[simulations] added uniform distribution

simulations/code/exprandom.m

@@ -1,8 +1,8 @@
-n = 10000;
+n = 200;                     % number of time intervals
-lambda = 50.0;                 % event rate
+rate = 50.0;                 % event rate
-t = exprnd(1.0/lambda, n, 1);  % exponentially distributed random numbers
+t = exprnd(1.0/rate, n, 1);  % exponentially distributed intervals
-m = mean(t);                   % mean time interval
+m = mean(t);                 % mean time interval
-s = std(t);                    % corresponding standard deviation
+s = std(t);                  % corresponding standard deviation
-fprintf('Generated n=%d exponentially distributed random time intervals\n', n);
+fprintf('Generated n=%d random time intervals with rate=%.0fHz:\n', n, rate);
-fprintf('  with rate=%.0fHz (-> mean=%.3fs, std=%.3fs)\n', lambda, 1.0/lambda, 1.0/lambda);
+fprintf('  Expected mean interval=%.3fs, std=%.3fs\n', 1.0/rate, 1.0/rate);
-fprintf('Measured mean=%.3fs, std=%.3fs\n', m, s);
+fprintf('  Measured mean interval=%.3fs, std=%.3fs\n', m, s);

simulations/code/exprandom.out

@@ -1,3 +1,3 @@
-Generated n=10000 exponentially distributed random time intervals
+Generated n=200 random time intervals with rate=50Hz:
-  with rate=50Hz (-> mean=0.020s, std=0.020s)
+  Expected mean interval=0.020s, std=0.020s
-Measured mean=0.020s, std=0.020s
+  Measured mean interval=0.020s, std=0.019s

simulations/code/unirandom.m

@@ -0,0 +1,16 @@
+n = 10000;                    % number of action potentials
+T = 100.0;                    % total time interval
+spikes = sort(T*rand(n, 1));  % sorted times of action potentials
+isis = diff(spikes);          % interspike intervals
+rate = n/T;                   % firing rate
+misi = mean(isis);            % mean interspike interval
+sisi = std(isis);             % and standard deviation
+fprintf('firing rate = %.1fHz\n', rate);
+fprintf('   mean ISI = %.1fms\n', 1000.0*misi);  % inverse of rate
+fprintf('    std ISI = %.1fms\n', 1000.0*sisi);  % same as mean
+hist(1000.0*isis, 50);        % exponential distribution
+xlabel('ISI [ms]');
+ylabel('count');
+
+
+

simulations/code/unirandom.out

@@ -0,0 +1,3 @@
+firing rate = 100.0Hz
+   mean ISI = 10.0ms
+    std ISI = 10.0ms

simulations/lecture/simulations.tex

@@ -79,24 +79,28 @@ script is run.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \section{Univariate data}
-The most basic type of simulation is to draw random numbers from a
+A basic type of stochastic simulations is to draw random numbers from
-given distribution like, for example, the normal distribution. This
+a given distribution like, for example, the normal distribution. This
-simulates repeated measurements of some quantity (e.g., weight of
+simulates repeated measurements of some quantity (weight of tigers,
-tigers or firing rate of neurons). Doing so we must specify from which
+firing rate of neurons). Doing so we must specify from which
 probability distribution the data should originate from and what are
 the parameters of that distribution (mean, standard deviation, shape
-parameters, ...). How to illustrate and quantify univariate data, no
+parameters). How to illustrate and quantify univariate data, no matter
-matter whether they have been actually measured or whether they have
+whether they have been actually measured or whether they have been
-been simulated as described in the following, is described in
+simulated as described in the following, is described in
 chapter~\ref{descriptivestatisticschapter}.
 
 \subsection{Normally distributed data}
-For drawing numbers $x_i$ from a normal distribution we use the
+For drawing numbers from a normal distribution
-\code{randn()} function. This function returns normally distributed
+\begin{equation}
-numbers $\xi_i$ with zero mean and unit standard deviation. For
+  p_{norm}(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}
-changing the standard deviation we need to multiply the returned data
+\end{equation}
-values with the required standard deviation $\sigma$. For changing the
+with zero mean and unit standard deviation we use the \code{randn()}
-mean we just add the desired mean $\mu$ to the random numbers:
+function. This function returns normally distributed numbers $\xi_i$
+with zero mean and unit standard deviation. For changing the standard
+deviation we need to multiply the returned data values with the
+required standard deviation $\sigma$. For changing the mean we just
+add the desired mean $\mu$ to the random numbers:
 \begin{equation}
   x_i = \sigma \xi_i + \mu
 \end{equation}
@@ -144,24 +148,44 @@ only defined for positive time intervals:
   \label{expdistribution}
   p_{exp}(t) = \lambda e^{-\lambda t}  \; , \quad t \ge 0, \; \lambda > 0
 \end{equation}
-The exponential distribution is parameterized by a single rate
+A single rate parameter $\lambda$ measured in Hertz parameterizes the
-parameter $\lambda$ measured in Hertz. It defines how often per time
+exponential distribution. It defines how often per time unit the event
-unit the event happens. Both the mean interval between the events and
+happens. Both the mean interval between the events and the
-the corresponding standard deviation equal the inverse rate.
+corresponding standard deviation equal the inverse rate.
 
 \begin{exercise}{exprandom.m}{exprandom.out}
-  Draw $n=10\,000$ random time intervals from an exponential
+  Draw $n=200$ random time intervals from an exponential distribution
-  distribution with rate $\lambda=50$\,Hz. Calculate the mean and the
+  with rate $\lambda=50$\,Hz. Calculate the mean and the standard
-  standard deviation of the random numbers and compare them with the
+  deviation of the random numbers and compare them with the expected
-  expected values.
+  values.
 \end{exercise}
 
-The gamma distribution (\code{gamrnd()}) phenomenologically describes
+The \code{rand()} function returns uniformly distributed random
-various types of interspike interval dsitributions
+numbers between zero and one
-(chapter~\ref{pointprocesseschapter}). scale and shape. exercise.
+\begin{equation}
+  p_{uni}(x) = \left\{\begin{array}{rcl} 1 & ; & 0 \le x < 1 \\
+                               0 & ; & \text{else} \end{array}\right.
+\end{equation}
+For random numbers $x_i$ uniformly distributed between $x_{\rm min}$
+and $x_{\rm max}$ we multiply by the desired range $x_{\rm max} -
+x_{\rm min}$ and add $x_{\rm min}$ to the numbers $u_i$ returned by
+\code{rand()}:
+\begin{equation}
+  x_i = (x_{\rm max} - x_{\rm min}) u_i + x_{\rm min}
+\end{equation}
+
+\begin{exercise}{unirandom.m}{unirandom.out}
+  Simulate the times of $n=10\,000$ action potentials occuring
+  randomly within a time interval of 100\,s using the \code{rand()}
+  function.  Sort the times using \code{sort()}, compute the time
+  intervals between the action potentials (interspike intervals, ISIs)
+  using the \code{diff()} function, plot a histogram of the interspike
+  intervals in milliseconds, and compute the mean and standard
+  deviation of the interspike intervals, also in milliseconds. Compare
+  the mean ISI with the firing rate (number of action potentials per
+  time).
+\end{exercise}
 
-  \code{rand()} between xmin
-and xmax.
 
 \subsection{Simulating probabilities}
 Simulating events that happen with some probability $P$ is also
@@ -177,6 +201,18 @@ the event does not occur.
   probability of $P=0.6$. Count the number of heads.
 \end{exercise}
 
+Random walk! Have a figure with a few 1D random walks and the
+increasing standard deviation as a sheded area behind.
+
+
+
+
+
+
+The gamma distribution (\code{gamrnd()}) phenomenologically describes
+various types of interspike interval dsitributions
+(chapter~\ref{pointprocesseschapter}). scale and shape. exercise.
+
 \subsection{Random integers}
 \code{randi()}