[pointprocesses] improved spike count and fano factor exercises

pointprocesses/lecture/pointprocesses.tex

@@ -239,19 +239,18 @@ describing univariate data sets of real numbers:
 
 \begin{exercise}{isihist.m}{}
   Implement a function \varcode{isihist()} that calculates the
-  normalized interspike interval histogram. The function should take
-  two input arguments; (i) a vector of interspike intervals and (ii)
-  the width of the bins used for the histogram. It returns the
+  normalized interspike interval histogram. The function should take a
+  vector of interspike intervals and the width of the bins to be used
+  for the histogram as input arguments. The function returns the
   probability density as well as the centers of the bins.
 \end{exercise}
 
 \begin{exercise}{plotisihist.m}{}
-  Implement a function that takes the returned values of
-  \varcode{isihist()} as input arguments and then plots the data. The
-  plot should show the histogram with the x-axis scaled to
-  milliseconds and should be annotated with the average ISI, the
-  standard deviation, and the coefficient of variation of the ISIs
-  (\figref{isihexamplesfig}).
+  Implement a function that uses the \varcode{isihist()} function from
+  the previous exercise to plot an ISI histogram. The plot shows the
+  histogram with the x-axis scaled to milliseconds, annotated with the
+  average ISI, the standard deviation, and the coefficient of
+  variation of the ISIs (\figref{isihexamplesfig}).
 \end{exercise}
 
 \subsection{Interval correlations}
@@ -332,8 +331,8 @@ characterized by the non-zero serial correlations
 
 \begin{exercise}{isiserialcorr.m}{}
   Implement a function \varcode{isiserialcorr()} that takes a vector
-  of interspike intervals as input argument and calculates the serial
-  correlations up to some maximum lag.
+  of interspike intervals as input and computes serial correlations up
+  to some maximum lag.
 \end{exercise}
 
 \begin{exercise}{plotisiserialcorr.m}{}
@@ -398,27 +397,10 @@ Because spike counts are unitless and positive numbers the
   \end{equation}
   is a commonly used measure for quantifying the variability of event
   counts relative to the mean number of events. In particular for
-  homogeneous Poisson processes the Fano factor equals one,
-  independently of the time window $W$.
+  homogeneous Poisson processes the Fano factor equals exactly one and
+  is independent of the time window $W$.
 \end{itemize}
 
-Note that all of these statistics depend in general on the chosen
-window length $W$. The average spike count, for example, grows
-linearly with $W$ for sufficiently large time windows: $\langle n
-\rangle = r W$, \eqnref{firingrate}. Doubling the counting window
-doubles the spike count. As does the spike-count variance
-(\figref{fanofig}). At smaller time windows the statistics of the
-event counts might depend on the particular duration of the counting
-window. There might be an optimal time window for which the variance
-of the spike count is minimal. The Fano factor plotted as a function
-of the time window illustrates such properties of point processes
-(\figref{fanofig}).
-
-This also has consequences for information transmission in neural
-systems. The lower the variance in spike count relative to the
-averaged count, the higher the signal-to-noise ratio at which
-information encoded in the mean spike count is transmitted.
-
 \begin{figure}[t]
   \includegraphics{fanoexamples}
   \titlecaption{\label{fanofig} Fano factor.}{Counting events in time
@@ -435,13 +417,52 @@ information encoded in the mean spike count is transmitted.
     interspike intervals (right).}
 \end{figure}
 
-\begin{exercise}{counthist.m}{}
-  Implement a function \varcode{counthist()} that calculates and plots
-  the distribution of spike counts observed in a certain time
-  window. The function should take two input arguments: a cell-array
-  of vectors containing the spike times in seconds observed in a
-  number of trials, and the duration of the time window that is used
-  to evaluate the counts.
+Note that all of these statistics depend in general on the chosen
+window length $W$ used for counting the events. The average spike
+count, for example, grows linearly with $W$ for sufficiently large
+time windows: $\langle n \rangle = r W$, \eqnref{firingrate}. Doubling
+the counting window doubles the spike count. As does the spike-count
+variance (\figref{fanofig}). At smaller time windows the statistics of
+the event counts might depend on the particular duration of the
+counting window. There could be an optimal time window for which the
+variance of the spike count is minimal. The Fano factor plotted as a
+function of the time window illustrates such properties of point
+processes in a single graph (\figref{fanofig}).
+
+This also has consequences for information transmission in neural
+systems. The membrane time constant of a post-synaptic neuron defines
+a counting window. If input spikes arrive within about one time
+constant they are added up. The lower the variance in spike count
+relative to the averaged count, the higher the signal-to-noise ratio
+at which information encoded in the mean spike count is
+transmitted. If the membrane time constant of the target neuron
+matches the counting window where the Fano factor is minimal, then
+information is potentially transmitted at the highest possible
+signal-to-noise ratio. For this reason, the Fano factor is used in the
+Neurosciences to quantify and analyze reliability of neuronal
+responses.
+
+\begin{exercise}{counts.m}{}
+  Write a function \varcode{counts()} that counts the number of spikes
+  in windows of given duration and returns the counts in a single
+  vector. Spike times are passed as a cell-array of vectors,
+  containing the spike times in seconds observed in a number of
+  trials, to the function.
+\end{exercise}
+
+\begin{exercise}{plotcounthist.m}{}
+  Write a function that takes a cell-array with spike times as input
+  and plots a normalized histogram of the spike counts counted in
+  windows of a given duration.
+\end{exercise}
+
+\begin{exercise}{plotfanofactor.m}{}
+  Write a function that takes a cell-array with spike times as input
+  and plots in one plot count variances a function of the
+  corresponding mean counts and in a second plot the Fano factor as a
+  function of the duration of the count window in logarithmic
+  scale. Two arguments of the function take the minimum and maximum
+  duration of the count window.
 \end{exercise}