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Started working on pointprocesses

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Jan Benda 2015-10-25 22:22:54 +01:00
parent d9252e9cbf
commit 857373c125
24 changed files with 896 additions and 692 deletions
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15
bootstrap/lecture/Makefile
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@ -1,22 +1,29 @@
BASENAME=bootstrap
PYFILES=$(wildcard *.py)
PYPDFFILES=$(PYFILES:.py=.pdf)
pdf : $(BASENAME)-chapter.pdf $(PYPDFFILES)
all : pdf
# script:
pdf : $(BASENAME)-chapter.pdf
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex
$(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 *~ $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out $(BASENAME).aux $(BASENAME).log
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
watch :
watchpdf :
while true; do ! make -q pdf && make pdf; sleep 0.5; done

15
likelihood/lecture/Makefile
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@ -1,22 +1,29 @@
BASENAME=likelihood
PYFILES=$(wildcard *.py)
PYPDFFILES=$(PYFILES:.py=.pdf)
pdf : $(BASENAME)-chapter.pdf $(PYPDFFILES)
all : pdf
# script:
pdf : $(BASENAME)-chapter.pdf
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex
$(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 *~ $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out $(BASENAME).aux $(BASENAME).log
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
watch :
watchpdf :
while true; do ! make -q pdf && make pdf; sleep 0.5; done

35
pointprocesses/exercises/Makefile Normal file
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BASENAME=pointprocesses
TEXFILES=$(wildcard $(BASENAME)??.tex)
EXERCISES=$(TEXFILES:.tex=.pdf)
SOLUTIONS=$(EXERCISES:pointprocesses%=pointprocesses-solutions%)
.PHONY: pdf exercises solutions watch watchexercises watchsolutions clean
pdf : $(SOLUTIONS) $(EXERCISES)
exercises : $(EXERCISES)
solutions : $(SOLUTIONS)
$(SOLUTIONS) : pointprocesses-solutions%.pdf : pointprocesses%.tex instructions.tex
{ echo "\\documentclass[answers,12pt,a4paper,pdftex]{exam}"; sed -e '1d' $<; } > $(patsubst %.pdf,%.tex,$@)
pdflatex -interaction=scrollmode $(patsubst %.pdf,%.tex,$@) | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $(patsubst %.pdf,%.tex,$@) || true
rm $(patsubst %.pdf,%,$@).[!p]*
$(EXERCISES) : %.pdf : %.tex instructions.tex
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
watch :
while true; do ! make -q pdf && make pdf; sleep 0.5; done
watchexercises :
while true; do ! make -q exercises && make exercises; sleep 0.5; done
watchsolutions :
while true; do ! make -q solutions && make solutions; sleep 0.5; done
clean :
rm -f *~ *.aux *.log *.out
cleanup : clean
rm -f $(SOLUTIONS) $(EXERCISES)

11
pointprocesses/exercises/instructions.tex Normal file
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\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}
\ifprintanswers%
\else
\fi

202
pointprocesses/exercises/pointprocesses01.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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pointprocesses/exercises/poisson.pdf
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160
pointprocesses/exercises/poisson.tex
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@ -1,160 +0,0 @@
\documentclass[addpoints,10pt]{exam}
\usepackage{url}
\usepackage{color}
\usepackage{hyperref}
\usepackage{graphicx}
\pagestyle{headandfoot}
\runningheadrule
\firstpageheadrule
\firstpageheader{Scientific Computing}{Homogeneous Poisson process}{Oct 27, 2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
\firstpagefooter{}{}{}
\runningfooter{}{}{}
\pointsinmargin
\bracketedpoints
%\printanswers
\shadedsolutions
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{listings}
\lstset{
basicstyle=\ttfamily,
numbers=left,
showstringspaces=false,
language=Matlab,
breaklines=true,
breakautoindent=true,
columns=flexible,
frame=single,
captionpos=t,
xleftmargin=2em,
xrightmargin=1em,
aboveskip=10pt,
%title=\lstname,
title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
}
\begin{document}
\sffamily
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{questions}
\question \textbf{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{simulations/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{simulations/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{simulations/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 simulations 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{simulations/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{simulations/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{simulations/fano.m}
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
\end{solution}
\end{parts}
\end{questions}
\end{document}

150
pointprocesses/lecture/Makefile
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@ -1,142 +1,70 @@
BASENAME=pointprocesses
TEXFILE=$(BASENAME).tex
DVIFILE=$(BASENAME).dvi
PSFILE=$(BASENAME).ps
PDFFILE=$(BASENAME).pdf
FOILSFILE=foils.pdf
THUMBNAILSFILE=thumbnails.pdf
HTMLBASENAME=$(BASENAME)h
HTMLTEXFILE=$(BASENAME)h.tex
HTMLDIR=$(BASENAME)h
PYFILES=$(wildcard *.py)
PYPDFFILES=$(PYFILES:.py=.pdf)
GPTFILES=$(wildcard *.gpt)
GPTTEXFILES=$(GPTFILES:.gpt=.tex)
all: ps pdf talk again watchps watchpdf foils thumbs html html1 epsfigs clean cleanup cleanplots help
.PHONY: epsfigs
all: pdf slides thumbs
# script:
pdf : $(BASENAME)-chapter.pdf
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex $(GPTTEXFILES) $(PYPDFFILES)
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
# slides:
slides: $(BASENAME)-slides.pdf
$(BASENAME)-slides.pdf : $(BASENAME)-slides.tex $(GPTTEXFILES) $(PYPDFFILES)
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
# thumbnails:
thumbs: $(THUMBNAILSFILE)
$(THUMBNAILSFILE): $(TEXFILE) $(GPTTEXFILES)
thumbs: $(BASENAME)-handout.pdf
$(BASENAME)-handout.pdf: $(BASENAME)-slides.tex $(GPTTEXFILES)
sed -e 's/setboolean{presentation}{true}/setboolean{presentation}{false}/; s/usepackage{crop}/usepackage[frame]{crop}/' $< > thumbsfoils.tex
pdflatex thumbsfoils | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex thumbsfoils || true
pdfnup --nup 2x4 --no-landscape --paper a4paper --trim "-1cm -1cm -1cm -1cm" --outfile $@ thumbsfoils.pdf '1-19'
pdfnup --nup 2x4 --no-landscape --paper a4paper --trim "-1cm -1cm -1cm -1cm" --outfile $@ thumbsfoils.pdf # 1-19
rm thumbsfoils.*
# transparencies:
foils: $(FOILSFILE)
$(FOILSFILE): $(TEXFILE) $(GPTTEXFILES)
sed -e 's/setboolean{presentation}{true}/setboolean{presentation}{false}/' $< > tfoils.tex
pdflatex tfoils | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex tfoils || true
pdfnup --nup 1x2 --orient portrait --trim "-1mm -1mm -1mm -1mm" --frame true --delta "1cm 1cm" --paper a4paper --outfile tfoils2.pdf tfoils.pdf
pdfnup --nup 1x1 --orient portrait --trim "-2cm -2cm -2cm -2cm" --paper a4paper --outfile $@ tfoils2.pdf
rm tfoils.* tfoils2.pdf
# talk:
talk: $(PDFFILE)
pdf: $(PDFFILE)
$(PDFFILE): $(TEXFILE) $(GPTTEXFILES)
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
# batchmode (no output, no stop on error)
# nonstopmode / scrollmode (no stop on error)
# errorstopmode (stop on error)
again :
pdflatex $(TEXFILE)
watchpdf :
while true; do ! make -q pdf && make pdf; sleep 0.5; done
# html
html : $(HTMLTEXFILE) $(GPTTEXFILES)
rm -f $(HTMLDIR)/*
htlatex $<
mkdir -p $(HTMLDIR)
mv $(HTMLBASENAME).html $(HTMLDIR)
mv $(HTMLBASENAME)*.* $(HTMLDIR)
mv z*.gif $(HTMLDIR)
cd $(HTMLDIR); for i in *.gif; do convert -page +0+0 $$i tmp.gif; mv tmp.gif $$i; done; rmtex $(HTMLBASENAME)
#$(HTMLTEXFILE) : $(TEXFILE) Makefile
# sed 's/setboolean{html}{false}/setboolean{html}{true}/; s/\\colorbox{white}{\(.*\)}/\1/g' $< > $@
html1 : $(HTMLTEXFILE) $(GPTTEXFILES)
latex2html -dir $(HTMLDIR) -mkdir -subdir -nonavigation -noinfo -image_type png -notransparent -white -split 0 $<
sed 's-<I>Date:</I>--' $(HTMLDIR)/$(HTMLDIR).html > tmp.html
cp tmp.html $(HTMLDIR)/index.html
mv tmp.html $(HTMLDIR)/$(HTMLDIR).html
$(HTMLTEXFILE) : $(TEXFILE)
sed '/^%nohtml/,/^%endnohtml/d; s/\\colorbox{white}{\(.*\)}/\1/g' $< > $@
# eps of all figures:
epsfigs:
mkdir -p epsfigs; \
for i in $(GPTFILES); do \
{ sed -n -e '1,/\\begin{document}/p' $(TEXFILE); echo "\texpicture{$${i%%.*}}"; echo "\end{document}"; } > tmp.tex; \
latex tmp.tex; \
dvips tmp.dvi; \
ps2eps tmp.ps; \
mv tmp.eps epsfigs/$${i%%.*}.eps; \
rm tmp.*; \
done
# plots:
%.tex: %.gpt whitestyles.gp
watchslides :
while true; do ! make -q slides && make slides; sleep 0.5; done
# python plots:
$(PYPDFFILES) : %.pdf: %.py
python $<
# gnuplot plots:
$(GPTTEXFILES) : %.tex: %.gpt whitestyles.gp
gnuplot whitestyles.gp $<
epstopdf $*.eps
clean :
rm -f *~
rmtex $(BASENAME)
rm -f $(GPTTEXFILES)
rm -f $(BASENAME).aux $(BASENAME).log
rm -f $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out
rm -f $(BASENAME)-slides.aux $(BASENAME)-slides.log $(BASENAME)-slides.out $(BASENAME)-slides.toc $(BASENAME)-slides.nav $(BASENAME)-slides.snm $(BASENAME)-slides.vrb
rm -f $(PYPDFFILES) $(GPTTEXFILES)
cleanup :
rm -f *~
rmtex $(BASENAME)
rm -f $(PSFILE) $(PDFFILE) $(FOILSFILE) $(THUMBNAILSFILE)
rm -f $(GPTTEXFILES)
rm -f -r $(HTMLDIR)
cleanplots :
sed -n -e '/\\begin{document}/,/\\end{document}/p' $(TEXFILE) | fgrep '\input{' | grep -v '^%' | sed 's/.*input{\(.*\).tex}.*/\1.gpt/' > plot.fls
mkdir -p unusedplots
for i in *.gp*; do \
grep -q $$i plot.fls || { grep -q $$i $$(<plot.fls) && echo $$i || mv $$i unusedplots; }; \
done >> plot.fls
for i in $$(<plot.fls); do \
sed "s/\([^'\" ]*\.dat\)/\n\1\n/g;" $$i | fgrep .dat; \
done | sort | uniq > dat.fls
mkdir -p unuseddata
for i in *.dat; do \
grep -q $$i dat.fls || mv $$i unuseddata; \
done
rm dat.fls plot.fls
cleanall : clean
rm -f $(BASENAME)-chapter.pdf $(BASENAME)-slides.pdf $(BASENAME)-handout.pdf
help :
@echo -e \
"make pdf: make the pdf file of the talk.\n"\
"make foils: make black&white postscript foils of the talk.\n"\
"make thumbs: make color thumbnails of the talk.\n"\
"make again: run latex and make the pdf file of the talk,\n"\
" no matter whether you changed the .tex file or not.\n\n"\
"make watchpdf: make the pdf file of the talk\n"\
"make pdf: make the pdf file of the script.\n"\
"make slides: make the pdf file of the slides.\n"\
"make thumbs: make color thumbnails of the talk.\n"\
"make watchpdf: make the pdf file of the script\n"\
" whenever the tex file is modified.\n"\
"make watchpdf: make the pdf file of the slides\n"\
" whenever the tex file is modified.\n"\
"make html: make a html version of the paper (in $(HTMLDIR)).\n\n"\
"make clean: remove all intermediate files,\n"\
" just leave the source files and the final .ps and .pdf files.\n"\
" just leave the source files and the final .pdf files.\n"\
"make cleanup: remove all intermediate files as well as\n"\
" the final .ps and .pdf files.\n"\
"make cleanplots: move all unused .gpt and .dat files\n"\
" into unusedplots/ and unuseddata/, respectively."
" the final .pdf files.\n"\

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\documentclass{beamer}
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[]{Scientific Computing --- Point Processes}
\author[]{Jan Benda}
\institute[]{Neuroethology}
\date[]{WS 14/15}
\titlegraphic{\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
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\frametitle{}
\vspace{-1cm}
\titlepage % erzeugt Titelseite
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Content}
\tableofcontents
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Point processes}
\begin{frame}
\frametitle{Point process}
\vspace{-3ex}
\texpicture{pointprocessscetchA}
A point process is a stochastic (or random) process that generates a sequence of events
at times $\{t_i\}$, $t_i \in \reZ$.
For each point process there is an underlying continuous-valued
process evolving in time. The associated point process occurs when
the underlying continuous process crosses a threshold.
Examples:
\begin{itemize}
\item Spikes/heartbeat: generated by the dynamics of the membrane potential of neurons/heart cells.
\item Earth quakes: generated by the pressure dynamics between the tectonic plates on either side of a geological fault line.
\item Onset of cricket/frogs/birds/... songs: generated by the dynamics of the state of a nervous system.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Point process}
\texpicture{pointprocessscetchB}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Homogeneous Poisson process}
\begin{frame}
\frametitle{Homogeneous Poisson process}
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 $\lambda$.
\includegraphics[width=1\textwidth]{poissonraster100hz}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Interval statistics}
\begin{frame}
\frametitle{Rate}
Rate of events $r$ (``spikes per time'') measured in Hertz.
\begin{itemize}
\item Number of events $N$ per observation time $W$: $r = \frac{N}{W}$
\item Without boundary effects: $r = \frac{N-1}{t_N-t_1}$
\item Inverse interval: $r = \frac{1}{\mu_{ISI}}$
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{(Interspike) interval statistics}
\begin{itemize}
\item Histogram $p(T)$ of intervals $T$. Normalized to $\int_0^{\infty} p(T) \; dT = 1$
\item Mean interval $\mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum\limits_{i=1}^n T_i$
\item Variance of intervals $\sigma_{ISI}^2 = \langle (T - \langle T \rangle)^2 \rangle$\vspace{1ex}
\item Coefficient of variation $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$
\item Diffusion coefficient $D_{ISI} = \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$
\vfill
\end{itemize}
\includegraphics[width=0.45\textwidth]{poissonisih100hz}\hfill
\includegraphics[width=0.45\textwidth]{lifisih16}
\end{frame}
\begin{frame}
\frametitle{Interval statistics of homogeneous Poisson process}
\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 $CV_{ISI} = 1$
\end{itemize}
\vfill
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
\end{frame}
\begin{frame}
\frametitle{Interval return maps}
Scatter plot between succeeding intervals separated by lag $k$.
\vfill
Poisson process $\lambda=100$\,Hz:
\includegraphics[width=1\textwidth]{poissonreturnmap100hz}\hfill
\end{frame}
\begin{frame}
\frametitle{Serial interval correlations}
Correlation coefficients between succeeding intervals separated by lag $k$:
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
\begin{itemize}
\item $\rho_0=1$ (correlation of each interval with itself).
\item Poisson process: $\rho_k =0$ for $k>0$ (renewal process!)
\end{itemize}
\vfill
\includegraphics[width=0.7\textwidth]{poissonserial100hz}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Count statistics}
\begin{frame}
\frametitle{Count statistics}
Histogram of number of events $N$ (counts) within observation window of duration $W$.
\vfill
\includegraphics[width=0.48\textwidth]{poissoncounthist100hz10ms}\hfill
\includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
\end{frame}
\begin{frame}
\frametitle{Count statistics of Poisson process}
Poisson distribution:
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
\vfill
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
\end{frame}
\begin{frame}
\frametitle{Count statistics --- Fano factor}
Statistics of number of events $N$ within observation window of duration $W$.
\begin{itemize}
\item Mean count: $\mu_N = \langle N \rangle$
\item Count variance: $\sigma_N^2 = \langle (N - \langle N \rangle)^2 \rangle$
\item Fano factor (variance divided by mean): $F = \frac{\sigma_N^2}{\mu_N}$
\item Poisson process: $F=1$
\end{itemize}
\vfill
Poisson process $\lambda=100$\,Hz:
\includegraphics[width=1\textwidth]{poissonfano100hz}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Integrate-and-fire models}
\begin{frame}
\frametitle{Integrate-and-fire models}
Leaky integrate-and-fire model (LIF):
\[ \tau \frac{dV}{dt} = -V + RI + D\xi \]
Whenever membrane potential $V(t)$ crosses the firing threshold $\theta$, a spike is emitted and
$V(t)$ is reset to $V_{reset}$.
\begin{itemize}
\item $\tau$: membrane time constant (typically 10\,ms)
\item $R$: input resistance (here 1\,mV (!))
\item $D\xi$: additive Gaussian white noise of strength $D$
\item $\theta$: firing threshold (here 10\,mV)
\item $V_{reset}$: reset potential (here 0\,mV)
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Integrate-and-fire models}
Discretization with time step $\Delta t$: $V(t) \rightarrow V_i,\;t_i = i \Delta t$.\\
Euler integration:
\begin{eqnarray*}
\frac{dV}{dt} & \approx & \frac{V_{i+1} - V_i}{\Delta t} \\
\Rightarrow \quad V_{i+1} & = & V_i + \Delta t \frac{-V_i+RI_i+\sqrt{2D\Delta t}N_i}{\tau}
\end{eqnarray*}
$N_i$ are normally distributed random numbers (Gaussian with zero mean and unit variance)
--- the $\sqrt{\Delta t}$ is for white noise.
\includegraphics[width=0.82\textwidth]{lifraster16}
\end{frame}
\begin{frame}
\frametitle{Interval statistics of LIF}
Interval distribution approaches Inverse Gaussian for large $I$:
\[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \]
where $\langle T \rangle$ is the mean interspike interval and $D$
is the diffusion coefficient.
\vfill
\includegraphics[width=0.45\textwidth]{lifisihdistr08}\hfill
\includegraphics[width=0.45\textwidth]{lifisihdistr16}
\end{frame}
\begin{frame}
\frametitle{Interval statistics of PIF}
For the perfect integrate-and-fire (PIF)
\[ \tau \frac{dV}{dt} = RI + D\xi \]
(the canonical model or supra-threshold firing on a limit cycle)\\
the Inverse Gaussian describes exactly the interspike interval distribution.
\vfill
\includegraphics[width=0.45\textwidth]{pifisihdistr01}\hfill
\includegraphics[width=0.45\textwidth]{pifisihdistr10}
\end{frame}
\begin{frame}
\frametitle{Interval return map of LIF}
LIF $I=15.7$:
\includegraphics[width=1\textwidth]{lifreturnmap16}
\end{frame}
\begin{frame}
\frametitle{Serial correlations of LIF}
LIF $I=15.7$:
\includegraphics[width=1\textwidth]{lifserial16}\\
Integrate-and-fire driven with white noise are still renewal processes!
\end{frame}
\begin{frame}
\frametitle{Count statistics of LIF}
LIF $I=15.7$:
\includegraphics[width=1\textwidth]{liffano16}\\
Fano factor is not one!
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Interval statistics of LIF with OU noise}
\begin{eqnarray*}
\tau \frac{dV}{dt} & = & -V + RI + U \\
\tau_{OU} \frac{dU}{dt} & = & - U + D\xi
\end{eqnarray*}
Ohrnstein-Uhlenbeck noise is lowpass filtered white noise.
\includegraphics[width=0.45\textwidth]{lifouisihdistr08-100ms}\hfill
\includegraphics[width=0.45\textwidth]{lifouisihdistr16-100ms}\\
More peaky than the inverse Gaussian!
\end{frame}
\begin{frame}
\frametitle{Interval return map of LIF with OU noise}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:
\includegraphics[width=1\textwidth]{lifoureturnmap16-100ms}
\end{frame}
\begin{frame}
\frametitle{Serial correlations of LIF with OU noise}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:
\includegraphics[width=1\textwidth]{lifouserial16-100ms}\\
OU-noise introduces positive interval correlations!
\end{frame}
\begin{frame}
\frametitle{Count statistics of LIF with OU noise}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:
\includegraphics[width=1\textwidth]{lifoufano16-100ms}\\
Fano factor increases with count window duration.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Interval statistics of LIF with adaptation}
\begin{eqnarray*}
\tau \frac{dV}{dt} & = & -V - A + RI + D\xi \\
\tau_{adapt} \frac{dA}{dt} & = & - A
\end{eqnarray*}
Adaptation $A$ with time constant $\tau_{adapt}$ and increment $\Delta A$ at spike.
\includegraphics[width=0.45\textwidth]{lifadaptisihdistr08-100ms}\hfill
\includegraphics[width=0.45\textwidth]{lifadaptisihdistr65-100ms}\\
Similar to LIF with white noise.
\end{frame}
\begin{frame}
\frametitle{Interval return map of LIF with adaptation}
LIF $I=10$, $\tau_{adapt}=100$\,ms:
\includegraphics[width=1\textwidth]{lifadaptreturnmap10-100ms}\\
Negative correlation at lag one.
\end{frame}
\begin{frame}
\frametitle{Serial correlations of LIF with adaptation}
LIF $I=10$, $\tau_{adapt}=100$\,ms:
\includegraphics[width=1\textwidth]{lifadaptserial10-100ms}\\
Adaptation with white noise introduces negative interval correlations!
\end{frame}
\begin{frame}
\frametitle{Count statistics of LIF with adaptation}
LIF $I=10$, $\tau_{adapt}=100$\,ms:
\includegraphics[width=1\textwidth]{lifadaptfano10-100ms}\\
Fano factor decreases with count window duration.
\end{frame}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non stationary}
\subsection{Inhomogeneous Poisson process}
\subsection{Firing rate}
\subsection{Instantaneous rate}
\subsection{Autocorrelation}
\subsection{Crosscorrelation}
\subsection{Joint PSTH}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Renewal process}
\subsection{Superthreshold firing}
\subsection{Subthreshold firing}
\section{Non-renewal processes}
\subsection{Bursting}
\subsection{Resonator}
\subsection{Standard distributions}
\subsubsection{Gamma}
\subsubsection{How to read ISI histograms}
refractoriness, poisson tail, sub-, supra-threshold, missed spikes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Correlation with stimulus}
\subsection{Tuning curve}
\subsection{Linear filter}
\subsection{Spatiotemporal receptive field}
\subsection{Generalized linear model}
\begin{frame}
\end{frame}

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pointprocesses/lecture/pointprocesses.tex
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\documentclass{beamer}
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[]{Scientific Computing --- Point Processes}
\author[]{Jan Benda}
\institute[]{Neuroethology}
\date[]{WS 14/15}
\titlegraphic{\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
%%%%% beamer %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>
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%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\lstset{
basicstyle=\ttfamily,
numbers=left,
showstringspaces=false,
language=Matlab,
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\titlepage % erzeugt Titelseite
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Content}
\tableofcontents
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Point processes}
\begin{frame}
\frametitle{Point process}
\vspace{-3ex}
\texpicture{pointprocessscetchA}
\chapter{\tr{Point processes}{Punktprozesse}}
A point process is a stochastic (or random) process that generates a sequence of events
at times $\{t_i\}$, $t_i \in \reZ$.
For each point process there is an underlying continuous-valued
process evolving in time. The associated point process occurs when
the underlying continuous process crosses a threshold.
Examples:
\begin{itemize}
\item Spikes/heartbeat: generated by the dynamics of the membrane potential of neurons/heart cells.
\item Earth quakes: generated by the pressure dynamics between the tectonic plates on either side of a geological fault line.
\item Onset of cricket/frogs/birds/... songs: generated by the dynamics of the state of a nervous system.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Point process}
\begin{figure}[t]
\texpicture{pointprocessscetchB}
\end{frame}
\caption{\label{pointprocessscetchfig}Ein Punktprozess ist eine
Abfolge von Zeitpunkten $t_i$ die auch durch die Intervalle
$T_i=t_{i+1}-t_i$ oder die Anzahl der Ereignisse $n_i$ beschrieben
werden kann. }
\end{figure}
Ein zeitlicher Punktprozess ist ein stochastischer Prozess der eine Abfolge von Ereignissen zu den Zeiten $\{t_i\}$, $t_i \in \reZ$ generiert.
Jeder Punktprozess wird durch einen sich in der Zeit kontinuierlichen
entwickelnden Prozess generiert. Wann immer dieser Prozess eine Schwelle \"uberschreitet
wird ein Ereigniss des Punktprozesses erzeugt. Zum Beispiel:
\begin{itemize}
\item Aktionspotentiale/Herzschlag: wird durch die Dynamik des
Membranpotentials eines Neurons/Herzzelle erzeugt.
\item Erdbeben: wird durch die Dynamik des Druckes zwischen
tektonischen Platten auf beiden Seiten einer geologischen Verwerfung
erzeugt.
\item Zeitpunkt eines Grillen/Frosch/Vogelgesangs: wird durch die
Dynamic des Nervensystems und des Muskelapparates erzeugt.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Homogeneous Poisson process}
\begin{frame}
\frametitle{Homogeneous Poisson process}
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 $\lambda$.
\includegraphics[width=1\textwidth]{poissonraster100hz}
\end{frame}
\section{Rate eines Punktprozesses}
Rate of events $r$ (``spikes per time'') measured in Hertz.
\begin{itemize}
\item Number of events $N$ per observation time $W$: $r = \frac{N}{W}$
\item Without boundary effects: $r = \frac{N-1}{t_N-t_1}$
\item Inverse interval: $r = \frac{1}{\mu_{ISI}}$
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Interval statistics}
\begin{frame}
\frametitle{Rate}
Rate of events $r$ (``spikes per time'') measured in Hertz.
\begin{itemize}
\item Number of events $N$ per observation time $W$: $r = \frac{N}{W}$
\item Without boundary effects: $r = \frac{N-1}{t_N-t_1}$
\item Inverse interval: $r = \frac{1}{\mu_{ISI}}$
\end{itemize}
\end{frame}
\section{Intervall Statistiken}
\begin{frame}
\frametitle{(Interspike) interval statistics}
\begin{itemize}
\item Histogram $p(T)$ of intervals $T$. Normalized to $\int_0^{\infty} p(T) \; dT = 1$
\item Mean interval $\mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum\limits_{i=1}^n T_i$
\item Variance of intervals $\sigma_{ISI}^2 = \langle (T - \langle T \rangle)^2 \rangle$\vspace{1ex}
\item Coefficient of variation $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$
\item Diffusion coefficient $D_{ISI} = \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$
\vfill
\end{itemize}
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{poissonisih100hz}\hfill
\includegraphics[width=0.45\textwidth]{lifisih16}
\end{frame}
\begin{frame}
\frametitle{Interval statistics of homogeneous Poisson process}
\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 $CV_{ISI} = 1$
\end{itemize}
\vfill
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
\end{frame}
\begin{frame}
\frametitle{Interval return maps}
Scatter plot between succeeding intervals separated by lag $k$.
\vfill
Poisson process $\lambda=100$\,Hz:
\includegraphics[width=1\textwidth]{poissonreturnmap100hz}\hfill
\end{frame}
\begin{frame}
\frametitle{Serial interval correlations}
Correlation coefficients between succeeding intervals separated by lag $k$:
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
\begin{itemize}
\item $\rho_0=1$ (correlation of each interval with itself).
\item Poisson process: $\rho_k =0$ for $k>0$ (renewal process!)
\end{itemize}
\vfill
\includegraphics[width=0.7\textwidth]{poissonserial100hz}
\end{frame}
\caption{\label{isihfig}Interspike-Intervall Histogramme von einem Poisson Prozess (links)
und einem Integrate-and-Fire Neuron (rechts).}
\end{figure}
\subsection{First order (Interspike) interval statistics}
\begin{itemize}
\item Histogram $p(T)$ of intervals $T$. Normalized to $\int_0^{\infty} p(T) \; dT = 1$
\item Mean interval $\mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum\limits_{i=1}^n T_i$
\item Variance of intervals $\sigma_{ISI}^2 = \langle (T - \langle T \rangle)^2 \rangle$\vspace{1ex}
\item Coefficient of variation $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$
\item Diffusion coefficient $D_{ISI} = \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$
\end{itemize}
\subsection{Interval return maps}
Scatter plot between succeeding intervals separated by lag $k$.
\begin{figure}[t]
\begin{minipage}[t]{0.49\textwidth}
LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
\includegraphics[width=1\textwidth]{lifadaptreturnmap10-100ms}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:\\
\includegraphics[width=1\textwidth]{lifoureturnmap16-100ms}
\end{minipage}
\caption{\label{returnmapfig}Interspike-Intervall return maps.}
\end{figure}
\subsection{Serial correlations of the intervals}
Correlation coefficients between succeeding intervals separated by lag $k$:
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
$\rho_0=1$ (correlation of each interval with itself).
\begin{figure}[t]
\begin{minipage}[t]{0.49\textwidth}
LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
\includegraphics[width=1\textwidth]{lifadaptserial10-100ms}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:\\
\includegraphics[width=1\textwidth]{lifouserial16-100ms}
\end{minipage}
\caption{\label{serialcorrfig}Serial correlations.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Count statistics}
\begin{frame}
\frametitle{Count statistics}
Histogram of number of events $N$ (counts) within observation window of duration $W$.
\vfill
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{poissoncounthist100hz10ms}\hfill
\includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
\end{frame}
\begin{frame}
\frametitle{Count statistics of Poisson process}
Poisson distribution:
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
\vfill
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
\end{frame}
\begin{frame}
\frametitle{Count statistics --- Fano factor}
Statistics of number of events $N$ within observation window of duration $W$.
\begin{itemize}
\item Mean count: $\mu_N = \langle N \rangle$
\item Count variance: $\sigma_N^2 = \langle (N - \langle N \rangle)^2 \rangle$
\item Fano factor (variance divided by mean): $F = \frac{\sigma_N^2}{\mu_N}$
\item Poisson process: $F=1$
\end{itemize}
\vfill
Poisson process $\lambda=100$\,Hz:
\includegraphics[width=1\textwidth]{poissonfano100hz}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Integrate-and-fire models}
\begin{frame}
\frametitle{Integrate-and-fire models}
Leaky integrate-and-fire model (LIF):
\[ \tau \frac{dV}{dt} = -V + RI + D\xi \]
Whenever membrane potential $V(t)$ crosses the firing threshold $\theta$, a spike is emitted and
$V(t)$ is reset to $V_{reset}$.
\begin{itemize}
\item $\tau$: membrane time constant (typically 10\,ms)
\item $R$: input resistance (here 1\,mV (!))
\item $D\xi$: additive Gaussian white noise of strength $D$
\item $\theta$: firing threshold (here 10\,mV)
\item $V_{reset}$: reset potential (here 0\,mV)
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Integrate-and-fire models}
Discretization with time step $\Delta t$: $V(t) \rightarrow V_i,\;t_i = i \Delta t$.\\
Euler integration:
\begin{eqnarray*}
\frac{dV}{dt} & \approx & \frac{V_{i+1} - V_i}{\Delta t} \\
\Rightarrow \quad V_{i+1} & = & V_i + \Delta t \frac{-V_i+RI_i+\sqrt{2D\Delta t}N_i}{\tau}
\end{eqnarray*}
$N_i$ are normally distributed random numbers (Gaussian with zero mean and unit variance)
--- the $\sqrt{\Delta t}$ is for white noise.
\includegraphics[width=0.82\textwidth]{lifraster16}
\end{frame}
\begin{frame}
\frametitle{Interval statistics of LIF}
Interval distribution approaches Inverse Gaussian for large $I$:
\[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \]
where $\langle T \rangle$ is the mean interspike interval and $D$
is the diffusion coefficient.
\vfill
\includegraphics[width=0.45\textwidth]{lifisihdistr08}\hfill
\includegraphics[width=0.45\textwidth]{lifisihdistr16}
\end{frame}
\begin{frame}
\frametitle{Interval statistics of PIF}
For the perfect integrate-and-fire (PIF)
\[ \tau \frac{dV}{dt} = RI + D\xi \]
(the canonical model or supra-threshold firing on a limit cycle)\\
the Inverse Gaussian describes exactly the interspike interval distribution.
\vfill
\includegraphics[width=0.45\textwidth]{pifisihdistr01}\hfill
\includegraphics[width=0.45\textwidth]{pifisihdistr10}
\end{frame}
\begin{frame}
\frametitle{Interval return map of LIF}
LIF $I=15.7$:
\includegraphics[width=1\textwidth]{lifreturnmap16}
\end{frame}
\begin{frame}
\frametitle{Serial correlations of LIF}
LIF $I=15.7$:
\includegraphics[width=1\textwidth]{lifserial16}\\
Integrate-and-fire driven with white noise are still renewal processes!
\end{frame}
\begin{frame}
\frametitle{Count statistics of LIF}
LIF $I=15.7$:
\includegraphics[width=1\textwidth]{liffano16}\\
Fano factor is not one!
\end{frame}
\caption{\label{countstatsfig}Count Statistik.}
\end{figure}
Histogram of number of events $N$ (counts) within observation window of duration $W$.
\subsection{Fano factor}
\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}
Statistics of number of events $N$ within observation window of duration $W$.
\begin{itemize}
\item Mean count: $\mu_N = \langle N \rangle$
\item Count variance: $\sigma_N^2 = \langle (N - \langle N \rangle)^2 \rangle$
\item Fano factor (variance divided by mean): $F = \frac{\sigma_N^2}{\mu_N}$
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Interval statistics of LIF with OU noise}
\begin{eqnarray*}
\tau \frac{dV}{dt} & = & -V + RI + U \\
\tau_{OU} \frac{dU}{dt} & = & - U + D\xi
\end{eqnarray*}
Ohrnstein-Uhlenbeck noise is lowpass filtered white noise.
\includegraphics[width=0.45\textwidth]{lifouisihdistr08-100ms}\hfill
\includegraphics[width=0.45\textwidth]{lifouisihdistr16-100ms}\\
More peaky than the inverse Gaussian!
\end{frame}
\begin{frame}
\frametitle{Interval return map of LIF with OU noise}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:
\includegraphics[width=1\textwidth]{lifoureturnmap16-100ms}
\end{frame}
\begin{frame}
\frametitle{Serial correlations of LIF with OU noise}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:
\includegraphics[width=1\textwidth]{lifouserial16-100ms}\\
OU-noise introduces positive interval correlations!
\end{frame}
\begin{frame}
\frametitle{Count statistics of LIF with OU noise}
LIF $I=15.7$, $\tau_{OU}=100$\,ms:
\includegraphics[width=1\textwidth]{lifoufano16-100ms}\\
Fano factor increases with count window duration.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Interval statistics of LIF with adaptation}
\begin{eqnarray*}
\tau \frac{dV}{dt} & = & -V - A + RI + D\xi \\
\tau_{adapt} \frac{dA}{dt} & = & - A
\end{eqnarray*}
Adaptation $A$ with time constant $\tau_{adapt}$ and increment $\Delta A$ at spike.
\includegraphics[width=0.45\textwidth]{lifadaptisihdistr08-100ms}\hfill
\includegraphics[width=0.45\textwidth]{lifadaptisihdistr65-100ms}\\
Similar to LIF with white noise.
\end{frame}
\begin{frame}
\frametitle{Interval return map of LIF with adaptation}
LIF $I=10$, $\tau_{adapt}=100$\,ms:
\includegraphics[width=1\textwidth]{lifadaptreturnmap10-100ms}\\
Negative correlation at lag one.
\end{frame}
\begin{frame}
\frametitle{Serial correlations of LIF with adaptation}
LIF $I=10$, $\tau_{adapt}=100$\,ms:
\includegraphics[width=1\textwidth]{lifadaptserial10-100ms}\\
Adaptation with white noise introduces negative interval correlations!
\end{frame}
\begin{frame}
\frametitle{Count statistics of LIF with adaptation}
LIF $I=10$, $\tau_{adapt}=100$\,ms:
\includegraphics[width=1\textwidth]{lifadaptfano10-100ms}\\
Fano factor decreases with count window duration.
\end{frame}
\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}
\end{document}
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 $\lambda$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non stationary}
\subsection{Inhomogeneous Poisson process}
\subsection{Firing rate}
\subsection{Instantaneous rate}
\subsection{Autocorrelation}
\subsection{Crosscorrelation}
\subsection{Joint PSTH}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Renewal process}
\subsection{Superthreshold firing}
\subsection{Subthreshold firing}
\section{Non-renewal processes}
\subsection{Bursting}
\subsection{Resonator}
\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}
\subsection{Standard distributions}
\subsubsection{Gamma}
\subsubsection{How to read ISI histograms}
refractoriness, poisson tail, sub-, supra-threshold, missed spikes
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Correlation with stimulus}
\subsection{Tuning curve}
\subsection{Linear filter}
\subsection{Spatiotemporal receptive field}
\subsection{Generalized linear model}
\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}
\begin{frame}
\end{frame}
Poisson distribution:
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]

4
pointprocesses/lecture/pointprocessscetchA.eps
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@ -1,7 +1,7 @@
%!PS-Adobe-2.0 EPSF-2.0
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@ -433,7 +433,7 @@ SDict begin [
/Author (jan)
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% /Keywords ()
/CreationDate (Sun Oct 26 14:09:12 2014)
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pointprocesses/lecture/pointprocessscetchB.eps
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9
scientificcomputing-script.tex
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@ -59,7 +59,8 @@
% figures:
\setlength{\fboxsep}{0pt}
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
\newcommand{\texinputpath}{}
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{\texinputpath#1.tex}}}
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
%\newcommand{\texpicture}[1]{}
@ -213,7 +214,7 @@
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\codepath\exercisesource}}}\medskip\stepcounter{theexercise}}
\graphicspath{{statistics/lecture/}{statistics/lecture/figures/}{bootstrap/lecture/}{bootstrap/lecture/figures/}{likelihood/lecture/}{likelihood/lecture/figures/}}
\graphicspath{{statistics/lecture/}{statistics/lecture/figures/}{bootstrap/lecture/}{bootstrap/lecture/figures/}{likelihood/lecture/}{likelihood/lecture/figures/}{pointprocesses/lecture/}{pointprocesses/lecture/figures/}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -233,4 +234,8 @@
\renewcommand{\codepath}{likelihood/code/}
\include{likelihood/lecture/likelihood}
\renewcommand{\codepath}{pointprocesses/code/}
\renewcommand{\texinputpath}{pointprocesses/lecture/}
\include{pointprocesses/lecture/pointprocesses}
\end{document}

15
statistics/lecture/Makefile
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@ -1,22 +1,29 @@
BASENAME=statistics
PYFILES=$(wildcard *.py)
PYPDFFILES=$(PYFILES:.py=.pdf)
pdf : $(BASENAME)-chapter.pdf $(PYPDFFILES)
all : pdf
# script:
pdf : $(BASENAME)-chapter.pdf
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex
$(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 *~ $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out $(BASENAME).aux $(BASENAME).log
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
watch :
watchpdf :
while true; do ! make -q pdf && make pdf; sleep 0.5; done

2
statistics/lecture/boxwhisker.py
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@ -43,5 +43,5 @@ ax.annotate('maximum',
ax.boxplot( x, whis=100.0 )
plt.tight_layout()
plt.savefig('boxwhisker.pdf')
plt.show()
#plt.show()

4
statistics/lecture/correlation.py
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@ -5,7 +5,6 @@ plt.xkcd()
fig = plt.figure( figsize=(6,5) )
n = 200
for k, r in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) :
print r
x = np.random.randn( n )
y = r*x + np.sqrt(1.0-r*r)*np.random.randn( n )
ax = fig.add_subplot( 2, 2, k+1 )
@ -30,5 +29,4 @@ for k, r in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) :
plt.tight_layout()
plt.savefig('correlation.pdf')
plt.show()
#plt.show()

3
statistics/lecture/diehistograms.py
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@ -28,5 +28,4 @@ ax.set_ylabel( 'Probability' )
ax.hist([x2, x1], bins, normed=True, color=['#FFCC00', '#FFFF66' ])
plt.tight_layout()
fig.savefig( 'diehistograms.pdf' )
plt.show()
#plt.show()

3
statistics/lecture/median.py
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@ -29,5 +29,4 @@ ax.plot(x,g, 'b', lw=4)
ax.plot([0.0, 0.0], [0.0, 0.45], 'k', lw=2 )
plt.tight_layout()
fig.savefig( 'median.pdf' )
plt.show()
#plt.show()

2
statistics/lecture/nonlincorrelation.py
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@ -39,4 +39,4 @@ ax.scatter( x, z )
plt.tight_layout()
plt.savefig('nonlincorrelation.pdf')
plt.show()
#plt.show()

2
statistics/lecture/pdfhistogram.py
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@ -35,5 +35,5 @@ ax.hist(r, 20, normed=True, color='#FFCC00')
plt.tight_layout()
fig.savefig( 'pdfhistogram.pdf' )
plt.show()
#plt.show()

3
statistics/lecture/pdfprobabilities.py
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@ -32,5 +32,4 @@ ax.fill_between( x[(x>x1)&(x<x2)], 0.0, g[(x>x1)&(x<x2)], color='#cc0000' )
ax.plot(x,g, 'b', lw=4)
plt.tight_layout()
fig.savefig( 'pdfprobabilities.pdf' )
plt.show()
#plt.show()

3
statistics/lecture/quartile.py
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@ -46,5 +46,4 @@ ax.plot([q[0], q[0]], [0.0, 0.4], 'k', lw=2 )
ax.plot([q[2], q[2]], [0.0, 0.4], 'k', lw=2 )
plt.tight_layout()
fig.savefig( 'quartile.pdf' )
plt.show()
#plt.show()