Jan Bs projects

projects/disclaimer.tex

@@ -5,7 +5,7 @@
       Each project has three elements that are graded: (i) the code,
       (ii) the slides/figures, and (iii) the presentation.
 
-      \vspace{.5cm}
+      \vspace{1ex}
 
       The {\bf code} and the {\bf presentation} should be uploaded to
       ILIAS at latest on Thursday, November 6th, 12:00h.
@@ -13,7 +13,7 @@
       your presentation as a pdf file. Bundle everything into a
       {\em single} zip-file.
 
-      \vspace{.5cm}
+      \vspace{1ex}
 
       The {\bf code} should be exectuable without any further
       adjustments from us. This means that you need to include all
@@ -25,11 +25,11 @@
       and comprehensible by third persons (use proper and consistent
       variable names).
 
-      \vspace{.5cm} \textbf{Please write your name and matriculation
+      \vspace{1ex} \textbf{Please write your name and matriculation
       number as a comment at the top of a script called \texttt{main.m}!}
       The \texttt{main.m} script then should call all your scripts.
 
-      \vspace{.5cm}
+      \vspace{1ex}
       
       The {\bf presentation} should be {\em at most} 10min long and be
       held in English. In the presentation you should (i) briefly

projects/project_fano_slope/Makefile

@@ -7,4 +7,4 @@ clean:
 	rm -f `basename *.tex .tex`.pdf
 
 zip: latex
-	zip `basename *.tex .tex`.zip *.pdf *.dat *.mat
+	zip `basename *.tex .tex`.zip *.pdf *.dat *.mat *.m

projects/project_fano_slope/fano_slope.tex

@@ -0,0 +1,133 @@
+\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 a neuron in response to
+  two different stimuli $I_1$ and $I_2$ (think of them, for example,
+  of two sound waves with different intensities $I_1$ and
+  $I_2$). Within an observation time of duration $W$ the neuron
+  responds stochastically with $n_i$ spikes.
+
+  How well can an upstream neuron discriminate the two stimuli based
+  on the spike counts $n_i$? How does this depend on the slope of the
+  tuning curve of the neural responses? How is this related to the
+  fano factor (the ratio between the variance and the mean of the
+  spike counts)?
+
+  \begin{parts}
+    \part The neuron is implemented in the file \texttt{lifboltzmanspikes.m}.
+    Call it with the following parameters:
+    \begin{lstlisting}
+      trials = 10;
+      tmax = 50.0;
+      Dnoise = 1.0;
+      imax = 25.0;
+      ithresh = 10.0;
+      slope=0.2;
+      input = 10.0;
+
+      spikes = lifboltzmanspikes( trials, input, tmax, Dnoise, imax, ithresh, slope );
+    \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.
+
+    For the two inputs use $I_1=10$ and $I_2=I_1 + 1$.
+
+    First, show two raster plots for the responses to the two differrent stimuli.
+
+    \part Measure the tuning curve of the neuron with respect to the input. That is,
+    compute the mean firing rate as a function of the input
+    strength. Find an appropriate range of input values.  Do this for
+    different values of the \texttt{slope} parameter (values between
+    0.1 and 2.0).
+
+    \part Generate histograms of the spike counts within $W=200$\,ms of the
+    responses to the two differrent stimuli $I_1$ and $I_2$. How do they depend on the slope
+    of the tuning curve of the neuron?
+
+    \part Think about a measure based on the spike count histograms that quantifies how well
+    the two stimuli can be distinguished based on the spike
+    counts. Plot the dependence of this measure as a function of the observation time $W$.
+
+    For which slopes can the two stimuli perfectly discriminated?
+
+    Hint: A possible readout is to set a threshold $n_{thresh}$ for
+    the observed spike count.  Any response smaller than the threshold
+    assumes that the stimulus was $I_1$, any response larger than the
+    threshold assumes that the stimulus was $I_2$. What is the
+    probability that the stimulus was indeed $I_1$ or $I_2$,
+    respectively? Find the threshold $n_{thresh}$ that
+    results in the best discrimination performance.
+
+    \part Also plot the Fano factor as a function of the slope. How is it related to the discriminability?
+
+    \uplevel{If you still have time you can continue with the following questions:}
+
+    \part You may change the difference between the two stimuli $I_1$ and $I_2$ 
+    as well as the intrinsic noise of the neuron via \texttt{Dnoise}
+    (change it in factors of ten, higher values will make the
+    responses more variable) and repeat your analysis.
+
+    \part For $I_1=10$ the mean firing is about $80$\,Hz. When changing the slope of the tuning curve
+    this firing rate may also change. Improve your analysis by finding for each slope the input
+    that results exactly in a firing rate of $80$\,Hz. Set $I_2$ on unit above $I_1$.
+
+    \part How does the dependence of the stimulus discrimination performance on the slope change
+    when you set both $I_1$ and $I_2$ such that they evoke $80$ and
+    $100$\,Hz firing rate, respectively?
+
+ \end{parts}
+
+\end{questions}
+
+\end{document}

projects/project_fano_slope/lifboltzmanspikes.m

@@ -0,0 +1,51 @@
+function spikes = lifboltzmanspikes( trials, input, tmaxdt, D, imax, ithresh, slope )
+% 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
+% imax: maximum output of boltzman
+% ithresh: threshold of boltzman input
+% slope: slope factor of boltzman input
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    end
+    if nargin < 5
+        imax = 20;
+    end
+    if nargin < 6
+        ithresh = 10;
+    end
+    if nargin < 7
+        slope = 1;
+    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
+    inb = imax./(1.0+exp(-slope.*(input - ithresh)));
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        times = [];
+        j = 1;
+        v = vreset;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            v = v + ( - v + noise(i) + inb(i))*dt/tau;
+            if v >= vthresh
+                v = vreset;
+                times(j) = i*dt;
+                j = j + 1;
+            end
+        end
+        spikes{k} = times;
+    end
+end

projects/project_fano_time/Makefile

@@ -0,0 +1,10 @@
+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_fano_time/fano_time.tex

@@ -0,0 +1,119 @@
+\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 a neuron in response to
+  two different stimuli $I_1$ and $I_2$ (think of them, for example,
+  of two sound waves with different intensities $I_1$ and
+  $I_2$). Within an observation time of duration $W$ the neuron
+  responds stochastically with $n_i$ spikes.
+
+  How well can an upstream neuron discriminate the two
+  stimuli based on the spike counts $n_i$? How does this depend on the
+  duration $W$ of the observation time? How is this related to the fano factor
+  (the ratio between the variance and the mean of the spike counts)?
+
+  \begin{parts}
+    \part The neuron is implemented in the file \texttt{lifadaptspikes.m}.
+    Call it with the following parameters:
+    \begin{lstlisting}
+      trials = 10;
+      tmax = 50.0;
+      input = 65.0; 
+      Dnoise = 0.1;
+      adapttau = 0.2;
+      adaptincr = 0.5;
+
+      spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr );
+    \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.
+
+    For the two inputs $I_1$ and $I_2$ use
+    \begin{lstlisting}
+      input = 65.0; % I_1
+      input = 75.0; % I_2
+    \end{lstlisting}
+
+    Show two raster plots for the responses to the two differrent stimuli.
+
+    \part Generate histograms of the spike counts within $W$ of the
+    responses to the two differrent stimuli. How do they depend on the observation time $W$
+    (use values between 1\,ms and 1\,s)?
+
+    \part Think about a measure based on the spike count histograms that quantifies how well
+    the two stimuli can be distinguished based on the spike
+    counts. Plot the dependence of this measure as a function of the observation time $W$.
+
+    For which observation times can the two stimuli perfectly discriminated?
+
+    Hint: A possible readout is to set a threshold $n_{thresh}$ for
+    the observed spike count.  Any response smaller than the threshold
+    assumes that the stimulus was $I_1$, any response larger than the
+    threshold assumes that the stimulus was $I_2$. What is the
+    probability that the stimulus was indeed $I_1$ or $I_2$,
+    respectively? For a given $W$ find the threshold $n_{thresh}$ that
+    results in the best discrimination performance.
+
+    \part Also plot the Fano factor as a function of $W$. How is it related to the discriminability?
+
+    \uplevel{If you still have time you can continue with the following question:}
+
+    \part You may change the two stimuli $I_1$ and $I_2$ and the intrinsic noise of the neuron via
+    \texttt{Dnoise} (change it in factors of ten, higher values will make the responses more variable)
+    and repeat your analysis.
+
+ \end{parts}
+
+\end{questions}
+
+\end{document}

projects/project_fano_time/lifadaptspikes.m

@@ -0,0 +1,53 @@
+function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr )
+% Generate spike times of a leaky integrate-and-fire neuron
+% with an adaptation current
+% 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
+% tauadapt: adaptation time constant
+% adaptincr: adaptation strength
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    end
+    if nargin < 5
+        tauadapt = 0.1;
+    end
+    if nargin < 6
+        adaptincr = 1.0;
+    end
+    vreset = 0.0;
+    vthresh = 10.0;
+    dt = 1e-4;
+    
+    if max( size( input ) ) == 1
+        input = input * ones( ceil( tmaxdt/dt ), 1 );
+    else
+        dt = tmaxdt;
+    end
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        times = [];
+        j = 1;
+        v = vreset;
+        a = 0.0;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            v = v + ( - v - a + noise(i) + input(i))*dt/tau;
+            a = a + ( - a )*dt/tauadapt;
+            if v >= vthresh
+                v = vreset;
+                a = a + adaptincr/tauadapt;
+                spiketime = i*dt;
+                if spiketime > 4.0*tauadapt
+                    times(j) = spiketime - 4.0*tauadapt;
+                    j = j + 1;
+                end
+            end
+        end
+        spikes{k} = times;
+    end
+end

projects/project_isicorrelations/Makefile

@@ -0,0 +1,10 @@
+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_isicorrelations/isicorrelations.tex

@@ -0,0 +1,109 @@
+\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 a neuron in response to
+  constant stimuli of intensity $I$ (think of that, for example,
+  of sound waves with intensities $I$). The neuron has an adaptatation
+  current that adapts the firing rate with a slow time constant down.
+
+  Explore the dependence of interspike interval correlations on the firing rate,
+  adaptation time constant and noise level.
+
+  \begin{parts}
+    \part The neuron is a neuron with an adaptation current. 
+    It is implemented in the file \texttt{lifadaptspikes.m}.  Call it
+    with the following parameters:
+    \begin{lstlisting}
+      trials = 10;
+      tmax = 50.0;
+      input = 10.0;  % the input I
+      Dnoise = 1e-2;  % noise strength
+      adapttau = 0.1;  % adaptation time constant in seconds
+      adaptincr = 0.5;  % adaptation strength
+
+      spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr );
+    \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, the noise strength via \texttt{Dnoise},
+    and the adaptation time constant via \texttt{adapttau}.
+
+    \part Measure the intensity-response curve of the neuron, that is the mean firing rate 
+    as a function of the input for a range of inputs from 0 to 120.
+
+    \part Compute the correlations between each interspike interval $T_i$ and the next one $T_{i+1}$
+    (serial interspike interval correlation at lag  1). Plot this correlation as a function of the 
+    firing rate by varying the input as in (a).
+
+    \part How does this dependence change for different values of the adaptation 
+    time constant \texttt{adapttau}?  Use values between 10\,ms and
+    1\,s for \texttt{adapttau}.
+
+    \part Determine the firing rate at which the minimum interspike interval correlation
+    occurs. How does the minimum correlation and this firing rate
+    depend on the adaptation time constant \texttt{adapttau}?
+
+    \part How do the results change if the level of the intrinsic noise \texttt{Dnoise} is modified?
+    Use values of 1e-4, 1e-3, 1e-2, 1e-1, and 1 for \texttt{Dnoise}.
+
+
+    \uplevel{If you still have time you can continue with the following question:}
+
+    \part How do the interspike interval distributions look like for the different noise levels
+    at some example values for the input and the adaptation time constant?
+
+ \end{parts}
+
+\end{questions}
+
+\end{document}

projects/project_isicorrelations/lifadaptspikes.m

@@ -0,0 +1,53 @@
+function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr )
+% Generate spike times of a leaky integrate-and-fire neuron
+% with an adaptation current
+% 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
+% tauadapt: adaptation time constant
+% adaptincr: adaptation strength
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    end
+    if nargin < 5
+        tauadapt = 0.1;
+    end
+    if nargin < 6
+        adaptincr = 1.0;
+    end
+    vreset = 0.0;
+    vthresh = 10.0;
+    dt = 1e-4;
+    
+    if max( size( input ) ) == 1
+        input = input * ones( ceil( tmaxdt/dt ), 1 );
+    else
+        dt = tmaxdt;
+    end
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        times = [];
+        j = 1;
+        v = vreset;
+        a = 0.0;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            v = v + ( - v - a + noise(i) + input(i))*dt/tau;
+            a = a + ( - a )*dt/tauadapt;
+            if v >= vthresh
+                v = vreset;
+                a = a + adaptincr/tauadapt;
+                spiketime = i*dt;
+                if spiketime > 4.0*tauadapt
+                    times(j) = spiketime - 4.0*tauadapt;
+                    j = j + 1;
+                end
+            end
+        end
+        spikes{k} = times;
+    end
+end

projects/project_isipdffit/Makefile

@@ -0,0 +1,10 @@
+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

@@ -0,0 +1,140 @@
+\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

@@ -0,0 +1,44 @@
+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

@@ -0,0 +1,44 @@
+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

projects/project_noiseficurves/Makefile

@@ -0,0 +1,10 @@
+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_noiseficurves/lifspikes.m

@@ -0,0 +1,38 @@
+function spikes = lifspikes( trials, input, tmaxdt, D )
+% 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
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    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;
+        v = vreset;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            v = v + ( - v + noise(i) + 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_noiseficurves/noiseficurves.tex

@@ -0,0 +1,91 @@
+\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 a neuron in response to
+  constant stimuli of intensity $I$ (think of that, for example,
+  of sound waves with intensities $I$).
+
+  Measure the tuning curve (also called the intensity-response curve) of the
+  neuron.  That is, what is the firing rate of the neuron's response
+  as a function of the input $I$. How does this depend on the level of
+  the intrinsic noise of the neuron?
+
+  \begin{parts}
+    \part The neuron is implemented in the file \texttt{lifspikes.m}.
+    Call it with the following parameters:
+    \begin{lstlisting}
+      trials = 10;
+      tmax = 50.0;
+      input = 10.0;  % the input I
+      Dnoise = 1.0;  % noise strength
+
+      spikes = lifspikes( trials, input, tmax, Dnoise );
+    \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, the noise strength via \texttt{Dnoise}.
+
+    \part First set the noise \texttt{Dnoise=0} (no noise). Compute and plot the firing rate
+    as a function of the input for inputs ranging from 0 to 20.
+
+    \part Do the same for various noise strength \texttt{Dnoise}. Use $D_{noise} = 1e-3$,
+    1e-2, and 1e-1. How does the intrinsic noise influence the response curve?
+
+    \part Show some interspike interval histograms for some interesting values of the input 
+    and the noise strength.
+
+ \end{parts}
+
+\end{questions}
+
+\end{document}

projects/project_template/Makefile

@@ -7,4 +7,4 @@ clean:
 	rm -f `basename *.tex .tex`.pdf
 
 zip: latex
-	zip `basename *.tex .tex`.zip *.pdf *.dat *.mat
+	zip `basename *.tex .tex`.zip *.pdf *.dat *.mat *.m