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@ -13,9 +13,9 @@ information. Analyzing the statistics of spike times and their
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relation to sensory stimuli or motor actions is central to
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neuroscientific research. With multi-electrode arrays it is nowadays
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possible to record from hundreds or even thousands of neurons
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simultaneously. The open challenge is how to analyze such data sets in
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order to understand how neural systems work. Let's start with the
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basics in this chapter.
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simultaneously. The open challenge is how to analyze such huge data
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sets in smart ways in order to gain insights into the way neural
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systems work. Let's start with the basics in this chapter.
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The result of the pre-processing of electrophysiological recordings
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are series of spike times, which are termed \enterm[spike train]{spike
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@ -27,15 +27,16 @@ process]{Punktprozess}{point processes}.
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\begin{figure}[bt]
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\includegraphics[width=1\textwidth]{rasterexamples}
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\titlecaption{\label{rasterexamplesfig}Raster plot.}{Raster plots of
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ten trials of data illustrating the times of action
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potentials. Each vertical stroke illustrates the time at which an
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action potential was observed. Each row displays the events of one
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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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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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a time-constant $\tau=100$\,ms, right).}
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\titlecaption{\label{rasterexamplesfig}Raster plots of spike
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trains.}{Raster plots of ten trials of data illustrating the times
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of action potentials. Each vertical stroke illustrates the time at
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which an action potential was observed. Each row displays the
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events of one trial. Shown is a stationary point process
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(homogeneous point process with a rate $\lambda=20$\;Hz, left) and
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an non-stationary point process with a rate that varies in time
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(noisy perfect integrate-and-fire neuron driven by
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Ornstein-Uhlenbeck noise with a time-constant $\tau=100$\,ms,
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right).}
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\end{figure}
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@ -59,9 +60,10 @@ process]{Punktprozess}{point processes}.
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\begin{figure}[tb]
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\includegraphics{pointprocesssketch}
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\titlecaption{\label{pointprocesssketchfig} Statistics of point
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processes.}{A point process is a sequence of instances in time
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$t_i$ that can be also characterized by inter-event intervals
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$T_i=t_{i+1}-t_i$ and event counts $n_i$.}
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processes.}{A temporal point process is a sequence of events in
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time, $t_i$, that can be also characterized by the corresponding
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inter-event intervals $T_i=t_{i+1}-t_i$ and event counts $n_i$,
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i.e. the number of events that occurred so far.}
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\end{figure}
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\noindent
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@ -77,30 +79,108 @@ plot in which each vertical line indicates the time of an event. The
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event from two different point processes are shown in
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\figref{rasterexamplesfig}. In addition to the event times, point
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processes can be described using the intervals $T_i=t_{i+1}-t_i$
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between successive events or the number of observed events within a
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certain time window $n_i$ (\figref{pointprocesssketchfig}).
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\begin{exercise}{rasterplot.m}{}
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Implement a function \varcode{rasterplot()} that displays the times of
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action potentials within the first \varcode{tmax} seconds in a raster
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plot. The spike times (in seconds) recorded in the individual trials
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are stored as vectors of times within a cell array.
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\end{exercise}
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between successive events or the number of observed events $n_i$
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within a certain time window (\figref{pointprocesssketchfig}).
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In \enterm[point process!stationary]{stationary} point processes the
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statistics does not change over time. In particular, the rate of the
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process, the number of events per time, is constant. The homogeneous
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point process shown in \figref{rasterexamplesfig} on the left is an
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example of a stationary point process. Although locally within each
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trial there are regions with many events per time and others with long
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intervals between events, the average number of events within a small
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time window over trials does not change in time. In the first sections
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of this chapter we introduce various statistics for characterizing
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stationary point processes.
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On the other hand, the example shown in \figref{rasterexamplesfig} on
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the right is a non-stationary point process. The rate of the events in
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all trials first decreases and then increases again. This common
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change in rate may have been caused by some sensory input to the
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neuron. How to estimate temporal changes in event rates (firing rates)
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is covered in section~\ref{nonstationarysec}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Homogeneous Poisson process}
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Before we are able to start analyzing point processes, we need some
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data. As before, because we are working with computers, we can easily
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simulate them. To get us started we here briefly introduce the
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homogeneous Poisson process. The Poisson process is to point processes
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what the Gaussian distribution is to the statistics of real-valued
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data. It is a standard against which everything is compared.
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In a \entermde[Poisson
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process!homogeneous]{Poissonprozess!homogener}{homogeneous Poisson
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process} events at every given time, that is within every small time
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window of width $\Delta t$ occur with the same constant
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probability. This probability is independent of absolute time and
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independent of any events occurring before. To observe an event right
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after an event is as likely as to observe an event at some specific
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time later on.
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The simplest way to simulate events of a homogeneous Poisson process
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is based on the exponential distribution \eqref{hompoissonexponential}
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of event intervals. We randomly draw intervals from this distribution
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and then sum them up to convert them to event times. The only
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parameter of a homogeneous Poisson process is its rate. It defines how
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many events per time are expected.
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Here is a function that generates several trials of a homogeneous
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Poisson process:
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\lstinputlisting[caption={hompoissonspikes.m}]{hompoissonspikes.m}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Raster plot}
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Let's generate some events with this function and display them in a
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raster plot as in \figref{rasterexamplesfig}. For the raster plot we
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need to draw for each event a line at the corresponding time of the
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event and at the height of the corresponding trial. You can go with a
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one for-loop through the trials and then with another for-loop through
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the event times and every time plot a two-point line. This, however,
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is very slow. The fastest way is to concatenate the coordinates of all
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strokes into large vectors, separate the events by \varcode{nan}
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entries, and pass this to a single call of \varcode{plot()}:
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\pageinputlisting[caption={rasterplot.m}]{rasterplot.m}
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Adapt this function to your needs and use it where ever possible to
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illustrate your spike train data to your readers. They appreciate
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seeing your raw data and being able to judge the data for themselves
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before you go on analyzing firing rates, interspike-interval
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correlations, etc.
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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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Let's start with the interval statistics of stationary point
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processes. The intervals $T_i=t_{i+1}-t_i$ between successive events
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are real positive numbers. In the context of action potentials they
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are 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. For analyzing event
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intervals we can use all the usual statistics that we know from
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describing univariate data sets of real numbers:
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\begin{figure}[t]
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\includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
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\includegraphics[width=0.96\textwidth]{isihexamples}
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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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histograms}{of the spike trains shown in
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\figref{rasterexamplesfig}. The intervals of the homogeneous
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Poisson process (left) are exponentially distributed according to
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\eqnref{hompoissonexponential} (red). Typically for many sensory
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neurons that fire more regularly is an ISI histogram like the one
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shown on the right. There is a preferred interval where the
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distribution peaks. The distribution falls off quickly towards
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smaller intervals, and very small intervals are absent, probably
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because of refractoriness of the spike generator. The tail of the
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distribution again approaches an exponential distribution as for
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the Poisson process.}
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\end{figure}
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\begin{exercise}{isis.m}{}
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@ -110,44 +190,99 @@ real-valued variables:
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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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(\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \;
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dT = 1$. Commonly referred to as the \enterm[interspike
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interval!histogram]{interspike interval histogram}. Its shape
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reveals many interesting aspects like locking or bursting that
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cannot be inferred from the mean or standard deviation. A particular
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reference is the exponential distribution of intervals
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\begin{equation}
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\label{hompoissonexponential}
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p_{exp}(T) = \lambda e^{-\lambda T}
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\end{equation}
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of a homogeneous Poisson spike train with rate $\lambda$.
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\item Mean interval
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\begin{equation}
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\label{meanisi}
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\mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum_{i=1}^n T_i
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\end{equation}
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The average time it takes from one event to the next. For stationary
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point processes the inverse of the mean interval is identical with
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the mean rate $\lambda$ (number of events per time, see below) of
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the process.
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\item Standard deviation of intervals
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\begin{equation}
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\label{stdisi}
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\sigma_{ISI} = \sqrt{\langle (T - \langle T \rangle)^2 \rangle}
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\end{equation}
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Periodically spiking neurons have little variability in their
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intervals, whereas many cortical neurons cover a wide range with
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their intervals. The standard deviation of homogeneous Poisson spike
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trains also equals the inverse rate. Whether the standard deviation
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of intervals is low or high, however, is better quantified by the
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\item \entermde[coefficient of
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variation]{Variationskoeffizient}{Coefficient of variation}, the
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standard deviation of the ISIs relative to their mean:
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\begin{equation}
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\label{cvisi}
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CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}
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\end{equation}
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Homogeneous Poisson spike trains have an $CV$ of exactly one. The
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lower the $CV$ the more regularly a neuron is firing. $CV$s larger than
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one are also possible in spike trains with small intervals separated
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by really long ones.
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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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Implement a function \varcode{isihist()} that calculates the
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|
normalized interspike interval histogram. The function should take
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|
two input arguments; (i) a vector of interspike intervals and (ii)
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the width of the bins used for the histogram. It 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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|
Implement a function that takes the returned 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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|
standard deviation, and the coefficient of variation of the ISIs
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(\figref{isihexamplesfig}).
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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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|
Intervals are not just numbers without an order, like weights of
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|
tigers. Intervals are temporally ordered and there could be temporal
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|
structure in the sequence of intervals. For example, short intervals
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|
could be followed by more longer ones, and vice versa. Such
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dependencies in the sequence of intervals do not show up in the
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|
interval histogram. We need additional measures to also quantify the
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|
temporal structure of the sequence of intervals.
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|
We can use the same techniques we know for visualizing and quantifying
|
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|
|
correlations in bivariate data sets, i.e. scatter plots and
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|
|
correlation coefficients. We form $(x,y)$ data pairs by taking the
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|
|
series of intervals $T_i$ as $x$-data values and pairing them with the
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|
$k$-th next intervals $T_{i+k}$ as $y$-data values. The parameter $k$
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is called \enterm{lag} (\determ{Verz\"ogerung}). For lag one we pair
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each interval with the next one. A \entermde{return map}{return map}
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illustrates dependencies between successive intervals by simply
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plotting the intervals $T_{i+k}$ against the intervals $T_i$ in a
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scatter plot (\figref{returnmapfig}). For Poisson spike trains there
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is no structure beyond the one expected from the exponential
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interspike interval distribution, hinting at neighboring interspike
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intervals being independent of each other. For the spike train based
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on an Ornstein-Uhlenbeck process the return map is more clustered
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along the diagonal, hinting at a positive correlation between
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succeeding intervals. That is, short intervals are more likely to be
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followed by short ones and long intervals more likely by long
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ones. This temporal structure was already clearly visible in each
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trial of the spike raster shown in \figref{rasterexamplesfig} on the
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right.
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\begin{figure}[tp]
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\includegraphics[width=1\textwidth]{serialcorrexamples}
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@ -165,22 +300,46 @@ maps are distinctly different \figref{returnmapfig}.
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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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Such dependencies can be further quantified by
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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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correlations}. These quantify the correlations between successive
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intervals by Pearson's correlation coefficients between the intervals
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$T_{i+k}$ and $T_i$ in dependence on lag $k$:
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\begin{equation}
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\label{serialcorrelation}
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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)
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\end{equation}
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The serial correlations $\rho_k$ are usually plotted against the lag
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$k$ for a small range of lags (\figref{returnmapfig}). $\rho_0=1$ is
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the correlation of each interval with itself and always equals one.
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If the serial correlations all equal zero, $\rho_k =0$ for $k>0$, then
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the length of an interval is independent of all the previous
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ones. Such a process is a \enterm{renewal process}
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(\determ{Erneuerungsprozess}). Each event, each action potential,
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erases the history. The occurrence of the next event is independent of
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what happened before. To a first approximation an action potential
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erases all information about the past from the membrane voltage and
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thus spike trains may approximate renewal processes.
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However, other variables like the intracellular calcium concentration
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or the states of slowly switching ion channels may carry information
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from one interspike interval to the next and thus introduce
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correlations between intervals. Such non-renewal dynamics is then
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characterized by the non-zero serial correlations
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(\figref{returnmapfig}).
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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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Implement a function \varcode{isiserialcorr()} that takes a vector
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of interspike intervals as input argument and calculates the serial
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correlations up to some maximum lag.
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\end{exercise}
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\begin{exercise}{plotisiserialcorr.m}{}
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Implement a function \varcode{plotisiserialcorr()} that takes a
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vector of interspike intervals as input argument and generates a
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plot of the serial correlations.
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\end{exercise}
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@ -189,11 +348,11 @@ with itself and is always 1.
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\begin{figure}[t]
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\includegraphics{countexamples}
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\titlecaption{\label{countstatsfig}Count statistics.}{The
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distribution of the number of events $k$ (blue) counted within
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windows of 20\,ms (left) or 200\,ms duration (right) of the
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|
homogeneous Poisson spike train with a rate of 20\,Hz shown in
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|
\figref{rasterexamplesfig}. For Poisson spike trains these
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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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@ -204,7 +363,7 @@ is the average number of spikes counted within some time interval $W$
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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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|
and is measured 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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@ -216,28 +375,44 @@ 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 Histogram of the counts $n_i$ appropriately normalized to
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|
probability distributions. For homogeneous Poisson spike trains with
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|
rate $\lambda$ the resulting probability distributions follow a
|
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|
Poisson distribution (\figref{countstatsfig}), where the probability
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of counting $k$ events within a time window $W$ is given by
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|
\begin{equation}
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|
\label{poissondist}
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P(k) = \frac{(\lambda W)^k e^{\lambda W}}{k!}
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|
\end{equation}
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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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|
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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|
by average count)
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|
\begin{equation}
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|
\label{fano}
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|
F = \frac{\sigma_n^2}{\mu_n}
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|
\end{equation}
|
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|
is a commonly used measure for quantifying the variability of event
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|
|
counts relative to the mean number of events. In particular for
|
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|
homogeneous Poisson processes the Fano factor equals one,
|
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|
|
independently of the time window $W$.
|
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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
|
|
|
|
|
particular duration of the counting window. There might be an optimal
|
|
|
|
|
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
|
|
|
|
|
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}).
|
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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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|
|
@ -246,126 +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
|
|
|
|
|
Ohrnstein-Uhlenbeck noise) has a minimum in the Fano factor, that
|
|
|
|
|
\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 (bottom).}
|
|
|
|
|
interspike intervals (right).}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
\begin{exercise}{counthist.m}{}
|
|
|
|
|
Implement a function \varcode{counthist()} that calculates and plots
|
|
|
|
|
the distribution of spike counts observed in a certain time
|
|
|
|
|
window. The function should take two input arguments: (i) a
|
|
|
|
|
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{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
|
|
|
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in one trial. Apply \eqnref{poissonintervals} to generate the event
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times.
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window. The function should take two input arguments: a cell-array
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of vectors containing the spike times in seconds observed in a
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number of trials, and the duration of the time window that is used
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to evaluate the counts.
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\end{exercise}
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\begin{figure}[t]
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\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
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\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
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\titlecaption{\label{hompoissoncountfig}Distribution of counts of a
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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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Time-dependent firing rate}
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\label{nonstationarysec}
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So far we have discussed stationary spike trains. The statistical properties
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of these did not change within the observation time (stationary point
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@ -533,6 +615,7 @@ relevate time-scale.
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\end{exercise}
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\section{Spike-triggered Average}
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\label{stasec}
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The graphical representation of the neuronal activity alone is not
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sufficient tot investigate the relation between the neuronal response
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