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[projects] fixed fano_slope

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projects/project_fano_slope/fano_slope.tex
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@ -9,63 +9,59 @@
\input{../instructions.tex} \input{../instructions.tex}
An important property of sensory systems is their ability to
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% discriminate similar stimuli. For example, discrimination of two
colors, light intensities, pitch of two tones, sound intensities, etc.
\begin{questions} Here we look at the level of a single neuron. What does it mean in
\question An important property of sensory systems is their ability terms of the neuron's $f$-$I$ curve (firing rate versus stimulus
to discriminate similar stimuli. For example, discrimination of two intensity) that two similar stimuli can be discriminated given the
colors, light intensities, pitch of two tones, sound intensities, spike train responses that have been evoked by the two stimuli?
etc. Here we look at the level of a single neuron. What does it
mean in terms of the neuron's $f$-$I$ curve (firing rate versus You are recording the activity of a neuron in response to two
stimulus intensity) that two similar stimuli can be discriminated different stimuli $I_1$ and $I_2$ (think of them, for example, of two
given the spike train responses that have been evoked by the two different sound intensities, $I_1$ and $I_2$, and the spiking activity
stimuli? of an auditory afferent). The neuron responds to a stimulus with a
number of spikes. You (an upstream neuron) can count the number of
You are recording the activity of a neuron in response to two spikes of this response within an observation time of duration
different stimuli $I_1$ and $I_2$ (think of them, for example, of $T=100$\,ms. For perfect discrimination, the number of spikes evoked
two different sound intensities, $I_1$ and $I_2$, and the spiking by the stronger stimulus within $T$ is always larger than for the
activity of an auditory afferent). The neuron responds to a stimulus smaller stimulus. The situation is more complicated, because the
with a number of spikes. You (an upstream neuron) can count the number of spikes evoked by one stimulus is not fixed but varies, such
number of spikes of this response within an observation time of that the number of spikes evoked by the stronger stimulus could happen
duration $T=100$\,ms. For perfect discrimination, the number of to be lower than the number of spikes evoked by the smaller stimulus.
spikes evoked by the stronger stimulus within $T$ is always larger
than for the smaller stimulus. The situation is more complicated, The central questions of this project are:
because the number of spikes evoked by one stimulus is not fixed but \begin{itemize}
varies, such that the number of spikes evoked by the stronger \item How can an upstream neuron discriminate two stimuli based on the
stimulus could happen to be lower than the number of spikes evoked spike counts $n$?
by the smaller stimulus. \item How does this depend on the gain of the neuron?
\end{itemize}
The central questions of this project are:
\begin{itemize} The neuron is implemented in the file \texttt{lifboltzmannspikes.m}.
\item How can an upstream neuron discriminate two stimuli based Call it with the following parameters:\vspace{-5ex}
on the spike counts $n$? \begin{lstlisting}
\item How does this depend on the gain of the neuron?
\end{itemize}
The neuron is implemented in the file \texttt{lifboltzmannspikes.m}.
Call it with the following parameters:\vspace{-5ex}
\begin{lstlisting}
trials = 10; trials = 10;
tmax = 50.0; tmax = 50.0;
gain = 0.1; gain = 0.1;
input = 10.0; input = 10.0;
spikes = lifboltzmanspikes(trials, input, tmax, gain); spikes = lifboltzmanspikes(trials, input, tmax, gain);
\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 elements, each being a vector of spike times (in seconds) computed for
for a duration of \texttt{tmax} seconds. The intensity of the a duration of \texttt{tmax} seconds. The intensity of the stimulus is
stimulus is set via the \texttt{input} variable. set via the \texttt{input} variable.
Think of calling the \texttt{lifboltzmannspikes()} function as a Think of calling the \texttt{lifboltzmannspikes()} function as a
simple way of doing an electrophysiological experiment. You are simple way of doing an electrophysiological experiment. You are
presenting a stimulus with an intensity $I$ that you set. The neuron presenting a stimulus with an 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 detecting the timepoints of the spikes in your recordings you get what
what the \texttt{lifboltzmannspikes()} function returns. In addition the \texttt{lifboltzmannspikes()} function returns. In addition you
you can record from different neurons with different properties can record from different neurons with different properties by setting
by setting the \texttt{gain} parameter to different values. the \texttt{gain} parameter to different values.
\begin{questions}
\question Spike counts of the responses
\begin{parts} \begin{parts}
\part Measure the tuning curve of the neuron with respect to the \part Measure the tuning curve of the neuron with respect to the
input. That is, compute the mean firing rate (number of spikes input. That is, compute the mean firing rate (number of spikes
@ -87,34 +83,40 @@ spikes = lifboltzmanspikes(trials, input, tmax, gain);
responses? responses?
\part Generate properly normalized histograms of the spike counts \part Generate properly normalized histograms of the spike counts
within $T$ (use $T=100$\,ms) of the spike responses to the two within windows of duration $T$ (use $T=100$\,ms) of the spike
different stimuli. Do the two histograms overlap? What does this responses to the two different stimuli.
mean for the discriminability of the two stimuli?
Do the two histograms overlap? What does this mean for the
discriminability of the two stimuli?
How do the histograms of the spike counts depend on the gain of How do the histograms of the spike counts depend on the gain of
the neuron? Plot them for the four different values of the gain the neuron? Plot them for the four different values of the gain
used in (a). used in (a).
\end{parts}
\question Discriminability of the responses
\begin{parts}
\part \label{discrmeasure} Think about a measure based on the \part \label{discrmeasure} Think about a measure based on the
spike-count histograms that quantifies how well the two stimuli spike-count histograms that quantifies how well the two stimuli
can be distinguished based on the spike counts. Plot the can be distinguished based on the spike counts.
dependence of this measure as a function of the gain of the
neuron.
%
For which gains can the two stimuli perfectly discriminated?
\underline{Hint:} A possible readout is to set a threshold \underline{Hint:} A possible readout is to set a threshold
$n_{thresh}$ for the observed spike count. Any response smaller $n_{thresh}$ for the observed spike count. Any response smaller
than the threshold assumes that the stimulus was $I_1$, any than the threshold assumes that the stimulus was $I_1$, any
response larger than the threshold assumes that the stimulus was response larger than the threshold assumes that the stimulus was
$I_2$. For a given $T$ find the threshold $n_{thresh}$ that $I_2$. For the given window $T$ find the threshold $n_{thresh}$ that
results in the best discrimination performance. How can you results in the best discrimination performance. How can you
quantify ``best discrimination'' performance? quantify ``best discrimination'' performance?
\part \label{gaindiscr} For which gains can the two stimuli perfectly discriminated?
Plot the dependence of this measure as a function of the gain of
the neuron.
\part Another way to quantify the discriminability of the spike \part Another way to quantify the discriminability of the spike
counts in response to the two stimuli is to apply an appropriate counts in response to the two stimuli is to apply an appropriate
statistical test and check for significant differences. How does statistical test and check for significant differences. How does
this compare to your findings from (\ref{discrmeasure})? this compare to your findings from (\ref{gaindiscr})?
\end{parts} \end{parts}