New exercise for point processes

pointprocesses/code/counthist.m

@@ -18,18 +18,17 @@ function [ counts, bins ] = counthist( spikes, w )
     end
     % histogram of spike counts:
     maxn = max( n );
-    [counts, bins ] = hist( n, 0:1:maxn+1 );
+    [counts, bins ] = hist( n, 0:1:maxn+10 );
     counts = counts / sum( counts );
     if nargout == 0
         bar( bins, counts );
         hold on;
         % Poisson distribution:
         rate = mean( r );
-        x = 0:1:20;
+        x = 0:1:maxn+10;
         l = rate*w;
         y = l.^x.*exp(-l)./factorial(x);
         plot( x, y, 'r', 'LineWidth', 3 );
-        xlim( [ 0 20 ] );
         hold off;
         xlabel( 'counts k' );
         ylabel( 'P(k)' );

pointprocesses/code/isihist.m

@@ -24,9 +24,9 @@ function isihist( isis, binwidth )
     misi = mean( isis );
     sdisi = std( isis );
     disi = sdisi^2.0/2.0/misi^3;
-    text( 0.5, 0.6, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' )
+    text( 0.95, 0.8, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized', 'HorizontalAlignment', 'right' )
-    text( 0.5, 0.5, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' )
+    text( 0.95, 0.7, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized', 'HorizontalAlignment', 'right' )
-    text( 0.5, 0.4, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' )
+    text( 0.95, 0.6, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized', 'HorizontalAlignment', 'right' )
     %text( 0.5, 0.3, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' )
 end
 

pointprocesses/code/lifadaptspikes.m

@@ -3,8 +3,8 @@ function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr
 % with an adaptation current
 % trials: the number of trials to be generated
 % input: the stimulus either as a single value or as a vector
-% tmaxdt: in case of a single value stimulus the duration of a trial
+% tmaxdt: in case of a single value stimulus: the duration of a trial
-%         in case of a vector as a stimulus the time step
+%         in case of a vector as a stimulus: the time step
 % D: the strength of additive white noise
 % tauadapt: adaptation time constant
 % adaptincr: adaptation strength

pointprocesses/code/lifspikes.m

@@ -2,8 +2,8 @@ function spikes = lifspikes( trials, input, tmaxdt, D )
 % Generate spike times of a leaky integrate-and-fire neuron
 % trials: the number of trials to be generated
 % input: the stimulus either as a single value or as a vector
-% tmaxdt: in case of a single value stimulus the duration of a trial
+% tmaxdt: in case of a single value stimulus: the duration of a trial
-%         in case of a vector as a stimulus the time step
+%         in case of a vector as a stimulus: the time step
 % D: the strength of additive white noise
     
     tau = 0.01;

pointprocesses/code/lifspikesoustim.m

@@ -0,0 +1,20 @@
+function spikes = lifspikesoustim(trials, tmax, D, Iou, Dou, tauou )
+% Generate spike times of a leaky integrate-and-fire neuron with frozen
+% Ohrnstein-Uhlenbeck stimulus
+% trials: the number of trials to be generated
+% tmax: the duration of a trial
+% D: the strength of additive white noise
+% Iou: the mean input
+% Dou: noise strength of the frozen OU noise
+% tauou: time constant of the OU noise
+    
+    dt = 1e-4;
+    input = zeros(round(tmax/dt), 1);
+    n = 0.0;
+    noise = sqrt(2.0*Dou)*randn(length(input), 1)/sqrt(dt);
+    for i=1:length(noise)
+        n = n + ( - n + noise(i))*dt/tauou;
+        input(i) = Iou + n;
+    end
+    spikes = lifspikes(trials, input, dt, D );
+end

pointprocesses/code/plotcounthist.m

@@ -0,0 +1,24 @@
+w = 0.1;
+cmax = 8;
+pmax = 0.5;
+subplot(1, 3, 1);
+counthist(poissonspikes, w);
+xlim([0 cmax])
+set(gca, 'XTick', 0:2:cmax)
+ylim([0 pmax])
+title('Poisson');
+
+subplot(1, 3, 2);
+counthist(pifouspikes, w);
+xlim([0 cmax])
+set(gca, 'XTick', 0:2:cmax)
+ylim([0 pmax])
+title('PIF OU');
+
+subplot(1, 3, 3);
+counthist(lifadaptspikes, w);
+xlim([0 cmax])
+set(gca, 'XTick', 0:2:cmax)
+ylim([0 pmax])
+title('LIF adapt');
+savefigpdf(gcf, 'counthist.pdf', 20, 7);

pointprocesses/code/plotisih.m

@@ -0,0 +1,19 @@
+maxisi = 300.0;
+subplot(1, 3, 1);
+poissonisis = isis(poissonspikes);
+isihist(poissonisis, 0.001);
+xlim([0, maxisi])
+title('Poisson');
+
+subplot(1, 3, 2);
+pifouisis = isis(pifouspikes);
+isihist(pifouisis, 0.001);
+xlim([0, maxisi])
+title('PIF OU');
+
+subplot(1, 3, 3);
+lifadaptisis = isis(lifadaptspikes);
+isihist(lifadaptisis, 0.001);
+xlim([0, maxisi])
+title('LIF adapt');
+savefigpdf(gcf, 'isihist.pdf', 20, 7);

pointprocesses/code/plotserialcorr.m

@@ -0,0 +1,17 @@
+maxlag = 10;
+rrange = [-0.5, 1.05];
+subplot(1, 3, 1);
+isiserialcorr(poissonisis, maxlag);
+ylim(rrange)
+title('Poisson');
+
+subplot(1, 3, 2);
+isiserialcorr(pifouisis, maxlag);
+ylim(rrange)
+title('PIF OU');
+
+subplot(1, 3, 3);
+isiserialcorr(lifadaptisis, maxlag);
+ylim(rrange)
+title('LIF adapt');
+savefigpdf(gcf, 'serialcorr.pdf', 20, 7);

pointprocesses/code/plotspikeraster.m

@@ -0,0 +1,13 @@
+subplot(1, 3, 1);
+spikeraster(poissonspikes, 1.0);
+title('Poisson');
+
+subplot(1, 3, 2);
+spikeraster(pifouspikes, 1.0);
+title('PIF OU');
+
+subplot(1, 3, 3);
+spikeraster(lifadaptspikes, 1.0);
+title('LIF adapt');
+
+savefigpdf(gcf, 'spikeraster.pdf', 15, 5);

pointprocesses/code/poissonspikes.m

@@ -14,7 +14,7 @@ function spikes = poissonspikes( trials, rate, tmax )
     end
     spikes = cell(trials, 1);
     for k=1:trials
-        x = rand( 1, round(tmax/dt) ); % uniform random numbers for each bin
+        x = rand(round(tmax/dt), 1); % uniform random numbers for each bin
         spikes{k} = find(x < p) * dt;
     end
 end

pointprocesses/code/psth.m

@@ -0,0 +1,14 @@
+function p = psth(spikes, dt, tmax)
+% plots a PSTH of the spikes with binwidth dt
+    t = 0.0:dt:tmax+dt;
+    p = zeros(1, length(t));
+    for k=1:length(spikes)
+        times = spikes{k};
+        [h, b] = hist(times, t);
+        p = p + h;
+    end
+    p = p/length(spikes)/dt;
+    t(end) = [];
+    p(end) = [];
+    plot(t, p);
+end

pointprocesses/code/spikeraster.m

@@ -1,15 +1,24 @@
-function spikeraster( spikes )
+function spikeraster(spikes, tmax)
 % Display a spike raster of the spike times given in spikes.
 % spikes: a cell array of vectors of spike times
+% tmax: plot spike raster upto tmax seconds
 
 ntrials = length(spikes);
 for k = 1:ntrials
-    times = 1000.0*spikes{k};  % conversion to ms
+    times = spikes{k};
+    times = times(times<tmax);
+    if tmax < 1.5
+        times = 1000.0*times; % conversion to ms
+    end
     for i = 1:length( times )
       line([times(i) times(i)],[k-0.4 k+0.4], 'Color', 'k' ); 
     end
 end
+if tmax < 1.5
     xlabel( 'Time [ms]' );
+else
+    xlabel( 'Time [s]' );
+end
 ylabel( 'Trials');
 ylim( [ 0.3 ntrials+0.7 ] )
 

pointprocesses/code/spikerate.m

@@ -0,0 +1,12 @@
+function r = spikerate(spikes, duration)
+% returns the average spike rate of the spikes
+% for the first duration seconds
+% spikes: a cell array of vectors of spike times
+
+    rates = zeros(length(spikes),1);
+    for k = 1:length(spikes)
+        times = spikes{k};
+        rates(k) = sum(times<duration)/duration;
+    end
+    r = mean(rates);
+end

pointprocesses/exercises/pointprocesses01.tex

@@ -11,11 +11,11 @@
 \usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
 \pagestyle{headandfoot}
 \ifprintanswers
-\newcommand{\stitle}{: L\"osungen}
+\newcommand{\stitle}{L\"osungen}
 \else
-\newcommand{\stitle}{}
+\newcommand{\stitle}{\"Ubung}
 \fi
-\header{{\bfseries\large \"Ubung 6\stitle}}{{\bfseries\large Statistik}}{{\bfseries\large 27. Oktober, 2015}}
+\header{{\bfseries\large \stitle}}{{\bfseries\large Punktprozesse}}{{\bfseries\large 27. Oktober, 2015}}
 \firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
 jan.benda@uni-tuebingen.de}
 \runningfooter{}{\thepage}{}
@@ -89,111 +89,96 @@ jan.benda@uni-tuebingen.de}
 \begin{questions}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\question \qt{Homogeneous Poisson process}
+  \question \qt{Statistik von Spiketrains}
-We use the Poisson process to generate spike trains on which we can test and imrpove some 
+  In Ilias findet ihr die Dateien \code{poisson.mat},
-standard analysis functions.
+  \code{pifou.mat}, und \code{lifadapt.mat}.  Jede dieser Dateien
-
+  enth\"alt mehrere Trials von Spiketrains von einer bestimmten Art
-A homogeneous Poisson process of rate $\lambda$ (measured in Hertz) is a point process
+  von Neuron. Die Spikezeiten sind in Sekunden gemessen.
-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 
+  Mit den folgenden Aufgaben wollen wir die Statistik der Spiketrains
-\[ P = \lambda \cdot \Delta t \]
+  der drei Neurone miteinander vergleichen.
-for sufficiently small $\Delta t$.
   \begin{parts}
-  
+    \part Lade die Spiketrains aus den drei Dateien. Achte darauf, dass sie verschiedene
-  \part Write a function that generates $n$ homogeneous Poisson spike trains of a given duration $T_{max}$
+    Variablennamen bekommen.
-  with rate $\lambda$.
     \begin{solution}
-    \lstinputlisting{hompoissonspikes.m}
+      \begin{lstlisting}
+        clear all
+        load poisson.mat
+        whos
+        poissonspikes = spikes;
+        load pifou.mat;
+        pifouspikes = spikes;
+        load lifadapt.mat;
+        lifadaptspikes = spikes;
+        clear spikes;
+      \end{lstlisting}
     \end{solution}
     
-  \part Using this function, generate a few trials and display them in a raster plot.
+    \part Schreibe eine Funktion, die die Spikezeiten der ersten
+    \code{tmax} Sekunden in einem Rasterplot visualisiert.  In jeder
+    Zeile des Rasterplots wird ein Spiketrain dargestellt. Jeder
+    einzelne Spike wird als senkrechte Linie zu der Zeit des
+    Auftretens des Spikes geplottet. Benutze die Funktion, um die
+    Spikeraster der ersten 1\,s der drei Neurone zu plotten.
     \begin{solution}
       \lstinputlisting{../code/spikeraster.m}
-    \begin{lstlisting}
+      \lstinputlisting{../code/plotspikeraster.m}
-      spikes = hompoissonspikes( 10, 100.0, 0.5 );
-      spikeraster( spikes )
-    \end{lstlisting}
       \mbox{}\\[-3ex]
-    \colorbox{white}{\includegraphics[width=0.7\textwidth]{poissonraster100hz}}
+      \colorbox{white}{\includegraphics[width=1\textwidth]{spikeraster}}
     \end{solution}
   
-  \part Write a function that extracts a single vector of interspike intervals
+    \part Schreibe eine Funktion, die einen einzigen Vektor mit den Interspike-Intervallen
-  from the spike times returned by the first function.
+    aller Trials von Spikezeiten zur\"uckgibt.
     \begin{solution}
       \lstinputlisting{../code/isis.m}
     \end{solution}
     
-  \part Write a function that plots the interspike-interval histogram
+    \part Schreibe eine Funktion, die ein normiertes Histogramm aus
-  from a vector of interspike intervals. The function should also
+    einem Vektor von Interspike-Intervallen, gegeben in Sekunden,
-  compute the mean, the standard deviation, and the CV of the intervals
+    berechnet und dieses mit richtiger Achsenbeschriftung plottet.  Die
-  and display the values in the plot.
+    Interspike-Intervalle sollen dabei in Millisekunden angegeben
-  \begin{solution}
+    werden. Die Funktion soll ausserdem den Mittelwert, die Standardabweichung,
-    \lstinputlisting{../code/isihist.m}
+    und den Variationskoeffizienten der Interspike Intervalle berechnen
-  \end{solution}
+    und diese im Plot mit angeben.
-  
-  \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
+    Benutze diese und die vorherige Funktion, um die Interspike-Intervall Verteilung
-  \[ p(T) = \lambda e^{-\lambda T} \]
+    der drei Neurone zu vergleichen.
-  for various rates $\lambda$.
     \begin{solution}
-    \lstinputlisting{hompoissonisih.m}
+      \lstinputlisting{../code/isihist.m}
-    \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}}
+      \lstinputlisting{../code/plotisih.m}
-    \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}}
+      \mbox{}\\[-3ex]
-  \end{solution}
+      \colorbox{white}{\includegraphics[width=1\textwidth]{isihist}}
-  
-  \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
+    \part Schreibe eine Funktion, die die Seriellen Korrelationen der
-  for a range of lags.
+    Interspike Intervalle f\"ur lags bis zu \code{maxlag} berechnet
-  The serial correlations $\rho_k$ at lag $k$ are defined as
+    und plottet.  Die Seriellen Korrelationen $\rho_k$ f\"ur lag $k$
-  \[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
+    der Interspike Intervalle $T_i$ sind wie folgt definiert:
-  Use this function to show that interspike intervals of Poisson spikes are independent.
+    \[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i -
+      \langle T \rangle) \rangle}{\langle (T_i - \langle T
+      \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm
+        var}(T_i)} = {\rm corrcoef}(T_{i+k}, T_i) \] Benutze dies Funktion,
+    um die Interspike Intervall Korrelationen der drei Neurone zu
+    vergleichen.
     \begin{solution}
       \lstinputlisting{../code/isiserialcorr.m}
-    \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonserial100hz}}
+      \lstinputlisting{../code/plotserialcorr.m}
+      \colorbox{white}{\includegraphics[width=1\textwidth]{serialcorr}}
     \end{solution}
   
-  \part Write a function that generates from spike times 
+    \part Schreibe eine Funktion, die aus Spikezeiten
-  a histogram of spike counts in a count window of given duration $W$.
+    Histogramme aus der Anzahl von Spikes, die in Fenstern gegebener L\"ange $W$
-  The function should also plot the Poisson distribution
+    gez\"ahlt werden, erzeugt und plottet. Zus\"atzlich soll die Funktion
-  \[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
+    die Poisson-Verteilung
-  for the rate $\lambda$ determined from the spike trains.
+    \[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \] mit der Rate
+    $\lambda$, die aus den Daten bestimmt werden kann, mit zu dem
+    Histogramm hineinzeichen.
     \begin{solution}
       \lstinputlisting{../code/counthist.m}
-    \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}}
+      \lstinputlisting{../code/plotcounthist.m}
-    \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}}
+      \colorbox{white}{\includegraphics[width=1\textwidth]{counthist}}
     \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}
   

pointprocesses/exercises/pointprocesses02.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}