412 lines
13 KiB
TeX
412 lines
13 KiB
TeX
\documentclass{beamer}
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%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\title[]{Scientific Computing --- Point Processes}
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\author[]{Jan Benda}
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\institute[]{Neuroethology}
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\date[]{WS 14/15}
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\titlegraphic{\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
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%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[english]{babel}
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\usepackage{amsmath}
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%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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language=Matlab,
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\begin{frame}[plain]
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\frametitle{}
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\vspace{-1cm}
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\titlepage % erzeugt Titelseite
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
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\frametitle{Content}
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\tableofcontents
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Point processes}
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\begin{frame}
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\frametitle{Point process}
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\vspace{-3ex}
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\includegraphics{pointprocesssketch}
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A point process is a stochastic (or random) process that generates a sequence of events
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at times $\{t_i\}$, $t_i \in \reZ$.
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For each point process there is an underlying continuous-valued
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process evolving in time. The associated point process occurs when
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the underlying continuous process crosses a threshold.
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Examples:
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\begin{itemize}
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\item Spikes/heartbeat: generated by the dynamics of the membrane potential of neurons/heart cells.
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\item Earth quakes: generated by the pressure dynamics between the tectonic plates on either side of a geological fault line.
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\item Onset of cricket/frogs/birds/... songs: generated by the dynamics of the state of a nervous system.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Point process}
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\includegraphics{pointprocesssketch}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Homogeneous Poisson process}
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\begin{frame}
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\frametitle{Homogeneous Poisson process}
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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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\includegraphics[width=1\textwidth]{poissonraster100hz}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Interval statistics}
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\begin{frame}
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\frametitle{Rate}
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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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\end{frame}
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\begin{frame}
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\frametitle{(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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\vfill
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\end{itemize}
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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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\end{frame}
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\begin{frame}
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\frametitle{Interval statistics of homogeneous Poisson process}
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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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\end{itemize}
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\vfill
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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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\end{frame}
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\begin{frame}
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\frametitle{Interval return maps}
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Scatter plot between succeeding intervals separated by lag $k$.
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\vfill
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Poisson process $\lambda=100$\,Hz:
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\includegraphics[width=1\textwidth]{poissonreturnmap100hz}\hfill
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\end{frame}
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\begin{frame}
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\frametitle{Serial interval correlations}
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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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\begin{itemize}
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\item $\rho_0=1$ (correlation of each interval with itself).
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\item Poisson process: $\rho_k =0$ for $k>0$ (renewal process!)
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\end{itemize}
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\vfill
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\includegraphics[width=0.7\textwidth]{poissonserial100hz}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Count statistics}
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\begin{frame}
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\frametitle{Count statistics}
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Histogram of number of events $N$ (counts) within observation window of duration $W$.
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\vfill
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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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\end{frame}
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\begin{frame}
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\frametitle{Count statistics of Poisson process}
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Poisson distribution:
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\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
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\vfill
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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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\end{frame}
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\begin{frame}
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\frametitle{Count statistics --- Fano factor}
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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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\item Poisson process: $F=1$
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\end{itemize}
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\vfill
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Poisson process $\lambda=100$\,Hz:
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\includegraphics[width=1\textwidth]{poissonfano100hz}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Integrate-and-fire models}
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\begin{frame}
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\frametitle{Integrate-and-fire models}
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Leaky integrate-and-fire model (LIF):
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\[ \tau \frac{dV}{dt} = -V + RI + D\xi \]
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Whenever membrane potential $V(t)$ crosses the firing threshold $\theta$, a spike is emitted and
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$V(t)$ is reset to $V_{reset}$.
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\begin{itemize}
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\item $\tau$: membrane time constant (typically 10\,ms)
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\item $R$: input resistance (here 1\,mV (!))
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\item $D\xi$: additive Gaussian white noise of strength $D$
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\item $\theta$: firing threshold (here 10\,mV)
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\item $V_{reset}$: reset potential (here 0\,mV)
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Integrate-and-fire models}
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Discretization with time step $\Delta t$: $V(t) \rightarrow V_i,\;t_i = i \Delta t$.\\
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Euler integration:
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\begin{eqnarray*}
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\frac{dV}{dt} & \approx & \frac{V_{i+1} - V_i}{\Delta t} \\
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\Rightarrow \quad V_{i+1} & = & V_i + \Delta t \frac{-V_i+RI_i+\sqrt{2D\Delta t}N_i}{\tau}
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\end{eqnarray*}
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$N_i$ are normally distributed random numbers (Gaussian with zero mean and unit variance)
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--- the $\sqrt{\Delta t}$ is for white noise.
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\includegraphics[width=0.82\textwidth]{lifraster16}
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\end{frame}
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\begin{frame}
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\frametitle{Interval statistics of LIF}
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Interval distribution approaches Inverse Gaussian for large $I$:
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\[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \]
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where $\langle T \rangle$ is the mean interspike interval and $D$
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is the diffusion coefficient.
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\vfill
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\includegraphics[width=0.45\textwidth]{lifisihdistr08}\hfill
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\includegraphics[width=0.45\textwidth]{lifisihdistr16}
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\end{frame}
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\begin{frame}
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\frametitle{Interval statistics of PIF}
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For the perfect integrate-and-fire (PIF)
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\[ \tau \frac{dV}{dt} = RI + D\xi \]
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(the canonical model or supra-threshold firing on a limit cycle)\\
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the Inverse Gaussian describes exactly the interspike interval distribution.
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\vfill
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\includegraphics[width=0.45\textwidth]{pifisihdistr01}\hfill
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\includegraphics[width=0.45\textwidth]{pifisihdistr10}
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\end{frame}
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\begin{frame}
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\frametitle{Interval return map of LIF}
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LIF $I=15.7$:
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\includegraphics[width=1\textwidth]{lifreturnmap16}
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\end{frame}
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\begin{frame}
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\frametitle{Serial correlations of LIF}
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LIF $I=15.7$:
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\includegraphics[width=1\textwidth]{lifserial16}\\
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Integrate-and-fire driven with white noise are still renewal processes!
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\end{frame}
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\begin{frame}
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\frametitle{Count statistics of LIF}
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LIF $I=15.7$:
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\includegraphics[width=1\textwidth]{liffano16}\\
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Fano factor is not one!
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
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\frametitle{Interval statistics of LIF with OU noise}
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\begin{eqnarray*}
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\tau \frac{dV}{dt} & = & -V + RI + U \\
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\tau_{OU} \frac{dU}{dt} & = & - U + D\xi
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\end{eqnarray*}
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Ohrnstein-Uhlenbeck noise is lowpass filtered white noise.
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\includegraphics[width=0.45\textwidth]{lifouisihdistr08-100ms}\hfill
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\includegraphics[width=0.45\textwidth]{lifouisihdistr16-100ms}\\
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More peaky than the inverse Gaussian!
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\end{frame}
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\begin{frame}
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\frametitle{Interval return map of LIF with OU noise}
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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{frame}
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\begin{frame}
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\frametitle{Serial correlations of LIF with OU noise}
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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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OU-noise introduces positive interval correlations!
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\end{frame}
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\begin{frame}
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\frametitle{Count statistics of LIF with OU noise}
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LIF $I=15.7$, $\tau_{OU}=100$\,ms:
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\includegraphics[width=1\textwidth]{lifoufano16-100ms}\\
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Fano factor increases with count window duration.
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
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\frametitle{Interval statistics of LIF with adaptation}
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\begin{eqnarray*}
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\tau \frac{dV}{dt} & = & -V - A + RI + D\xi \\
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\tau_{adapt} \frac{dA}{dt} & = & - A
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\end{eqnarray*}
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Adaptation $A$ with time constant $\tau_{adapt}$ and increment $\Delta A$ at spike.
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\includegraphics[width=0.45\textwidth]{lifadaptisihdistr08-100ms}\hfill
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\includegraphics[width=0.45\textwidth]{lifadaptisihdistr65-100ms}\\
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Similar to LIF with white noise.
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\end{frame}
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\begin{frame}
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\frametitle{Interval return map of LIF with adaptation}
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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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Negative correlation at lag one.
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\end{frame}
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\begin{frame}
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\frametitle{Serial correlations of LIF with adaptation}
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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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Adaptation with white noise introduces negative interval correlations!
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\end{frame}
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\begin{frame}
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\frametitle{Count statistics of LIF with adaptation}
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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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Fano factor decreases with count window duration.
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\end{frame}
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\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Non stationary}
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\subsection{Inhomogeneous Poisson process}
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\subsection{Firing rate}
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\subsection{Instantaneous rate}
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\subsection{Autocorrelation}
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\subsection{Crosscorrelation}
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\subsection{Joint PSTH}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Renewal process}
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\subsection{Superthreshold firing}
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\subsection{Subthreshold firing}
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\section{Non-renewal processes}
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\subsection{Bursting}
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\subsection{Resonator}
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\subsection{Standard distributions}
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\subsubsection{Gamma}
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\subsubsection{How to read ISI histograms}
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refractoriness, poisson tail, sub-, supra-threshold, missed spikes
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Correlation with stimulus}
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\subsection{Tuning curve}
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\subsection{Linear filter}
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\subsection{Spatiotemporal receptive field}
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\subsection{Generalized linear model}
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\begin{frame}
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\end{frame}
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