added all my stuff

pointprocesses/exercises/hompoissonisih.m

@@ -0,0 +1,18 @@
+% generate spike times:
+rate = 20.0;
+spikes = hompoissonspikes( 10, rate, 50.0 );
+% isi histogram:
+isivec = isis( spikes );
+isihist( isivec );
+hold on
+% theoretical density:
+xmax = 5.0/rate;
+x = 0:0.0001:xmax;
+y = rate*exp(-rate*x);
+plot( 1000.0*x, y, 'r', 'LineWidth', 3 );
+% plot details:
+title( sprintf( 'Poisson spike trains, rate=%g Hz, nisi=%d', rate, length( isivec ) ) )
+xlim( [ 0.0 1000.0*xmax ] )
+ylim( [ 0.0 1.1*rate ] )
+legend( 'data', 'poisson' )
+hold off

pointprocesses/exercises/hompoissonisistats.m

@@ -0,0 +1,46 @@
+rates = 1:1:100;
+avisi = [];
+sdisi = [];
+cvisi = [];
+
+for rate = rates
+    spikes = hompoissonspikes( 10, rate, 100.0 );
+    isivec = isis( spikes );
+    av = mean( isivec );
+    sd = std( isivec );
+    cv = sd/av;
+    avisi = [ avisi av ];
+    sdisi = [ sdisi sd ];
+    cvisi = [ cvisi cv ];
+end
+
+f = figure;
+subplot( 1, 3, 1 );
+scatter( rates, 1000.0*avisi, 'b', 'filled' );
+hold on;
+plot( rates, 1000.0./rates, 'r' );
+hold off;
+xlabel( 'Rate \lambda [Hz]' );
+ylim( [ 0 1000 ] );
+title( 'Mean ISI [ms]' );
+legend( 'simulation', 'theory 1/\lambda' );
+
+subplot( 1, 3, 2 );
+scatter( rates, 1000.0*sdisi, 'b', 'filled' );
+hold on;
+plot( rates, 1000.0./rates, 'r' );
+hold off;
+xlabel( 'Rate \lambda [Hz]' );
+ylim( [ 0 1000 ] )
+title( 'Standard deviation ISI [ms]' );
+legend( 'simulation', 'theory 1/\lambda' );
+
+subplot( 1, 3, 3 );
+scatter( rates, cvisi, 'b', 'filled' );
+hold on;
+plot( rates, ones( size( rates ) ), 'r' );
+hold off;
+xlabel( 'Rate \lambda [Hz]' );
+ylim( [ 0 2 ] )
+title( 'CV' );
+legend( 'simulation', 'theory' );

pointprocesses/exercises/hompoissonspikes.m

@@ -0,0 +1,19 @@
+function spikes = hompoissonspikes( trials, rate, tmax )
+% Generate spike times of a homogeneous poisson process
+% trials: number of trials that should be generated
+% rate: the rate of the Poisson process in Hertz
+% tmax: the duration of each trial in seconds
+% returns a cell array of vectors of spike times
+
+    dt = 3.33e-5;
+    p = rate*dt;
+    if p > 0.2
+        p = 0.2
+        dt = p/rate;
+    end
+    x = rand( trials, ceil(tmax/dt) );
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        spikes{k} = find( x(k,:) >= 1.0-p ) * dt;
+    end
+end

pointprocesses/exercises/iafisistats.tex

@@ -0,0 +1,142 @@
+\documentclass[addpoints,10pt]{exam}
+\usepackage{url}
+\usepackage{color}
+\usepackage{hyperref}
+\usepackage{graphicx}
+
+\pagestyle{headandfoot}
+\runningheadrule
+\firstpageheadrule
+
+\firstpageheader{Scientific Computing}{Integrate-and-fire models}{Oct 28, 2014}
+%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
+\firstpagefooter{}{}{}
+\runningfooter{}{}{}
+\pointsinmargin
+\bracketedpoints
+
+%\printanswers
+\shadedsolutions
+
+\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
+
+%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{
+ basicstyle=\ttfamily,
+ numbers=left,
+ showstringspaces=false,
+ language=Matlab,
+ breaklines=true,
+ breakautoindent=true,
+ columns=flexible,
+ frame=single,
+ captionpos=t,
+ xleftmargin=2em,
+ xrightmargin=1em,
+ aboveskip=10pt,
+ %title=\lstname,
+ title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
+ }
+
+
+\begin{document}
+
+\sffamily
+%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{questions}
+  \question \textbf{Statistics of integrate-and-fire neurons}
+  For the following use different variants of the leaky integrate-and-fire models provided in \texttt{lifspikes.m}, 
+  \texttt{lifouspikes.m}, and \texttt{lifadaptspikes.m} do generate some spike train data.
+  Use the functions you wrote for the Poisson process to analyze the statistics of the spike trains.
+  \begin{parts}
+    \part Generate a few trials of the two models for two different inputs
+    that result in qualitatively different spike trains and display
+    them in a raster plot. Decide for a noise strength (good values to try are 0.001, 0.01, 0.1, 1).
+    \begin{solution}
+      \begin{lstlisting}
+spikes = pifspikes( 10, 1.0, 0.5, 0.01 );
+%spikes = pifspikes( 10, 10.0, 0.5, 0.01 );
+%spikes = lifspikes( 10, 11.0, 0.5, 0.001 );
+%spikes = lifspikes( 10, 15.0, 0.5, 0.001 );
+spikeraster( spikes )
+      \end{lstlisting}
+      \mbox{}\\[-3ex]
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifraster02}}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifraster10}}\\
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifraster10}}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifraster15}}
+    \end{solution}
+
+
+    \part The inverse Gaussian describes the interspike interval distribution of a PIF driven with white noise:
+    \[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \]
+    where $\langle T \rangle$ is the mean interspike interval and 
+    \[ D = \frac{\langle(T - \langle T \rangle)^2\rangle}{2 \langle T \rangle^3} \]
+    is the diffusion coefficient (variance of the interspike intervals
+    $T$ divided by two times the mean cubed). Show in two plots how
+    this distribution depends on $\langle T \rangle$ and $D$.
+    \begin{solution}
+      \lstinputlisting{simulations/inversegauss.m}
+      \lstinputlisting{simulations/inversegaussplot.m}
+      \colorbox{white}{\includegraphics[width=0.98\textwidth]{inversegauss}}
+    \end{solution}
+
+    \part Extent your function plotting an interspike interval histogram
+    to also report the diffusion coefficient $D$.
+    \begin{solution}
+      \begin{lstlisting}
+...
+% annotation:
+misi = mean( isis );
+sdisi = std( isis );
+disi = sdisi^2.0/2.0/misi^3;
+text( 0.6, 0.7, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' )
+text( 0.6, 0.6, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' )
+text( 0.6, 0.5, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' )
+text( 0.6, 0.4, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' )
+...
+      \end{lstlisting}
+    \end{solution}
+
+    \part Compare intersike interval histograms obtained from the LIF and PIF models with the inverse Gaussian.
+    \begin{solution}
+      \lstinputlisting{simulations/lifisih.m}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih01}}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih10}}\\
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih08}}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih16}}
+    \end{solution}
+
+    \part Plot the firing rate (inverse mean interspike interval),
+    mean interspike interval, the corresponding standard deviation,
+    CV, and diffusion coefficient as a function of the input to the LIF
+    and the PIF with noise strength set to 0.01.
+    \begin{solution}
+      \lstinputlisting{simulations/lifisistats.m}
+      Leaky integrate-and-fire:\\
+      \colorbox{white}{\includegraphics[width=0.8\textwidth]{lifisistats}}\\
+      Perfect integrate-and-fire:\\
+      \colorbox{white}{\includegraphics[width=0.8\textwidth]{pifisistats}}
+    \end{solution}
+
+    \part Plot the firing rate as a function of input of the LIF and the PIF for various values
+    of the noise strength.
+    \begin{solution}
+      \lstinputlisting{simulations/lifficurves.m}
+      Leaky integrate-and-fire:\\
+      \colorbox{white}{\includegraphics[width=0.7\textwidth]{lifficurves}}\\
+      Perfect integrate-and-fire:\\
+      \colorbox{white}{\includegraphics[width=0.7\textwidth]{pifficurves}}
+    \end{solution}
+
+    \part Use the functions for computing serial correlations, count statistics and fano factors
+    to further explore the statistics of the integrate-and-fire models!
+
+  \end{parts}
+  
+\end{questions}
+
+
+\end{document}

pointprocesses/exercises/poisson.tex

@@ -0,0 +1,160 @@
+\documentclass[addpoints,10pt]{exam}
+\usepackage{url}
+\usepackage{color}
+\usepackage{hyperref}
+\usepackage{graphicx}
+
+\pagestyle{headandfoot}
+\runningheadrule
+\firstpageheadrule
+
+\firstpageheader{Scientific Computing}{Homogeneous Poisson process}{Oct 27, 2014}
+%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
+\firstpagefooter{}{}{}
+\runningfooter{}{}{}
+\pointsinmargin
+\bracketedpoints
+
+%\printanswers
+\shadedsolutions
+
+\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
+
+%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{
+ basicstyle=\ttfamily,
+ numbers=left,
+ showstringspaces=false,
+ language=Matlab,
+ breaklines=true,
+ breakautoindent=true,
+ columns=flexible,
+ frame=single,
+ captionpos=t,
+ xleftmargin=2em,
+ xrightmargin=1em,
+ aboveskip=10pt,
+ %title=\lstname,
+ title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
+ }
+
+
+\begin{document}
+
+\sffamily
+%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{questions}
+  \question \textbf{Homogeneous Poisson process}
+  We use the Poisson process to generate spike trains on which we can test and imrpove some 
+  standard analysis functions.
+
+  A homogeneous Poisson process of rate $\lambda$ (measured in Hertz) is a point process
+  where the probability of an event is independent of time $t$ and independent of previous events.
+  The probability $P$ of an event within a bin of width $\Delta t$ is 
+  \[ P = \lambda \cdot \Delta t \]
+  for sufficiently small $\Delta t$.
+  \begin{parts}
+
+    \part Write a function that generates $n$ homogeneous Poisson spike trains of a given duration $T_{max}$
+    with rate $\lambda$.
+    \begin{solution}
+      \lstinputlisting{hompoissonspikes.m}
+    \end{solution}
+
+    \part Using this function, generate a few trials and display them in a raster plot.
+    \begin{solution}
+      \lstinputlisting{simulations/spikeraster.m}
+      \begin{lstlisting}
+spikes = hompoissonspikes( 10, 100.0, 0.5 );
+spikeraster( spikes )
+      \end{lstlisting}
+      \mbox{}\\[-3ex]
+      \colorbox{white}{\includegraphics[width=0.7\textwidth]{poissonraster100hz}}
+    \end{solution}
+
+    \part Write a function that extracts a single vector of interspike intervals
+    from the spike times returned by the first function.
+    \begin{solution}
+      \lstinputlisting{simulations/isis.m}
+    \end{solution}
+
+    \part Write a function that plots the interspike-interval histogram
+    from a vector of interspike intervals. The function should also
+    compute the mean, the standard deviation, and the CV of the intervals
+    and display the values in the plot.
+    \begin{solution}
+      \lstinputlisting{simulations/isihist.m}
+    \end{solution}
+
+    \part Compute histograms for Poisson spike trains with rate
+    $\lambda=100$\,Hz. Play around with $T_{max}$ and $n$ and the bin width
+    (start with 1\,ms) of the histogram.
+    How many
+    interspike intervals do you approximately need to get a ``nice''
+    histogram? How long do you need to record from the neuron?
+    \begin{solution}
+      About 5000 intervals for 25 bins. This corresponds to a $5000 / 100\,\hertz = 50\,\second$ recording
+      of a neuron firing with 100\,\hertz.
+    \end{solution}
+
+    \part Compare the histogram with the true distribution of intervals $T$ of the Poisson process
+    \[ p(T) = \lambda e^{-\lambda T} \]
+    for various rates $\lambda$.
+    \begin{solution}
+      \lstinputlisting{hompoissonisih.m}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}}
+    \end{solution}
+
+    \part What happens if you make the bin width of the histogram smaller than $\Delta t$ 
+    used for generating the Poisson spikes?
+    \begin{solution}
+      The bins between the discretization have zero entries. Therefore
+      the other ones become higher than they should be.
+    \end{solution}
+
+    \part Plot the mean interspike interval, the corresponding standard deviation, and the CV
+    as a function of the rate $\lambda$ of the Poisson process.
+    Compare the simulations with the theoretical expectations for the dependence on $\lambda$.
+    \begin{solution}
+      \lstinputlisting{hompoissonisistats.m}
+      \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}}
+    \end{solution}
+
+    \part Write a function that computes serial correlations for the interspike intervals
+    for a range of lags.
+    The serial correlations $\rho_k$ at lag $k$ are defined as
+    \[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
+    Use this function to show that interspike intervals of Poisson spikes are independent.
+    \begin{solution}
+      \lstinputlisting{simulations/isiserialcorr.m}
+      \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonserial100hz}}
+    \end{solution}
+
+    \part Write a function that generates from spike times 
+    a histogram of spike counts in a count window of given duration $W$.
+    The function should also plot the Poisson distribution
+    \[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
+    for the rate $\lambda$ determined from the spike trains.
+    \begin{solution}
+      \lstinputlisting{simulations/counthist.m}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}}
+      \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}}
+    \end{solution}
+
+    \part Write a function that computes mean count, variance of count and the corresponding Fano factor
+    for a range of count window durations. The function should generate tow plots: one plotting
+    the count variance against the mean, the other one the Fano factor as a function of the window duration.
+    \begin{solution}
+      \lstinputlisting{simulations/fano.m}
+      \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
+    \end{solution}
+
+  \end{parts}
+  
+\end{questions}
+
+
+\end{document}