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Merge branch 'master' of https://whale.am28.uni-tuebingen.de/git/teaching/scientificComputing

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Jan Grewe 2020-01-26 16:14:47 +01:00
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105
pointprocesses/lecture/pointprocessscetchA.eps
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0.00 0.00 0.00 C 528 460 M 0.00 0.00 0.00 C 528 460 M
-63 0 V -63 0 V
stroke stroke
LTb LTb
LCb setrgbcolor LCb setrgbcolor
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0.00 0.00 0.00 C 528 652 M 0.00 0.00 0.00 C 528 652 M
-63 0 V -63 0 V
stroke stroke
LTb LTb
LCb setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 528 844 M 0.00 0.00 0.00 C 528 844 M
-63 0 V -63 0 V
stroke stroke
LTb LTb
LCb setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 528 1036 M 0.00 0.00 0.00 C 528 1036 M
-63 0 V -63 0 V
stroke stroke
LTb LTb
LCb setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 528 1228 M 0.00 0.00 0.00 C 528 1228 M
-63 0 V -63 0 V
stroke stroke
LTb LTb
LCb setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 2.000 UL 0.00 0.00 0.00 C 2.000 UL
LTb LTb
LCb setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 528 1276 M 0.00 0.00 0.00 C 528 1276 M
528 220 L 528 220 L
5801 0 R 5801 0 R
@ -646,9 +711,7 @@ LCb setrgbcolor
-5801 0 R -5801 0 R
1.000 UP 1.000 UP
stroke stroke
LTb
LCb setrgbcolor LCb setrgbcolor
LTb
1.000 UL 1.000 UL
LTb LTb
gsave 6208 268 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill gsave 6208 268 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
@ -662,9 +725,11 @@ gsave 6208 268 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
stroke stroke
2.000 UL 2.000 UL
LTb LTb
0.00 0.00 0.00 C 3.000 UL 0.00 0.00 0.00 C % Begin plot #1
LT0 3.000 UL
LC0 setrgbcolor LTb
LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 528 268 M 0.00 0.00 0.00 C 528 268 M
382 0 V 382 0 V
0 96 R 0 96 R
@ -685,11 +750,15 @@ LC0 setrgbcolor
1035 0 V 1035 0 V
0 96 R 0 96 R
533 0 V 533 0 V
1.500 UP
stroke stroke
LTw
% End plot #1
% Begin plot #2
1.500 UP
2.000 UL 2.000 UL
LT0 LTb
LC0 setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 910 268 CircleF 0.00 0.00 0.00 C 910 268 CircleF
1412 364 CircleF 1412 364 CircleF
2123 460 CircleF 2123 460 CircleF
@ -699,10 +768,14 @@ LC0 setrgbcolor
4033 844 CircleF 4033 844 CircleF
4650 940 CircleF 4650 940 CircleF
5685 1036 CircleF 5685 1036 CircleF
LTw
% End plot #2
% Begin plot #3
1.000 UP 1.000 UP
2.000 UL 2.000 UL
LT0 LTb
LC0 setrgbcolor LCb setrgbcolor
[] 0 setdash
1.00 1.00 1.00 C 910 268 CircleF 1.00 1.00 1.00 C 910 268 CircleF
1412 364 CircleF 1412 364 CircleF
2123 460 CircleF 2123 460 CircleF
@ -712,10 +785,14 @@ LC0 setrgbcolor
4033 844 CircleF 4033 844 CircleF
4650 940 CircleF 4650 940 CircleF
5685 1036 CircleF 5685 1036 CircleF
LTw
% End plot #3
% Begin plot #4
1.500 UP 1.500 UP
2.000 UL 2.000 UL
LT0 LTb
LC0 setrgbcolor LCb setrgbcolor
[] 0 setdash
0.00 0.00 0.00 C 910 364 CircleF 0.00 0.00 0.00 C 910 364 CircleF
1412 460 CircleF 1412 460 CircleF
2123 556 CircleF 2123 556 CircleF
@ -725,10 +802,15 @@ LC0 setrgbcolor
4033 940 CircleF 4033 940 CircleF
4650 1036 CircleF 4650 1036 CircleF
5685 1132 CircleF 5685 1132 CircleF
1.000 UP LTw
% End plot #4
2.000 UL 2.000 UL
LTb LTb
LCb setrgbcolor LCb setrgbcolor
[] 0 setdash
1.000 UP
2.000 UL
LTb
0.00 0.00 0.00 C stroke 0.00 0.00 0.00 C stroke
grestore grestore
end end

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pointprocesses/lecture/pointprocessscetchB.pdf
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25
projects/README
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@ -1,10 +1,21 @@
How to make a new project How to make a new project
------------------------- -------------------------
Copy `project_template/` to your `project_NAME/` and adapt according to your needs.
Rename `template.tex` to `NAME.tex` and write questions. - Copy `project_template/` to your `project_NAME/` and adapt according to your needs.
Put data that are needed for the project into the `data/` subfolder. - Rename `template.tex` to `NAME.tex` and write questions.
Put your solution into the `code/` subfolder. - Put code needed for the project into the project's root directory.
Don't forget to add the project files to git (`git add FILENAMES`). - Put data that are needed for the project into the `data/` subfolder.
- Put your solution into the `solution/` subfolder.
- Put code that is needed to generate some data for the project,
but which is not part of the project, into the `code/` subfolder.
- Don't forget to add the project files to git (`git add FILENAMES`).
Upload projects to Ilias
------------------------
Simply upload ALL zip files into one folder or Uebungseinheit.
Provide an additional file that links project names to students.
Projects Projects
@ -12,7 +23,7 @@ Projects
1) project_activation_curve 1) project_activation_curve
medium medium
Write questions also normalize activation curve to maximum.
2) project_adaptation_fit 2) project_adaptation_fit
OK, medium OK, medium
@ -34,7 +45,6 @@ OK, medium-difficult
7) project_ficurves 7) project_ficurves
OK, medium OK, medium
Maybe add correlation test or fit statistics
8) project_lif 8) project_lif
OK, difficult OK, difficult
@ -42,7 +52,6 @@ no statistics
9) project_mutualinfo 9) project_mutualinfo
OK, medium OK, medium
Example code is missing
10) project_noiseficurves 10) project_noiseficurves
OK, simple-medium OK, simple-medium

16
projects/project_activation_curve/solution/ivcurve.m Normal file
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@ -0,0 +1,16 @@
function [vsteps, peakcurrents] = ivcurve(vsteps, time, currents, tmax)
peakcurrents = zeros(1, length(vsteps));
for k = 1:length(peakcurrents)
c = currents((time>0.0)&(time<tmax), k);
minc = min(c);
maxc = max(c);
if abs(minc) > maxc
peakcurrents(k) = minc;
else
peakcurrents(k) = maxc;
end
end
end

50
projects/project_activation_curve/solution/main.m Normal file
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@ -0,0 +1,50 @@
%% plot data:
x = load('../data/WT_01.mat');
wtdata = x.data;
plotcurrents(wtdata.t, wtdata.I);
x = load('../data/A1622D_01.mat');
addata = x.data;
plotcurrents(addata.t, addata.I);
%% I-V curve:
[wtsteps, wtpeaks] = ivcurve(wtdata.steps, wtdata.t, wtdata.I, 100.0);
[adsteps, adpeaks] = ivcurve(addata.steps, addata.t, addata.I, 100.0);
figure();
plot(wtsteps, wtpeaks, '-b');
hold on;
plot(adsteps, adpeaks, '-r');
hold off;
%% reversal potential:
wtE = reversalpotential(wtsteps, wtpeaks);
adE = reversalpotential(adsteps, adpeaks);
%% activation curve:
wtg = wtpeaks./(wtsteps - wtE);
adg = adpeaks./(adsteps - adE);
wtinfty = wtg(wtsteps<40.0)/mean(wtg((wtsteps>=20.0)&(wtsteps<=40.0)));
adinfty = adg(adsteps<40.0)/mean(adg((adsteps>=20.0)&(adsteps<=40.0)));
wtsteps = wtsteps(wtsteps<40.0);
adsteps = adsteps(adsteps<40.0);
figure();
plot(wtsteps, wtinfty, '-b');
hold on;
plot(adsteps, adinfty, '-r');
%% boltzmann fit:
bf = @(p, v) 1.0./(1.0+exp(-p(1)*(v - p(2))));
p = lsqcurvefit(bf, [1.0, -40.0], wtsteps, wtinfty);
wtfit = bf(p, wtsteps);
p = lsqcurvefit(bf, [1.0, -40.0], adsteps, adinfty);
adfit = bf(p, adsteps);
plot(wtsteps, wtfit, '-b');
plot(wtsteps, adfit, '-r');
hold off;

13
projects/project_activation_curve/solution/plotcurrents.m Normal file
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@ -0,0 +1,13 @@
function plotcurrents(time, currents)
figure();
hold on;
for k = 1:size(currents, 2)
plot(time, currents(:, k))
end
hold off;
xlabel('Time [ms]')
ylabel('Current')
end

4
projects/project_activation_curve/solution/reversalpotential.m Normal file
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@ -0,0 +1,4 @@
function E = reversalpotential(vsteps, currents)
p = polyfit(vsteps((vsteps>=20.0)&(vsteps<50.0)), currents((vsteps>=20.0)&(vsteps<50.0)), 1);
E = -p(2)/p(1);
end

95
projects/project_mutualinfo/mutualinfo.tex
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@ -11,6 +11,49 @@
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
The mutual information is a measure from information theory that is
used in neuroscience to quantify, for example, how much information a
spike train carries about a sensory stimulus. It quantifies the
dependence of an output $y$ (e.g. a spike train) on some input $x$
(e.g. a sensory stimulus).
The probability of each of $n$ input values $x = {x_1, x_2, ... x_n}$
is given by the corresponding probabilty distribution $P(x)$. The entropy
\begin{equation}
\label{entropy}
H[x] = - \sum_{x} P(x) \log_2 P(x)
\end{equation}
is a measure for the surprise of getting a specific value of $x$. For
example, if from two possible values '1' and '2', the probability of
getting a '1' is close to one ($P(1) \approx 1$) then the probability
of getting a '2' is close to zero ($P(2) \approx 0$). For this case
the entropy, the surprise level, is almost zero, because both $0 \log
0 = 0$ and $1 \log 1 = 0$. It is not surprising at all that you almost
always get a '1'. The entropy is largest for equally likely outcomes
of $x$. If getting a '1' or a '2' is equally likely then you will be
most surprised by each new number you get, because you can not predict
them.
Mutual information measures information transmitted between an input
and an output. It is computed from the probability distributions of
the input, $P(x)$, the output $P(y)$ and their joint distribution
$P(x,y)$:
\begin{equation}
\label{mi}
I[x:y] = \sum_{x}\sum_{y} P(x,y) \log_2\frac{P(x,y)}{P(x)P(y)}
\end{equation}
where the sums go over all possible values of $x$ and $y$. The mutual
information can be also expressed in terms of entropies. Mutual
information is the entropy of the outputs $y$ reduced by the entropy
of the outputs given the input:
\begin{equation}
\label{mientropy}
I[x:y] = E[y] - E[x|y]
\end{equation}
The following project is meant to explore the concept of mutual
information with the help of a simple example.
\begin{questions} \begin{questions}
\question A subject was presented two possible objects for a very \question A subject was presented two possible objects for a very
brief time ($50$\,ms). The task of the subject was to report which of brief time ($50$\,ms). The task of the subject was to report which of
@ -19,40 +62,56 @@
object was reported by the subject. object was reported by the subject.
\begin{parts} \begin{parts}
\part Plot the data appropriately. \part Plot the raw data (no sums or probabilities) appropriately.
\part Compute and plot the probability distributions of presented
and reported objects.
\part Compute a 2-d histogram that shows how often different \part Compute a 2-d histogram that shows how often different
combinations of reported and presented came up. combinations of reported and presented came up.
\part Normalize the histogram such that it sums to one (i.e. make \part Normalize the histogram such that it sums to one (i.e. make
it a probability distribution $P(x,y)$ where $x$ is the presented it a probability distribution $P(x,y)$ where $x$ is the presented
object and $y$ is the reported object). Compute the probability object and $y$ is the reported object).
distributions $P(x)$ and $P(y)$ in the same way.
\part Use that probability distribution to compute the mutual \part Use the computed probability distributions to compute the mutual
information information \eqref{mi} that the answers provide about the
\[ I[x:y] = \sum_{x\in\{1,2\}}\sum_{y\in\{1,2\}} P(x,y) actually presented object.
\log_2\frac{P(x,y)}{P(x)P(y)}\]
that the answers provide about the actually presented object.
The mutual information is a measure from information theory that is \part Use a permutation test to compute the $95\%$ confidence
used in neuroscience to quantify, for example, how much information interval for the mutual information estimate in the dataset from
a spike train carries about a sensory stimulus. {\tt decisions.mat}. Does the measured mutual information indicate
signifikant information transmission?
\part What is the maximally achievable mutual information (try to \end{parts}
find out by generating your own dataset which naturally should
yield maximal information)?
\part Use bootstrapping (permutation test) to compute the $95\%$ \question What is the maximally achievable mutual information?
confidence interval for the mutual information estimate in the
dataset from {\tt decisions.mat}.
\begin{parts}
\part Show this numerically by generating your own datasets which
naturally should yield maximal information. Consider different
distributions of $P(x)$.
\part Compare the maximal mutual information with the corresponding
entropy \eqref{entropy}.
\end{parts} \end{parts}
\end{questions} \question What is the minimum possible mutual information?
This is the mutual information between an output is independent of the
input.
How is the joint distribution $P(x,y)$ related to the marginls
$P(x)$ and $P(y)$ if $x$ and $y$ are independent? What is the value
of the logarithm in eqn.~\eqref{mi} in this case? So what is the
resulting value for the mutual information?
\end{questions}
Hint: You may encounter a problem when computing the mutual
information whenever $P(x,y)$ equals zero. For treating this special
case think about (plot it) what the limit of $x \log x$ is for $x$
approaching zero. Use this information to fix the computation of the
mutual information.
\end{document} \end{document}

8
projects/project_mutualinfo/solution/mi.m Normal file
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@ -0,0 +1,8 @@
function I = mi(nxy)
pxy = nxy / sum(nxy(:));
px = sum(nxy, 2) / sum(nxy(:));
py = sum(nxy, 1) / sum(nxy(:));
pi = pxy .* log2(pxy./(px*py));
pi(nxy == 0) = 0.0;
I = sum(pi(:));
end

90
projects/project_mutualinfo/solution/mutualinfo.m Normal file
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@ -0,0 +1,90 @@
%% load data:
x = load('../data/decisions.mat');
presented = x.presented;
reported = x.reported;
%% plot data:
figure()
plot(presented, 'ob', 'markersize', 10, 'markerfacecolor', 'b');
hold on;
plot(reported, 'or', 'markersize', 5, 'markerfacecolor', 'r');
hold off
ylim([0.5, 2.5])
p1 = sum(presented == 1);
p2 = sum(presented == 2);
r1 = sum(reported == 1);
r2 = sum(reported == 2);
figure()
bar([p1, p2, r1, r2]);
set(gca, 'XTickLabel', {'p1', 'p2', 'r1', 'r2'});
%% histogram:
nxy = zeros(2, 2);
for x = [1, 2]
for y = [1, 2]
nxy(x, y) = sum((presented == x) & (reported == y));
end
end
figure()
bar3(nxy)
set(gca, 'XTickLabel', {'p1', 'p2'});
set(gca, 'YTickLabel', {'r1', 'r2'});
%% normalized histogram:
pxy = nxy / sum(nxy(:));
figure()
imagesc(pxy)
px = sum(nxy, 2) / sum(nxy(:));
py = sum(nxy, 1) / sum(nxy(:));
%% mutual information:
miv = mi(nxy);
%% permutation:
np = 10000;
mis = zeros(np, 1);
for k = 1:np
ppre = presented(randperm(length(presented)));
prep = reported(randperm(length(reported)));
pnxy = zeros(2, 2);
for x = [1, 2]
for y = [1, 2]
pnxy(x, y) = sum((ppre == x) & (prep == y));
end
end
mis(k) = mi(pnxy);
end
alpha = sum(mis>miv)/length(mis);
fprintf('signifikance: %g\n', alpha);
bins = [0.0:0.025:0.4];
hist(mis, bins)
hold on;
plot([miv, miv], [0, np/10], '-r')
hold off;
xlabel('MI')
ylabel('Count')
%% maximum MI:
n = 100000;
pxs = [0:0.01:1.0];
mis = zeros(length(pxs), 1);
for k = 1:length(pxs)
p = rand(n, 1);
nxy = zeros(2, 2);
nxy(1, 1) = sum(p<pxs(k));
nxy(2, 2) = length(p) - nxy(1, 1);
mis(k) = mi(nxy);
%nxy(1, 2) = 0;
%nxy(2, 1) = 0;
%mi(nxy)
end
figure();
plot(pxs, mis);
hold on;
plot([px(1), px(1)], [0, 1], '-r')
hold off;
xlabel('p(x=1)')
ylabel('Max MI=Entropy')

119
projects/project_noiseficurves/noiseficurves.tex
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@ -9,49 +9,50 @@
\input{../instructions.tex} \input{../instructions.tex}
\begin{questions} You are recording the activity of neurons that differ in the strength
\question You are recording the activity of a neuron in response to of their intrinsic noise in response to constant stimuli of intensity
constant stimuli of intensity $I$ (think of that, for example, $I$ (think of that, for example, as a current $I$ injected via a
as a current $I$ injected via a patch-electrode into the neuron). patch-electrode into the neuron).
Measure the tuning curve (also called the intensity-response curve) of the We first characterize the neurons by their tuning curves (also called
neuron. That is, what is the mean firing rate of the neuron's response intensity-response curve). That is, what is the mean firing rate of
as a function of the constant input current $I$? the neuron's response as a function of the constant input current $I$?
How does the intensity-response curve of a neuron depend on the In the second part we demonstrate how intrinsic noise can be useful
level of the intrinsic noise of the neuron? for encoding stimuli on the example of the so called ``subthreshold
stochastic resonance''.
How can intrinsic noise be usefull for encoding stimuli?
The neuron is implemented in the file \texttt{lifspikes.m}. Call it
The neuron is implemented in the file \texttt{lifspikes.m}. Call it with the following parameters:\\[-7ex]
with the following parameters:\\[-7ex] \begin{lstlisting}
\begin{lstlisting} trials = 10;
trials = 10; tmax = 50.0;
tmax = 50.0; current = 10.0; % the constant input current I
current = 10.0; % the constant input current I Dnoise = 1.0; % noise strength
Dnoise = 1.0; % noise strength spikes = lifspikes(trials, current, tmax, Dnoise);
spikes = lifspikes(trials, current, tmax, Dnoise); \end{lstlisting}
\end{lstlisting} The returned \texttt{spikes} is a cell array with \texttt{trials}
The returned \texttt{spikes} is a cell array with \texttt{trials} elements, each being a vector of spike times (in seconds) computed for
elements, each being a vector of spike times (in seconds) computed a duration of \texttt{tmax} seconds. The input current is set via the
for a duration of \texttt{tmax} seconds. The input current is set \texttt{current} variable, the strength of the intrinsic noise via
via the \texttt{current} variable, the strength of the intrinsic \texttt{Dnoise}. If \texttt{current} is a single number, then an input
noise via \texttt{Dnoise}. If \texttt{current} is a single number, current of that intensity is simulated for \texttt{tmax}
then an input current of that intensity is simulated for seconds. Alternatively, \texttt{current} can be a vector containing an
\texttt{tmax} seconds. Alternatively, \texttt{current} can be a input current that changes in time. In this case, \texttt{tmax} is
vector containing an input current that changes in time. In this ignored, and you have to provide a value for the input current for
case, \texttt{tmax} is ignored, and you have to provide a value every 0.0001\,seconds.
for the input current for every 0.0001\,seconds.
Think of calling the \texttt{lifspikes()} function as a simple way of
Think of calling the \texttt{lifspikes()} function as a simple way doing an electrophysiological experiment. You are presenting a
of doing an electrophysiological experiment. You are presenting a stimulus with a constant intensity $I$ that you set. The neuron
stimulus with a constant intensity $I$ that you set. The neuron responds to this stimulus, and you record this response. After
responds to this stimulus, and you record this response. After detecting the timepoints of the spikes in your recordings you get what
detecting the timepoints of the spikes in your recordings you get the \texttt{lifspikes()} function returns. In addition you can record
what the \texttt{lifspikes()} function returns. In addition you from different neurons with different noise properties by setting the
can record from different neurons with different noise properties \texttt{Dnoise} parameter to different values.
by setting the \texttt{Dnoise} parameter to different values.
\begin{questions}
\question Tuning curves
\begin{parts} \begin{parts}
\part First set the noise \texttt{Dnoise=0} (no noise). Compute \part First set the noise \texttt{Dnoise=0} (no noise). Compute
and plot the neuron's $f$-$I$ curve, i.e. the mean firing rate and plot the neuron's $f$-$I$ curve, i.e. the mean firing rate
@ -64,37 +65,43 @@ spikes = lifspikes(trials, current, tmax, Dnoise);
\part Compute the $f$-$I$ curves of neurons with various noise \part Compute the $f$-$I$ curves of neurons with various noise
strengths \texttt{Dnoise}. Use for example $D_{noise} = 10^{-3}$, strengths \texttt{Dnoise}. Use for example $D_{noise} = 10^{-3}$,
$10^{-2}$, and $10^{-1}$. $10^{-2}$, and $10^{-1}$. Depending on the resulting curves you
might want to try additional noise levels.
How does the intrinsic noise influence the response curve? How does the intrinsic noise level influence the tuning curves?
What are possible sources of this intrinsic noise? What are possible sources of this intrinsic noise?
\part Show spike raster plots and interspike interval histograms \part Show spike raster plots and interspike interval histograms
of the responses for some interesting values of the input and the of the responses for some interesting values of the input and the
noise strength. For example, you might want to compare the noise strength. For example, you might want to compare the
responses of the four different neurons to the same input, or by responses of the different neurons to the same input, or by the
the same resulting mean firing rate. same resulting mean firing rate.
How do the responses differ? How do the responses differ?
\end{parts}
\part Let's now use as an input to the neuron a 1\,s long sine \question Subthreshold stochastic resonance
wave $I(t) = I_0 + A \sin(2\pi f t)$ with offset current $I_0$,
amplitude $A$, and frequency $f$. Set $I_0=5$, $A=4$, and
$f=5$\,Hz.
Do you get a response of the noiseless ($D_{noise}=0$) neuron? Let's now use as an input to the neuron a 1\,s long sine wave $I(t)
= I_0 + A \sin(2\pi f t)$ with offset current $I_0$, amplitude $A$,
and frequency $f$. Set $I_0=5$, $A=4$, and $f=5$\,Hz.
What happens if you increase the noise strength? \begin{parts}
\part Do you get a response of the noiseless ($D_{noise}=0$) neuron?
What happens at really large noise strengths? \part What happens if you increase the noise strength?
Generate some example plots that illustrate your findings. \part What happens at really large noise strengths?
Explain the encoding of the sine wave based on your findings \part Generate some example plots that illustrate your findings.
\part Explain the encoding of the sine wave based on your findings
regarding the $f$-$I$ curves. regarding the $f$-$I$ curves.
\end{parts} \part Why is this phenomenon called ``subthreshold stochastic resonance''?
\end{parts}
\end{questions} \end{questions}

2
projects/project_power_analysis/power_analysis.tex
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@ -1,6 +1,6 @@
\documentclass[a4paper,12pt,pdftex]{exam} \documentclass[a4paper,12pt,pdftex]{exam}
\newcommand{\ptitle}{EOD waveform} \newcommand{\ptitle}{Power analysis}
\input{../header.tex} \input{../header.tex}
\firstpagefooter{Supervisor: Peter Pilz}{phone: 29 74835}% \firstpagefooter{Supervisor: Peter Pilz}{phone: 29 74835}%
{email: peter.pilz@uni-tuebingen.de} {email: peter.pilz@uni-tuebingen.de}