diff --git a/projects/project_fano_slope/fano_slope.tex b/projects/project_fano_slope/fano_slope.tex index a204e9a..2a9ac20 100644 --- a/projects/project_fano_slope/fano_slope.tex +++ b/projects/project_fano_slope/fano_slope.tex @@ -9,63 +9,59 @@ \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? - - 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. - - 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 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} +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. + +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 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; tmax = 50.0; gain = 0.1; 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. - - 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. +\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. + +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. +\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}