From 857373c125c5dc97d3b76bb6102dec03ab88f4a5 Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Sun, 25 Oct 2015 22:22:54 +0100
Subject: [PATCH] Started working on pointprocesses
---
bootstrap/lecture/Makefile | 15 +-
likelihood/lecture/Makefile | 15 +-
pointprocesses/exercises/Makefile | 35 ++
pointprocesses/exercises/instructions.tex | 11 +
pointprocesses/exercises/pointprocesses01.tex | 202 +++++++
pointprocesses/exercises/poisson.pdf | Bin 81657 -> 0 bytes
pointprocesses/exercises/poisson.tex | 160 ------
pointprocesses/lecture/Makefile | 154 ++---
.../lecture/pointprocesses-slides.tex | 412 ++++++++++++++
pointprocesses/lecture/pointprocesses.tex | 530 +++++-------------
.../lecture/pointprocessscetchA.eps | 4 +-
.../lecture/pointprocessscetchA.pdf | Bin 2765 -> 2765 bytes
.../lecture/pointprocessscetchB.eps | 4 +-
.../lecture/pointprocessscetchB.pdf | Bin 4673 -> 4673 bytes
scientificcomputing-script.tex | 9 +-
statistics/lecture/Makefile | 15 +-
statistics/lecture/boxwhisker.py | 2 +-
statistics/lecture/correlation.py | 4 +-
statistics/lecture/diehistograms.py | 3 +-
statistics/lecture/median.py | 3 +-
statistics/lecture/nonlincorrelation.py | 2 +-
statistics/lecture/pdfhistogram.py | 2 +-
statistics/lecture/pdfprobabilities.py | 3 +-
statistics/lecture/quartile.py | 3 +-
24 files changed, 896 insertions(+), 692 deletions(-)
create mode 100644 pointprocesses/exercises/Makefile
create mode 100644 pointprocesses/exercises/instructions.tex
create mode 100644 pointprocesses/exercises/pointprocesses01.tex
delete mode 100644 pointprocesses/exercises/poisson.pdf
delete mode 100644 pointprocesses/exercises/poisson.tex
create mode 100644 pointprocesses/lecture/pointprocesses-slides.tex
diff --git a/bootstrap/lecture/Makefile b/bootstrap/lecture/Makefile
index f7f02ba..de4182d 100644
--- a/bootstrap/lecture/Makefile
+++ b/bootstrap/lecture/Makefile
@@ -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 $(BASENAME)-chapter.pdf
-watch :
+watchpdf :
while true; do ! make -q pdf && make pdf; sleep 0.5; done
diff --git a/likelihood/lecture/Makefile b/likelihood/lecture/Makefile
index 4e6367b..42b8e3d 100644
--- a/likelihood/lecture/Makefile
+++ b/likelihood/lecture/Makefile
@@ -1,22 +1,29 @@
BASENAME=likelihood
+
PYFILES=$(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 $(BASENAME)-chapter.pdf
-watch :
+watchpdf :
while true; do ! make -q pdf && make pdf; sleep 0.5; done
diff --git a/pointprocesses/exercises/Makefile b/pointprocesses/exercises/Makefile
new file mode 100644
index 0000000..0b63fb9
--- /dev/null
+++ b/pointprocesses/exercises/Makefile
@@ -0,0 +1,35 @@
+BASENAME=pointprocesses
+TEXFILES=$(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)
diff --git a/pointprocesses/exercises/instructions.tex b/pointprocesses/exercises/instructions.tex
new file mode 100644
index 0000000..96ac4bc
--- /dev/null
+++ b/pointprocesses/exercises/instructions.tex
@@ -0,0 +1,11 @@
+\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
diff --git a/pointprocesses/exercises/pointprocesses01.tex b/pointprocesses/exercises/pointprocesses01.tex
new file mode 100644
index 0000000..b4d927c
--- /dev/null
+++ b/pointprocesses/exercises/pointprocesses01.tex
@@ -0,0 +1,202 @@
+\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}
\ No newline at end of file
diff --git a/pointprocesses/exercises/poisson.pdf b/pointprocesses/exercises/poisson.pdf
deleted file mode 100644
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diff --git a/pointprocesses/exercises/poisson.tex b/pointprocesses/exercises/poisson.tex
deleted file mode 100644
index 4b8f0f0..0000000
--- a/pointprocesses/exercises/poisson.tex
+++ /dev/null
@@ -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}
diff --git a/pointprocesses/lecture/Makefile b/pointprocesses/lecture/Makefile
index 7a243a5..4ac656e 100644
--- a/pointprocesses/lecture/Makefile
+++ b/pointprocesses/lecture/Makefile
@@ -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)
-
-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
+ rm -f *~
+ 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)
+
+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"\
diff --git a/pointprocesses/lecture/pointprocesses-slides.tex b/pointprocesses/lecture/pointprocesses-slides.tex
new file mode 100644
index 0000000..ff49cb2
--- /dev/null
+++ b/pointprocesses/lecture/pointprocesses-slides.tex
@@ -0,0 +1,412 @@
+\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>
+{
+ \usetheme{Singapore}
+ \setbeamercovered{opaque}
+ \usecolortheme{tuebingen}
+ \setbeamertemplate{navigation symbols}{}
+ \usefonttheme{default}
+ \useoutertheme{infolines}
+ % \useoutertheme{miniframes}
+}
+
+%\AtBeginSection[]
+%{
+% \begin{frame}<beamer>
+% \begin{center}
+% \Huge \insertsectionhead
+% \end{center}
+% \end{frame}
+%}
+
+\setbeamertemplate{blocks}[rounded][shadow=true]
+\setcounter{tocdepth}{1}
+
+%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[english]{babel}
+\usepackage{amsmath}
+\usepackage{bm}
+\usepackage{pslatex} % nice font for pdf file
+%\usepackage{multimedia}
+
+\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}}
+
+%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{graphicx}
+\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}}
+
+%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{
+ basicstyle=\ttfamily,
+ 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,
+ captionpos=b,
+ xleftmargin=1em,
+ xrightmargin=1em,
+ aboveskip=10pt
+ }
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+
+\begin{frame}[plain]
+ \frametitle{}
+ \vspace{-1cm}
+ \titlepage % erzeugt Titelseite
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
diff --git a/pointprocesses/lecture/pointprocesses.tex b/pointprocesses/lecture/pointprocesses.tex
index ff49cb2..58f3cd1 100644
--- a/pointprocesses/lecture/pointprocesses.tex
+++ b/pointprocesses/lecture/pointprocesses.tex
@@ -1,412 +1,168 @@
-\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>
-{
- \usetheme{Singapore}
- \setbeamercovered{opaque}
- \usecolortheme{tuebingen}
- \setbeamertemplate{navigation symbols}{}
- \usefonttheme{default}
- \useoutertheme{infolines}
- % \useoutertheme{miniframes}
-}
-
-%\AtBeginSection[]
-%{
-% \begin{frame}<beamer>
-% \begin{center}
-% \Huge \insertsectionhead
-% \end{center}
-% \end{frame}
-%}
-
-\setbeamertemplate{blocks}[rounded][shadow=true]
-\setcounter{tocdepth}{1}
-
-%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\usepackage[english]{babel}
-\usepackage{amsmath}
-\usepackage{bm}
-\usepackage{pslatex} % nice font for pdf file
-%\usepackage{multimedia}
-
-\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}}
-
-%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\usepackage{graphicx}
-\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}}
-
-%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\usepackage{listings}
-\lstset{
- basicstyle=\ttfamily,
- 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,
- captionpos=b,
- xleftmargin=1em,
- xrightmargin=1em,
- aboveskip=10pt
- }
-
-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{document}
-
-\begin{frame}[plain]
- \frametitle{}
- \vspace{-1cm}
- \titlepage % erzeugt Titelseite
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\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}
+\chapter{\tr{Point processes}{Punktprozesse}}
-\begin{frame}
- \frametitle{Point process}
+\begin{figure}[t]
\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}
+ \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{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{Intervall Statistiken}
+
+\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}
+ \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\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}
+\section{\tr{Homogeneous Poisson process}{Homogener Poisson Prozess}}
-\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}
+\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).
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Non stationary}
-\subsection{Inhomogeneous Poisson process}
-\subsection{Firing rate}
-\subsection{Instantaneous rate}
-\subsection{Autocorrelation}
-\subsection{Crosscorrelation}
-\subsection{Joint PSTH}
+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{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:
+\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
\ No newline at end of file
diff --git a/pointprocesses/lecture/pointprocessscetchA.eps b/pointprocesses/lecture/pointprocessscetchA.eps
index 95a12ae..83dc344 100644
--- a/pointprocesses/lecture/pointprocessscetchA.eps
+++ b/pointprocesses/lecture/pointprocessscetchA.eps
@@ -1,7 +1,7 @@
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diff --git a/scientificcomputing-script.tex b/scientificcomputing-script.tex
index 306a84a..8cbde10 100644
--- a/scientificcomputing-script.tex
+++ b/scientificcomputing-script.tex
@@ -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}
diff --git a/statistics/lecture/Makefile b/statistics/lecture/Makefile
index 445913e..ec84405 100644
--- a/statistics/lecture/Makefile
+++ b/statistics/lecture/Makefile
@@ -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
diff --git a/statistics/lecture/boxwhisker.py b/statistics/lecture/boxwhisker.py
index 1209f7e..38a0dcb 100644
--- a/statistics/lecture/boxwhisker.py
+++ b/statistics/lecture/boxwhisker.py
@@ -43,5 +43,5 @@ ax.annotate('maximum',
ax.boxplot( x, whis=100.0 )
plt.tight_layout()
plt.savefig('boxwhisker.pdf')
-plt.show()
+#plt.show()
diff --git a/statistics/lecture/correlation.py b/statistics/lecture/correlation.py
index db317c7..2fa85d4 100644
--- a/statistics/lecture/correlation.py
+++ b/statistics/lecture/correlation.py
@@ -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()
diff --git a/statistics/lecture/diehistograms.py b/statistics/lecture/diehistograms.py
index d5e0380..a8832e0 100644
--- a/statistics/lecture/diehistograms.py
+++ b/statistics/lecture/diehistograms.py
@@ -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()
diff --git a/statistics/lecture/median.py b/statistics/lecture/median.py
index 2bf420c..a1231f9 100644
--- a/statistics/lecture/median.py
+++ b/statistics/lecture/median.py
@@ -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()
diff --git a/statistics/lecture/nonlincorrelation.py b/statistics/lecture/nonlincorrelation.py
index c0ca723..498d985 100644
--- a/statistics/lecture/nonlincorrelation.py
+++ b/statistics/lecture/nonlincorrelation.py
@@ -39,4 +39,4 @@ ax.scatter( x, z )
plt.tight_layout()
plt.savefig('nonlincorrelation.pdf')
-plt.show()
+#plt.show()
diff --git a/statistics/lecture/pdfhistogram.py b/statistics/lecture/pdfhistogram.py
index 039b524..a61460e 100644
--- a/statistics/lecture/pdfhistogram.py
+++ b/statistics/lecture/pdfhistogram.py
@@ -35,5 +35,5 @@ ax.hist(r, 20, normed=True, color='#FFCC00')
plt.tight_layout()
fig.savefig( 'pdfhistogram.pdf' )
-plt.show()
+#plt.show()
diff --git a/statistics/lecture/pdfprobabilities.py b/statistics/lecture/pdfprobabilities.py
index 6481da1..bcbaa09 100644
--- a/statistics/lecture/pdfprobabilities.py
+++ b/statistics/lecture/pdfprobabilities.py
@@ -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()
diff --git a/statistics/lecture/quartile.py b/statistics/lecture/quartile.py
index 34b9960..dbdd202 100644
--- a/statistics/lecture/quartile.py
+++ b/statistics/lecture/quartile.py
@@ -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()