teaching/scientificComputing
Archived
3
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

[pointprocesses] improved chapter

Browse Source
This commit is contained in:
Jan Benda
2021-01-18 13:13:35 +01:00
parent 0f0dfafd56
commit 20332e2013
10 changed files with 270 additions and 192 deletions
Download Patch File Download Diff File Expand all files Collapse all files

40
pointprocesses/lecture/fanoexamples.py
View File

@@ -91,22 +91,38 @@ def plot_count_fano(ax1, ax2, spikes):
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=2.0))
plot_count_fano(axs[0,0], axs[0,1], homspikes)
axs[0,0].text(0.1, 0.95, 'Poisson', transform=axs[0,0].transAxes)
axs[0,0].set_xlabel('')
axs[0,1].set_xlabel('')
axs[0,0].xaxis.set_major_formatter(mpt.NullFormatter())
axs[0,1].xaxis.set_major_formatter(mpt.NullFormatter())
plot_count_fano(axs[1,0], axs[1,1], inhspikes)
axs[1,1].axhline(1.0, **lsGrid)
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(0.51, 0.58, 'Fano factor', va='center', rotation='vertical')
plot_fano(ax1, homspikes)
ax1.set_ylabel('Fano factor')
ax1.text(0.1, 0.95, 'Poisson', transform=ax1.transAxes)
plot_fano(ax2, inhspikes)
ax2.axhline(1.0, **lsGrid)
ax2.text(0.1, 0.95, 'OU noise', transform=ax2.transAxes)
plt.savefig('fanoexamples.pdf')
plt.close()

326
pointprocesses/lecture/pointprocesses.tex
View File

@@ -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
order to understand how neural systems work. Let's start with the
basics in this chapter.
simultaneously. The open challenge is how to analyze such huge data
sets in smart ways in order to gain insights into the way neural
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
ten trials of data illustrating the times of action
potentials. Each vertical stroke illustrates the time at which an
action potential was observed. Each row displays the events of one
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 Ornstein-Uhlenbeck noise with
a time-constant $\tau=100$\,ms, right).}
\titlecaption{\label{rasterexamplesfig}Raster plots of spike
trains.}{Raster plots of ten trials of data illustrating the times
of action potentials. Each vertical stroke illustrates the time at
which an action potential was observed. Each row displays the
events of one 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
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
$t_i$ that can be also characterized by inter-event intervals
$T_i=t_{i+1}-t_i$ and event counts $n_i$.}
processes.}{A temporal point process is a sequence of events in
time, $t_i$, that can be also characterized by the corresponding
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
@@ -77,75 +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
certain time window $n_i$ (\figref{pointprocesssketchfig}).
between successive events or the number of observed events $n_i$
within a certain time window (\figref{pointprocesssketchfig}).
\begin{exercise}{rasterplot.m}{}
Implement a function \varcode{rasterplot()} that displays the times of
action potentials within the first \varcode{tmax} seconds in a raster
plot. The spike times (in seconds) recorded in the individual trials
are stored as vectors of times within a cell array.
\end{exercise}
In \enterm[point process!stationary]{stationary} point processes the
statistics does not change over time. In particular, the rate of the
process, the number of events per time, is constant. The homogeneous
point process shown in \figref{rasterexamplesfig} on the left is an
example of a stationary point process. Although locally within each
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}
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}.
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} 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 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.
In an \entermde[Poisson process!inhomogeneous]{Poissonprozess!inhomogener}{inhomogeneous Poisson
process}, however, the rate $\lambda$ depends on time: $\lambda =
\lambda(t)$.
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.
\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}
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.
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
Let's start with the interval statistics of stationary point
processes. 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:
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}{}
@@ -168,19 +203,25 @@ describing the statistics of real-valued variables:
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 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:
@@ -188,28 +229,29 @@ describing the statistics of real-valued variables:
\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.
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
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
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 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
Implement a function that takes the returned 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.
standard deviation, and the coefficient of variation of the ISIs
(\figref{isihexamplesfig}).
\end{exercise}
\subsection{Interval correlations}
@@ -227,19 +269,20 @@ 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}.
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}
@@ -259,17 +302,17 @@ raster shown in \figref{rasterexamplesfig}.
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 $k$:
correlations}. These quantify the correlations between successive
intervals by Pearson's 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.
$k$ 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
@@ -282,15 +325,21 @@ 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}).
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
interspike intervals as input argument and calculates the serial
correlation. The function should further plot the serial
correlation.
Implement a function \varcode{isiserialcorr()} that takes a vector
of interspike intervals as input argument and calculates the serial
correlations up to some maximum lag.
\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}
@@ -326,10 +375,11 @@ 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
\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!}
@@ -338,7 +388,7 @@ counts can be quantified in the usual ways:
\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)
@@ -351,7 +401,6 @@ Because spike counts are unitless and positive numbers, the
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
@@ -372,33 +421,33 @@ 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).}
\titlecaption{\label{fanofig} Fano factor.}{Counting events in time
windows of given duration and then dividing the variance of the
counts by their mean results in the Fano factor. Here, the Fano
factor is plotted as a function of the duration of the window used
to count events. For Poisson spike trains the variance always
equals the mean counts and consequently the Fano factor equals one
irrespective of the count window (left). 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 (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
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.
window. The function should take two input arguments: a cell-array
of vectors containing the spike times in seconds observed in a
number of trials, and the duration of the time window that is used
to evaluate the counts.
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
@@ -566,6 +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

2
pointprocesses/lecture/rasterexamples.py
View File

@@ -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)