Merge branch 'master' of https://whale.am28.uni-tuebingen.de/git/teaching/scientificComputing

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}}}
 

pointprocesses/code/counthist.m

@@ -1,16 +1,11 @@
-function [counts, bins] = counthist(spikes, w)
+function counthist(spikes, w)
-% Compute and plot histogram of spike counts.
+% Plot histogram of spike counts.
 %
-% [counts, bins] = counthist(spikes, w)
+% counthist(spikes, w)
 %
 % Arguments:
 %   spikes: a cell array of vectors of spike times in seconds
-%   w: observation window duration in seconds for computing the counts
+%   w: duration of window in seconds for computing the counts
-%
-% Returns:
-%   counts: the histogram of counts normalized to probabilities
-%   bins: the bin centers for the histogram
-
     % collect spike counts:
     tmax = spikes{1}(end);
     n = [];
@@ -21,12 +16,12 @@ function [counts, bins] = counthist(spikes, w)
 %         for tk = 0:w:tmax-w
 %             nn = sum((times >= tk) & (times < tk+w));
 %             %nn = length(find((times >= tk) & (times < tk+w)));
-%             n = [n nn];
+%             n = [n, nn];
 %         end
 %     alternative 2: use the hist() function to do that!
         tbins = 0.5*w:w:tmax-0.5*w;
         nn = hist(times, tbins);
-        n = [n nn];
+        n = [n, nn];
     end
 
     % histogram of spike counts:
@@ -36,9 +31,7 @@ function [counts, bins] = counthist(spikes, w)
     counts = counts / sum(counts);
 
     % plot:
-    if nargout == 0
     bar(bins, counts);
     xlabel('counts k');
     ylabel('P(k)');
 end
-end

pointprocesses/code/isihist.m

@@ -8,7 +8,7 @@ function [pdf, centers] = isihist(isis, binwidth)
 %   binwidth: optional width in seconds to be used for the isi bins
 %
 % Returns:
-%   pdf: vector with probability density of interspike intervals in Hz
+%   pdf: vector with pdf of interspike intervals in Hertz
 %   centers: vector with centers of interspikeintervalls in seconds
 
     if nargin < 2

pointprocesses/code/isis.m

@@ -5,11 +5,9 @@ function isivec = isis(spikes)
 %
 % Arguments:
 %   spikes: a cell array of vectors of spike times in seconds
-%   isivec: a column vector with all the interspike intervalls
 %
 % Returns:
 %   isivec: a column vector with all the interspike intervalls
-
     isivec = [];
     for k = 1:length(spikes)
         difftimes = diff(spikes{k});

pointprocesses/code/isiserialcorr.m

@@ -1,15 +1,15 @@
-function isicorr = isiserialcorr(isivec, maxlag)
+function [isicorr, lags] = isiserialcorr(isivec, maxlag)
 % serial correlation of interspike intervals
 %
 % isicorr = isiserialcorr(isivec, maxlag)
 %
 % Arguments:
 %   isivec: vector of interspike intervals in seconds
-%   maxlag: the maximum lag in seconds
+%   maxlag: the maximum lag
 %
 % Returns:
 %   isicorr: vector with the serial correlations for lag 0 to maxlag
-
+%   lags: vector with the lags corresponding to isicorr
     lags = 0:maxlag;
     isicorr = zeros(size(lags));
     for k = 1:length(lags)
@@ -21,14 +21,4 @@ function isicorr = isiserialcorr(isivec, maxlag)
             isicorr(k) = corr(isivec(1:end-lag), isivec(lag+1:end));
         end
     end
-    
-    if nargout == 0
-        % plot:
-        plot(lags, isicorr, '-b');
-        hold on;
-        scatter(lags, isicorr, 50.0, 'b', 'filled');
-        hold off;
-        xlabel('Lag k')
-        ylabel('\rho_k')
-    end
 end

pointprocesses/code/plotisiserialcorr.m

@@ -0,0 +1,16 @@
+function isicorr = plotisiserialcorr(isivec, maxlag)
+% plot serial correlation of interspike intervals
+%
+% plotisiserialcorr(isivec, maxlag)
+%
+% Arguments:
+%   isivec: vector of interspike intervals in seconds
+%   maxlag: the maximum lag
+    [isicorr, lags] = isiserialcorr(isivec, maxlag);
+    plot(lags, isicorr, '-b');
+    hold on;
+    scatter(lags, isicorr, 20.0, 'b', 'filled');
+    hold off;
+    xlabel('Lag k')
+    ylabel('\rho_k')
+end

pointprocesses/code/rasterplot.m

@@ -2,21 +2,35 @@ function rasterplot(spikes, tmax)
 % Display a spike raster of the spike times given in spikes.
 %
 % rasterplot(spikes, tmax)
+%
+% Arguments:
 %   spikes: a cell array of vectors of spike times in seconds
 %   tmax: plot spike raster up to tmax seconds
-
+    in_msecs = tmax < 1.5
+    spiketimes = [];
+    trials = [];
     ntrials = length(spikes);
     for k = 1:ntrials
         times = spikes{k};
         times = times(times<tmax);
-        if tmax < 1.5
+        if in_msecs
             times = 1000.0*times;               % conversion to ms
         end
-        for i = 1:length( times )
+        % (x,y) pairs for start and stop of stroke
-            line([times(i) times(i)],[k-0.4 k+0.4], 'Color', 'k'); 
+        % plus nan separating strokes:
-        end
+        spiketimes = [spiketimes, ...
+                       [times(:)'; times(:)'; times(:)'*nan]];
+        trials = [trials, ...
+                   [ones(1, length(times)) * (k-0.4); ...
+                    ones(1, length(times)) * (k+0.4); ...
+                    ones(1, length(times)) * nan]];
     end
-    if tmax < 1.5
+    % convert matrices into simple vectors of (x,y) pairs:
+    spiketimes = spiketimes(:);
+    trials = trials(:);
+    % plotting this is lightning fast:
+    plot(spiketimes, trials, 'k')
+    if in_msecs
         xlabel('Time [ms]');
         xlim([0.0 1000.0*tmax]);
     else

pointprocesses/lecture/countexamples.py

@@ -5,8 +5,8 @@ from plotstyle import *
 
 
 rate = 20.0
-trials = 10
+trials = 20
-duration = 200.0
+duration = 100.0
 
 
 def hompoisson(rate, trials, duration) :
@@ -28,7 +28,7 @@ def plot_count_hist(ax, spikes, win, pmax):
         counts.extend(c)
     cb = np.arange(0.0, 15.5, 1.0)
     h, b = np.histogram(counts, cb, density=True)
-    ax.bar(b[:-1], h, bar_fac, **fsA)
+    ax.bar(b[:-1], h, bar_fac, align='center', **fsA)
     ax.plot(cb, poisson.pmf(cb, rate*win), **lsBm)
     ax.plot(cb, poisson.pmf(cb, rate*win), **psBm)
     ax.text(0.9, 0.9, 'T=%.0fms' % (1000.0*win), transform=ax.transAxes, ha='right')
@@ -44,6 +44,8 @@ if __name__ == "__main__":
     fig, (ax1, ax2) = plt.subplots(1, 2)
     fig.subplots_adjust(**adjust_fs(fig, top=0.5, right=1.5))
     plot_count_hist(ax1, spikes, 0.02, 0.7)
-    plot_count_hist(ax2, spikes, 0.2, 0.25)
+    ax1.set_yticks(np.arange(0.0, 0.7, 0.2))
+    plot_count_hist(ax2, spikes, 0.2, 0.22)
+    ax2.set_yticks(np.arange(0.0, 0.25, 0.1))
     plt.savefig('countexamples.pdf')
     plt.close()

pointprocesses/lecture/fanoexamples.py

@@ -76,32 +76,53 @@ def plot_count_fano(ax1, ax2, spikes):
             counts.extend(c)
         mean_counts[k] = np.mean(counts)
         var_counts[k] = np.var(counts)
-    ax1.plot(mean_counts, var_counts, **lsA)
+    ax1.plot(mean_counts, var_counts, zorder=100, **lsA)
     ax1.set_xlabel('Mean count')
     ax1.set_xlim(0.0, 20.0)
     ax1.set_ylim(0.0, 20.0)
+    ax1.set_xticks(np.arange(0.0, 21.0, 10.0))
+    ax1.set_yticks(np.arange(0.0, 21.0, 10.0))
     ax2.plot(1000.0*wins, var_counts/mean_counts, **lsB)
     ax2.set_xlabel('Window', 'ms')
     ax2.set_ylim(0.0, 1.2)
     ax2.set_xscale('log')
     ax2.set_xticks([10, 100, 1000])
     ax2.set_xticklabels(['10', '100', '1000'])
+    ax2.xaxis.set_minor_locator(mpt.NullLocator())
+    ax2.set_yticks(np.arange(0.0, 1.2, 0.5))
+
+
+def plot_fano(ax, spikes):
+    wins = np.logspace(-2, 0.0, 200)
+    mean_counts = np.zeros(len(wins))
+    var_counts = np.zeros(len(wins))
+    for k, win in enumerate(wins):
+        counts = []
+        for times in spikes:
+            c, _ = np.histogram(times, np.arange(0.0, duration, win))
+            counts.extend(c)
+        mean_counts[k] = np.mean(counts)
+        var_counts[k] = np.var(counts)
+    ax.plot(1000.0*wins, var_counts/mean_counts, **lsB)
+    ax.set_xlabel('Window', 'ms')
+    ax.set_ylim(0.0, 1.2)
+    ax.set_xscale('log')
+    ax.set_xticks([10, 100, 1000])
+    ax.set_xticklabels(['10', '100', '1000'])
+    ax.xaxis.set_minor_locator(mpt.NullLocator())
+    ax.set_yticks(np.arange(0.0, 1.2, 0.5))
 
     
 if __name__ == "__main__":
     homspikes = hompoisson(rate, trials, duration)
     inhspikes = oupifspikes(rate, trials, duration, dt, 0.3, drate, tau)
-    fig, axs = plt.subplots(2, 2)
+    fig, (ax1, ax2) = plt.subplots(1, 2)
-    fig.subplots_adjust(**adjust_fs(fig, top=0.5, right=1.5))
+    fig.subplots_adjust(**adjust_fs(fig, top=0.5, right=2.0))
-    plot_count_fano(axs[0,0], axs[0,1], homspikes)
+    plot_fano(ax1, homspikes)
-    axs[0,0].text(0.1, 0.95, 'Poisson', transform=axs[0,0].transAxes)
+    ax1.set_ylabel('Fano factor')
-    axs[0,0].set_xlabel('')
+    ax1.text(0.1, 0.95, 'Poisson', transform=ax1.transAxes)
-    axs[0,1].set_xlabel('')
+    plot_fano(ax2, inhspikes)
-    axs[0,0].xaxis.set_major_formatter(mpt.NullFormatter())
+    ax2.axhline(1.0, **lsGrid)
-    axs[0,1].xaxis.set_major_formatter(mpt.NullFormatter())
+    ax2.text(0.1, 0.95, 'OU noise', transform=ax2.transAxes)
-    plot_count_fano(axs[1,0], axs[1,1], inhspikes)
-    axs[1,0].text(0.1, 0.95, 'OU noise', transform=axs[1,0].transAxes)
-    fig.text(0.01, 0.58, 'Count variance', va='center', rotation='vertical')
-    fig.text(0.53, 0.58, 'Fano factor', va='center', rotation='vertical')
     plt.savefig('fanoexamples.pdf')
     plt.close()

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)
 
     

pointprocesses/lecture/pointprocesses.tex

@@ -13,9 +13,9 @@ information. Analyzing the statistics of spike times and their
 relation to sensory stimuli or motor actions is central to
 neuroscientific research. With multi-electrode arrays it is nowadays
 possible to record from hundreds or even thousands of neurons
-simultaneously. The open challenge is how to analyze such data sets in
+simultaneously. The open challenge is how to analyze such huge data
-order to understand how neural systems work. Let's start with the
+sets in smart ways in order to gain insights into the way neural
-basics in this chapter.
+systems work. Let's start with the basics in this chapter.
 
 The result of the pre-processing of electrophysiological recordings
 are series of spike times, which are termed \enterm[spike train]{spike
@@ -27,15 +27,16 @@ process]{Punktprozess}{point processes}.
 
 \begin{figure}[bt]
   \includegraphics[width=1\textwidth]{rasterexamples}
-  \titlecaption{\label{rasterexamplesfig}Raster plot.}{Raster plots of
+  \titlecaption{\label{rasterexamplesfig}Raster plots of spike
-    ten trials of data illustrating the times of action
+    trains.}{Raster plots of ten trials of data illustrating the times
-    potentials. Each vertical stroke illustrates the time at which an
+    of action potentials. Each vertical stroke illustrates the time at
-    action potential was observed. Each row displays the events of one
+    which an action potential was observed. Each row displays the
-    trial. Shown is a stationary point process (homogeneous point
+    events of one trial. Shown is a stationary point process
-    process with a rate $\lambda=20$\;Hz, left) and an non-stationary
+    (homogeneous point process with a rate $\lambda=20$\;Hz, left) and
-    point process with a rate that varies in time (noisy perfect
+    an non-stationary point process with a rate that varies in time
-    integrate-and-fire neuron driven by Ohrnstein-Uhlenbeck noise with
+    (noisy perfect integrate-and-fire neuron driven by
-    a time-constant $\tau=100$\,ms, right).}
+    Ornstein-Uhlenbeck noise with a time-constant $\tau=100$\,ms,
+    right).}
 \end{figure}
 
 
@@ -59,9 +60,10 @@ process]{Punktprozess}{point processes}.
 \begin{figure}[tb]
   \includegraphics{pointprocesssketch}
   \titlecaption{\label{pointprocesssketchfig} Statistics of point
-    processes.}{A point process is a sequence of instances in time
+    processes.}{A temporal point process is a sequence of events in
-    $t_i$ that can be also characterized by inter-event intervals
+    time, $t_i$, that can be also characterized by the corresponding
-    $T_i=t_{i+1}-t_i$ and event counts $n_i$.}
+    inter-event intervals $T_i=t_{i+1}-t_i$ and event counts $n_i$,
+    i.e. the number of events that occurred so far.}
 \end{figure}
 
 \noindent
@@ -77,30 +79,108 @@ plot in which each vertical line indicates the time of an event. The
 event from two different point processes are shown in
 \figref{rasterexamplesfig}.  In addition to the event times, point
 processes can be described using the intervals $T_i=t_{i+1}-t_i$
-between successive events or the number of observed events within a
+between successive events or the number of observed events $n_i$
-certain time window $n_i$ (\figref{pointprocesssketchfig}).
+within a certain time window (\figref{pointprocesssketchfig}).
-
+
-\begin{exercise}{rasterplot.m}{}
+In \enterm[point process!stationary]{stationary} point processes the
-  Implement a function \varcode{rasterplot()} that displays the times of
+statistics does not change over time. In particular, the rate of the
-  action potentials within the first \varcode{tmax} seconds in a raster
+process, the number of events per time, is constant. The homogeneous
-  plot. The spike times (in seconds) recorded in the individual trials
+point process shown in \figref{rasterexamplesfig} on the left is an
-  are stored as vectors of times within a cell array.
+example of a stationary point process. Although locally within each
-\end{exercise}
+trial there are regions with many events per time and others with long
+intervals between events, the average number of events within a small
+time window over trials does not change in time. In the first sections
+of this chapter we introduce various statistics for characterizing
+stationary point processes.
+
+On the other hand, the example shown in \figref{rasterexamplesfig} on
+the right is a non-stationary point process. The rate of the events in
+all trials first decreases and then increases again. This common
+change in rate may have been caused by some sensory input to the
+neuron. How to estimate temporal changes in event rates (firing rates)
+is covered in section~\ref{nonstationarysec}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Homogeneous Poisson process}
+
+Before we are able to start analyzing point processes, we need some
+data. As before, because we are working with computers, we can easily
+simulate them. To get us started we here briefly introduce the
+homogeneous Poisson process. The Poisson process is to point processes
+what the Gaussian distribution is to the statistics of real-valued
+data. It is a standard against which everything is compared.
+
+In a \entermde[Poisson
+process!homogeneous]{Poissonprozess!homogener}{homogeneous Poisson
+  process} events at every given time, that is within every small time
+window of width $\Delta t$ occur with the same constant
+probability. This probability is independent of absolute time and
+independent of any events occurring before. To observe an event right
+after an event is as likely as to observe an event at some specific
+time later on.
+
+The simplest way to simulate events of a homogeneous Poisson process
+is based on the exponential distribution \eqref{hompoissonexponential}
+of event intervals. We randomly draw intervals from this distribution
+and then sum them up to convert them to event times. The only
+parameter of a homogeneous Poisson process is its rate. It defines how
+many events per time are expected.
+
+Here is a function that generates several trials of a homogeneous
+Poisson process:
+
+\lstinputlisting[caption={hompoissonspikes.m}]{hompoissonspikes.m}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+\section{Raster plot}
+
+Let's generate some events with this function and display them in a
+raster plot as in \figref{rasterexamplesfig}. For the raster plot we
+need to draw for each event a line at the corresponding time of the
+event and at the height of the corresponding trial. You can go with a
+one for-loop through the trials and then with another for-loop through
+the event times and every time plot a two-point line. This, however,
+is very slow. The fastest way is to concatenate the coordinates of all
+strokes into large vectors, separate the events by \varcode{nan}
+entries, and pass this to a single call of \varcode{plot()}:
+
+\pageinputlisting[caption={rasterplot.m}]{rasterplot.m}
+
+Adapt this function to your needs and use it where ever possible to
+illustrate your spike train data to your readers. They appreciate
+seeing your raw data and being able to judge the data for themselves
+before you go on analyzing firing rates, interspike-interval
+correlations, etc.
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \section{Interval statistics}
 
-The intervals $T_i=t_{i+1}-t_i$ between successive events are real
+Let's start with the interval statistics of stationary point
-positive numbers. In the context of action potentials they are
+processes.  The intervals $T_i=t_{i+1}-t_i$ between successive events
-referred to as \entermde{Interspikeintervalle}{interspike
+are real positive numbers. In the context of action potentials they
-  intervals}. The statistics of interspike intervals are described
+are referred to as \entermde[interspike
-using common measures for describing the statistics of stochastic
+interval]{Interspikeintervall}{interspike intervals}, in short
-real-valued variables:
+\entermde[ISI|see{interspike
+  interval}]{ISI|see{Interspikeintervall}}{ISI}s.  For analyzing event
+intervals we can use all the usual statistics that we know from
+describing univariate data sets of real numbers:
 
 \begin{figure}[t]
-  \includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
+  \includegraphics[width=0.96\textwidth]{isihexamples}
   \titlecaption{\label{isihexamplesfig}Interspike-interval
-    histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
+    histograms}{of the spike trains shown in
+    \figref{rasterexamplesfig}.  The intervals of the homogeneous
+    Poisson process (left) are exponentially distributed according to
+    \eqnref{hompoissonexponential} (red). Typically for many sensory
+    neurons that fire more regularly is an ISI histogram like the one
+    shown on the right. There is a preferred interval where the
+    distribution peaks. The distribution falls off quickly towards
+    smaller intervals, and very small intervals are absent, probably
+    because of refractoriness of the spike generator. The tail of the
+    distribution again approaches an exponential distribution as for
+    the Poisson process.}
 \end{figure}
 
 \begin{exercise}{isis.m}{}
@@ -110,44 +190,99 @@ real-valued variables:
   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
+  (\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \;
-  = 1$.
+  dT = 1$. Commonly referred to as the \enterm[interspike
-\item Average interval: $\mu_{ISI} = \langle T \rangle =
+  interval!histogram]{interspike interval histogram}. Its shape
-  \frac{1}{n}\sum\limits_{i=1}^n T_i$.
+  reveals many interesting aspects like locking or bursting that
-\item Standard deviation of the interspike intervals: $\sigma_{ISI} = \sqrt{\langle (T - \langle T
+  cannot be inferred from the mean or standard deviation. A particular
-    \rangle)^2 \rangle}$\vspace{1ex}
+  reference is the exponential distribution of intervals
-\item \entermde[coefficient of variation]{Variationskoeffizient}{Coefficient of variation}:
+  \begin{equation}
-  $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$.
+    \label{hompoissonexponential}
-\item \entermde[diffusion coefficient]{Diffusionskoeffizient}{Diffusion coefficient}: $D_{ISI} =
+    p_{exp}(T) = \lambda e^{-\lambda T}
-  \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
+  \end{equation}
+  of a homogeneous Poisson spike train with rate $\lambda$.
+\item Mean interval
+  \begin{equation}
+    \label{meanisi}
+    \mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum_{i=1}^n T_i
+  \end{equation}
+  The average time it takes from one event to the next. For stationary
+  point processes 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
+  \begin{equation}
+    \label{stdisi}
+    \sigma_{ISI} = \sqrt{\langle (T - \langle T \rangle)^2 \rangle}
+  \end{equation}
+  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 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 a neuron is firing. $CV$s 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
+  Implement a function \varcode{isihist()} that calculates the
-  interspike interval histogram. The function should take two input
+  normalized interspike interval histogram. The function should take
-  arguments; (i) a vector of interspike intervals and (ii) the width
+  two input arguments; (i) a vector of interspike intervals and (ii)
-  of the bins used for the histogram. It further returns the
+  the width of the bins used for the histogram. It 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
+  Implement a function that takes the returned values of
-  \varcode{isiHist()} as input arguments and then plots the data. The
+  \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.
+  standard deviation, and the coefficient of variation of the ISIs
+  (\figref{isihexamplesfig}).
 \end{exercise}
 
 \subsection{Interval correlations}
-So called \entermde[return map]{return map}{return maps} are used to
+Intervals are not just numbers without an order, like weights of
-illustrate interdependencies between successive interspike
+tigers.  Intervals are temporally ordered and there could be temporal
-intervals. The return map plots the delayed interval $T_{i+k}$ against
+structure in the sequence of intervals. For example, short intervals
-the interval $T_i$. The parameter $k$ is called the \enterm{lag}
+could be followed by more longer ones, and vice versa. Such
-(\determ{Verz\"ogerung}) $k$. Stationary and non-stationary return
+dependencies in the sequence of intervals do not show up in the
-maps are distinctly different \figref{returnmapfig}.
+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}
+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 each
+trial of the spike raster shown in \figref{rasterexamplesfig} on the
+right.
 
 \begin{figure}[tp]
   \includegraphics[width=1\textwidth]{serialcorrexamples}
@@ -165,22 +300,46 @@ maps are distinctly different \figref{returnmapfig}.
     the mean firing rate of the spike trains.}
 \end{figure}
 
-Such dependencies can be further quantified using the
+Such dependencies can be further quantified by
 \entermde[correlation!serial]{Korrelation!serielle}{serial
-  correlations} \figref{returnmapfig}. The serial correlation is the
+  correlations}. These quantify the correlations between successive
-correlation coefficient of the intervals $T_i$ and the intervals
+intervals by Pearson's correlation coefficients between the intervals
-delayed by the lag $T_{i+k}$:
+$T_{i+k}$ and $T_i$ in dependence on lag $k$:
-\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} 
+\begin{equation}
-= {\rm corr}(T_{i+k}, T_i) \] The resulting correlation coefficient
+  \label{serialcorrelation}
-$\rho_k$ is usually plotted against the 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)} 
-\figref{returnmapfig}. $\rho_0=1$ is the correlation of each interval
+= {\rm corr}(T_{i+k}, T_i)
-with itself and is always 1.
+\end{equation}
+The serial correlations $\rho_k$ are usually plotted against the lag
+$k$ for a 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 introduce
+correlations between intervals. Such non-renewal dynamics is then
+characterized by the non-zero serial correlations
+(\figref{returnmapfig}).
 
 \begin{exercise}{isiserialcorr.m}{}
-  Implement a function \varcode{isiserialcorr()} that takes a vector of
+  Implement a function \varcode{isiserialcorr()} that takes a vector
-  interspike intervals as input argument and calculates the serial
+  of interspike intervals as input argument and calculates the serial
-  correlation. The function should further plot the serial
+  correlations up to some maximum lag.
-  correlation.
+\end{exercise}
+
+\begin{exercise}{plotisiserialcorr.m}{}
+  Implement a function \varcode{plotisiserialcorr()} that takes a
+  vector of interspike intervals as input argument and generates a
+  plot of the serial correlations.
 \end{exercise}
 
 
@@ -189,11 +348,11 @@ with itself and is always 1.
 
 \begin{figure}[t]
   \includegraphics{countexamples}
-  \titlecaption{\label{countstatsfig}Count statistics.}{The
+  \titlecaption{\label{countstatsfig}Count statistics.}{Probability
-    distribution of the number of events $k$ (blue) counted within
+    distributions of counting $k$ events $k$ (blue) within windows of
-    windows of 20\,ms (left) or 200\,ms duration (right) of the
+    20\,ms (left) or 200\,ms duration (right) of a homogeneous Poisson
-    homogeneous Poisson spike train with a rate of 20\,Hz shown in
+    spike train with a rate of 20\,Hz
-    \figref{rasterexamplesfig}. For Poisson spike trains these
+    (\figref{rasterexamplesfig}). For Poisson spike trains these
     distributions are given by Poisson distributions (red).}
 \end{figure}
 
@@ -204,7 +363,7 @@ is the average number of spikes counted within some time interval $W$
   \label{firingrate}
   r = \frac{\langle n \rangle}{W}
 \end{equation}
-and is neasured in Hertz. The average of the spike counts is taken
+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
@@ -216,28 +375,44 @@ 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 Histogram of the counts $n_i$ appropriately normalized to
+  probability distributions. 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
+Because spike counts are unitless and positive numbers the
 \begin{itemize}
 \item \entermde{Fano Faktor}{Fano factor} (variance of counts divided
-  by average count): $F = \frac{\sigma_n^2}{\mu_n}$.
+  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
-Note that all of these statistics depend on the chosen window length
+window length $W$. The average spike count, for example, grows
-$W$. The average spike count, for example, grows linearly with $W$ for
+linearly with $W$ for sufficiently large time windows: $\langle n
-sufficiently large time windows: $\langle n \rangle = r W$,
+\rangle = r W$, \eqnref{firingrate}. Doubling the counting window
-\eqnref{firingrate}. Doubling the counting window doubles the spike
+doubles the spike count. As does the spike-count variance
-count. As does the spike-count variance (\figref{fanofig}). At smaller
+(\figref{fanofig}). At smaller time windows the statistics of the
-time windows the statistics of the event counts might depend on the
+event counts might depend on the particular duration of the counting
-particular duration of the counting window. There might be an optimal
+window. There might be an optimal time window for which the variance
-time window for which the variance of the spike count is minimal. The
+of the spike count is minimal. The Fano factor plotted as a function
-Fano factor plotted as a function of the time window illustrates such
+of the time window illustrates such properties of point processes
-properties of point processes (\figref{fanofig}).
+(\figref{fanofig}).
 
 This also has consequences for information transmission in neural
 systems. The lower the variance in spike count relative to the
@@ -246,126 +421,33 @@ information encoded in the mean spike count is transmitted.
 
 \begin{figure}[t]
   \includegraphics{fanoexamples}
-  \titlecaption{\label{fanofig}
+  \titlecaption{\label{fanofig} Fano factor.}{Counting events in time
-    Count variance and Fano factor.}{Variance of event counts as a
+    windows of given duration and then dividing the variance of the
-    function of mean counts obtained by varying the duration of the
+    counts by their mean results in the Fano factor. Here, the Fano
-    count window (left). Dividing the count variance by the respective
+    factor is plotted as a function of the duration of the window used
-    mean results in the Fano factor that can be plotted as a function
+    to count events.  For Poisson spike trains the variance always
-    of the count window (right). For Poisson spike trains the variance
+    equals the mean counts and consequently the Fano factor equals one
-    always equals the mean counts and consequently the Fano factor
+    irrespective of the count window (left). A spike train with
-    equals one irrespective of the count window (top). A spike train
+    positive correlations between interspike intervals (caused by an
-    with positive correlations between interspike intervals (caused by
+    Ornstein-Uhlenbeck process) has a minimum in the Fano factor, that
-    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).}
+    interspike intervals (right).}
 \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
+  window. The function should take two input arguments: a cell-array
-  cell-array of vectors containing the spike times in seconds observed
+  of vectors containing the spike times in seconds observed in a
-  in a number of trials, and (ii) the duration of the time window that
+  number of trials, and the duration of the time window that is used
-  is used to evaluate the counts.
+  to evaluate the counts.
-\end{exercise}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Homogeneous Poisson process}
-
-The Gaussian distribution is, because of the central limit theorem,
-the standard distribution for continuous measures. The equivalent in
-the realm of point processes is the
-\entermde[distribution!Poisson]{Verteilung!Poisson-}{Poisson distribution}.
-
-In a \entermde[Poisson process!homogeneous]{Poissonprozess!homogener}{homogeneous Poisson
-  process} 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 event within
-a small time window of width $\Delta t$ is given by
-\begin{equation}
-  \label{hompoissonprob}
-  P = \lambda \cdot \Delta t \; .
-\end{equation}
-
-In an \entermde[Poisson process!inhomogeneous]{Poissonprozess!inhomogener}{inhomogeneous Poisson
-  process}, however, the rate $\lambda$ depends on time: $\lambda =
-\lambda(t)$.
-
-\begin{exercise}{poissonspikes.m}{}
-  Implement a function \varcode{poissonspikes()} that uses a homogeneous
-  Poisson process to generate events at a given rate for a certain
-  duration and a number of trials. The rate should be given in Hertz
-  and the duration of the trials is given in seconds. The function
-  should return the event times in a cell-array. Each entry in this
-  array represents the events observed in one trial. Apply
-  \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
-  for a certain duration and a number of trials. The rate should be
-  given in Hertz and the duration of the trials is given in
-  seconds. The function should return the event times in a
-  cell-array. Each entry in this array represents the events observed
-  in one trial. Apply \eqnref{poissonintervals} to generate the event
-  times.
-\end{exercise}
-
-\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}.}
-\end{figure}
-
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Time-dependent firing rate}
+\label{nonstationarysec}
 
 So far we have discussed stationary spike trains. The statistical properties
 of these did not change within the observation time (stationary point
@@ -533,6 +615,7 @@ relevate time-scale.
 \end{exercise}
 
 \section{Spike-triggered Average}
+\label{stasec}
 
 The graphical representation of the neuronal activity alone is not
 sufficient tot investigate the relation between the neuronal response

pointprocesses/lecture/rasterexamples.py

@@ -85,7 +85,7 @@ def plot_inhomogeneous_spikes(ax):
 
 
 if __name__ == "__main__":
-    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=cm_size(figure_width, 0.5*figure_width))
+    fig, (ax1, ax2) = plt.subplots(1, 2)
     fig.subplots_adjust(**adjust_fs(fig, left=4.0, right=1.0, top=1.2))
     plot_homogeneous_spikes(ax1)
     plot_inhomogeneous_spikes(ax2)

scientificcomputing-script.tex

@@ -116,9 +116,10 @@
 \includechapter{pointprocesses}
 
 % add a chapter on simulating point-processes/spike trains
-% (Poisson spike trains, integrate-and-fire models)
+% (Poisson spike trains, LMP models, integrate-and-fire models, full HH like models)
 
 % add chapter on information theory, mutual information, stimulus reconstruction, coherence
+% move STA here!
 
 % add a chapter on Bayesian inference (the Neuroscience of it and a
 % bit of application for statistical problems).