[simulations] added random walk

scientificcomputing-script.tex

@@ -46,6 +46,10 @@
 % - data analysis skills way beyond basic statistical test are at the
 %   core of modern neuroscience
 % - *understanding* of basic concepts of data analysis concepts is important
+% - math is the universal language of natural sciences, including neuroscience.
+%   it is the most precise way to express relation.
+%   We therefore expose you to some mathematical equations 
+%   that do not go beyond what you learned at school.
 % - most concepts are also quite relevant for the brain itself!
 % - modern approaches:
 %   * open source science (open data, open code, open algorithmns in contrast 

simulations/lecture/simulations.tex

@@ -160,6 +160,13 @@ corresponding standard deviation equal the inverse rate.
   values.
 \end{exercise}
 
+
+\subsection{Uniformly distributed random numbers}
+
+\begin{figure}[t]
+  Exponential distribution of ISIs.
+\end{figure}
+
 The \code{rand()} function returns uniformly distributed random
 numbers between zero and one
 \begin{equation}
@@ -201,12 +208,40 @@ 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.
 
+\subsection{Random walks}
+A basic concept for stochastic models is the random walk. A walker
+starts at some initial position $x_0$. It then takes steps to the
+right, $x_n = x_{n-1} + 1$, with some probability $P_r$ or to the
+left, $x_n = x_{n-1} - 1$, with probability $P_l = 1-P_r$.
 
+For a symmetric random walk, the probabilities to step to the left or
+to the right are the same, $P_r = P_l = \frac{1}{2}$. The average
+position of many walkers is then independent of the iteration step and
+equals the initial position $x_0$. The standard deviation of the
+walker positions, however, grows with the square root of the number of
+iteration steps $n$: $\sigma_{x_n} = \sqrt{n}$.
 
+\begin{figure}[tb]
+  Have a figure with a few 1D random walks and the increasing standard
+  deviation as a shaded area behind.  And with absorbing boundary.
+\end{figure}
 
+The sub-threshold membrane potential of a neuron can be modeled as a
+one-dimensional random walk. The thousands of excitatory and
+inhibitory synaptic inputs of a principal neuron in the brain make the
+membrane potential to randomly step upwards or downwards. Once the
+membrane potential hits a firing threshold an action potential is
+generated and the membrane potential is reset to the resting
+potential. The firing threshold in this context is often called an
+``absorbing boundary''. The time it takes from the initial position to
+cross the threshold is the ``first-passage time''. For neurons this is
+the interspike interval. For symmetric random walks the first-passage
+times are again exponentially distributed. ??? Is that so ???
+
+% Gerstein and Mandelbrot 1964
+
+Higher dimensions, Brownian motion.
 
 
 The gamma distribution (\code{gamrnd()}) phenomenologically describes