[pointprocesses] improved chapter and added rasterplot()

chapter.mk

@@ -9,7 +9,8 @@ pythonplots : $(PYPDFFILES)
 
 $(PYPDFFILES) : %.pdf: %.py
 	echo $$(which python)
-	python3 $<
+	#python3 $<
+	python $<
 
 cleanpythonplots :
 	rm -f $(PYPDFFILES)

pointprocesses/code/plotspikeraster.m

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

pointprocesses/code/rasterplot.m

@@ -1,7 +1,7 @@
-function spikeraster(spikes, tmax)
+function rasterplot(spikes, tmax)
 % Display a spike raster of the spike times given in spikes.
 %
-% spikeraster(spikes, tmax)
+% rasterplot(spikes, tmax)
 %   spikes: a cell array of vectors of spike times in seconds
 %   tmax: plot spike raster upto tmax seconds
 

pointprocesses/lecture/pointprocesses-chapter.tex

@@ -18,7 +18,7 @@
 
 \section{TODO}
 \begin{itemize}
-\item Add spikeraster function
+\item Explain difference stationary versus non-stationary point process
 \item Multitrial firing rates
 \item Choice of bin width for PSTH, kernel width, also in relation sto
   stimulus time scale

pointprocesses/lecture/pointprocesses.tex

@@ -2,17 +2,15 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \chapter{Spiketrain analysis}
 
-\selectlanguage{english}
+\enterm[Action potentials]{action potentials} (\enterm{spikes}) are
-
+the carriers of information in the nervous system. Thereby it is the
-\enterm[Actionspotentials]{Actionspotentials} (\enterm{spikes}) are
+time at which the spikes are generated that is of importance for
-the carriers of information in the nervous system. Thereby it is
+information transmission. The waveform of the action potential is
-mainly the time at which the spikes are generated that is of
+largely stereotyped and therefore does not carry information.
-importance. The waveform of the action potential is largely
+
-stereotyped and does not carry information.
+The result of the pre-processing of electrophysiological recordings are
-
+series of spike times, which are termed \enterm{spiketrains}. If
-The result of the processing of electrophysiological recordings are
+measurements are repeated we get several \enterm{trials} of
-series of spike times, which are then termed \enterm{spiketrains}. If
-measurements are repeated we yield several \enterm{trials} of
 spiketrains (\figref{rasterexamplesfig}).
 
 Spiketrains are times of events, the action potentials. The analysis
@@ -21,34 +19,32 @@ of these leads into the realm of the so called \enterm[point
 
 \begin{figure}[ht]
   \includegraphics[width=1\textwidth]{rasterexamples}
-  \titlecaption{\label{rasterexamplesfig}Raster-plot.}{Raster-plot of
+  \titlecaption{\label{rasterexamplesfig}Raster-plot.}{Raster-plots of
-    ten realizations of a stationary point process (homogeneous point
+    ten trials of data illustrating the times of the action
-    process with a rate $\lambda=20$\;Hz, left) and an inhomogeneous
+    potentials. Each vertical dash illustrates the time at which an
-    point process (perfect integrate-and-fire neuron dirven by
+    action potential was observed. Each line displays the events of
-    Ohrnstein-Uhlenbeck noise with a time-constant $\tau=100$\,ms,
+    one trial. Shown is a stationary point process (left, homogeneous
-    right). Each vertical dash illustrates the time at which the
+    point process with a rate $\lambda=20$\;Hz, left) and an
-    action potential was observed. Each line represents the event of
+    non-stationary point process (right, perfect integrate-and-fire
-    each trial.}
+    neuron dirven by Ohrnstein-Uhlenbeck noise with a time-constant
+    $\tau=100$\,ms, right).}
 \end{figure}
 
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Point processes}
 
-A temporal \enterm{point process} is a stochastic process that
-generates a sequence of events at times $\{t_i\}$, $t_i \in
-\reZ$.
-
 \begin{ibox}{Examples of point processes}
   Every point process is generated by a temporally continuously
   developing process. An event is generated whenever this process
-  reaches a certain threshold. For example:
+  crosses some threshold. For example:\vspace{-1ex}
   \begin{itemize}
   \item Action potentials/heart beat: created by the dynamics of the
     neuron/sinoatrial node
   \item Earthquake: defined by the dynamics of the pressure between
     tectonical plates.
-  \item Evoked communication calls in crickets/frogs/birds: shaped by
+  \item Communication calls in crickets/frogs/birds: shaped by
-    the dynamics of nervous system and the muscle appartus.
+    the dynamics of the nervous system and the muscle appartus.
   \end{itemize}
 \end{ibox}
 
@@ -60,43 +56,51 @@ generates a sequence of events at times $\{t_i\}$, $t_i \in
     $T_i=t_{i+1}-t_i$ and the number of events $n_i$. }
 \end{figure}
 
-In the neurosciences, the statistics of point processes is of
+A temporal \enterm{point process} is a stochastic process that
-importance since the timing of the neuronal events (the action
+generates a sequence of events at times $\{t_i\}$, $t_i \in \reZ$.  In
-potentials) is crucial for information transmission and can be treated
+the neurosciences, the statistics of point processes is of importance
-as such a process.
+since the timing of neuronal events (action potentials, post-synaptic
-
+potentials, events in EEG or local-field recordings, etc.) is crucial
-Point processes can be described using the intervals between
+for information transmission and can be treated as such a process.
-successive events $T_i=t_{i+1}-t_i$ and the number of observed events
+
-within a certain time window $n_i$ (\figref{pointprocessscetchfig}).
+The events of a point process can be illustrated by means of a raster
-
+plot in which each vertical line indicates the time of an event. The
-The events originating from a point process can be illustrated in form
+event from two different point processes are shown in
-of a scatter- or raster plot in which each vertical line indicates the
+\figref{rasterexamplesfig}.  Point processes can be described using
-time of an event. The event from two different point processes are
+the intervals between successive events $T_i=t_{i+1}-t_i$ and the
-shown in \figref{rasterexamplesfig}.
+number of observed events within a certain time window $n_i$
+(\figref{pointprocessscetchfig}).
+
+\begin{exercise}{rasterplot.m}{}
+  Implement a function \code{rasterplot()} that displays the times of
+  action potentials within the first \code{tmax} seconds in a raster
+  plot. The spike times (in seconds) recorded in the individual trials
+  are stored as vectors of times within a \codeterm{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 \enterm{interspike intervals}. The statistics of these
+referred to as \enterm{interspike intervals}. The statistics of
-are described using the common measures.
+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
-    histogram}{of the spikes depicted in \figref{rasterexamplesfig}.}
+    histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
 \end{figure}
 
 \begin{exercise}{isis.m}{}
   Implement a function \code{isis()} that calculates the interspike
   intervals from several spike trains. The function should return a
-  single vector of intervals. The action potentials recorded in the
+  single vector of intervals. The spike times (in seconds) of each
-  individual trials are stored as vectors of spike times within a
+  trial are stored as vectors within a \codeterm{cell-array}.
-  \codeterm{cell-array}. Spike times are given in seconds.
 \end{exercise}
 
-\subsection{First order interval statistics}
+%\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
@@ -138,10 +142,17 @@ and non-stationary return maps are distinctly different
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{returnmapexamples}
   \includegraphics[width=1\textwidth]{serialcorrexamples}
-  \titlecaption{\label{returnmapfig}Interspike interval analyses of a
+  \titlecaption{\label{returnmapfig}Interspike interval
-    stationary and a non-stationary pointprocess.}{Upper plots show the
+    correlations}{of the spike trains shown in
-    return maps and the lower panels depict the serial correlation of
+    \figref{rasterexamplesfig}. Upper panels show the return maps and
-    successive intervals separated by the lag $k$.}
+    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}. 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 \enterm{serial
@@ -170,17 +181,18 @@ with itself and is always 1.
 %   \includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
 %   \titlecaption{\label{countstatsfig}Count Statistik.}{}
 % \end{figure}
-The number of events $n_i$ (counts) in a time window $i$ of the duration $W$
+Counting the number of events $n_i$ (counts) in time windows $i$ of duration $W$
 yields positive integer random numbers that are commonly quantified
 using the following measures:
 \begin{itemize}
 \item Histogram of the counts $n_i$.
-\item Average number of events: $\mu_N = \langle n \rangle$.
+\item Average number of counts: $\mu_n = \langle n \rangle$.
-\item Variance of the counts: $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
+\item Variance of counts: $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
-\item \determ{Fano Faktor} (The variance divided by the average): $F = \frac{\sigma_n^2}{\mu_n}$.
+\item \determ{Fano Factor} (variance of counts divided by average count): $F = \frac{\sigma_n^2}{\mu_n}$.
 \end{itemize}
-And in particular the average firing rate $r$ (spike count per time interval
+Of particular interest is the average firing rate $r$ (spike count per
-, \determ{Feuerrate}) that is given in Hertz \sindex[term]{Feuerrate!mittlere Rate}
+time interval , \determ{Feuerrate}) that is given in Hertz
+\sindex[term]{firing rate!average rate}
 \begin{equation}
   \label{firingrate}
   r = \frac{\langle n \rangle}{W} \; .
@@ -204,30 +216,30 @@ And in particular the average firing rate $r$ (spike count per time interval
   the distribution of spike counts observed in a certain time
   window. The function should take two input arguments: (i) a
   \codeterm{cell-array} of vectors containing the spike times in
-  seconds observed in a number of trials and (ii) the duration of the
+  seconds observed in a number of trials, and (ii) the duration of the
-  time window that is used to evaluate the counts.
+  time window that is used to evaluate the counts.\pagebreak[4]
 \end{exercise}
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Homogeneous Poisson process}
 
-The Gaussian distribution is, due to the central limit theorem, the
+The Gaussian distribution is, because of the central limit theorem,
-standard for continuous measures. The equivalent in the realm of point
+the standard distribution for continuous measures. The equivalent in
-processes is the \enterm{Poisson distribution}.
+the realm of point processes is the \enterm{Poisson distribution}.
 
 In a \enterm[Poisson process!homogeneous]{homogeneous Poisson process}
-the events occur at a fixed rate $\lambda=\text{const.}$ and are
+the events occur at a fixed rate $\lambda=\text{const}$ and are
 independent of both the time $t$ and occurrence of previous events
-(\figref{hompoissonfig}). The probability of observing an even within a
+(\figref{hompoissonfig}). The probability of observing an event within
-small time window of width $\Delta t$ is given by
+a small time window of width $\Delta t$ is given by
 \begin{equation}
   \label{hompoissonprob}
   P = \lambda \cdot \Delta t \; .
 \end{equation}
 
 In an \enterm[Poisson process!inhomogeneous]{inhomogeneous Poisson
-  process}, however, the rate $\lambda$ depends on the time: $\lambda =
+  process}, however, the rate $\lambda$ depends on time: $\lambda =
 \lambda(t)$.
 
 \begin{exercise}{poissonspikes.m}{}
@@ -249,7 +261,11 @@ In an \enterm[Poisson process!inhomogeneous]{inhomogeneous Poisson
 \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.}{}
+  \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:
@@ -267,7 +283,10 @@ The homogeneous Poisson process has the following properties:
   process is also called a \enterm{renewal process}.
 \item The number of events $k$ within a temporal window of duration
   $W$ is Poisson distributed:
-\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
+\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}
@@ -286,8 +305,11 @@ The homogeneous Poisson process has the following properties:
 \begin{figure}[t]
   \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
   \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
-  \titlecaption{\label{hompoissoncountfig}Count statistics of Poisson
+  \titlecaption{\label{hompoissoncountfig}Distribution of counts of a
-    spiketrains.}{}
+    Poisson spiketrain.}{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}.}
 \end{figure}
 
 
@@ -312,7 +334,7 @@ closely.
 \begin{figure}[tp]
   \includegraphics[width=\columnwidth]{firingrates}
   \titlecaption{Estimating the time-dependent firing
-    rate.}{\textbf{A)} Rasterplot depicting the spiking activity of a
+    rate.}{\textbf{A)} Rasterplot showing the spiking activity of a
     neuron. \textbf{B)} Firing rate calculated from the
     \emph{instantaneous rate}. \textbf{C)} classical PSTH with the
     \emph{binning} method. \textbf{D)} Firing rate estimated by
@@ -529,5 +551,3 @@ kernel.
   the same size as the original stimulus contained in file
   \file{sta\_data.mat}.
 \end{exercise}
-
-\selectlanguage{english}

pointprocesses/lecture/pointprocessscetch.gpt

@@ -1,4 +1,4 @@
-set term epslatex size 11.4cm, 6.5cm
+set term epslatex size 11.4cm, 6.4cm
 set out 'pointprocessscetch.tex'
 
 set border 0

pointprocesses/lecture/pointprocessscetchA.eps

@@ -1,7 +1,7 @@
 %!PS-Adobe-2.0 EPSF-2.0
 %%Title: pointprocessscetchA.tex
 %%Creator: gnuplot 4.6 patchlevel 4
-%%CreationDate: Tue Nov 28 10:02:49 2017
+%%CreationDate: Mon Nov 26 17:57:49 2018
 %%DocumentFonts: 
 %%BoundingBox: 50 50 373 135
 %%EndComments
@@ -433,7 +433,7 @@ SDict begin [
   /Author (benda)
 %  /Producer (gnuplot)
 %  /Keywords ()
-  /CreationDate (Tue Nov 28 10:02:49 2017)
+  /CreationDate (Mon Nov 26 17:57:49 2018)
   /DOCINFO pdfmark
 end
 } ifelse

pointprocesses/lecture/pointprocessscetchB.eps

@@ -1,7 +1,7 @@
 %!PS-Adobe-2.0 EPSF-2.0
 %%Title: pointprocessscetchB.tex
 %%Creator: gnuplot 4.6 patchlevel 4
-%%CreationDate: Tue Nov 28 10:02:49 2017
+%%CreationDate: Mon Nov 26 17:57:49 2018
 %%DocumentFonts: 
 %%BoundingBox: 50 50 373 237
 %%EndComments
@@ -433,7 +433,7 @@ SDict begin [
   /Author (benda)
 %  /Producer (gnuplot)
 %  /Keywords ()
-  /CreationDate (Tue Nov 28 10:02:49 2017)
+  /CreationDate (Mon Nov 26 17:57:49 2018)
   /DOCINFO pdfmark
 end
 } ifelse