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[projects] fixed fano_slope

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Jan Benda 2021-01-31 22:55:40 +01:00
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projects/project_fano_slope/fano_slope.tex
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@ -9,37 +9,31 @@
\input{../instructions.tex}
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{questions}
\question An important property of sensory systems is their ability
to discriminate similar stimuli. For example, discrimination of two
colors, light intensities, pitch of two tones, sound intensities,
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
stimulus intensity) that two similar stimuli can be discriminated
given the spike train responses that have been evoked by the two
stimuli?
An important property of sensory systems is their ability to
discriminate similar stimuli. For example, discrimination of two
colors, light intensities, pitch of two tones, sound intensities, 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 stimulus
intensity) that two similar stimuli can be discriminated given the
spike train responses that have been evoked by the two stimuli?
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 different sound intensities, $I_1$ and $I_2$, and the spiking
activity of an auditory afferent). The neuron responds to a stimulus
with a number of spikes. You (an upstream neuron) can count the
number of spikes of this response within an observation time of
duration $T=100$\,ms. For perfect discrimination, the number of
spikes evoked by the stronger stimulus within $T$ is always larger
than for the smaller stimulus. The situation is more complicated,
because the number of spikes evoked by one stimulus is not fixed but
varies, such that the number of spikes evoked by the stronger
stimulus could happen to be lower than the number of spikes evoked
by the smaller stimulus.
different stimuli $I_1$ and $I_2$ (think of them, for example, of two
different sound intensities, $I_1$ and $I_2$, and the spiking activity
of an auditory afferent). The neuron responds to a stimulus with a
number of spikes. You (an upstream neuron) can count the number of
spikes of this response within an observation time of duration
$T=100$\,ms. For perfect discrimination, the number of spikes evoked
by the stronger stimulus within $T$ is always larger than for the
smaller stimulus. The situation is more complicated, because the
number of spikes evoked by one stimulus is not fixed but varies, such
that the number of spikes evoked by the stronger stimulus could happen
to be lower than the number of spikes evoked by the smaller stimulus.
The central questions of this project are:
\begin{itemize}
\item How can an upstream neuron discriminate two stimuli based
on the spike counts $n$?
\item How can an upstream neuron discriminate two stimuli based on the
spike counts $n$?
\item How does this depend on the gain of the neuron?
\end{itemize}
@ -53,19 +47,21 @@ input = 10.0;
spikes = lifboltzmanspikes(trials, input, tmax, gain);
\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 intensity of the
stimulus is set via the \texttt{input} variable.
elements, each being a vector of spike times (in seconds) computed for
a duration of \texttt{tmax} seconds. The intensity of the stimulus is
set via the \texttt{input} variable.
Think of calling the \texttt{lifboltzmannspikes()} function as a
simple way of doing an electrophysiological experiment. You are
presenting a stimulus with an 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{lifboltzmannspikes()} function returns. In addition
you can record from different neurons with different properties
by setting the \texttt{gain} parameter to different values.
detecting the timepoints of the spikes in your recordings you get what
the \texttt{lifboltzmannspikes()} function returns. In addition you
can record from different neurons with different properties by setting
the \texttt{gain} parameter to different values.
\begin{questions}
\question Spike counts of the responses
\begin{parts}
\part Measure the tuning curve of the neuron with respect to the
input. That is, compute the mean firing rate (number of spikes
@ -87,34 +83,40 @@ spikes = lifboltzmanspikes(trials, input, tmax, gain);
responses?
\part Generate properly normalized histograms of the spike counts
within $T$ (use $T=100$\,ms) of the spike responses to the two
different stimuli. Do the two histograms overlap? What does this
mean for the discriminability of the two stimuli?
within windows of duration $T$ (use $T=100$\,ms) of the spike
responses to the two different 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
the neuron? Plot them for the four different values of the gain
used in (a).
\end{parts}
\question Discriminability of the responses
\begin{parts}
\part \label{discrmeasure} 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 gain of the
neuron.
%
For which gains can the two stimuli perfectly discriminated?
can be distinguished based on the spike counts.
\underline{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
$I_2$. 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
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
counts in response to the two stimuli is to apply an appropriate
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}