diff --git a/header.tex b/header.tex
index 2f2360d..73288b8 100644
--- a/header.tex
+++ b/header.tex
@@ -222,11 +222,26 @@
   {\vspace{-2ex}\lstset{#1}\noindent\minipage[t]{1\linewidth}}%
   {\endminipage}
 
-%%%%% english, german, code and file terms: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%% english and german terms: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \usepackage{ifthen}
 
+% \enterm[en-index]{term}: Typeset term and add it to the index of english terms
+%
+% \determ[de-index]{term}: Typeset term and add it to the index of german terms
+%
+% \entermde[en-index]{de-index}{term}: Typeset term and add it to the index of english terms
+%                                      and de-index to the index of german terms
+%
+% how to specificy an index:
+% \enterm{term}                                 - just put term into the index
+% \enterm[en-index]{term}                       - typeset term and put en-index into the index
+% \enterm[statistics!mean]{term}                - mean is a subentry of statistics
+% \enterm[statistics!average|see{statistics!mean}]{term} - cross reference to statistics mean
+% \enterm[statistics@\textbf{statistics}]{term} - put index at statistics but use 
+%                                               \textbf{statistics} for typesetting in the index
+
 % \enterm[english index entry]{<english term>}
-% typeset the term in italics and add it (or the optional argument) to
+% typeset the term in italics and add it (or rather the optional argument) to
 % the english index.
 \newcommand{\enterm}[2][]{\textit{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[enterm]{#2}}{\protect\sindex[enterm]{#1}}}
 
diff --git a/pointprocesses/lecture/isihexamples.py b/pointprocesses/lecture/isihexamples.py
index 9d0f7c0..c3e2d87 100644
--- a/pointprocesses/lecture/isihexamples.py
+++ b/pointprocesses/lecture/isihexamples.py
@@ -93,6 +93,8 @@ def plot_hom_isih(ax):
     ax.set_ylim(0.0, 31.0)
     ax.set_xticks(np.arange(0.0, 151.0, 50.0))
     ax.set_yticks(np.arange(0.0, 31.0, 10.0))
+    tt = np.linspace(0.0, 0.15, 100)
+    ax.plot(1000.0*tt, rate*np.exp(-rate*tt), **lsB)
     plotisih(ax, isis(homspikes), 0.005)
 
     
diff --git a/pointprocesses/lecture/pointprocesses.tex b/pointprocesses/lecture/pointprocesses.tex
index 65947c3..f1a57f9 100644
--- a/pointprocesses/lecture/pointprocesses.tex
+++ b/pointprocesses/lecture/pointprocesses.tex
@@ -34,7 +34,7 @@ process]{Punktprozess}{point processes}.
     trial. Shown is a stationary point process (homogeneous point
     process with a rate $\lambda=20$\;Hz, left) and an non-stationary
     point process with a rate that varies in time (noisy perfect
-    integrate-and-fire neuron driven by Ohrnstein-Uhlenbeck noise with
+    integrate-and-fire neuron driven by Ornstein-Uhlenbeck noise with
     a time-constant $\tau=100$\,ms, right).}
 \end{figure}
 
@@ -87,190 +87,6 @@ certain time window $n_i$ (\figref{pointprocesssketchfig}).
   are stored as vectors of times within a cell array.
 \end{exercise}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-\section{Interval statistics}
-
-The intervals $T_i=t_{i+1}-t_i$ between successive events are real
-positive numbers. In the context of action potentials they are
-referred to as \entermde{Interspikeintervalle}{interspike
-  intervals}. The statistics of interspike intervals are described
-using common measures for describing the statistics of stochastic
-real-valued variables:
-
-\begin{figure}[t]
-  \includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
-  \titlecaption{\label{isihexamplesfig}Interspike-interval
-    histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
-\end{figure}
-
-\begin{exercise}{isis.m}{}
-  Implement a function \varcode{isis()} that calculates the interspike
-  intervals from several spike trains. The function should return a
-  single vector of intervals. The spike times (in seconds) of each
-  trial are stored as vectors within a cell-array.
-\end{exercise}
-
-%\subsection{First order interval statistics}
-\begin{itemize}
-\item Probability density $p(T)$ of the intervals $T$
-  (\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \; dT
-  = 1$.
-\item Average interval: $\mu_{ISI} = \langle T \rangle =
-  \frac{1}{n}\sum\limits_{i=1}^n T_i$.
-\item Standard deviation of the interspike intervals: $\sigma_{ISI} = \sqrt{\langle (T - \langle T
-    \rangle)^2 \rangle}$\vspace{1ex}
-\item \entermde[coefficient of variation]{Variationskoeffizient}{Coefficient of variation}:
-  $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$.
-\item \entermde[diffusion coefficient]{Diffusionskoeffizient}{Diffusion coefficient}: $D_{ISI} =
-  \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
-\end{itemize}
-
-\begin{exercise}{isihist.m}{}
-  Implement a function \varcode{isiHist()} that calculates the normalized
-  interspike interval histogram. The function should take two input
-  arguments; (i) a vector of interspike intervals and (ii) the width
-  of the bins used for the histogram. It further returns the
-  probability density as well as the centers of the bins.
-\end{exercise}
-
-\begin{exercise}{plotisihist.m}{}
-  Implement a function that takes the return values of
-  \varcode{isiHist()} as input arguments and then plots the data. The
-  plot should show the histogram with the x-axis scaled to
-  milliseconds and should be annotated with the average ISI, the
-  standard deviation and the coefficient of variation.
-\end{exercise}
-
-\subsection{Interval correlations}
-So called \entermde[return map]{return map}{return maps} are used to
-illustrate interdependencies between successive interspike
-intervals. The return map plots the delayed interval $T_{i+k}$ against
-the interval $T_i$. The parameter $k$ is called the \enterm{lag}
-(\determ{Verz\"ogerung}) $k$. Stationary and non-stationary return
-maps are distinctly different \figref{returnmapfig}.
-
-\begin{figure}[tp]
-  \includegraphics[width=1\textwidth]{serialcorrexamples}
-  \titlecaption{\label{returnmapfig}Interspike-interval
-    correlations}{of the spike trains shown in
-    \figref{rasterexamplesfig}. Upper panels show the return maps and
-    lower panels the serial correlations of successive intervals
-    separated by lag $k$. All the interspike intervals of the
-    stationary spike trains are independent of each other --- this is
-    a so called \enterm{renewal process}
-    (\determ{Erneuerungsprozess}). In contrast, the ones of the
-    non-stationary spike trains show positive correlations that decay
-    for larger lags.  The positive correlations in this example are
-    caused by a common stimulus that slowly increases and decreases
-    the mean firing rate of the spike trains.}
-\end{figure}
-
-Such dependencies can be further quantified using the
-\entermde[correlation!serial]{Korrelation!serielle}{serial
-  correlations} \figref{returnmapfig}. The serial correlation is the
-correlation coefficient of the intervals $T_i$ and the intervals
-delayed by the lag $T_{i+k}$:
-\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} 
-= {\rm corr}(T_{i+k}, T_i) \] The resulting correlation coefficient
-$\rho_k$ is usually plotted against the lag $k$
-\figref{returnmapfig}. $\rho_0=1$ is the correlation of each interval
-with itself and is always 1.
-
-\begin{exercise}{isiserialcorr.m}{}
-  Implement a function \varcode{isiserialcorr()} that takes a vector of
-  interspike intervals as input argument and calculates the serial
-  correlation. The function should further plot the serial
-  correlation.
-\end{exercise}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Count statistics}
-
-\begin{figure}[t]
-  \includegraphics{countexamples}
-  \titlecaption{\label{countstatsfig}Count statistics.}{Probability
-    distributions of counting $k$ events $k$ (blue) within windows of
-    20\,ms (left) or 200\,ms duration (right) of a homogeneous Poisson
-    spike train with a rate of 20\,Hz
-    (\figref{rasterexamplesfig}). For Poisson spike trains these
-    distributions are given by Poisson distributions (red).}
-\end{figure}
-
-The most commonly used measure for characterizing spike trains is the
-\enterm[firing rate!average]{average firing rate}. The firing rate $r$
-is the average number of spikes counted within some time interval $W$
-\begin{equation}
-  \label{firingrate}
-  r = \frac{\langle n \rangle}{W}
-\end{equation}
-and is neasured in Hertz. The average of the spike counts is taken
-over trials. For stationary spike trains (no change in statistics, in
-particular the firing rate, over time), the firing rate based on the
-spike count equals the inverse average interspike interval
-$1/\mu_{ISI}$.
-
-The firing rate based on an averaged spike counts is one example of
-many statistics based on event counts. Stationary spike trains can be
-split into many segments $i$, each of duration $W$, and the number of
-events $n_i$ in each of the segments can be counted. The integer event
-counts can be quantified in the usual ways:
-\begin{itemize}
-\item Histogram of the counts $n_i$ (\figref{countstatsfig}).
-\item Average number of counts: $\mu_n = \langle n \rangle$.
-\item Variance of counts:
-  $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
-\end{itemize}
-Because spike counts are unitless and positive numbers, the
-\begin{itemize}
-\item \entermde{Fano Faktor}{Fano factor} (variance of counts divided
-  by average count): $F = \frac{\sigma_n^2}{\mu_n}$.
-\end{itemize}
-is an additional measure quantifying event counts.
-
-Note that all of these statistics depend on the chosen window length
-$W$. The average spike count, for example, grows linearly with $W$ for
-sufficiently large time windows: $\langle n \rangle = r W$,
-\eqnref{firingrate}. Doubling the counting window doubles the spike
-count. As does the spike-count variance (\figref{fanofig}). At smaller
-time windows the statistics of the event counts might depend on the
-particular duration of the counting window. There might be an optimal
-time window for which the variance of the spike count is minimal. The
-Fano factor plotted as a function of the time window illustrates such
-properties of point processes (\figref{fanofig}).
-
-This also has consequences for information transmission in neural
-systems. The lower the variance in spike count relative to the
-averaged count, the higher the signal-to-noise ratio at which
-information encoded in the mean spike count is transmitted.
-
-\begin{figure}[t]
-  \includegraphics{fanoexamples}
-  \titlecaption{\label{fanofig}
-    Count variance and Fano factor.}{Variance of event counts as a
-    function of mean counts obtained by varying the duration of the
-    count window (left). Dividing the count variance by the respective
-    mean results in the Fano factor that can be plotted as a function
-    of the count window (right). For Poisson spike trains the variance
-    always equals the mean counts and consequently the Fano factor
-    equals one irrespective of the count window (top). A spike train
-    with positive correlations between interspike intervals (caused by
-    Ohrnstein-Uhlenbeck noise) has a minimum in the Fano factor, that
-    is an analysis window for which the relative count variance is
-    minimal somewhere close to the correlation time scale of the
-    interspike intervals (bottom).}
-\end{figure}
-
-\begin{exercise}{counthist.m}{}
-  Implement a function \varcode{counthist()} that calculates and plots
-  the distribution of spike counts observed in a certain time
-  window. The function should take two input arguments: (i) a
-  cell-array of vectors containing the spike times in seconds observed
-  in a number of trials, and (ii) the duration of the time window that
-  is used to evaluate the counts.
-\end{exercise}
-
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Homogeneous Poisson process}
 
@@ -303,45 +119,6 @@ In an \entermde[Poisson process!inhomogeneous]{Poissonprozess!inhomogener}{inhom
   \eqnref{hompoissonprob} to generate the event times.
 \end{exercise}
 
-\begin{figure}[t]
-  \includegraphics[width=1\textwidth]{poissonraster100hz}
-  \titlecaption{\label{hompoissonfig}Rasterplot of spikes of a
-    homogeneous Poisson process with a rate $\lambda=100$\,Hz.}{}
-\end{figure}
-
-\begin{figure}[t]
-  \includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
-  \includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
-  \titlecaption{\label{hompoissonisihfig}Distribution of interspike
-    intervals of two Poisson processes.}{The processes differ in their
-    rate (left: $\lambda=20$\,Hz, right: $\lambda=100$\,Hz). The red
-    lines indicate the corresponding exponential interval distribution
-    \eqnref{poissonintervals}.}
-\end{figure}
-
-The homogeneous Poisson process has the following properties:
-\begin{itemize}
-\item Intervals $T$ are exponentially distributed (\figref{hompoissonisihfig}):
-  \begin{equation}
-    \label{poissonintervals}
-    p(T) = \lambda e^{-\lambda T} \; .
-  \end{equation}
-\item The average interval is $\mu_{ISI} = \frac{1}{\lambda}$ .
-\item The variance of the intervals is $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$ .
-\item Thus, the coefficient of variation is always $CV_{ISI} = 1$ .
-\item The serial correlation is $\rho_k =0$ for $k>0$, since the
-  occurrence of an event is independent of all previous events. Such a
-  process is also called a \enterm{renewal process} (\determ{Erneuerungsprozess}).
-\item The number of events $k$ within a temporal window of duration
-  $W$ is Poisson distributed:
-\begin{equation}
-  \label{poissoncounts}
-  P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!}
-\end{equation}
-(\figref{hompoissoncountfig})
-\item The Fano Faktor is always $F=1$ .
-\end{itemize}
-
 \begin{exercise}{hompoissonspikes.m}{}
   Implement a function \varcode{hompoissonspikes()} that uses a
   homogeneous Poisson process to generate spike events at a given rate
@@ -353,16 +130,272 @@ The homogeneous Poisson process has the following properties:
   times.
 \end{exercise}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+\section{Interval statistics}
+
+The intervals $T_i=t_{i+1}-t_i$ between successive events are real
+positive numbers. In the context of action potentials they are
+referred to as \entermde[interspike
+interval]{Interspikeintervall}{interspike intervals}, in short
+\entermde[ISI|see{interspike
+  interval}]{ISI|see{Interspikeintervall}}{ISI}s. The statistics of
+interspike intervals are described using common measures for
+describing the statistics of real-valued variables:
+
 \begin{figure}[t]
-  \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
-  \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
-  \titlecaption{\label{hompoissoncountfig}Distribution of counts of a
-    Poisson spike train.}{The count statistics was generated for two
-    different windows of width $W=10$\,ms (left) and width $W=100$\,ms
-    (right).  The red line illustrates the corresponding Poisson
-    distribution \eqnref{poissoncounts}.}
+  \includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
+  \titlecaption{\label{isihexamplesfig}Interspike-interval
+    histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
 \end{figure}
 
+\begin{exercise}{isis.m}{}
+  Implement a function \varcode{isis()} that calculates the interspike
+  intervals from several spike trains. The function should return a
+  single vector of intervals. The spike times (in seconds) of each
+  trial are stored as vectors within a cell-array.
+\end{exercise}
+
+\begin{itemize}
+\item Probability density $p(T)$ of the intervals $T$
+  (\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \;
+  dT = 1$. Commonly referred to as the \enterm[interspike
+  interval!histogram]{interspike interval histogram}. Its shape
+  reveals many interesting aspects like locking or bursting that
+  cannot be inferred from the mean or standard deviation. A particular
+  reference is the exponential distribution of intervals
+  \begin{equation}
+    \label{hompoissonexponential}
+    p_{exp}(T) = \lambda e^{-\lambda T}
+  \end{equation}
+  of a homogeneous Poisson spike train with rate $\lambda$.
+\item Mean interval: $\mu_{ISI} = \langle T \rangle =
+  \frac{1}{n}\sum\limits_{i=1}^n T_i$. The average time it takes from
+  one event to the next. The inverse of the mean interval is identical
+  with the mean rate $\lambda$ (number of events per time, see below)
+  of the process.
+\item Standard deviation of intervals: $\sigma_{ISI} = \sqrt{\langle
+    (T - \langle T \rangle)^2 \rangle}$. Periodically spiking neurons
+  have little variability in their intervals, whereas many cortical
+  neurons cover a wide range with their intervals. The standard
+  deviation of homogeneous Poisson spike trains, $\sigma_{ISI} =
+  \frac{1}{\lambda}$, also equals the inverse rate.  Whether the
+  standard deviation of intervals is low or high, however, is better
+  quantified by the
+\item \entermde[coefficient of
+  variation]{Variationskoeffizient}{Coefficient of variation}, the
+  standard deviation of the ISIs relative to their mean:
+  \begin{equation}
+    \label{cvisi}
+    CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}
+  \end{equation}
+  Homogeneous Poisson spike trains have an CV of exactly one. The
+  lower the CV the more regularly firing a neuron is firing. CVs
+  larger than one are also possible in spike trains with small
+  intervals separated by really long ones.
+%\item \entermde[diffusion coefficient]{Diffusionskoeffizient}{Diffusion coefficient}: $D_{ISI} =
+%  \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
+\end{itemize}
+
+\begin{exercise}{isihist.m}{}
+  Implement a function \varcode{isiHist()} that calculates the normalized
+  interspike interval histogram. The function should take two input
+  arguments; (i) a vector of interspike intervals and (ii) the width
+  of the bins used for the histogram. It further returns the
+  probability density as well as the centers of the bins.
+\end{exercise}
+
+\begin{exercise}{plotisihist.m}{}
+  Implement a function that takes the return values of
+  \varcode{isiHist()} as input arguments and then plots the data. The
+  plot should show the histogram with the x-axis scaled to
+  milliseconds and should be annotated with the average ISI, the
+  standard deviation and the coefficient of variation.
+\end{exercise}
+
+\subsection{Interval correlations}
+Intervals are not just numbers without an order, like weights of
+tigers.  Intervals are temporally ordered and there could be temporal
+structure in the sequence of intervals. For example, short intervals
+could be followed by more longer ones, and vice versa. Such
+dependencies in the sequence of intervals do not show up in the
+interval histogram.  We need additional measures to also quantify the
+temporal structure of the sequence of intervals.
+
+We can use the same techniques we know for visualizing and quantifying
+correlations in bivariate data sets, i.e. scatter plots and
+correlation coefficients. We form $(x,y)$ data pairs by taking the
+series of intervals $T_i$ as $x$-data values and pairing them with the
+$k$-th next intervals $T_{i+k}$ as $y$-data values. The parameter $k$
+is called \enterm{lag} (\determ{Verz\"ogerung}). For lag one we pair
+each interval with the next one. A \entermde[return map]{return
+  map}{Return map} illustrates dependencies between successive
+intervals by simply plotting the intervals $T_{i+k}$ against the
+intervals $T_i$ in a scatter plot (\figref{returnmapfig}). For Poisson
+spike trains there is no structure beyond the one expected from the
+exponential interspike interval distribution, hinting at neighboring
+interspike intervals being independent of each other. For the spike
+train based on an Ornstein-Uhlenbeck process the return map is more
+clustered along the diagonal, hinting at a positive correlation
+between succeeding intervals. That is, short intervals are more likely
+to be followed by short ones and long intervals more likely by long
+ones. This temporal structure was already clearly visible in the spike
+raster shown in \figref{rasterexamplesfig}.
+
+\begin{figure}[tp]
+  \includegraphics[width=1\textwidth]{serialcorrexamples}
+  \titlecaption{\label{returnmapfig}Interspike-interval
+    correlations}{of the spike trains shown in
+    \figref{rasterexamplesfig}. Upper panels show the return maps and
+    lower panels the serial correlations of successive intervals
+    separated by lag $k$. All the interspike intervals of the
+    stationary spike trains are independent of each other --- this is
+    a so called \enterm{renewal process}
+    (\determ{Erneuerungsprozess}). In contrast, the ones of the
+    non-stationary spike trains show positive correlations that decay
+    for larger lags.  The positive correlations in this example are
+    caused by a common stimulus that slowly increases and decreases
+    the mean firing rate of the spike trains.}
+\end{figure}
+
+Such dependencies can be further quantified by
+\entermde[correlation!serial]{Korrelation!serielle}{serial
+  correlations}. These are the correlation coefficients between the
+intervals $T_{i+k}$ and $T_i$ in dependence on lag $k$:
+\begin{equation}
+  \label{serialcorrelation}
+  \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} 
+= {\rm corr}(T_{i+k}, T_i)
+\end{equation}
+The serial correlations $\rho_k$ are usually plotted against the lag
+$k$ for a range small range of lags
+(\figref{returnmapfig}). $\rho_0=1$ is the correlation of each
+interval with itself and always equals one.
+
+If the serial correlations all equal zero, $\rho_k =0$ for $k>0$, then
+the length of an interval is independent of all the previous
+ones. Such a process is a \enterm{renewal process}
+(\determ{Erneuerungsprozess}). Each event, each action potential,
+erases the history. The occurrence of the next event is independent of
+what happened before. To a first approximation an action potential
+erases all information about the past from the membrane voltage and
+thus spike trains may approximate renewal processes.
+
+However, other variables like the intracellular calcium concentration
+or the states of slowly switching ion channels may carry information
+from one interspike interval to the next and thus introducing
+correlations. Such non-renewal dynamics can then be described by the
+non-zero serial correlations (\figref{returnmapfig}).
+
+\begin{exercise}{isiserialcorr.m}{}
+  Implement a function \varcode{isiserialcorr()} that takes a vector of
+  interspike intervals as input argument and calculates the serial
+  correlation. The function should further plot the serial
+  correlation.
+\end{exercise}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Count statistics}
+
+\begin{figure}[t]
+  \includegraphics{countexamples}
+  \titlecaption{\label{countstatsfig}Count statistics.}{Probability
+    distributions of counting $k$ events $k$ (blue) within windows of
+    20\,ms (left) or 200\,ms duration (right) of a homogeneous Poisson
+    spike train with a rate of 20\,Hz
+    (\figref{rasterexamplesfig}). For Poisson spike trains these
+    distributions are given by Poisson distributions (red).}
+\end{figure}
+
+The most commonly used measure for characterizing spike trains is the
+\enterm[firing rate!average]{average firing rate}. The firing rate $r$
+is the average number of spikes counted within some time interval $W$
+\begin{equation}
+  \label{firingrate}
+  r = \frac{\langle n \rangle}{W}
+\end{equation}
+and is measured in Hertz. The average of the spike counts is taken
+over trials. For stationary spike trains (no change in statistics, in
+particular the firing rate, over time), the firing rate based on the
+spike count equals the inverse average interspike interval
+$1/\mu_{ISI}$.
+
+The firing rate based on an averaged spike counts is one example of
+many statistics based on event counts. Stationary spike trains can be
+split into many segments $i$, each of duration $W$, and the number of
+events $n_i$ in each of the segments can be counted. The integer event
+counts can be quantified in the usual ways:
+\begin{itemize}
+\item Histogram of the counts $n_i$. For homogeneous Poisson spike
+  trains with rate $\lambda$ the resulting probability distributions
+  follow a Poisson distribution (\figref{countstatsfig}), where the
+  probability of counting $k$ events within a time window $W$ is given by
+  \begin{equation}
+    \label{poissondist}
+    P(k) = \frac{(\lambda W)^k e^{\lambda W}}{k!}
+  \end{equation}
+\item Average number of counts: $\mu_n = \langle n \rangle$.
+\item Variance of counts:
+  $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
+\end{itemize}
+Because spike counts are unitless and positive numbers, the
+\begin{itemize}
+\item \entermde{Fano Faktor}{Fano factor} (variance of counts divided
+  by average count)
+  \begin{equation}
+    \label{fano}
+    F = \frac{\sigma_n^2}{\mu_n}
+  \end{equation}
+  is a commonly used measure for quantifying the variability of event
+  counts relative to the mean number of events. In particular for
+  homogeneous Poisson processes the Fano factor equals one,
+  independently of the time window $W$.
+\end{itemize}
+is an additional measure quantifying event counts.
+
+Note that all of these statistics depend in general on the chosen
+window length $W$. The average spike count, for example, grows
+linearly with $W$ for sufficiently large time windows: $\langle n
+\rangle = r W$, \eqnref{firingrate}. Doubling the counting window
+doubles the spike count. As does the spike-count variance
+(\figref{fanofig}). At smaller time windows the statistics of the
+event counts might depend on the particular duration of the counting
+window. There might be an optimal time window for which the variance
+of the spike count is minimal. The Fano factor plotted as a function
+of the time window illustrates such properties of point processes
+(\figref{fanofig}).
+
+This also has consequences for information transmission in neural
+systems. The lower the variance in spike count relative to the
+averaged count, the higher the signal-to-noise ratio at which
+information encoded in the mean spike count is transmitted.
+
+\begin{figure}[t]
+  \includegraphics{fanoexamples}
+  \titlecaption{\label{fanofig}
+    Count variance and Fano factor.}{Variance of event counts as a
+    function of mean counts obtained by varying the duration of the
+    count window (left). Dividing the count variance by the respective
+    mean results in the Fano factor that can be plotted as a function
+    of the count window (right). For Poisson spike trains the variance
+    always equals the mean counts and consequently the Fano factor
+    equals one irrespective of the count window (top). A spike train
+    with positive correlations between interspike intervals (caused by
+    an Ornstein-Uhlenbeck process) has a minimum in the Fano factor,
+    that is an analysis window for which the relative count variance
+    is minimal somewhere close to the correlation time scale of the
+    interspike intervals (bottom).}
+\end{figure}
+
+\begin{exercise}{counthist.m}{}
+  Implement a function \varcode{counthist()} that calculates and plots
+  the distribution of spike counts observed in a certain time
+  window. The function should take two input arguments: (i) a
+  cell-array of vectors containing the spike times in seconds observed
+  in a number of trials, and (ii) the duration of the time window that
+  is used to evaluate the counts.
+\end{exercise}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Time-dependent firing rate}