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[pointprocesses] better incorporated Poisson spike train

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Jan Benda 2021-01-17 23:57:06 +01:00
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19
header.tex
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@ -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}}}

2
pointprocesses/lecture/isihexamples.py
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@ -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)

313
pointprocesses/lecture/pointprocesses.tex
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@ -34,7 +34,7 @@ process]{Punktprozess}{point processes}.
trial. Shown is a stationary point process (homogeneous point
process with a rate $\lambda=20$\;Hz, left) and an non-stationary
point process with a rate that varies in time (noisy perfect
integrate-and-fire neuron driven by Ohrnstein-Uhlenbeck noise with
integrate-and-fire neuron driven by Ornstein-Uhlenbeck noise with
a time-constant $\tau=100$\,ms, right).}
\end{figure}
@ -87,15 +87,60 @@ certain time window $n_i$ (\figref{pointprocesssketchfig}).
are stored as vectors of times within a cell array.
\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{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{Interspikeintervalle}{interspike
intervals}. The statistics of interspike intervals are described
using common measures for describing the statistics of stochastic
real-valued variables:
referred to as \entermde[interspike
interval]{Interspikeintervall}{interspike intervals}, in short
\entermde[ISI|see{interspike
interval}]{ISI|see{Interspikeintervall}}{ISI}s. The statistics of
interspike intervals are described using common measures for
describing the statistics of real-valued variables:
\begin{figure}[t]
\includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
@ -110,19 +155,45 @@ 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 $\int_0^{\infty} p(T) \; dT
= 1$.
\item Average interval: $\mu_{ISI} = \langle T \rangle =
\frac{1}{n}\sum\limits_{i=1}^n T_i$.
\item Standard deviation of the interspike intervals: $\sigma_{ISI} = \sqrt{\langle (T - \langle T
\rangle)^2 \rangle}$\vspace{1ex}
\item \entermde[coefficient of variation]{Variationskoeffizient}{Coefficient of variation}:
$CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$.
\item \entermde[diffusion coefficient]{Diffusionskoeffizient}{Diffusion coefficient}: $D_{ISI} =
\frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
(\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \;
dT = 1$. Commonly referred to as the \enterm[interspike
interval!histogram]{interspike interval histogram}. Its shape
reveals many interesting aspects like locking or bursting that
cannot be inferred from the mean or standard deviation. A particular
reference is the exponential distribution of intervals
\begin{equation}
\label{hompoissonexponential}
p_{exp}(T) = \lambda e^{-\lambda T}
\end{equation}
of a homogeneous Poisson spike train with rate $\lambda$.
\item Mean interval: $\mu_{ISI} = \langle T \rangle =
\frac{1}{n}\sum\limits_{i=1}^n T_i$. The average time it takes from
one event to the next. The inverse of the mean interval is identical
with the mean rate $\lambda$ (number of events per time, see below)
of the process.
\item Standard deviation of intervals: $\sigma_{ISI} = \sqrt{\langle
(T - \langle T \rangle)^2 \rangle}$. Periodically spiking neurons
have little variability in their intervals, whereas many cortical
neurons cover a wide range with their intervals. The standard
deviation of homogeneous Poisson spike trains, $\sigma_{ISI} =
\frac{1}{\lambda}$, also equals the inverse rate. Whether the
standard deviation of intervals is low or high, however, is better
quantified by the
\item \entermde[coefficient of
variation]{Variationskoeffizient}{Coefficient of variation}, the
standard deviation of the ISIs relative to their mean:
\begin{equation}
\label{cvisi}
CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}
\end{equation}
Homogeneous Poisson spike trains have an CV of exactly one. The
lower the CV the more regularly firing a neuron is firing. CVs
larger than one are also possible in spike trains with small
intervals separated by really long ones.
%\item \entermde[diffusion coefficient]{Diffusionskoeffizient}{Diffusion coefficient}: $D_{ISI} =
% \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
\end{itemize}
\begin{exercise}{isihist.m}{}
@ -142,12 +213,33 @@ real-valued variables:
\end{exercise}
\subsection{Interval correlations}
So called \entermde[return map]{return map}{return maps} are used to
illustrate interdependencies between successive interspike
intervals. The return map plots the delayed interval $T_{i+k}$ against
the interval $T_i$. The parameter $k$ is called the \enterm{lag}
(\determ{Verz\"ogerung}) $k$. Stationary and non-stationary return
maps are distinctly different \figref{returnmapfig}.
Intervals are not just numbers without an order, like weights of
tigers. Intervals are temporally ordered and there could be temporal
structure in the sequence of intervals. For example, short intervals
could be followed by more longer ones, and vice versa. Such
dependencies in the sequence of intervals do not show up in the
interval histogram. We need additional measures to also quantify the
temporal structure of the sequence of intervals.
We can use the same techniques we know for visualizing and quantifying
correlations in bivariate data sets, i.e. scatter plots and
correlation coefficients. We form $(x,y)$ data pairs by taking the
series of intervals $T_i$ as $x$-data values and pairing them with the
$k$-th next intervals $T_{i+k}$ as $y$-data values. The parameter $k$
is called \enterm{lag} (\determ{Verz\"ogerung}). For lag one we pair
each interval with the next one. A \entermde[return map]{return
map}{Return map} illustrates dependencies between successive
intervals by simply plotting the intervals $T_{i+k}$ against the
intervals $T_i$ in a scatter plot (\figref{returnmapfig}). For Poisson
spike trains there is no structure beyond the one expected from the
exponential interspike interval distribution, hinting at neighboring
interspike intervals being independent of each other. For the spike
train based on an Ornstein-Uhlenbeck process the return map is more
clustered along the diagonal, hinting at a positive correlation
between succeeding intervals. That is, short intervals are more likely
to be followed by short ones and long intervals more likely by long
ones. This temporal structure was already clearly visible in the spike
raster shown in \figref{rasterexamplesfig}.
\begin{figure}[tp]
\includegraphics[width=1\textwidth]{serialcorrexamples}
@ -165,16 +257,34 @@ 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
correlation coefficient of the intervals $T_i$ and the intervals
delayed by the lag $T_{i+k}$:
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)}
= {\rm corr}(T_{i+k}, T_i) \] The resulting correlation coefficient
$\rho_k$ is usually plotted against the lag $k$
\figref{returnmapfig}. $\rho_0=1$ is the correlation of each interval
with itself and is always 1.
correlations}. These are the correlation coefficients between the
intervals $T_{i+k}$ and $T_i$ in dependence on lag $k$:
\begin{equation}
\label{serialcorrelation}
\rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)}
= {\rm corr}(T_{i+k}, T_i)
\end{equation}
The serial correlations $\rho_k$ are usually plotted against the lag
$k$ for a range small range of lags
(\figref{returnmapfig}). $\rho_0=1$ is the correlation of each
interval with itself and always equals one.
If the serial correlations all equal zero, $\rho_k =0$ for $k>0$, then
the length of an interval is independent of all the previous
ones. Such a process is a \enterm{renewal process}
(\determ{Erneuerungsprozess}). Each event, each action potential,
erases the history. The occurrence of the next event is independent of
what happened before. To a first approximation an action potential
erases all information about the past from the membrane voltage and
thus spike trains may approximate renewal processes.
However, other variables like the intracellular calcium concentration
or the states of slowly switching ion channels may carry information
from one interspike interval to the next and thus introducing
correlations. Such non-renewal dynamics can then be described by the
non-zero serial correlations (\figref{returnmapfig}).
\begin{exercise}{isiserialcorr.m}{}
Implement a function \varcode{isiserialcorr()} that takes a vector of
@ -204,7 +314,7 @@ 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
@ -216,7 +326,14 @@ 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 $n_i$. 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:
$\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
@ -224,20 +341,29 @@ counts can be quantified in the usual ways:
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 on the chosen window length
$W$. The average spike count, for example, grows linearly with $W$ for
sufficiently large time windows: $\langle n \rangle = r W$,
\eqnref{firingrate}. Doubling the counting window doubles the spike
count. As does the spike-count variance (\figref{fanofig}). At smaller
time windows the statistics of the event counts might depend on the
particular duration of the counting window. There might be an optimal
time window for which the variance of the spike count is minimal. The
Fano factor plotted as a function of the time window illustrates such
properties of point processes (\figref{fanofig}).
Note that all of these statistics depend in general on the chosen
window length $W$. The average spike count, for example, grows
linearly with $W$ for sufficiently large time windows: $\langle n
\rangle = r W$, \eqnref{firingrate}. Doubling the counting window
doubles the spike count. As does the spike-count variance
(\figref{fanofig}). At smaller time windows the statistics of the
event counts might depend on the particular duration of the counting
window. There might be an optimal time window for which the variance
of the spike count is minimal. The Fano factor plotted as a function
of the time window illustrates such properties of point processes
(\figref{fanofig}).
This also has consequences for information transmission in neural
systems. The lower the variance in spike count relative to the
@ -255,9 +381,9 @@ information encoded in the mean spike count is transmitted.
always equals the mean counts and consequently the Fano factor
equals one irrespective of the count window (top). A spike train
with positive correlations between interspike intervals (caused by
Ohrnstein-Uhlenbeck noise) has a minimum in the Fano factor, that
is an analysis window for which the relative count variance is
minimal somewhere close to the correlation time scale of the
an Ornstein-Uhlenbeck process) has a minimum in the Fano factor,
that is an analysis window for which the relative count variance
is minimal somewhere close to the correlation time scale of the
interspike intervals (bottom).}
\end{figure}
@ -271,99 +397,6 @@ information encoded in the mean spike count is transmitted.
\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}