168 lines
6.6 KiB
TeX
168 lines
6.6 KiB
TeX
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\chapter{\tr{Point processes}{Punktprozesse}}
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\begin{figure}[t]
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\texpicture{pointprocessscetchB}
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\caption{\label{pointprocessscetchfig}Ein Punktprozess ist eine
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Abfolge von Zeitpunkten $t_i$ die auch durch die Intervalle
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$T_i=t_{i+1}-t_i$ oder die Anzahl der Ereignisse $n_i$ beschrieben
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werden kann. }
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\end{figure}
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Ein zeitlicher Punktprozess ist ein stochastischer Prozess der eine Abfolge von Ereignissen zu den Zeiten $\{t_i\}$, $t_i \in \reZ$ generiert.
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Jeder Punktprozess wird durch einen sich in der Zeit kontinuierlichen
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entwickelnden Prozess generiert. Wann immer dieser Prozess eine Schwelle \"uberschreitet
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wird ein Ereigniss des Punktprozesses 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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Dynamic des Nervensystems und des Muskelapparates erzeugt.
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\end{itemize}
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\section{Rate eines Punktprozesses}
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Rate of events $r$ (``spikes per time'') measured in Hertz.
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\begin{itemize}
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\item Number of events $N$ per observation time $W$: $r = \frac{N}{W}$
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\item Without boundary effects: $r = \frac{N-1}{t_N-t_1}$
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\item Inverse interval: $r = \frac{1}{\mu_{ISI}}$
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\end{itemize}
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\section{Intervall Statistiken}
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\begin{figure}[t]
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\includegraphics[width=0.45\textwidth]{poissonisih100hz}\hfill
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\includegraphics[width=0.45\textwidth]{lifisih16}
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\caption{\label{isihfig}Interspike-Intervall Histogramme von einem Poisson Prozess (links)
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und einem Integrate-and-Fire Neuron (rechts).}
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\end{figure}
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\subsection{First order (Interspike) interval statistics}
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\begin{itemize}
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\item Histogram $p(T)$ of intervals $T$. Normalized to $\int_0^{\infty} p(T) \; dT = 1$
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\item Mean interval $\mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum\limits_{i=1}^n T_i$
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\item Variance of intervals $\sigma_{ISI}^2 = \langle (T - \langle T \rangle)^2 \rangle$\vspace{1ex}
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\item Coefficient of variation $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$
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\item Diffusion coefficient $D_{ISI} = \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$
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\end{itemize}
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\subsection{Interval return maps}
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Scatter plot between succeeding intervals separated by lag $k$.
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\begin{figure}[t]
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\begin{minipage}[t]{0.49\textwidth}
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LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
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\includegraphics[width=1\textwidth]{lifadaptreturnmap10-100ms}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.49\textwidth}
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LIF $I=15.7$, $\tau_{OU}=100$\,ms:\\
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\includegraphics[width=1\textwidth]{lifoureturnmap16-100ms}
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\end{minipage}
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\caption{\label{returnmapfig}Interspike-Intervall return maps.}
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\end{figure}
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\subsection{Serial correlations of the intervals}
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Correlation coefficients between succeeding intervals separated by lag $k$:
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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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$\rho_0=1$ (correlation of each interval with itself).
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\begin{figure}[t]
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\begin{minipage}[t]{0.49\textwidth}
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LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
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\includegraphics[width=1\textwidth]{lifadaptserial10-100ms}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.49\textwidth}
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LIF $I=15.7$, $\tau_{OU}=100$\,ms:\\
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\includegraphics[width=1\textwidth]{lifouserial16-100ms}
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\end{minipage}
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\caption{\label{serialcorrfig}Serial correlations.}
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\end{figure}
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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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\caption{\label{countstatsfig}Count Statistik.}
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\end{figure}
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Histogram of number of events $N$ (counts) within observation window of duration $W$.
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\subsection{Fano factor}
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\begin{figure}[t]
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\begin{minipage}[t]{0.49\textwidth}
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Poisson process $\lambda=100$\,Hz:\\
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\includegraphics[width=1\textwidth]{poissonfano100hz}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.49\textwidth}
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LIF $I=10$, $\tau_{adapt}=100$\,ms:\\
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\includegraphics[width=1\textwidth]{lifadaptfano10-100ms}
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\end{minipage}
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\caption{\label{fanofig}Fano factor.}
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\end{figure}
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Statistics of number of events $N$ within observation window of duration $W$.
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\begin{itemize}
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\item Mean count: $\mu_N = \langle N \rangle$
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\item Count variance: $\sigma_N^2 = \langle (N - \langle N \rangle)^2 \rangle$
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\item Fano factor (variance divided by mean): $F = \frac{\sigma_N^2}{\mu_N}$
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\end{itemize}
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\section{\tr{Homogeneous Poisson process}{Homogener Poisson Prozess}}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{poissonraster100hz}
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\caption{\label{hompoissonfig}Rasterplot von Poisson-Spikes.}
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\end{figure}
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The probability $p(t)\delta t$ of an event occuring at time $t$
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is independent of $t$ and independent of any previous event
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(independent of event history).
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The probability $P$ for an event occuring within a time bin of width $\Delta t$
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is
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\[ P=\lambda \cdot \Delta t \]
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for a Poisson process with rate $\lambda$.
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\subsection{Statistics of homogeneous Poisson process}
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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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\caption{\label{hompoissonisihfig}Interspike interval histograms of poisson spike train.}
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\end{figure}
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\begin{itemize}
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\item Exponential distribution of intervals $T$: $p(T) = \lambda e^{-\lambda T}$
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\item Mean interval $\mu_{ISI} = \frac{1}{\lambda}$
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\item Variance of intervals $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$
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\item Coefficient of variation $CV_{ISI} = 1$
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\item Serial correlation $\rho_k =0$ for $k>0$ (renewal process!)
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\item Fano factor $F=1$
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\end{itemize}
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\subsection{Count statistics of Poisson process}
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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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\caption{\label{hompoissoncountfig}Count statistics of poisson spike train.}
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\end{figure}
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Poisson distribution:
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\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \] |