[pointprocesses] better incorporated Poisson spike train
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header.tex
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header.tex
@ -222,11 +222,26 @@
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{\vspace{-2ex}\lstset{#1}\noindent\minipage[t]{1\linewidth}}%
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{\vspace{-2ex}\lstset{#1}\noindent\minipage[t]{1\linewidth}}%
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{\endminipage}
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%%%%% english, german, code and file terms: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%% english and german terms: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{ifthen}
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\usepackage{ifthen}
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% \enterm[english index entry]{<english term>}
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\newcommand{\enterm}[2][]{\textit{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[enterm]{#2}}{\protect\sindex[enterm]{#1}}}
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\newcommand{\enterm}[2][]{\textit{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[enterm]{#2}}{\protect\sindex[enterm]{#1}}}
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@ -93,6 +93,8 @@ def plot_hom_isih(ax):
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ax.set_ylim(0.0, 31.0)
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ax.set_ylim(0.0, 31.0)
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ax.set_xticks(np.arange(0.0, 151.0, 50.0))
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ax.set_xticks(np.arange(0.0, 151.0, 50.0))
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ax.set_yticks(np.arange(0.0, 31.0, 10.0))
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ax.set_yticks(np.arange(0.0, 31.0, 10.0))
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tt = np.linspace(0.0, 0.15, 100)
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ax.plot(1000.0*tt, rate*np.exp(-rate*tt), **lsB)
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plotisih(ax, isis(homspikes), 0.005)
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plotisih(ax, isis(homspikes), 0.005)
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@ -34,7 +34,7 @@ process]{Punktprozess}{point processes}.
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trial. Shown is a stationary point process (homogeneous point
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trial. Shown is a stationary point process (homogeneous point
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process with a rate $\lambda=20$\;Hz, left) and an non-stationary
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process with a rate $\lambda=20$\;Hz, left) and an non-stationary
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point process with a rate that varies in time (noisy perfect
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point process with a rate that varies in time (noisy perfect
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integrate-and-fire neuron driven by Ohrnstein-Uhlenbeck noise with
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integrate-and-fire neuron driven by Ornstein-Uhlenbeck noise with
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a time-constant $\tau=100$\,ms, right).}
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a time-constant $\tau=100$\,ms, right).}
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\end{figure}
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\end{figure}
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@ -87,190 +87,6 @@ certain time window $n_i$ (\figref{pointprocesssketchfig}).
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are stored as vectors of times within a cell array.
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are stored as vectors of times within a cell array.
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\end{exercise}
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Interval statistics}
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The intervals $T_i=t_{i+1}-t_i$ between successive events are real
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positive numbers. In the context of action potentials they are
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referred to as \entermde{Interspikeintervalle}{interspike
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intervals}. The statistics of interspike intervals are described
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using common measures for describing the statistics of stochastic
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real-valued variables:
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\begin{figure}[t]
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\includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
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\titlecaption{\label{isihexamplesfig}Interspike-interval
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histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
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\end{figure}
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\begin{exercise}{isis.m}{}
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Implement a function \varcode{isis()} that calculates the interspike
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intervals from several spike trains. The function should return a
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single vector of intervals. The spike times (in seconds) of each
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trial are stored as vectors within a cell-array.
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\end{exercise}
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%\subsection{First order interval statistics}
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\begin{itemize}
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\item Probability density $p(T)$ of the intervals $T$
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(\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \; dT
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= 1$.
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\item Average interval: $\mu_{ISI} = \langle T \rangle =
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\frac{1}{n}\sum\limits_{i=1}^n T_i$.
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\item Standard deviation of the interspike intervals: $\sigma_{ISI} = \sqrt{\langle (T - \langle T
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\rangle)^2 \rangle}$\vspace{1ex}
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\item \entermde[coefficient of variation]{Variationskoeffizient}{Coefficient of variation}:
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$CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$.
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\item \entermde[diffusion coefficient]{Diffusionskoeffizient}{Diffusion coefficient}: $D_{ISI} =
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\frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
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\end{itemize}
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\begin{exercise}{isihist.m}{}
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Implement a function \varcode{isiHist()} that calculates the normalized
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interspike interval histogram. The function should take two input
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arguments; (i) a vector of interspike intervals and (ii) the width
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of the bins used for the histogram. It further returns the
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probability density as well as the centers of the bins.
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\end{exercise}
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\begin{exercise}{plotisihist.m}{}
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Implement a function that takes the return values of
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\varcode{isiHist()} as input arguments and then plots the data. The
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plot should show the histogram with the x-axis scaled to
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milliseconds and should be annotated with the average ISI, the
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standard deviation and the coefficient of variation.
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\end{exercise}
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\subsection{Interval correlations}
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So called \entermde[return map]{return map}{return maps} are used to
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illustrate interdependencies between successive interspike
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intervals. The return map plots the delayed interval $T_{i+k}$ against
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the interval $T_i$. The parameter $k$ is called the \enterm{lag}
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(\determ{Verz\"ogerung}) $k$. Stationary and non-stationary return
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maps are distinctly different \figref{returnmapfig}.
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\begin{figure}[tp]
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\includegraphics[width=1\textwidth]{serialcorrexamples}
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\titlecaption{\label{returnmapfig}Interspike-interval
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correlations}{of the spike trains shown in
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\figref{rasterexamplesfig}. Upper panels show the return maps and
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lower panels the serial correlations of successive intervals
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separated by lag $k$. All the interspike intervals of the
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stationary spike trains are independent of each other --- this is
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a so called \enterm{renewal process}
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(\determ{Erneuerungsprozess}). In contrast, the ones of the
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non-stationary spike trains show positive correlations that decay
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for larger lags. The positive correlations in this example are
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caused by a common stimulus that slowly increases and decreases
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the mean firing rate of the spike trains.}
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\end{figure}
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Such dependencies can be further quantified using the
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\entermde[correlation!serial]{Korrelation!serielle}{serial
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correlations} \figref{returnmapfig}. The serial correlation is the
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correlation coefficient of the intervals $T_i$ and the intervals
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delayed by the lag $T_{i+k}$:
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\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)}
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= {\rm corr}(T_{i+k}, T_i) \] The resulting correlation coefficient
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$\rho_k$ is usually plotted against the lag $k$
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\figref{returnmapfig}. $\rho_0=1$ is the correlation of each interval
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with itself and is always 1.
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\begin{exercise}{isiserialcorr.m}{}
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Implement a function \varcode{isiserialcorr()} that takes a vector of
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interspike intervals as input argument and calculates the serial
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correlation. The function should further plot the serial
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correlation.
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Count statistics}
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\begin{figure}[t]
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\includegraphics{countexamples}
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\titlecaption{\label{countstatsfig}Count statistics.}{Probability
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distributions of counting $k$ events $k$ (blue) within windows of
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20\,ms (left) or 200\,ms duration (right) of a homogeneous Poisson
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spike train with a rate of 20\,Hz
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(\figref{rasterexamplesfig}). For Poisson spike trains these
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distributions are given by Poisson distributions (red).}
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\end{figure}
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The most commonly used measure for characterizing spike trains is the
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\enterm[firing rate!average]{average firing rate}. The firing rate $r$
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is the average number of spikes counted within some time interval $W$
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\begin{equation}
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\label{firingrate}
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r = \frac{\langle n \rangle}{W}
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\end{equation}
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and is neasured in Hertz. The average of the spike counts is taken
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over trials. For stationary spike trains (no change in statistics, in
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particular the firing rate, over time), the firing rate based on the
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spike count equals the inverse average interspike interval
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$1/\mu_{ISI}$.
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The firing rate based on an averaged spike counts is one example of
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many statistics based on event counts. Stationary spike trains can be
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split into many segments $i$, each of duration $W$, and the number of
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events $n_i$ in each of the segments can be counted. The integer event
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counts can be quantified in the usual ways:
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\begin{itemize}
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\item Histogram of the counts $n_i$ (\figref{countstatsfig}).
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\item Average number of counts: $\mu_n = \langle n \rangle$.
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\item Variance of counts:
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$\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
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\end{itemize}
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Because spike counts are unitless and positive numbers, the
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\begin{itemize}
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\item \entermde{Fano Faktor}{Fano factor} (variance of counts divided
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by average count): $F = \frac{\sigma_n^2}{\mu_n}$.
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\end{itemize}
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is an additional measure quantifying event counts.
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Note that all of these statistics depend on the chosen window length
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$W$. The average spike count, for example, grows linearly with $W$ for
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sufficiently large time windows: $\langle n \rangle = r W$,
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\eqnref{firingrate}. Doubling the counting window doubles the spike
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count. As does the spike-count variance (\figref{fanofig}). At smaller
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time windows the statistics of the event counts might depend on the
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particular duration of the counting window. There might be an optimal
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time window for which the variance of the spike count is minimal. The
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Fano factor plotted as a function of the time window illustrates such
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properties of point processes (\figref{fanofig}).
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This also has consequences for information transmission in neural
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systems. The lower the variance in spike count relative to the
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averaged count, the higher the signal-to-noise ratio at which
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information encoded in the mean spike count is transmitted.
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\begin{figure}[t]
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\includegraphics{fanoexamples}
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\titlecaption{\label{fanofig}
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Count variance and Fano factor.}{Variance of event counts as a
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function of mean counts obtained by varying the duration of the
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count window (left). Dividing the count variance by the respective
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mean results in the Fano factor that can be plotted as a function
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of the count window (right). For Poisson spike trains the variance
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always equals the mean counts and consequently the Fano factor
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equals one irrespective of the count window (top). A spike train
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with positive correlations between interspike intervals (caused by
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Ohrnstein-Uhlenbeck noise) has a minimum in the Fano factor, that
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is an analysis window for which the relative count variance is
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minimal somewhere close to the correlation time scale of the
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interspike intervals (bottom).}
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\end{figure}
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\begin{exercise}{counthist.m}{}
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Implement a function \varcode{counthist()} that calculates and plots
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the distribution of spike counts observed in a certain time
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window. The function should take two input arguments: (i) a
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cell-array of vectors containing the spike times in seconds observed
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in a number of trials, and (ii) the duration of the time window that
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is used to evaluate the counts.
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Homogeneous Poisson process}
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\section{Homogeneous Poisson process}
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@ -303,45 +119,6 @@ In an \entermde[Poisson process!inhomogeneous]{Poissonprozess!inhomogener}{inhom
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\eqnref{hompoissonprob} to generate the event times.
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\eqnref{hompoissonprob} to generate the event times.
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\end{exercise}
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\end{exercise}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{poissonraster100hz}
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\titlecaption{\label{hompoissonfig}Rasterplot of spikes of a
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homogeneous Poisson process with a rate $\lambda=100$\,Hz.}{}
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\end{figure}
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\begin{figure}[t]
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\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
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\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
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\titlecaption{\label{hompoissonisihfig}Distribution of interspike
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intervals of two Poisson processes.}{The processes differ in their
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rate (left: $\lambda=20$\,Hz, right: $\lambda=100$\,Hz). The red
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lines indicate the corresponding exponential interval distribution
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\eqnref{poissonintervals}.}
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\end{figure}
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The homogeneous Poisson process has the following properties:
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\begin{itemize}
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\item Intervals $T$ are exponentially distributed (\figref{hompoissonisihfig}):
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\begin{equation}
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\label{poissonintervals}
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p(T) = \lambda e^{-\lambda T} \; .
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\end{equation}
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\item The average interval is $\mu_{ISI} = \frac{1}{\lambda}$ .
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\item The variance of the intervals is $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$ .
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\item Thus, the coefficient of variation is always $CV_{ISI} = 1$ .
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\item The serial correlation is $\rho_k =0$ for $k>0$, since the
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occurrence of an event is independent of all previous events. Such a
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process is also called a \enterm{renewal process} (\determ{Erneuerungsprozess}).
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\item The number of events $k$ within a temporal window of duration
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$W$ is Poisson distributed:
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\begin{equation}
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\label{poissoncounts}
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P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!}
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\end{equation}
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(\figref{hompoissoncountfig})
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\item The Fano Faktor is always $F=1$ .
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\end{itemize}
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\begin{exercise}{hompoissonspikes.m}{}
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\begin{exercise}{hompoissonspikes.m}{}
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Implement a function \varcode{hompoissonspikes()} that uses a
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Implement a function \varcode{hompoissonspikes()} that uses a
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homogeneous Poisson process to generate spike events at a given rate
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homogeneous Poisson process to generate spike events at a given rate
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@ -353,16 +130,272 @@ The homogeneous Poisson process has the following properties:
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times.
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times.
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\end{exercise}
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Interval statistics}
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The intervals $T_i=t_{i+1}-t_i$ between successive events are real
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positive numbers. In the context of action potentials they are
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referred to as \entermde[interspike
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interval]{Interspikeintervall}{interspike intervals}, in short
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\entermde[ISI|see{interspike
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interval}]{ISI|see{Interspikeintervall}}{ISI}s. The statistics of
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interspike intervals are described using common measures for
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describing the statistics of real-valued variables:
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\begin{figure}[t]
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\begin{figure}[t]
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\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
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\includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
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\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
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\titlecaption{\label{isihexamplesfig}Interspike-interval
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\titlecaption{\label{hompoissoncountfig}Distribution of counts of a
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histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
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Poisson spike train.}{The count statistics was generated for two
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different windows of width $W=10$\,ms (left) and width $W=100$\,ms
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(right). The red line illustrates the corresponding Poisson
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distribution \eqnref{poissoncounts}.}
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||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{exercise}{isis.m}{}
|
||||||
|
Implement a function \varcode{isis()} that calculates the interspike
|
||||||
|
intervals from several spike trains. The function should return a
|
||||||
|
single vector of intervals. The spike times (in seconds) of each
|
||||||
|
trial are stored as vectors within a cell-array.
|
||||||
|
\end{exercise}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item Probability density $p(T)$ of the intervals $T$
|
||||||
|
(\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}{}
|
||||||
|
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
|
||||||
|
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
|
||||||
|
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.
|
||||||
|
\end{exercise}
|
||||||
|
|
||||||
|
\subsection{Interval correlations}
|
||||||
|
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}
|
||||||
|
\titlecaption{\label{returnmapfig}Interspike-interval
|
||||||
|
correlations}{of the spike trains shown in
|
||||||
|
\figref{rasterexamplesfig}. Upper panels show the return maps and
|
||||||
|
lower panels the serial correlations of successive intervals
|
||||||
|
separated by lag $k$. All the interspike intervals of the
|
||||||
|
stationary spike trains are independent of each other --- this is
|
||||||
|
a so called \enterm{renewal process}
|
||||||
|
(\determ{Erneuerungsprozess}). In contrast, the ones of the
|
||||||
|
non-stationary spike trains show positive correlations that decay
|
||||||
|
for larger lags. The positive correlations in this example are
|
||||||
|
caused by a common stimulus that slowly increases and decreases
|
||||||
|
the mean firing rate of the spike trains.}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
Such dependencies can be further quantified 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 $k$:
|
||||||
|
\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
|
||||||
|
interspike intervals as input argument and calculates the serial
|
||||||
|
correlation. The function should further plot the serial
|
||||||
|
correlation.
|
||||||
|
\end{exercise}
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Count statistics}
|
||||||
|
|
||||||
|
\begin{figure}[t]
|
||||||
|
\includegraphics{countexamples}
|
||||||
|
\titlecaption{\label{countstatsfig}Count statistics.}{Probability
|
||||||
|
distributions of counting $k$ events $k$ (blue) within windows of
|
||||||
|
20\,ms (left) or 200\,ms duration (right) of a homogeneous Poisson
|
||||||
|
spike train with a rate of 20\,Hz
|
||||||
|
(\figref{rasterexamplesfig}). For Poisson spike trains these
|
||||||
|
distributions are given by Poisson distributions (red).}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The most commonly used measure for characterizing spike trains is the
|
||||||
|
\enterm[firing rate!average]{average firing rate}. The firing rate $r$
|
||||||
|
is the average number of spikes counted within some time interval $W$
|
||||||
|
\begin{equation}
|
||||||
|
\label{firingrate}
|
||||||
|
r = \frac{\langle n \rangle}{W}
|
||||||
|
\end{equation}
|
||||||
|
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
|
||||||
|
$1/\mu_{ISI}$.
|
||||||
|
|
||||||
|
The firing rate based on an averaged spike counts is one example of
|
||||||
|
many statistics based on event counts. Stationary spike trains can be
|
||||||
|
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
|
||||||
|
\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
|
||||||
|
\begin{itemize}
|
||||||
|
\item \entermde{Fano Faktor}{Fano factor} (variance of counts divided
|
||||||
|
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
|
||||||
|
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
|
||||||
|
averaged count, the higher the signal-to-noise ratio at which
|
||||||
|
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).}
|
||||||
|
\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.
|
||||||
|
\end{exercise}
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
\section{Time-dependent firing rate}
|
\section{Time-dependent firing rate}
|
||||||
|
|||||||
Reference in New Issue
Block a user