Jan Bs projects

projects/project_isipdffit/Makefile

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+latex:
+	pdflatex *.tex > /dev/null
+	pdflatex *.tex > /dev/null
+
+clean:
+	rm -rf *.log *.aux *.zip *.out auto
+	rm -f `basename *.tex .tex`.pdf
+
+zip: latex
+	zip `basename *.tex .tex`.zip *.pdf *.dat *.mat *.m

projects/project_isipdffit/isipdffit.tex

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+\documentclass[addpoints,10pt]{exam}
+\usepackage{url}
+\usepackage{color}
+\usepackage{hyperref}
+
+\pagestyle{headandfoot}
+\runningheadrule
+\firstpageheadrule
+\firstpageheader{Scientific Computing}{Project Assignment}{11/05/2014
+  -- 11/06/2014}
+%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
+\firstpagefooter{}{}{}
+\runningfooter{}{}{}
+\pointsinmargin
+\bracketedpoints
+
+%\printanswers
+%\shadedsolutions
+
+%%%%% 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}
+%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
+\sffamily
+% \begin{flushright}
+% \gradetable[h][questions]
+% \end{flushright}
+
+\begin{center}
+  \input{../disclaimer.tex}
+\end{center}
+
+%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{questions}
+  \question You are recording the activity of two neurons in response to
+  a constant stimulus $I$ (think of it, for example,
+  of a sound wave with intensity $I$).
+
+  For different inputs $I$ the interspike interval ($T$) distribution looks
+  quite different. You want to compare these distributions to
+  the following three standard distributions.
+
+  The first is the exponential distribution of a Poisson spike train:
+  \begin{equation}
+    \label{exppdf}
+    p_{exp}(T) = \lambda e^{-\lambda T}
+  \end{equation}
+  where $\lambda$ is the mean firing rate of the response.
+
+  The second distribution is the inverse Gaussian:
+  \begin{equation}
+    \label{invgauss}
+    p_\mathrm{ig}(T) = \frac{1}{\sqrt{4 \pi D T^{3}}} \exp \left[ - \frac{(T - \mu)^{2} }{4 D T \mu^{2}} \right]
+  \end{equation}
+  where $\mu$ is the mean interspike interval and
+  $D=\textrm{var}(T)/(2\mu^3)$ is the so called diffusion coefficient.
+
+  The third one was derived for neurons driven with colored noise:
+  \begin{equation}\label{pcn}
+    p_\mathrm{cn}(T)=\frac{1}{2\tau\sqrt{4\pi\epsilon\gamma_1^3}}\exp\left[-\frac{(T-\mu)^2}{4\epsilon\tau^2\gamma_1}\right]\left\{\frac{[(\mu-T)\gamma_2+2\gamma_1\tau]^2}{2\gamma_1\tau^2}-\epsilon(\gamma_2^2-2\gamma_1e^{-T/\tau})\right\} 
+  \end{equation}
+  with $\gamma_1(T)=T/\tau+e^{-T/\tau}-1$, $\gamma_2(T)=1-e^{-T/\tau}$
+  and correlation time of the colored noise $\tau$.  
+  Eq.~(\ref{pcn}) thus has the three parameter $\mu$, $\epsilon>0$, and $\tau$.
+
+  The squared coefficient of variation (standard deviation of the
+  interspike intervals divided by their mean) of the density
+  eq.~(\ref{pcn}) is given by
+  \begin{equation}
+    \label{cvpcn}
+    C_V^2=\frac{2}{\delta}\left[\epsilon\left(1-\frac{1-e^{-\delta}}{\delta}\right)+\epsilon^2\left(e^{-\delta}+\frac{(1-e^{-\delta})(1-2e^{-\delta})}{\delta}\right)\right]
+  \end{equation}
+  with $\delta=\mu/\tau$. 
+
+  \begin{parts}
+    \part The two neurons are implemented in the files \texttt{pifouspikes.m}
+    and \texttt{lifouspikes.m}.
+    Call them with the following parameters:
+    \begin{lstlisting}
+      trials = 10;
+      tmax = 50.0;
+      input = 10.0;  % the input I
+      Dnoise = 1.0;  % noise strength
+      outau = 1.0;   % correlation time of the noise in seconds
+
+      spikes = pifouspikes( trials, input, tmax, Dnoise, outau );
+    \end{lstlisting}
+    The returned \texttt{spikes} is a cell array with \texttt{trials} elements, each being a vector
+    of spike times (in seconds) computed for a duration of \texttt{tmax} seconds.
+    The input is set via the \texttt{input} variable.
+
+    \part Find for both model neurons the inputs $I_i$ required to make the fire with a mean rate
+    of 10, 20, 50, and 100\,Hz. 
+
+    \part Compute interspike interval distributions of the two model neurons for these inputs $I_i$.
+
+    \part Compare the interspike interval distributions with the exponential 
+    distribution eq.~(\ref{exppdf}) and the inverse Gaussian
+    eq.~(\ref{invgauss}) by measuring their parameters from the
+    interspike intervals. How well do theu describe the real
+    distributions for the different conditions?
+
+    \part Also fit eq.~(\ref{pcn}) to the data. Here you need to apply a non-linear fit algorithm.
+
+    How well does this function describe the data?
+
+    Compare the fitted value for $\tau$ with the one used for the model (\texttt{outau}).
+
+
+    \uplevel{If you still have time you can continue with the following question:}
+
+    \part Compare the measured coefficient of variation with eq.~(\ref{cvpcn}).
+
+    \part Repeat your analysis for different values of the intrinsic noise strengh of the neurons
+    \texttt{Dnoise}. Increase or decrease it in factors of ten.
+
+ \end{parts}
+
+\end{questions}
+
+\end{document}

projects/project_isipdffit/lifouspikes.m

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+function spikes = lifouspikes( trials, input, tmaxdt, D, outau )
+% Generate spike times of a leaky integrate-and-fire neuron
+% trials: the number of trials to be generated
+% input: the stimulus either as a single value or as a vector
+% tmaxdt: in case of a single value stimulus the duration of a trial
+%         in case of a vector as a stimulus the time step
+% D: the strength of additive white noise
+% outau: time constant of the colored noise
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    end
+    if nargin < 5
+        outau = 1.0;
+    end
+    vreset = 0.0;
+    vthresh = 10.0;
+    dt = 1e-4;
+    
+    if length( input ) == 1
+        input = input * ones( ceil( tmaxdt/dt ), 1 );
+    else
+        dt = tmaxdt;
+    end
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        times = [];
+        j = 1;
+        n = 0.0;
+        v = vreset;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            n = n + ( - n + noise(i))*dt/outau;
+            v = v + ( - v + n + input(i))*dt/tau;
+            if v >= vthresh
+                v = vreset;
+                times(j) = i*dt;
+                j = j + 1;
+            end
+        end
+        spikes{k} = times;
+    end
+end

projects/project_isipdffit/pifouspikes.m

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+function spikes = pifouspikes( trials, input, tmaxdt, D, outau )
+% Generate spike times of a perfect integrate-and-fire neuron
+% trials: the number of trials to be generated
+% input: the stimulus either as a single value or as a vector
+% tmaxdt: in case of a single value stimulus the duration of a trial
+%         in case of a vector as a stimulus the time step
+% D: the strength of additive white noise
+% outau: time constant of the colored noise
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    end
+    if nargin < 5
+        outau = 1.0;
+    end
+    vreset = 0.0;
+    vthresh = 10.0;
+    dt = 1e-4;
+    
+    if length( input ) == 1
+        input = input * ones( ceil( tmaxdt/dt ), 1 );
+    else
+        dt = tmaxdt;
+    end
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        times = [];
+        j = 1;
+        n = 0.0;
+        v = vreset;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            n = n + ( - n + noise(i))*dt/outau;
+            v = v + ( n + input(i))*dt/tau;
+            if v >= vthresh
+                v = vreset;
+                times(j) = i*dt;
+                j = j + 1;
+            end
+        end
+        spikes{k} = times;
+    end
+end