Jan Bs projects

projects/project_fano_time/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_fano_time/fano_time.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 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

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+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