbildung als pdf.
|
||||
\item Ladet den Stimulus aus dem Ilias Ordner und benutzt die
|
||||
\verb+subplot+ Funktion um den Stimulus zu der neuronalen
|
||||
Aktivit\"at zu plotten.
|
||||
\end{enumerate}
|
||||
\begin{frame}
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Feurerrate als Funktion der Zeit --- Faltungsmethode ---}
|
||||
\textbf{Vorteile:}
|
||||
\begin{enumerate}
|
||||
\item Sehr ``nat\"urliche'' erscheinende Darstellung.
|
||||
\item Sehr gutes Verhalten im Fourrier Raum.
|
||||
\end{enumerate}
|
||||
|
||||
\textbf{Nachteile:}
|
||||
\begin{enumerate}
|
||||
\item Relativ rechenintensiv.
|
||||
\item Macht Annahmen \"uber die Zeitskalen neuronaler Verarbeitung.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[plain]
|
||||
\huge{3. Analyse der Beziehung zwischen Stimulus und Antwort}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Spike-Triggered-Average}
|
||||
\begin{itemize}
|
||||
\item[] Die Antworten darzustellen ist gut und sch\"on, aber was sagt es uns?\pause
|
||||
\item[] Idealerweise wollen wir die Antworten in Beziehung zum
|
||||
hervorrufenden Stimulus setzen.
|
||||
\item[] Eine Methode ist der sogenannte \textbf{Spike-Triggered-Average} (STA).
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Spike-Triggered-Average}
|
||||
|
||||
Der STA stellt den (mittleren) Stimulus dar, der zu einem
|
||||
Aktionspotential gef\"uhrt hat:
|
||||
|
||||
\begin{equation}
|
||||
STA(\tau) = \frac{1}{\langle n \rangle} \left\langle \displaystyle\sum_{i=1}^{n}{s(t_i - \tau)} \right\rangle
|
||||
\end{equation}
|
||||
Wobei: $\tau$ ist eine bestimmte Zeit relativ zur Zeit eines
|
||||
Aktionspotentials, $t_i$ ist der Zeitpunkt eines APs, $s(t)$ ist der
|
||||
Stimulus.\\
|
||||
|
||||
Leider m\"ussen wir das selbst implementieren...
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Spike-Triggered-Average}
|
||||
\large{Algorithmus:}
|
||||
\normalsize
|
||||
\begin{enumerate}
|
||||
\item Der \textbf{STA} ist der mittlere Stimulus, der zu einem Aktionspotential f\"uhrt.
|
||||
\item F\"ur jeden Spike wird ein entsprechender Abschnitt um die Zeit des Spikes herausgeschnitten.
|
||||
@ -471,4 +599,45 @@ saveas(fig, 'spike_detection.pdf', 'pdf')
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{Spike-Triggered-Average}
|
||||
\vspace{-1em}
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=0.6\columnwidth]{images/sta}
|
||||
\end{figure}
|
||||
\pause
|
||||
\vspace{-0.5em}
|
||||
Welche Information liefert der \textbf{STA}?
|
||||
\small
|
||||
\begin{enumerate}
|
||||
\item Gibt es eine Beziehung zwischen Stimulus und Antwort?\pause
|
||||
\item Gibt es eine Verz\"ogerung zwischen Stimulus und Antwort? Wie
|
||||
gro{\ss} ist diese?\pause
|
||||
\item Wie weit h\"angt das Auftreten eines Aktionspotentials von der
|
||||
Vergangenheit ab? \pause
|
||||
\item Kann die Zelle in die Zukunft sehen?
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
|
||||
\begin{frame} [fragile]
|
||||
\frametitle{Spiketrain Analyse}
|
||||
\framesubtitle{\"Ubung}
|
||||
\begin{enumerate}
|
||||
\item Berechnet die Feuerrate eines Neurons mit einer der drei Methoden.
|
||||
\item Die Abbildung soll f\"ur eine einspaltige Abbildung im
|
||||
\textit{Journal of Neuroscience} geeignet sein
|
||||
(\url{http://www.jneurosci.org/site/misc/ifa_illustrations.xhtml}).
|
||||
\item Erzeugt/ver\"andert/erweitert das Programm zum plotten so, dass
|
||||
die Abbildung automatisch erstellt und gespeichert wird.
|
||||
\item Speichert die Abbildung als pdf.
|
||||
\item Ladet den Stimulus aus dem Ilias Ordner und benutzt die
|
||||
\verb+subplot+ Funktion um den Stimulus zu der neuronalen
|
||||
Aktivit\"at zu plotten.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
@ -237,6 +237,9 @@
|
||||
|
||||
\renewcommand{\codepath}{pointprocesses/code/}
|
||||
\renewcommand{\texinputpath}{pointprocesses/lecture/}
|
||||
%\include{pointprocesses/lecture/pointprocesses}
|
||||
\include{pointprocesses/lecture/pointprocesses}
|
||||
|
||||
\renewcommand{\codepath}{designpattern/code/}
|
||||
\include{designpattern/lecture/designpattern}
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -1,9 +0,0 @@
|
||||
function gradient = lsq_gradient_sigmoid(parameter, x, y)
|
||||
h = 1e-6;
|
||||
|
||||
gradient = zeros(size(parameter));
|
||||
for i = 1:length(parameter)
|
||||
parameter_h = parameter;
|
||||
parameter_h(i) = parameter_h(i) + h;
|
||||
gradient(i) = (lsq_sigmoid_error(parameter_h, x, y) - lsq_sigmoid_error(parameter, x, y)) / h;
|
||||
end
|
||||
@ -1,8 +0,0 @@
|
||||
function error = lsq_sigmoid_error(parameter, x, y)
|
||||
% p(1) the amplitude
|
||||
% p(2) the slope
|
||||
% p(3) the x-shift
|
||||
% p(4) the y-shift
|
||||
|
||||
y_est = parameter(1)./(1+ exp(-parameter(2) .* (x - parameter(3)))) + parameter(4);
|
||||
error = mean((y_est - y).^2);
|
||||
@ -1,44 +0,0 @@
|
||||
|
||||
|
||||
%% fit the sigmoid
|
||||
|
||||
clear
|
||||
close all
|
||||
|
||||
load('iv_curve.mat')
|
||||
|
||||
figure()
|
||||
plot(voltage, current, 'o')
|
||||
xlabel('voltate [mV]')
|
||||
ylabel('current [pA]')
|
||||
|
||||
% amplitude, slope, x-shift, y-shift
|
||||
%parameter = [10 0.25 -50, 2.5];
|
||||
parameter = [20 0.5 -50, 2.5];
|
||||
|
||||
eps = 0.1;
|
||||
% do the descent
|
||||
gradient = [];
|
||||
steps = 0;
|
||||
error = [];
|
||||
|
||||
while isempty(gradient) || norm(gradient) > 0.01
|
||||
steps = steps + 1;
|
||||
gradient = lsq_gradient_sigmoid(parameter, voltage, current);
|
||||
error(steps) = lsq_sigmoid_error(parameter, voltage, current);
|
||||
parameter = parameter - eps .* gradient;
|
||||
end
|
||||
plot(1:steps, error)
|
||||
|
||||
disp('gradient descent done!')
|
||||
disp(strcat('final position: ', num2str(parameter)))
|
||||
disp(strcat('final error: ', num2str(error(end))))
|
||||
|
||||
%% use fminsearch
|
||||
parameter = [10 0.5 -50, 2.5];
|
||||
|
||||
objective_function = @(p)lsq_sigmoid_error(p, voltage, current);
|
||||
param = fminunc(objective_function, parameter);
|
||||
disp(param)
|
||||
param1 = fminsearch(objective_function, parameter);
|
||||
disp(param1)
|
||||
Reference in New Issue
Block a user