[pointprocesses] updated some figures

pointprocesses/lecture/isihexamples.py

@@ -55,11 +55,11 @@ def plotisih( ax, isis, binwidth=None ) :
             binwidth = 5e-4
     h, b = np.histogram(isis, np.arange(0.0, np.max(isis)+binwidth, binwidth), density=True)
     ax.text(0.9, 0.85, 'rate={:.0f}Hz'.format(1.0/np.mean(isis)), ha='right', transform=ax.transAxes)
-    ax.text(0.9, 0.75, 'mean={:.0f}ms'.format(1000.0*np.mean(isis)), ha='right', transform=ax.transAxes)
+    ax.text(0.9, 0.7, 'mean={:.0f}ms'.format(1000.0*np.mean(isis)), ha='right', transform=ax.transAxes)
-    ax.text(0.9, 0.65, 'CV={:.2f}'.format(np.std(isis)/np.mean(isis)), ha='right', transform=ax.transAxes)
+    ax.text(0.9, 0.55, 'CV={:.2f}'.format(np.std(isis)/np.mean(isis)), ha='right', transform=ax.transAxes)
     ax.set_xlabel('ISI', 'ms')
     ax.set_ylabel('p(ISI)', '1/s')
-    ax.bar( 1000.0*b[:-1], h, bar_fac*1000.0*np.diff(b), facecolor=colors['blue'])
+    ax.bar( 1000.0*b[:-1], h, bar_fac*1000.0*np.diff(b), **fsA)
 
 # parameter:
 rate = 20.0
@@ -85,16 +85,18 @@ x[x<0.0] = 0.0
 inhspikes = pifspikes(x, trials, dt, D=0.3)
 
 fig, (ax1, ax2) = plt.subplots(1, 2)
-fig.subplots_adjust(**adjust_fs(fig, top=1.5))
+fig.subplots_adjust(**adjust_fs(fig, top=0.5, right=1.5))
-ax1.set_title('stationary')
+ax1.set_xlim(0.0, 150.0)
-ax1.set_xlim(0.0, 200.0)
+ax1.set_ylim(0.0, 31.0)
-ax1.set_ylim(0.0, 40.0)
+ax1.set_xticks(np.arange(0.0, 151.0, 50.0))
-plotisih(ax1, isis(homspikes))
+ax1.set_yticks(np.arange(0.0, 31.0, 10.0))
+plotisih(ax1, isis(homspikes), 0.005)
 
-ax2.set_title('non-stationary')
+ax2.set_xlim(0.0, 150.0)
-ax2.set_xlim(0.0, 200.0)
+ax2.set_ylim(0.0, 31.0)
-ax2.set_ylim(0.0, 40.0)
+ax2.set_xticks(np.arange(0.0, 151.0, 50.0))
-plotisih(ax2, isis(inhspikes))
+ax2.set_yticks(np.arange(0.0, 31.0, 10.0))
+plotisih(ax2, isis(inhspikes), 0.005)
 
 plt.savefig('isihexamples.pdf')
 plt.close()

pointprocesses/lecture/pointprocesses.tex

@@ -1,36 +1,41 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Spiketrain analysis}
+\chapter{Spike-train analysis}
 \label{pointprocesseschapter}
-\exercisechapter{Spiketrain analysis}
+\exercisechapter{Spike-train analysis}
 
 \entermde[action potential]{Aktionspotential}{Action potentials}
-(\enterm[spike|seealso{action potential}]{spikes}) are the carriers of
+(\enterm[spike|seealso{action potential}]{spikes}) carry information
-information in the nervous system. Thereby it is the time at which the
+within neural systems. More precisely, the times at which action
-spikes are generated that is of importance for information
+potentials are generated contain the information.  The waveform of the
-transmission. The waveform of the action potential is largely
+action potential is largely stereotyped and therefore conveys no
-stereotyped and therefore does not carry information.
+information. Analyzing the statistics of spike times and their
-
+relation to sensory stimuli or motor actions is central to
-The result of the pre-processing of electrophysiological recordings are
+neuroscientific research. With multi-electrode arrays it is nowadays
-series of spike times, which are termed \enterm{spiketrains}. If
+possible to record from hundreds or even thousands of neurons
-measurements are repeated we get several \enterm{trials} of
+simultaneously. The open challenge is how to analyze such data sets in
-spiketrains (\figref{rasterexamplesfig}).
+order to understand how neural systems work. Let's start with the
-
+basics in this chapter.
-Spiketrains are times of events, the action potentials. Analyzing
+
-spike trains leads into the realm of the so called \entermde[point
+The result of the pre-processing of electrophysiological recordings
-  process]{Punktprozess}{point processes}.
+are series of spike times, which are termed \enterm[spike train]{spike
-
+  trains}. If measurements are repeated we get several \enterm{trials}
-\begin{figure}[ht]
+of spike trains (\figref{rasterexamplesfig}).  Spike trains are lists
+of times of events, the action potentials. Analyzing spike trains
+leads into the realm of the statistics of so called \entermde[point
+process]{Punktprozess}{point processes}.
+
+\begin{figure}[bt]
   \includegraphics[width=1\textwidth]{rasterexamples}
-  \titlecaption{\label{rasterexamplesfig}Raster-plot.}{Raster-plots of
+  \titlecaption{\label{rasterexamplesfig}Raster plot.}{Raster plots of
-    ten trials of data illustrating the times of the action
+    ten trials of data illustrating the times of action
-    potentials. Each vertical dash illustrates the time at which an
+    potentials. Each vertical stroke illustrates the time at which an
-    action potential was observed. Each line displays the events of
+    action potential was observed. Each row displays the events of one
-    one trial. Shown is a stationary point process (homogeneous point
+    trial. Shown is a stationary point process (homogeneous point
     process with a rate $\lambda=20$\;Hz, left) and an non-stationary
-    point process (perfect integrate-and-fire neuron driven by
+    point process with a rate that varies in time (noisy perfect
-    Ohrnstein-Uhlenbeck noise with a time-constant $\tau=100$\,ms,
+    integrate-and-fire neuron driven by Ohrnstein-Uhlenbeck noise with
-    right).}
+    a time-constant $\tau=100$\,ms, right).}
 \end{figure}
 
 
@@ -43,7 +48,7 @@ spike trains leads into the realm of the so called \entermde[point
   crosses some threshold. For example:\vspace{-1ex}
   \begin{itemize}
   \item Action potentials/heart beat: created by the dynamics of the
-    neuron/sinoatrial node
+    neuron/sinoatrial node.
   \item Earthquake: defined by the dynamics of the pressure between
     tectonical plates.
   \item Communication calls in crickets/frogs/birds: shaped by
@@ -59,21 +64,21 @@ spike trains leads into the realm of the so called \entermde[point
     $T_i=t_{i+1}-t_i$ and the number of events $n_i$. }
 \end{figure}
 
+\noindent
 A temporal \entermde{Punktprozess}{point process} is a stochastic
-process that generates a sequence of events at times $\{t_i\}$, $t_i
+process that generates a sequence of events at times $\{t_i\}$.  In
-\in \reZ$.  In the neurosciences, the statistics of point processes is
+the neurosciences, the statistics of point processes is of importance
-of importance since the timing of neuronal events (action potentials,
+since the timing of neuronal events (action potentials, post-synaptic
-post-synaptic potentials, events in EEG or local-field recordings,
+potentials, events in EEG or local-field recordings, etc.) is crucial
-etc.) is crucial for information transmission and can be treated as
+for information transmission and can be treated as such a process.
-such a process.
 
 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
 event from two different point processes are shown in
-\figref{rasterexamplesfig}.  Point processes can be described using
+\figref{rasterexamplesfig}.  In addition to the event times, point
-the intervals between successive events $T_i=t_{i+1}-t_i$ and the
+processes can be described using the intervals $T_i=t_{i+1}-t_i$
-number of observed events within a certain time window $n_i$
+between successive events or the number of observed events within a
-(\figref{pointprocessscetchfig}).
+certain time window $n_i$ (\figref{pointprocessscetchfig}).
 
 \begin{exercise}{rasterplot.m}{}
   Implement a function \varcode{rasterplot()} that displays the times of
@@ -94,7 +99,7 @@ real-valued variables:
 
 \begin{figure}[t]
   \includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
-  \titlecaption{\label{isihexamplesfig}Interspike interval
+  \titlecaption{\label{isihexamplesfig}Interspike-interval
     histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
 \end{figure}
 
@@ -144,7 +149,7 @@ 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}[t]
+\begin{figure}[tp]
   \includegraphics[width=1\textwidth]{returnmapexamples}
   \includegraphics[width=1\textwidth]{serialcorrexamples}
   \titlecaption{\label{returnmapfig}Interspike interval
@@ -314,7 +319,7 @@ The homogeneous Poisson process has the following properties:
   \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
   \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
   \titlecaption{\label{hompoissoncountfig}Distribution of counts of a
-    Poisson spiketrain.}{The count statistics was generated for two
+    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}.}
@@ -324,7 +329,7 @@ The homogeneous Poisson process has the following properties:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Time-dependent firing rate}
 
-So far we have discussed stationary spiketrains. The statistical properties
+So far we have discussed stationary spike trains. The statistical properties
 of these did not change within the observation time (stationary point
 processes). Most commonly, however, this is not the case. A sensory
 neuron, for example, might respond to a stimulus by modulating its
@@ -351,7 +356,7 @@ justifications, their pros- and cons.
 
 \begin{figure}[tp]
   \includegraphics[width=\columnwidth]{isimethod}
-  \titlecaption{Instantaneous firing rate.}{The recorded spiketrain
+  \titlecaption{Instantaneous firing rate.}{The recorded spike train
     (top).  Arrows illustrate the interspike intervals and numbers
     give the intervals in milliseconds. The inverse of the interspike
     intervals is the \emph{instantaneous firing rate}
@@ -412,7 +417,7 @@ methods make an assumption about the relevant observation time-scale
 \begin{figure}[tp]
   \includegraphics[width=\columnwidth]{binmethod}
   \titlecaption{Estimating the PSTH using the binning method.}{The
-    same spiketrain as shown in \figref{instratefig} (top). Vertical
+    same spike train as shown in \figref{instratefig} (top). Vertical
     gray lines indicate the borders between adjacent bins in which the
     number of action potentials is counted (red numbers). The firing
     rate is then the histogram normalized to the binwidth
@@ -448,8 +453,8 @@ time-scale.
 \begin{figure}[tp]
   \includegraphics[width=\columnwidth]{convmethod}
   \titlecaption{Estimating the firing rate using the convolution
-    method.}{The same spiketrain as in \figref{instratefig} (top). The
+    method.}{The same spike train as in \figref{instratefig} (top). The
-    convolution of the spiketrain with a kernel replaces each spike
+    convolution of the spike train with a kernel replaces each spike
     event with the kernel (red). A Gaussian kernel is used here, but
     other kernels are also possible. If the kernel is properly
     normalized the firing rate results directly form the superposition

pointprocesses/lecture/rasterexamples.py

@@ -1,5 +1,6 @@
 import numpy as np
 import matplotlib.pyplot as plt
+from plotstyle import *
 
 def hompoisson(rate, trials, duration) :
     spikes = []
@@ -8,8 +9,8 @@ def hompoisson(rate, trials, duration) :
         t = 0.0
         while t < duration :
             t += np.random.exponential(1/rate)
-            times.append( t )
+            times.append(t)
-        spikes.append( times )
+        spikes.append(times)
     return spikes
 
 def inhompoisson(rate, trials, dt) :
@@ -18,7 +19,7 @@ def inhompoisson(rate, trials, dt) :
     for k in range(trials) :
         x = np.random.rand(len(rate))
         times = dt*np.nonzero(x<p)[0]
-        spikes.append( times )
+        spikes.append(times)
     return spikes
 
 
@@ -36,7 +37,7 @@ def pifspikes(input, trials, dt, D=0.1) :
             if v >= vthresh :
                 v = vreset
                 times.append(k*dt)
-        spikes.append( times )
+        spikes.append(times)
     return spikes
 
 # parameter:
@@ -64,23 +65,22 @@ x[x<0.0] = 0.0
 # pif spike trains:
 inhspikes = pifspikes(x, trials, dt, D=0.3)
 
-fig = plt.figure( figsize=(9,4) )
+fig, (ax1, ax2) = plt.subplots(1, 2, figsize=cm_size(figure_width, 0.5*figure_width))
-ax = fig.add_subplot(1, 2, 1)
+fig.subplots_adjust(**adjust_fs(fig, left=4.0, right=1.0, top=1.2))
-ax.set_title('stationary')
-ax.set_xlim(0.0, duration)
-ax.set_ylim(-0.5, trials-0.5)
-ax.set_xlabel('Time [s]')
-ax.set_ylabel('Trials')
-ax.eventplot(homspikes, colors=[[0, 0, 0]], linelength=0.8)
 
-ax = fig.add_subplot(1, 2, 2)
+ax1.set_title('stationary')
-ax.set_title('non-stationary')
+ax1.set_xlim(0.0, duration)
-ax.set_xlim(0.0, duration)
+ax1.set_ylim(-0.5, trials-0.5)
-ax.set_ylim(-0.5, trials-0.5)
+ax1.set_xlabel('Time [s]')
-ax.set_xlabel('Time [s]')
+ax1.set_ylabel('Trial')
-ax.set_ylabel('Trials')
+ax1.eventplot(homspikes, colors=[lsA['color']], linelength=0.8, lw=1)
-ax.eventplot(inhspikes, colors=[[0, 0, 0]], linelength=0.8)
+
+ax2.set_title('non-stationary')
+ax2.set_xlim(0.0, duration)
+ax2.set_ylim(-0.5, trials-0.5)
+ax2.set_xlabel('Time [s]')
+ax2.set_ylabel('Trial')
+ax2.eventplot(inhspikes, colors=[lsA['color']], linelength=0.8, lw=1)
 
-plt.tight_layout()
 plt.savefig('rasterexamples.pdf')
 plt.close()

pointprocesses/lecture/returnmapexamples.py

@@ -40,19 +40,21 @@ def pifspikes(input, trials, dt, D=0.1) :
         spikes.append( times )
     return spikes
 
+
 def isis( spikes ) :
     isi = []
     for k in range(len(spikes)) :
         isi.extend(np.diff(spikes[k]))
     return np.array( isi )
 
-def plotreturnmap(ax, isis, lag=1, max=None) :
+
+def plotreturnmap(ax, isis, lag=1, max=1.0) :
     ax.set_xlabel(r'ISI$_i$', 'ms')
     ax.set_ylabel(r'ISI$_{i+1}$', 'ms')
-    if max != None :
+    ax.set_xlim(0.0, 1000.0*max)
-        ax.set_xlim(0.0, 1000.0*max)
+    ax.set_ylim(0.0, 1000.0*max)
-        ax.set_ylim(0.0, 1000.0*max)
+    isiss = isis[isis<max]
-    ax.scatter(1000.0*isis[:-lag], 1000.0*isis[lag:], c=colors['blue'])
+    ax.plot(1000.0*isiss[:-lag], 1000.0*isiss[lag:], clip_on=False, **psAm)
 
 # parameter:
 rate = 20.0
@@ -79,11 +81,14 @@ inhspikes = pifspikes(x, trials, dt, D=0.3)
 
 fig, (ax1, ax2) = plt.subplots(1, 2)
 fig.subplots_adjust(**adjust_fs(fig, left=6.5, top=1.5))
-ax1.set_title('stationary')
 plotreturnmap(ax1, isis(homspikes), 1, 0.3)
+ax1.set_xticks(np.arange(0.0, 301.0, 100.0))
+ax1.set_yticks(np.arange(0.0, 301.0, 100.0))
 
-ax2.set_title('non-stationary')
 plotreturnmap(ax2, isis(inhspikes), 1, 0.3)
+ax2.set_ylabel('')
+ax2.set_xticks(np.arange(0.0, 301.0, 100.0))
+ax2.set_yticks(np.arange(0.0, 301.0, 100.0))
 
 plt.savefig('returnmapexamples.pdf')
 plt.close()

pointprocesses/lecture/serialcorrexamples.py

@@ -13,6 +13,7 @@ def hompoisson(rate, trials, duration) :
         spikes.append( times )
     return spikes
 
+
 def inhompoisson(rate, trials, dt) :
     spikes = []
     p = rate*dt
@@ -55,7 +56,9 @@ def plotserialcorr(ax, isis, maxlag=10) :
     ax.set_ylabel(r'ISI correlation $\rho_k$')
     ax.set_xlim(0.0, maxlag)
     ax.set_ylim(-1.0, 1.0)
-    ax.plot(lags, corr, '.-', markersize=15, c=colors['blue'])
+    ax.plot([0, 10], [0.0, 0.0], **lsGrid)
+    ax.plot(lags, corr, clip_on=False, zorder=100, **lpsAm)
+
     
 # parameter:
 rate = 20.0
@@ -87,6 +90,7 @@ plotserialcorr(ax1, isis(homspikes))
 ax1.set_ylim(-0.2, 1.0)
 
 plotserialcorr(ax2, isis(inhspikes))
+ax2.set_ylabel('')
 ax2.set_ylim(-0.2, 1.0)
 
 plt.savefig('serialcorrexamples.pdf')