381 lines
10 KiB
TeX
381 lines
10 KiB
TeX
\documentclass{beamer}
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\usepackage{xcolor}
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\usepackage{listings}
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\usepackage{pgf}
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%\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade}
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%\usepackage{multimedia}
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\usepackage[english]{babel}
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\usepackage{movie15}
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\usepackage[latin1]{inputenc}
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\usepackage{times}
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\usepackage{amsmath}
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\usepackage{bm}
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\usepackage[T1]{fontenc}
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\usepackage[scaled=.90]{helvet}
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\usepackage{scalefnt}
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\usepackage{tikz}
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\usepackage{ textcomp }
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\usepackage{soul}
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\usepackage{hyperref}
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\definecolor{lightblue}{rgb}{.7,.7,1.}
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\definecolor{mygreen}{rgb}{0,1.,0}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>
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{
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\usetheme{Singapore}
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\setbeamercovered{opaque}
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\usecolortheme{tuebingen}
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\setbeamertemplate{navigation symbols}{}
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\usefonttheme{default}
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\useoutertheme{infolines}
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% \useoutertheme{miniframes}
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}
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\AtBeginSection[]
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{
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\begin{frame}<beamer>
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\begin{center}
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\Huge \insertsectionhead
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\end{center}
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% \frametitle{\insertsectionhead}
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% \tableofcontents[currentsection,hideothersubsections]
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\end{frame}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
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\setbeamertemplate{blocks}[rounded][shadow=true]
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\title[]{Introduction to Scientific Computing -- \\
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Cross-Correlation, Spike--Triggered--Average and Reverse Reconstruction}
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\author[]{Jan Grewe\\Abteilung f\"ur Neuroethologie\\
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Universit\"at T\"ubingen}
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\institute[Introduction to Scientific Computing]{}
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\date{2014/10/13 - 2014/11/07}
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%\logo{\pgfuseimage{../../resources/UT_BM_Rot_RGB.pdf}}
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\subject{Einf\"uhrung in wissenschaftliche Datenverarbeitung}
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\vspace{1em}
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\titlegraphic{
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\includegraphics[width=0.5\linewidth]{../../resources/UT_WBMW_Rot_RGB}
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}
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%%%%%%%%%% configuration for code
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\lstset{
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basicstyle=\ttfamily,
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numbers=left,
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showstringspaces=false,
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language=Matlab,
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commentstyle=\itshape\color{darkgray},
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keywordstyle=\color{blue},
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stringstyle=\color{green},
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backgroundcolor=\color{blue!10},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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captionpos=b,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\mycite}[1]{
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\begin{flushright}
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\tiny \color{black!80} #1
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\end{flushright}
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}
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\input{../../latex/environments.tex}
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\makeatother
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\begin{document}
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\begin{frame}[plain]
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\frametitle{}
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\vspace{-1cm}
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\titlepage % erzeugt Titelseite
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\end{frame}
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\begin{frame}[plain]
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\frametitle{Rekapitulation}
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\begin{enumerate}
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\item PSTH\pause
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\end{enumerate}
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\end{frame}
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\begin{frame}
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\frametitle{Introduction to scientific computing}
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\frametitle{Menue}
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\begin{enumerate}
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\item Cross-correlation
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\item Estimation of the Spike--Triggered--Average -- STA
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\item Reverse reconstruction
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\end{enumerate}
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\end{frame}
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\begin{frame}[plain]
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\huge{1. Recapitulation: PSTH}
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\end{frame}
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\begin{frame}
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\frametitle{Recapitulation}
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\only<1>{
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\framesubtitle{Displaying the neuronal response over time - Rasterplot}
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\begin{figure}
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\centering
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\includegraphics[width=0.375\columnwidth]{images/rasterplot}
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\end{figure}
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}
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\only<2>{
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\framesubtitle{Displaying the neuronal response over time - PSTH}
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\begin{figure}
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\centering
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\includegraphics[width=0.5\columnwidth]{images/conv}
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\end{figure}
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}
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\end{frame}
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\begin{frame}[plain]
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\huge{2. Relating stimulus and response}
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\end{frame}
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\begin{frame}
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\frametitle{Relating stimulus and response}
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\framesubtitle{How can we relate the response to the stimulus?}
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\begin{figure}
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\centering
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\includegraphics[height=0.9\textheight]{images/conv_stim}
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\end{figure}
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\end{frame}
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\begin{frame}
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\frametitle{Relating stimulus and response}
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\framesubtitle{Cross--correlation}
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The cross - correlation for (time) discrete data is defined as:
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\begin{equation}
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Q_{rs}(\tau) = \frac{1}{2\cdot N-|\tau|} \displaystyle\sum^{T}_{\tau=-T}{r(t) \cdot s(t-\tau)} \cdot \Delta \tau
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\end{equation}
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\pause
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Don't worry, this is already implemented in MATLAB!
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Cross--correlation}
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\footnotesize
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\begin{lstlisting}
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[c,lags] = xcorr(stimulus, response, 10000, 'unbiased');
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fig = figure();
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set(fig, 'PaperUnits', 'centimeters');
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set(fig, 'PaperSize', [11.7 9.0]);
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set(fig, 'PaperPosition',[0.0 0.0 11.7 9.0]);
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set(fig,'Color', 'white')
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plot(lags/sample_rate, c)
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xlabel('lag [s]')
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ylabel('correlation')
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\end{lstlisting}
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\end{frame}
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\begin{frame}
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\frametitle{Relating stimulus and response}
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\framesubtitle{Cross--correlation}
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\begin{figure}
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\includegraphics[width=0.5\linewidth]{images/correlation}
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\end{figure}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Cross--correlation - Exercises}
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\begin{enumerate}
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\item calculate the cross-correlation between two vectors of random
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numbers.
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\item Calculate the cross-correlation between one of these vectors
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and itself (auto-correlation).
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\item Calculate the cross-correlation between one vector and a
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time-shifted version of itself (use \verb+circshift+ to do this).
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\end{enumerate}
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\textbf{Note:} Select max\_lag to be less than 10\% of the length of
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your vectors!
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\begin{enumerate}
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\item Create the cross correlation of the p-unit data and stimulus.
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\item \textbf{Note:} you have to convert the spike\_times to a PSTH!
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\item Find out the position of the correlation peak.
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\item What does this tell you?
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\end{enumerate}
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\end{frame}
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\begin{frame}[plain]
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\huge{2. Spike--Triggered--Average --- STA}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Spike--Triggered--Average --- STA}
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The \textbf{STA} is the average stimulus that led to a spike in the
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neuron.
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\begin{equation}
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STA(\tau) = \left\langle \frac{1}{n} \displaystyle\sum^{i=1}_{n}{s(t_i - \tau)} \right\rangle
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\end{equation}
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Sadly, we need to implement this ourselves.\newline\newline\pause
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\vspace{1em}
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\noindent
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\textbf{Algorithm:}
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\begin{enumerate}
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\item For each spike a snippet is taken from the respective stimulus.
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\item The snippets are averaged.
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\end{enumerate}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Spike--Triggered--Average -- STA}
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\vspace{-1em}
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\begin{figure}
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\centering
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\includegraphics[width=0.65\columnwidth]{images/sta}
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\end{figure}
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\pause
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\vspace{-0.5em}
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What does the \textbf{STA} tell us?
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\begin{enumerate}
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\item Is there a relation between stimulus and response? \pause
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\item Is there a lag between them and how large is it? \pause
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\item How far in the past does a neuron encode? \pause
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\item Can it see into the future?
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\end{enumerate}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{STA -- Exercises}
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\textbf{Exercise:}
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\begin{enumerate}
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\item Write a function \verb+sta(x, y, count, sample_rate)+ that
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takes the stimulus (x), the response (y, as spike times), the
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number (count) of sampling points it should cut out from the
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stimulus and the sampling\_rate to convert from times to
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indices.
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\item \textbf{Beware:} sometimes the spike\_time may be too close
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to the beginning or the end of the stimulus to cut out enough
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data.
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\item Calculate the STA for P-unt data.
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\end{enumerate}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Spike--Triggered--Average -- STA}
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What does the \textbf{STA} tell us?
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\begin{figure}
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\centering
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\includegraphics[width=0.25\columnwidth]{images/sta}
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\end{figure}
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\begin{enumerate}
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\item Is there a relation between stimulus and response?\pause
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\item Is there a lag between them and how large is it?\pause
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\item How far in the past does a neuron encode?\pause
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\item Can it see into the future?
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\end{enumerate}
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\end{frame}
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\begin{frame}[plain]
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\huge{3. Reverse reconstruction using the \textbf{STA}}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Reverse reconstruction using the \textbf{STA}}
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\textbf{Basic idea:}\newline
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\begin{itemize}
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\item The \textbf{STA} is the average stimlus that led to a spike. \pause
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\item The other way round: Whenever we observe a spike, the stimulus
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was likely to have looked like the \textbf{STA}.\pause
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\item Thus, we should be able to reconstruct how the stimulus looked
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like to evoke the observed response.
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Relating stimulus and response}
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\framesubtitle{Reverse reconstruction using the \textbf{STA}}
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\textbf{Algorithm:}\newline
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\begin{enumerate}
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\item Estimate the \textbf{STA}. \pause
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\item Create a binary representation of the spike train (vector of
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\verb+zeros+ in which each \textbf{1} signals the occurence of a
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spike).\pause
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\item Use the STA as a Kernel (compare to the spike convolution
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method to create the PSTH) and convolve (\verb+conv+) it with the
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\textbf{STA}.
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\end{enumerate}
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\textbf{Exercise:}
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\begin{enumerate}
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\item Reconstruct the stimulus from the P-unit data.
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\item Plot original and reconstructed stimulus into the same plot.
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\item Are they similar? In which instances are they different?
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\end{enumerate}
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\end{frame}
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\end{document}
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\begin{frame}[plain]
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\huge{4. Spike--Triggered--Covariance \textbf{STC}}
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\end{frame}
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\begin{frame}
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\frametitle{Relating stimulus and response}
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\framesubtitle{Spike--Triggered--Covariance \textbf{STC}}
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\textbf{Problem:}
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\begin{itemize}
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\item The \textbf{STA} is a linear filter.
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\item It can only reveal linear relationships between stimulus and response.\pause
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\end{itemize}
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\vspace{1em}
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Further relations can be recovered using the Spike-triggered-covariance.\pause
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\begin{equation}
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STC = \frac{1}{N-1} \cdot \displaystyle\sum_{n=1}^{N}[\overrightarrow{s}(t_n) - STA]
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[\overrightarrow{s}(t_n) - STA]^T
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\end{equation}
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where:\\ $N$ is the total number of spikes,
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$\overrightarrow{s}(t_n)$ the stimulus snippet for the $n^{th}$
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spike, and $STA$ the Spike-Triggered-Average.
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\end{frame}
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\begin{frame}
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\frametitle{Relating stimulus and response}
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\framesubtitle{Spike--Triggered--Covariance}
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\begin{itemize}
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\item
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\end{itemize}
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\end{frame}
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\end{document}
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