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[simulations] added random walk neuron figures

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This commit is contained in:
Jan Benda 2020-12-29 23:45:46 +01:00
parent 2bac236768
commit 16aa2b8f4b
5 changed files with 150 additions and 43 deletions
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2
plotstyle.py
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@ -52,6 +52,8 @@ lsSpine = {'c': colors['black'], 'linestyle': '-', 'linewidth': 1, 'clip_on': Fa
lsGrid = {'c': colors['gray'], 'linestyle': '--', 'linewidth': 1}
lsMarker = {'c': colors['black'], 'linestyle': '-', 'linewidth': 2}
psMarker = dict({'color': colors['black'], 'linestyle': 'none'}, **largemarker)
# line (ls), point (ps), and fill styles (fs).
# Each style is derived from a main color as indicated by the capital letter.

58
simulations/lecture/randomwalkneuron.py Normal file
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@ -0,0 +1,58 @@
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from plotstyle import *
def random_walk(n, p, rng):
steps = rng.rand(n)
steps[steps>=1.0-p] = 1
steps[steps<1.0-p] = -1
x = np.hstack(((0.0,), np.cumsum(steps)))
return x
def random_walk_neuron(n, p, thresh, rng):
x = random_walk(n, p, rng)
spikes = []
j = -1
i = 1
while i > 0:
if j > 0:
spikes.append(j+i)
j += i + 1
x[j:] -= x[j]
i = np.argmax(x[j:] >= thresh - 0.1)
return x, spikes
if __name__ == "__main__":
fig = plt.figure()
spec = gridspec.GridSpec(nrows=1, ncols=2, width_ratios=[2, 1], wspace=0.4,
**adjust_fs(fig, left=6.5, right=1.5))
ax = fig.add_subplot(spec[0, 0])
rng = np.random.RandomState(52281)
p = 0.6
thresh = 10.0
nmax = 500
n = np.arange(0.0, nmax+1, 1.0)
x, spikes = random_walk_neuron(nmax, p, thresh, rng)
ax.axhline(0.0, **lsGrid)
ax.axhline(thresh, **lsAm)
ax.plot(n, x, **lsBm)
for tspike in spikes:
ax.plot([tspike, tspike], [12.0, 16.0], **lsC)
ax.set_xlabel('Time')
ax.set_ylabel('Potential')
ax.set_xlim(0, nmax)
ax.set_ylim(-10, 17)
ax.set_yticks(np.arange(-10, 11, 10))
ax = fig.add_subplot(spec[0, 1])
nmax = 100000
x, spikes = random_walk_neuron(nmax, p, thresh, rng)
isis = np.diff(spikes)
ax.hist(isis, np.arange(0.0, 151.0, 10.0), **fsAs)
ax.set_xlabel('ISI')
ax.set_ylabel('Count')
ax.set_xticks(np.arange(0, 151, 50))
ax.set_yticks(np.arange(0, 401, 100))
fig.savefig("randomwalkneuron.pdf")
plt.close()

32
simulations/lecture/randomwalkone.py
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@ -4,31 +4,39 @@ from plotstyle import *
def random_walk(n, p, rng):
steps = rng.rand(n)
steps[steps>=p] = 1
steps[steps<p] = -1
steps[steps>=1.0-p] = 1
steps[steps<1.0-p] = -1
x = np.hstack(((0.0,), np.cumsum(steps)))
return x
if __name__ == "__main__":
fig, ax = plt.subplots()
fig.subplots_adjust(**adjust_fs(fig, right=1.0))
rng = np.random.RandomState(52281)
nmax = 80
def plot_random_walk(ax, nmax, p, rng, ymin=-20.0, ymax=20.0):
nn = np.linspace(0.0, nmax, 200)
ax.fill_between(nn, np.sqrt(nn), -np.sqrt(nn), **fsAa)
ax.axhline(0.0, **lsAm)
m = 2*p-1
v = 4*p*(1-p)
ax.fill_between(nn, m*nn+np.sqrt(nn*v), m*nn-np.sqrt(nn*v), **fsAa)
ax.plot([0.0, nmax], [0.0, m*nmax], **lsAm)
lcs = [colors['red'], colors['orange'],
colors['yellow'], colors['green']]
n = np.arange(0.0, nmax+1, 1.0)
for k in range(12):
x = random_walk(nmax, 0.5, rng)
x = random_walk(nmax, p, rng)
ls = dict(**lsAm)
ls['color'] = lcs[k%len(lcs)]
ax.plot(n, x, **ls)
ax.set_xlabel('Iteration $n$')
ax.set_ylabel('Position $x_n$')
ax.set_xlim(0, nmax)
ax.set_ylim(-20, 20)
ax.set_yticks(np.arange(-20, 21, 10))
ax.set_ylim(ymin, ymax)
ax.set_yticks(np.arange(ymin, ymax+1.0, 10.0))
if __name__ == "__main__":
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=cm_size(figure_width, 2.0*figure_height))
fig.subplots_adjust(**adjust_fs(fig, right=1.0))
rng = np.random.RandomState(52281)
plot_random_walk(ax1, 80, 0.5, rng)
ax1.text(5.0, 20.0, 'symmetric', va='center')
plot_random_walk(ax2, 80, 0.6, rng, -10.0, 30.0)
ax2.text(5.0, 30.0, 'with drift', va='center')
fig.savefig("randomwalkone.pdf")
plt.close()

1
simulations/lecture/randomwalktwo.py
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@ -27,6 +27,7 @@ if __name__ == "__main__":
ls = dict(**lsAm)
ls['color'] = lcs[k%len(lcs)]
ax.plot(x, y, **ls)
ax.plot([0], [0], **psMarker)
ax.set_xlabel('Position $x_n$')
ax.set_ylabel('Position $y_n$')
ax.set_xlim(-30, 30)

100
simulations/lecture/simulations.tex
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@ -212,31 +212,69 @@ the event does not occur.
\end{exercise}
\subsection{Random walks}
A basic concept for stochastic models is the random walk. A walker
starts at some initial position $x_0$. At every iteration $n$ it takes
a step $\delta_n$: $x_n = x_{n-1} + \delta_n$. The walker either steps
to the right ($\delta_n = +1$) with some probability $P_r$ or to the
left ($\delta_n = -1$) with probability $P_l = 1-P_r$.
For a symmetric random walk (\figref{randomwalkonefig}), 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 variance of each step is $\sigma_{\delta_n} = 1$. Since
each step is independent of the steps before, the variances of each
step add up such that the total variance of the walker position at
step $n$ equals $\sigma_{x_n}^2 = n$. The standard deviation of the
walker positions grows with the square root of the number of iteration
steps $n$: $\sigma_{x_n} = \sqrt{n}$.
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()}
\section{Random walks}
A basic concept for stochastic models is the \enterm{random walk}. A
walker starts at some initial position $x_0$. At every iteration $n$
it takes a step $\delta_n$: $x_n = x_{n-1} + \delta_n$. The walker
either steps to the right ($\delta_n = +1$) with some probability $p$
or to the left ($\delta_n = -1$) with probability $q = 1-p$.
\begin{figure}[tb]
\includegraphics{randomwalkone}
\titlecaption{\label{randomwalkonefig} Random walk in one
dimension.}{Twelve symmetric random walks starting at $x_0=0$. The
ensemble average of the position of the random walkers is zero
(blue line) and the standard deviation increases with the square
root of the number of steps taken (shaded area).}
dimension.}{Twelve random walks starting at $x_0=0$. The ensemble
average of the position of the random walkers marked by the blue
line and the corresponding standard deviation by the shaded
area. The standard deviation increases with the square root of the
number of steps taken. The top plot shows a symmetric random walk
where the probability of a step to the right equals the one for a
step to the left ($p=q=\frac{1}{2}$). The bottom plot shows a
random walk with drift. Here the probability of taking a step to
the right ($p=0.6$) is larger than the probability for a step to
the left ($q=0.4$).}
\end{figure}
For a \enterm[random walk!symmetric]{symmetric random walk}
(\figref{randomwalkonefig} top), the probabilities for a step to the
left or to the right are the same, $p = q = \frac{1}{2}$. The average
position of many walkers is then independent of the iteration step and
equals the initial position $x_0$. The variance of each step is
$\sigma_{\delta_n}^2 = p \cdot 1^2 + q \cdot(-1)^2 = 1$. Since each
step is independent of the steps before, the variances of all the
steps add up such that the total variance of the walker position at
step $n$ equals $\sigma_{x_n}^2 = \sum_{i=1}^n \sigma_{\delta_n}^2 =
n$. The standard deviation of the walker positions grows with the
square root of the number of iteration steps $n$: $\sigma_{x_n} =
\sqrt{n}$.
\enterm[random walk!with drift]{Random walks with drift} are biased to
one direction (\figref{randomwalkonefig} bottom). The probability to
step to the left does not equal the probability to step to the right,
$p \ne q$. Consequently, the average step is no longer zero. Instead
it equals $\mu_{\delta_n} = p \cdot 1 + q \cdot (-1) = 2p - 1$. The
average position grows proportional to the number of iterations:
$\mu_{x_n} = n(2p - 1)$. The variance of each step turns out to be
$\sigma_{\delta_n}^2 = 4pq$ (this indeed equals one for
$p=q=\frac{1}{2}$). Again, because of the independence of the steps,
the variance of the position at the $n$-th iteration, $\sigma_{x_n}^2
= 4pqn$, is proportional to $n$ and the standard deviation,
$\sigma_{x_n} = 2\sqrt{pqn}$, grows with the square root of $n$.
\begin{exercise}{}{}
Random walk exercise.
\end{exercise}
\begin{figure}[tb]
\includegraphics{randomwalkneuron}
\titlecaption{\label{randomwalkneuronfig} Random walk spiking neuron.}{.}
\end{figure}
The sub-threshold membrane potential of a neuron can be modeled as a
@ -253,23 +291,23 @@ times are again exponentially distributed. ??? Is that so ???
% Gerstein and Mandelbrot 1964
Higher dimensions (\figref{randomwalktwofig}), Brownian motion. Crystal growth.
\begin{exercise}{}{}
Random walk neuron exercise.
\end{exercise}
\begin{figure}[tb]
\includegraphics{randomwalktwo}
\titlecaption{\label{randomwalktwofig} Random walk in two
dimensions.}{Fivehundred steps of five random walkers that in each
iteration take a step either up, down, to the left, or to the
right.}
dimensions.}{Fivehundred steps of five random walkers starting at
the origin (black dot) and taking steps of distance one either up,
down, to the left, or to the right with equal probability.}
\end{figure}
Higher dimensions (\figref{randomwalktwofig}), Brownian motion. Crystal growth.
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()}
\begin{exercise}{}{}
Random walk in 2D exercise.
\end{exercise}
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