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[projects] updated mutual information and noisy ficurves

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105
projects/project_mutualinfo/mutualinfo.tex
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%%%%%%%%%%%%%% 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}
\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
@ -19,50 +62,56 @@
object was reported by the subject.
\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
combinations of reported and presented came up.
\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
object and $y$ is the reported object). Compute the probability
distributions $P(x)$ and $P(y)$ in the same way.
\part Use that probability distribution to compute the mutual
information
\[ I[x:y] = \sum_{x\in\{1,2\}}\sum_{y\in\{1,2\}} P(x,y)
\log_2\frac{P(x,y)}{P(x)P(y)}\]
that the answers provide about the actually presented object.
The mutual information is a measure from information theory that is
used in neuroscience to quantify, for example, how much information
a spike train carries about a sensory stimulus.
\part What is the maximally achievable mutual information?
Show this numerically by generating your own datasets which
naturally should yield maximal information. Consider different
distributions of $P(x)$.
Here 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.
object and $y$ is the reported object).
\part Use the computed probability distributions to compute the mutual
information \eqref{mi} that the answers provide about the
actually presented object.
\part Use a permutation test to compute the $95\%$ confidence
interval for the mutual information estimate in the dataset from
{\tt decisions.mat}. Does the measured mutual information indicate
signifikant information transmission?
\end{parts}
\question What is the maximally achievable mutual information?
\begin{parts}
\part Show this numerically by generating your own datasets which
naturally should yield maximal information. Consider different
distributions of $P(x)$.
\end{questions}
\part Compare the maximal mutual information with the corresponding
entropy \eqref{entropy}.
\end{parts}
\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}

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projects/project_noiseficurves/noiseficurves.tex
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@ -9,49 +9,50 @@
\input{../instructions.tex}
\begin{questions}
\question You are recording the activity of a neuron in response to
constant stimuli of intensity $I$ (think of that, for example,
as a current $I$ injected via a patch-electrode into the neuron).
Measure the tuning curve (also called the intensity-response curve) of the
neuron. That is, what is the mean firing rate of 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
level of the intrinsic noise of the neuron?
How can intrinsic noise be usefull for encoding stimuli?
The neuron is implemented in the file \texttt{lifspikes.m}. Call it
with the following parameters:\\[-7ex]
\begin{lstlisting}
trials = 10;
tmax = 50.0;
current = 10.0; % the constant input current I
Dnoise = 1.0; % noise strength
spikes = lifspikes(trials, current, 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 current is set
via the \texttt{current} variable, the strength of the intrinsic
noise via \texttt{Dnoise}. If \texttt{current} is a single number,
then an input current of that intensity is simulated for
\texttt{tmax} seconds. Alternatively, \texttt{current} can be a
vector containing an input current that changes in time. In this
case, \texttt{tmax} is ignored, and you have to provide a value
for the input current for every 0.0001\,seconds.
Think of calling the \texttt{lifspikes()} function as a simple way
of doing an electrophysiological experiment. You are presenting a
stimulus with a constant intensity $I$ that you set. The neuron
responds to this stimulus, and you record this response. After
detecting the timepoints of the spikes in your recordings you get
what the \texttt{lifspikes()} function returns. In addition you
can record from different neurons with different noise properties
by setting the \texttt{Dnoise} parameter to different values.
You are recording the activity of neurons that differ in the strength
of their intrinsic noise in response to constant stimuli of intensity
$I$ (think of that, for example, as a current $I$ injected via a
patch-electrode into the neuron).
We first characterize the neurons by their tuning curves (also called
intensity-response curve). That is, what is the mean firing rate of
the neuron's response as a function of the constant input current $I$?
In the second part we demonstrate how intrinsic noise can be useful
for encoding stimuli on the example of the so called ``subthreshold
stochastic resonance''.
The neuron is implemented in the file \texttt{lifspikes.m}. Call it
with the following parameters:\\[-7ex]
\begin{lstlisting}
trials = 10;
tmax = 50.0;
current = 10.0; % the constant input current I
Dnoise = 1.0; % noise strength
spikes = lifspikes(trials, current, 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 current is set via the
\texttt{current} variable, the strength of the intrinsic noise via
\texttt{Dnoise}. If \texttt{current} is a single number, then an input
current of that intensity is simulated for \texttt{tmax}
seconds. Alternatively, \texttt{current} can be a vector containing an
input current that changes in time. In this case, \texttt{tmax} is
ignored, and you have to provide a value for the input current for
every 0.0001\,seconds.
Think of calling the \texttt{lifspikes()} function as a simple way of
doing an electrophysiological experiment. You are presenting a
stimulus with a constant intensity $I$ that you set. The neuron
responds to this stimulus, and you record this response. After
detecting the timepoints of the spikes in your recordings you get what
the \texttt{lifspikes()} function returns. In addition you can record
from different neurons with different noise properties by setting the
\texttt{Dnoise} parameter to different values.
\begin{questions}
\question Tuning curves
\begin{parts}
\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
@ -64,37 +65,43 @@ spikes = lifspikes(trials, current, tmax, Dnoise);
\part Compute the $f$-$I$ curves of neurons with various noise
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?
\part Show spike raster plots and interspike interval histograms
of the responses for some interesting values of the input and the
noise strength. For example, you might want to compare the
responses of the four different neurons to the same input, or by
the same resulting mean firing rate.
responses of the different neurons to the same input, or by the
same resulting mean firing rate.
How do the responses differ?
\end{parts}
\question Subthreshold stochastic resonance
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.
\part 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.
Do you get a response of the noiseless ($D_{noise}=0$) neuron?
\begin{parts}
\part Do you get a response of the noiseless ($D_{noise}=0$) neuron?
What happens if you increase the noise strength?
\part What happens if you increase the noise strength?
What happens at really large noise strengths?
\part What happens at really large noise strengths?
Generate some example plots that illustrate your findings.
\part Generate some example plots that illustrate your findings.
Explain the encoding of the sine wave based on your findings
\part Explain the encoding of the sine wave based on your findings
regarding the $f$-$I$ curves.
\end{parts}
\part Why is this phenomenon called ``subthreshold stochastic resonance''?
\end{parts}
\end{questions}