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@ -1,186 +1,186 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\chapter{Analyse von Spiketrains}
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\chapter{Spiketrain analysis}
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\selectlanguage{ngerman}
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\selectlanguage{english}
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\determ[Aktionspotential]{Aktionspotentiale} (\enterm{spikes}) sind die Tr\"ager der
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Information in Nervensystemen. Dabei ist in erster Linie nur der
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Zeitpunkt des Auftretens eines Aktionspotentials von Bedeutung. Die
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genaue Form des Aktionspotentials spielt keine oder nur eine
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untergeordnete Rolle.
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\enterm[Actionspotentials]{Actionspotentials} (\enterm{spikes}) are
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the carriers of information in the nervous system. Thereby it is
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mainly the time at which the spikes are generated that is of
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importance. The waveform of the action potential is largely
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stereotyped and does not carry information.
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Nach etwas Vorverarbeitung haben elektrophysiologische Messungen
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deshalb Listen von Spikezeitpunkten als Ergebniss --- sogenannte
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\enterm{spiketrains}. Diese Messungen k\"onnen wiederholt werden und
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es ergeben sich mehrere \enterm{trials} von Spiketrains
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(\figref{rasterexamplesfig}).
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The result of the processing of electrophysiological recordings are
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series of spike times, which are then termed \enterm{spiketrains}. If
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measurements are repeated we yield several \enterm{trials} of
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spiketrains (\figref{rasterexamplesfig}).
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Spiketrains sind Zeitpunkte von Ereignissen --- den Aktionspotentialen
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--- und deren Analyse f\"allt daher in das Gebiet der Statistik von
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sogenannten \determ[Punktprozess]{Punktprozessen}.
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Spiketrains are times of events, the action potentials. The analysis
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of these leads into the realm of the so called \enterm[point
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process]{point processes}.
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\begin{figure}[ht]
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\includegraphics[width=1\textwidth]{rasterexamples}
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\titlecaption{\label{rasterexamplesfig}Raster-Plot.}{Raster-Plot von
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jeweils 10 Realisierungen eines station\"arenen Punktprozesses
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(homogener Poisson Prozess mit Rate $\lambda=20$\;Hz, links) und
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eines nicht-station\"aren Punktprozesses (perfect
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integrate-and-fire Neuron getrieben mit Ohrnstein-Uhlenbeck
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Rauschen mit Zeitkonstante $\tau=100$\,ms, rechts). Jeder
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vertikale Strich markiert den Zeitpunkt eines Ereignisses.
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Jede Zeile zeigt die Ereignisse eines trials.}
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\titlecaption{\label{rasterexamplesfig}Raster-plot.}{Raster-plot of
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ten realizations of a stationary point process (homogeneous point
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process with a rate $\lambda=20$\;Hz, left) and an inhomogeneous
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point process (perfect integrate-and-fire neuron dirven by
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Ohrnstein-Uhlenbeck noise with a time-constant $\tau=100$\,ms,
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right). Each vertical dash illustrates the time at which the
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action potential was observed. Each line represents the event of
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each trial.}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Punktprozesse}
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\section{Point processes}
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Ein zeitlicher Punktprozess (\enterm{point process}) ist ein
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stochastischer Prozess, der eine Abfolge von Ereignissen zu den Zeiten
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$\{t_i\}$, $t_i \in \reZ$, generiert.
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A temporal \enterm{point process} is a stochastic process that
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generates a sequence of events at times $\{t_i\}$, $t_i \in
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\reZ$.
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\begin{ibox}{Beispiele von Punktprozessen}
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Jeder Punktprozess wird durch einen sich in der Zeit kontinuierlich
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entwickelnden Prozess generiert. Wann immer dieser Prozess eine
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Schwelle \"uberschreitet wird ein Ereigniss des Punktprozesses
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erzeugt. Zum Beispiel:
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\begin{itemize}
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\item Aktionspotentiale/Herzschlag: wird durch die Dynamik des
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Membranpotentials eines Neurons/Herzzelle erzeugt.
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\item Erdbeben: wird durch die Dynamik des Druckes zwischen
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tektonischen Platten auf beiden Seiten einer geologischen Verwerfung
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erzeugt.
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\item Zeitpunkt eines Grillen/Frosch/Vogelgesangs: wird durch die
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Dynamik des Nervensystems und des Muskelapparates erzeugt.
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\end{itemize}
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\begin{ibox}{Examples of point processes}
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Every point process is generated by a temporally continuously
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developing process. An event is generated whenever this process
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reaches a certain threshold. For example:
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\begin{itemize}
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\item Action potentials/heart beat: created by the dynamics of the
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neuron/sinoatrial node
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\item Earthquake: defined by the dynamics of the pressure between
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tectonical plates.
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\item Evoked communication calls in crickets/frogs/birds: shaped by
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the dynamics of nervous system and the muscle appartus.
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\end{itemize}
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\end{ibox}
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\begin{figure}[t]
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\texpicture{pointprocessscetch}
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\titlecaption{\label{pointprocessscetchfig} Statistik von
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Punktprozessen.}{Ein Punktprozess ist eine Abfolge von
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Zeitpunkten $t_i$ die auch durch die Intervalle $T_i=t_{i+1}-t_i$
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oder die Anzahl der Ereignisse $n_i$ beschrieben werden kann. }
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\titlecaption{\label{pointprocessscetchfig} 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 characterized through the inter-event-intervals
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$T_i=t_{i+1}-t_i$ and the number of events $n_i$. }
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\end{figure}
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F\"ur die Neurowissenschaften ist die Statistik der Punktprozesse
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besonders wichtig, da die Zeitpunkte der Aktionspotentiale als
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zeitlicher Punktprozess betrachtet werden k\"onnen und entscheidend
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f\"ur die Informations\"ubertragung sind.
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In the neurosciences, the statistics of point processes is of
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importance since the timing of the neuronal events (the action
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potentials) is crucial for information transmission and can be treated
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as such a process.
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Bei Punktprozessen k\"onnen wir die Zeitpunkte $t_i$ ihres Auftretens,
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die Intervalle zwischen diesen Zeitpunkten $T_i=t_{i+1}-t_i$, sowie
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die Anzahl der Ereignisse $n_i$ bis zu einer bestimmten Zeit betrachten
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(\figref{pointprocessscetchfig}).
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Zwei Punktprozesse mit verschiedenen Eigenschaften sind in
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\figref{rasterexamplesfig} als Rasterplot dargestellt, bei dem die
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Zeitpunkte der Ereignisse durch senkrechte Striche markiert werden.
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Point processes can be described using the intervals between
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successive events $T_i=t_{i+1}-t_i$ and the number of observed events
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within a certain time window $n_i$ (\figref{pointprocessscetchfig}).
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The events originating from a point process can be illustrated in form
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of a scatter- or raster plot in which each vertical line indicates the
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time of an event. The event from two different point processes are
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shown in \figref{rasterexamplesfig}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Intervallstatistik}
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\section{Intervalstatistics}
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Die Intervalle $T_i=t_{i+1}-t_i$ zwischen aufeinanderfolgenden
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Ereignissen sind reelle, positive Zahlen. Bei Aktionspotentialen
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heisen die Intervalle auch \determ{Interspikeintervalle}
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(\enterm{interspike intervals}). Deren Statistik kann mit den
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\"ublichen Gr\"o{\ss}en beschrieben werden.
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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 \enterm{interspike intervals}. The statistics of these
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are described using the common measures.
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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}Interspikeintervall Histogramme}{der in
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\figref{rasterexamplesfig} gezeigten Spikes.}
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\titlecaption{\label{isihexamplesfig}Interspike interval
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histogram}{of the spikes depicted in \figref{rasterexamplesfig}.}
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\end{figure}
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\begin{exercise}{isis.m}{}
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Schreibe eine Funktion \code{isis()}, die aus mehreren trials von Spiketrains die
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Interspikeintervalle bestimmt und diese in einem Vektor
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zur\"uckgibt. Jeder trial der Spiketrains ist ein Vektor mit den
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Spikezeiten gegeben in Sekunden als Element in einem \codeterm{cell-array}.
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Implement a function \code{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 action potentials recorded in the
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individual trials are stored as vectors of spike times within a
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\codeterm{cell-array}. Spike times are given in seconds.
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\end{exercise}
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\subsection{Intervallstatistik erster Ordnung}
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\subsection{First order interval statistics}
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\begin{itemize}
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\item Wahrscheinlichkeitsdichte $p(T)$ der Intervalle $T$
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(\figref{isihexamplesfig}). Normiert auf $\int_0^{\infty} p(T) \; dT
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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 Mittleres Intervall: $\mu_{ISI} = \langle T \rangle =
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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 Standardabweichung der Intervalle: $\sigma_{ISI} = \sqrt{\langle (T - \langle T
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\rangle)^2 \rangle}$\vspace{1ex}
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\item \determ{Variationskoeffizient} (\enterm{coefficient of variation}): $CV_{ISI} =
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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 \enterm{Coefficient of variation}: $CV_{ISI} =
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\frac{\sigma_{ISI}}{\mu_{ISI}}$.
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\item \determ{Diffusionskoeffizient} (\enterm{diffusion coefficient}): $D_{ISI} =
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\item \enterm{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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Schreibe eine Funktion \code{isiHist()}, die einen Vektor mit Interspikeintervallen
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entgegennimmt und daraus ein normiertes Histogramm der Interspikeintervalle
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berechnet.
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Implement a function \code{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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Schreibe eine Funktion, die die Histogrammdaten der Funktion
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\code{isiHist()} entgegennimmt, um das Histogramm zu plotten. Im
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Plot sollen die Interspikeintervalle in Millisekunden aufgetragen
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werden. Das Histogramm soll zus\"atzlich mit Mittelwert,
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Standardabweichung und Variationskoeffizient der
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Interspikeintervalle annotiert werden.
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Implement a function that takes the return values of
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\code{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{Korrelationen der Intervalle}
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In \enterm{return maps} werden die um das \enterm{lag} $k$ verz\"ogerten
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Intervalle $T_{i+k}$ gegen die Intervalle $T_i$ geplottet. Dies macht
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m\"ogliche Abh\"angigkeiten von aufeinanderfolgenden Intervallen
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sichtbar.
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\subsection{Interval correlations}
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So called \enterm{return maps} are used to illustrate
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interdependencies between successive interspike intervals. The return
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map plots the delayed interval $T_{i+k}$ against the interval
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$T_i$. The parameter $k$ is called the \enterm{lag} $k$. Stationary
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and non-stationary return maps are distinctly different
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\figref{returnmapfig}.
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{returnmapexamples}
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\includegraphics[width=1\textwidth]{serialcorrexamples}
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\titlecaption{\label{returnmapfig}Interspikeintervall return maps und
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serielle Korrelationen}{zwischen aufeinander folgenden Intervallen
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im Abstand des Lags $k$.}
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\titlecaption{\label{returnmapfig}Interspike interval analyses of a
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stationary and a non-stationary pointprocess.}{Upper plots show the
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return maps and the lower panels depict the serial correlation of
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successive intervals separated by the lag $k$.}
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\end{figure}
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Solche Ab\"angigkeiten werden durch die \determ{serielle
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Korrelationen} (\enterm{serial correlations}) der Intervalle
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quantifiziert. Das ist der \determ{Korrelationskoeffizient} zwischen
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aufeinander folgenden Intervallen getrennt durch lag $k$:
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Such dependencies can be further quantified using the \enterm{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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\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)}
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= {\rm corr}(T_{i+k}, T_i) \]
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\"Ublicherweise wird die Korrelation $\rho_k$ gegen den Lag $k$
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aufgetragen (\figref{returnmapfig}). $\rho_0=1$ (Korrelation jedes
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Intervalls mit sich selber).
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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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Schreibe eine Funktion \code{isiserialcorr()}, die einen Vektor mit Interspikeintervallen
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entgegennimmt und daraus die seriellen Korrelationen berechnet und plottet.
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\pagebreak[4]
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Implement a function \code{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. \pagebreak[4]
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Z\"ahlstatistik}
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\section{Count statistics}
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% \begin{figure}[t]
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% \includegraphics[width=0.48\textwidth]{poissoncounthist100hz10ms}\hfill
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% \includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
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% \titlecaption{\label{countstatsfig}Count Statistik.}{}
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% \end{figure}
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Die Anzahl der Ereignisse $n_i$ in Zeifenstern $i$ der
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L\"ange $W$ ergeben ganzzahlige, positive Zufallsvariablen, die meist
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durch folgende Sch\"atzer charakterisiert werden:
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The number of events $n_i$ (counts) in a time window $i$ of the duration $W$
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yields positive integer random numbers that are commonly quantified
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using the following measures:
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\begin{itemize}
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\item Histogramm der counts $n_i$.
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\item Mittlere Anzahl von Ereignissen: $\mu_N = \langle n \rangle$.
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\item Varianz der Anzahl: $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
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\item \determ{Fano Faktor} (Varianz geteilt durch Mittelwert): $F = \frac{\sigma_n^2}{\mu_n}$.
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\item Histogram of the counts $n_i$.
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\item Average number of events: $\mu_N = \langle n \rangle$.
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\item Variance of the counts: $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
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\item \determ{Fano Faktor} (The variance divided by the average): $F = \frac{\sigma_n^2}{\mu_n}$.
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\end{itemize}
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Insbesondere ist die mittlere Rate der Ereignisse $r$ (Spikes pro
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Zeit, \determ{Feuerrate}) gemessen in Hertz \sindex[term]{Feuerrate!mittlere Rate}
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And in particular the average firing rate $r$ (spike count per time interval
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, \determ{Feuerrate}) that is given in Hertz \sindex[term]{Feuerrate!mittlere Rate}
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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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@ -200,110 +200,114 @@ Zeit, \determ{Feuerrate}) gemessen in Hertz \sindex[term]{Feuerrate!mittlere Rat
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% \end{figure}
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\begin{exercise}{counthist.m}{}
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Schreibe eine Funktion \code{counthist()}, die aus mehreren trials
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von Spiketrains die Verteilung der Anzahl der Spikes in Fenstern
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einer der Funktion \"ubergegebenen Breite bestimmt, das Histogramm
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plottet und zur\"uckgibt. Jeder trial der Spiketrains ist ein Vektor
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mit den Spikezeiten gegeben in Sekunden als Element in einem
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\codeterm{cell-array}.
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Implement a function \code{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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\codeterm{cell-array} of vectors containing the spike times in
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seconds observed in a number of trials and (ii) the duration of the
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time window that is used to evaluate the counts.
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Homogener Poisson Prozess}
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F\"ur kontinuierliche Me{\ss}gr\"o{\ss}en ist die Normalverteilung
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u.a. wegen dem Zentralen Grenzwertsatz die Standardverteilung. Eine
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\"ahnliche Rolle spielt bei Punktprozessen der \determ{Poisson
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Prozess}.
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Beim \determ[Poisson Prozess!homogener]{homogenen Poisson Prozess}
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treten Ereignisse mit einer festen Rate $\lambda=\text{const.}$ auf
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und sind unabh\"angig von der Zeit $t$ und unabh\"angig von den
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Zeitpunkten fr\"uherer Ereignisse (\figref{hompoissonfig}). Die
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Wahrscheinlichkeit zu irgendeiner Zeit ein Ereigniss in einem kleinen
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Zeitfenster der Breite $\Delta t$ zu bekommen ist
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\section{Homogeneous Poisson process}
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The Gaussian distribution is, due to the central limit theorem, the
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standard for continuous measures. The equivalent in the realm of point
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processes is the \enterm{Poisson distribution}.
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In a \enterm[Poisson process!homogeneous]{homogeneous Poisson process}
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the events occur at a fixed rate $\lambda=\text{const.}$ and are
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independent of both the time $t$ and occurrence of previous events
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(\figref{hompoissonfig}). The probability of observing an even within a
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small time window of width $\Delta t$ is given by
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\begin{equation}
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\label{hompoissonprob}
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P = \lambda \cdot \Delta t \; .
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P = \lambda \cdot \Delta t \; .
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\end{equation}
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Beim \determ[Poisson Prozess!inhomogener]{inhomogenen Poisson Prozess}
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h\"angt die Rate $\lambda$ von der Zeit ab: $\lambda = \lambda(t)$.
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In an \enterm[Poisson process!inhomogeneous]{inhomogeneous Poisson
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process}, however, the rate $\lambda$ depends on the time: $\lambda =
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\lambda(t)$.
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\begin{exercise}{poissonspikes.m}{}
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Schreibe eine Funktion \code{poissonspikes()}, die die Spikezeiten
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eines homogenen Poisson-Prozesses mit gegebener Rate in Hertz f\"ur
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eine Anzahl von trials gegebener maximaler L\"ange in Sekunden in
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einem \codeterm{cell-array} zur\"uckgibt. Benutze \eqnref{hompoissonprob}
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um die Spikezeiten zu bestimmen.
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Implement a function \code{poissonspikes()} that uses a homogeneous
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Poisson process to generate events at a given rate for a certain
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duration and a number of trials. The rate should be given in Hertz
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and the duration of the trials is given in seconds. The function
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should return the event times in a cell-array. Each entry in this
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array represents the events observed in one trial. Apply
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\eqnref{hompoissonprob} to generate the event times.
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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 von Spikes eines homogenen
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|
Poisson Prozesses mit $\lambda=100$\,Hz.}{}
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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}Interspikeintervallverteilungen
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|
zweier Poissonprozesse.}{}
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\titlecaption{\label{hompoissonisihfig}Distribution of interspike intervals of two Poisson processes.}{}
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|
\end{figure}
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Der homogene Poissonprozess hat folgende Eigenschaften:
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|
The homogeneous Poisson process has the following properties:
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|
|
\begin{itemize}
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|
\item Die Intervalle $T$ sind exponentiell verteilt (\figref{hompoissonisihfig}):
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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 Das mittlere Intervall ist $\mu_{ISI} = \frac{1}{\lambda}$ .
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|
\item Die Varianz der Intervalle ist $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$ .
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|
\item Der Variationskoeffizient ist also immer $CV_{ISI} = 1$ .
|
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|
\item Die \determ[serielle Korrelationen]{seriellen Korrelationen}
|
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|
|
$\rho_k =0$ f\"ur $k>0$, da das Auftreten der Ereignisse
|
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|
|
unabh\"angig von der Vorgeschichte ist. Ein solcher Prozess wird
|
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|
|
auch \determ{Erneuerungsprozess} genannt (\enterm{renewal process}).
|
|
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|
|
\item Die Anzahl der Ereignisse $k$ innerhalb eines Fensters der
|
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|
|
L\"ange W ist \determ[Poisson-Verteilung]{Poissonverteilt}:
|
|
|
|
|
\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}.
|
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|
|
\item The number of events $k$ within a temporal window of duration
|
|
|
|
|
$W$ is Poisson distributed:
|
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|
|
|
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
|
|
|
|
|
(\figref{hompoissoncountfig})
|
|
|
|
|
\item Der \determ{Fano Faktor} ist immer $F=1$ .
|
|
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|
|
(\figref{hompoissoncountfig})
|
|
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|
|
\item The Fano Faktor is always $F=1$ .
|
|
|
|
|
\end{itemize}
|
|
|
|
|
|
|
|
|
|
\begin{exercise}{hompoissonspikes.m}{}
|
|
|
|
|
Schreibe eine Funktion \code{hompoissonspikes()}, die die Spikezeiten
|
|
|
|
|
eines homogenen Poisson-Prozesses mit gegebener Rate in Hertz f\"ur
|
|
|
|
|
eine Anzahl von trials gegebener maximaler L\"ange in Sekunden in
|
|
|
|
|
einem \codeterm{cell-array} zur\"uckgibt. Benutze die exponentiell-verteilten
|
|
|
|
|
Interspikeintervalle \eqnref{poissonintervals}, um die Spikezeiten zu erzeugen.
|
|
|
|
|
Implement a function \code{hompoissonspikes()} that uses a
|
|
|
|
|
homogeneous Poisson process to generate spike events at a given rate
|
|
|
|
|
for a certain duration and a number of trials. The rate should be
|
|
|
|
|
given in Hertz and the duration of the trials is given in
|
|
|
|
|
seconds. The function should return the event times in a
|
|
|
|
|
cell-array. Each entry in this array represents the events observed
|
|
|
|
|
in one trial. Apply \eqnref{poissonintervals} to generate the event
|
|
|
|
|
times.
|
|
|
|
|
\end{exercise}
|
|
|
|
|
|
|
|
|
|
\begin{figure}[t]
|
|
|
|
|
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
|
|
|
|
|
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
|
|
|
|
|
\titlecaption{\label{hompoissoncountfig}Z\"ahlstatistik von Poisson Spikes.}{}
|
|
|
|
|
\titlecaption{\label{hompoissoncountfig}Count statistics of Poisson
|
|
|
|
|
spiketrains.}{}
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
\section{Zeitabh\"angige Feuerraten}
|
|
|
|
|
|
|
|
|
|
Bisher haben wir station\"are Spiketrains betrachtet, deren Statistik
|
|
|
|
|
sich innerhalb der Analysezeit nicht ver\"andert (station\"are
|
|
|
|
|
Punktprozesse). Meistens jedoch \"andert sich die Statistik der
|
|
|
|
|
Spiketrains eines Neurons mit der Zeit. Z.B. kann ein sensorisches
|
|
|
|
|
Neuron auf einen Reiz hin mit einer erh\"ohten Feuerrate antworten
|
|
|
|
|
(nichtstation\"arer Punktprozess).
|
|
|
|
|
|
|
|
|
|
Wie die mittlere Anzahl der Spikes sich mit der Zeit ver\"andert, die
|
|
|
|
|
\determ{Feuerrate} $r(t)$, ist die wichtigste Gr\"o{\ss}e bei
|
|
|
|
|
nicht-station\"aren Spiketrains. Die Einheit der Feuerrate ist Hertz,
|
|
|
|
|
also Anzahl Aktionspotentiale pro Sekunde. Es gibt verschiedene
|
|
|
|
|
Methoden diese zu bestimmen. Drei solcher Methoden sind in Abbildung
|
|
|
|
|
\ref{psthfig} dargestellt. Alle Methoden haben ihre Berechtigung und
|
|
|
|
|
ihre Vor- und Nachteile. Im folgenden werden die drei Methoden aus
|
|
|
|
|
Abbildung \ref{psthfig} n\"aher erl\"autert.
|
|
|
|
|
\section{Time-dependent firing rate}
|
|
|
|
|
|
|
|
|
|
So far we discussed stationary spiketrains. The statistical properties
|
|
|
|
|
of these did not change within the observation time (stationary point
|
|
|
|
|
processes. Most commonly, however, this is not the case. A sensory
|
|
|
|
|
neuron, for example, might respond to a stimulus by modulating its
|
|
|
|
|
firing rate (non-stationary point process).
|
|
|
|
|
|
|
|
|
|
How the firing rate $r(t)$ changes over time is the most important
|
|
|
|
|
measure, when analyzing non-stationary spike trains. The unit of the
|
|
|
|
|
firing rate is Hertz, i.e. the number of action potentials per
|
|
|
|
|
second. There are different ways to estimate the firing rate and three
|
|
|
|
|
of these methods will are illustrated in \figref{psthfig}. All of
|
|
|
|
|
these have their own justifications and pros- and cons. In the
|
|
|
|
|
following we will discuss the methods shown in \figref{psthfig} more
|
|
|
|
|
closely.
|
|
|
|
|
|
|
|
|
|
\begin{figure}[tp]
|
|
|
|
|
\includegraphics[width=\columnwidth]{firingrates}
|
|
|
|
|
|
