Fixed plotting Makefile nad pathes.
Copied psth_sta text and scripts to pointprocess chapter.
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2
Makefile
2
Makefile
@ -1,7 +1,7 @@
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BASENAME=scientificcomputing-script
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SUBDIRS=designpattern statistics bootstrap regression likelihood pointprocesses
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#programming spike_trains
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#programming
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SUBTEXS=$(foreach subd, $(SUBDIRS), $(subd)/lecture/$(subd).tex)
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pdf : chapters $(BASENAME).pdf
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10
header.tex
10
header.tex
@ -170,14 +170,15 @@
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\newcommand{\koZ}{\mathds{C}}
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%%%%% code/matlab commands: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{textcomp}
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\newcommand{\code}[1]{\setlength{\fboxsep}{0.5ex}\colorbox{blue!10}{\texttt{#1}}}
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\newcommand{\matlab}{MATLAB}
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\newcommand{\matlab}{MATLAB$^{\copyright}$}
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\newcommand{\matlabfun}[1]{(\tr{\matlab{}-function}{\matlab-Funktion} \setlength{\fboxsep}{0.5ex}\colorbox{blue!10}{\texttt{#1}})}
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%%%%% exercises environment: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% usage:
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%
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% \begin{exercise}[somecode.m]
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% \begin{exercise}{somecode.m}{someoutput.out}
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% Write a function that computes the median.
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% \end{exercise}
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%
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@ -187,6 +188,8 @@
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% Write a function that computes the median.
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% Listing 1: somecode.m
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% the listing
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% Output:
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% content of someoutput.out
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%
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% Innerhalb der exercise Umgebung ist enumerate umdefiniert, um (a), (b), (c), .. zu erzeugen.
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\usepackage{ifthen}
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@ -194,8 +197,7 @@
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\newcounter{maxexercise}
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\setcounter{maxexercise}{10000} % show listings up to exercise maxexercise
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\newcounter{exercise}[chapter]
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\setcounter{exercise}{0}
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\newcommand{\theexercise}{\arabic{exercise}}
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\renewcommand{\theexercise}{\arabic{exercise}}
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\newcommand{\codepath}{}
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\newenvironment{exercise}[2]%
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{\newcommand{\exercisesource}{#1}%
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@ -11,7 +11,7 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\include{psth_sta}
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\include{plotting}
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\end{document}
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268
pointprocesses/lecture/firingrates.py
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268
pointprocesses/lecture/firingrates.py
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import numpy as np
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import matplotlib.pyplot as plt
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import scipy.io as spio
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import scipy.stats as spst
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import scipy as sp
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from IPython import embed
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def set_axis_fontsize(axis, label_size, tick_label_size=None, legend_size=None):
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"""
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Sets axis, tick label and legend font sizes to the desired size.
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:param axis: the axes object
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:param label_size: the size of the axis label
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:param tick_label_size: the size of the tick labels. If None, lable_size is used
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:param legend_size: the size of the font used in the legend.If None, the label_size is used.
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"""
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if not tick_label_size:
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tick_label_size = label_size
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if not legend_size:
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legend_size = label_size
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axis.xaxis.get_label().set_fontsize(label_size)
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axis.yaxis.get_label().set_fontsize(label_size)
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for tick in axis.xaxis.get_major_ticks() + axis.yaxis.get_major_ticks():
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tick.label.set_fontsize(tick_label_size)
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l = axis.get_legend()
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if l:
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for t in l.get_texts():
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t.set_fontsize(legend_size)
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def get_instantaneous_rate(times, max_t=30., dt=1e-4):
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time = np.arange(0., max_t, dt)
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indices = np.asarray(times / dt, dtype=int)
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intervals = np.diff(np.hstack(([0], times)))
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inst_rate = np.zeros(time.shape)
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for i, index in enumerate(indices[1:]):
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inst_rate[indices[i-1]:indices[i]] = 1/intervals[i]
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return time, inst_rate
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def plot_isi_rate(spike_times, max_t=30, dt=1e-4):
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times = np.squeeze(spike_times[0][0])[:50000]
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time, rate = get_instantaneous_rate(times, max_t=50000*dt)
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rates = np.zeros((len(rate), len(spike_times)))
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for i in range(len(spike_times)):
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_, rates[:, i] = get_instantaneous_rate(np.squeeze(spike_times[i][0])[:50000],
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max_t=50000*dt)
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avg_rate = np.mean(rates, axis=1)
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rate_std = np.std(rates, axis=1)
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fig = plt.figure()
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ax1 = fig.add_subplot(311)
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ax2 = fig.add_subplot(312)
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ax3 = fig.add_subplot(313)
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ax1.vlines(times[times < (50000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
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ax1.set_ylabel("skpikes", fontsize=12)
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set_axis_fontsize(ax1, 12)
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ax1.set_xlim([0, 5])
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ax2.plot(time, rate, label="instantaneous rate, trial 1")
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ax2.set_ylabel("firing rate [Hz]", fontsize=12)
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ax2.legend(fontsize=12)
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set_axis_fontsize(ax2, 12)
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ax3.fill_between(time, avg_rate+rate_std, avg_rate-rate_std, color="dodgerblue",
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alpha=0.5, label="standard deviation")
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ax3.plot(time, avg_rate, label="average rate")
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ax3.set_xlabel("times [s]", fontsize=12)
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ax3.set_ylabel("firing rate [Hz]", fontsize=12)
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ax3.legend(fontsize=12)
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ax3.set_ylim([0, 450])
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set_axis_fontsize(ax3, 12)
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fig.set_size_inches(15, 10)
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fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95)
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fig.set_facecolor("white")
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fig.savefig("figures/instantaneous_rate.png")
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plt.close()
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def get_binned_rate(times, bin_width=0.05, max_t=30., dt=1e-4):
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time = np.arange(0., max_t, dt)
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bins = np.arange(0., max_t, bin_width)
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bin_indices = bins / dt
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hist, _ = sp.histogram(times, bins)
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rate = np.zeros(time.shape)
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for i, b in enumerate(bin_indices[1:]):
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rate[bin_indices[i-1]:b] = hist[i-1]/bin_width
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return time, rate
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def plot_bin_rate(spike_times, bin_width, max_t=30, dt=1e-4):
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times = np.squeeze(spike_times[0][0])
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time, rate = get_binned_rate(times)
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rates = np.zeros((len(rate), len(spike_times)))
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for i in range(len(spike_times)):
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_, rates[:, i] = get_binned_rate(np.squeeze(spike_times[i][0]))
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avg_rate = np.mean(rates, axis=1)
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rate_std = np.std(rates, axis=1)
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fig = plt.figure()
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ax1 = fig.add_subplot(311)
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ax2 = fig.add_subplot(312)
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ax3 = fig.add_subplot(313)
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ax1.vlines(times[times < (100000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
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ax1.set_ylabel("skpikes", fontsize=12)
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ax1.set_xlim([0, 5])
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set_axis_fontsize(ax1, 12)
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ax2.plot(time, rate, label="binned rate, trial 1")
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ax2.set_ylabel("firing rate [Hz]", fontsize=12)
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ax2.legend(fontsize=12)
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ax2.set_xlim([0, 5])
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set_axis_fontsize(ax2, 12)
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ax3.fill_between(time, avg_rate+rate_std, avg_rate-rate_std, color="dodgerblue",
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alpha=0.5, label="standard deviation")
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ax3.plot(time, avg_rate, label="average rate")
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ax3.set_xlabel("times [s]", fontsize=12)
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ax3.set_ylabel("firing rate [Hz]", fontsize=12)
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ax3.legend(fontsize=12)
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ax3.set_xlim([0, 5])
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ax3.set_ylim([0, 450])
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set_axis_fontsize(ax3, 12)
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fig.set_size_inches(15, 10)
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fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95)
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fig.set_facecolor("white")
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fig.savefig("figures/binned_rate.png")
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plt.close()
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def get_convolved_rate(times, sigma, max_t=30., dt=1.e-4):
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time = np.arange(0., max_t, dt)
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kernel = spst.norm.pdf(np.arange(-8*sigma, 8*sigma, dt),loc=0,scale=sigma)
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indices = np.asarray(times/dt, dtype=int)
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rate = np.zeros(time.shape)
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rate[indices] = 1.;
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conv_rate = np.convolve(rate, kernel, mode="same")
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return time, conv_rate
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def plot_conv_rate(spike_times, sigma=0.05, max_t=30, dt=1e-4):
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times = np.squeeze(spike_times[0][0])
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time, rate = get_convolved_rate(times, sigma)
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rates = np.zeros((len(rate), len(spike_times)))
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for i in range(len(spike_times)):
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_, rates[:, i] = get_convolved_rate(np.squeeze(spike_times[i][0]), sigma)
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avg_rate = np.mean(rates, axis=1)
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rate_std = np.std(rates, axis=1)
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fig = plt.figure()
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ax1 = fig.add_subplot(311)
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ax2 = fig.add_subplot(312)
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ax3 = fig.add_subplot(313)
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ax1.vlines(times[times < (100000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
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ax1.set_ylabel("skpikes", fontsize=12)
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ax1.set_xlim([0, 5])
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set_axis_fontsize(ax1, 12)
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ax2.plot(time, rate, label="convolved rate, trial 1")
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ax2.set_ylabel("firing rate [Hz]", fontsize=12)
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ax2.legend(fontsize=12)
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ax2.set_xlim([0, 5])
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set_axis_fontsize(ax2, 12)
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ax3.fill_between(time, avg_rate+rate_std, avg_rate-rate_std, color="dodgerblue",
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alpha=0.5, label="standard deviation")
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ax3.plot(time, avg_rate, label="average rate")
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ax3.set_xlabel("times [s]", fontsize=12)
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ax3.set_ylabel("firing rate [Hz]", fontsize=12)
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ax3.legend(fontsize=12)
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ax3.set_xlim([0, 5])
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ax3.set_ylim([0, 450])
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set_axis_fontsize(ax3, 12)
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fig.set_size_inches(15, 10)
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fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95)
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fig.set_facecolor("white")
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fig.savefig("figures/convolved_rate.png")
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plt.close()
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def plot_comparison(spike_times, bin_width, sigma, max_t=30., dt=1e-4):
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times = np.squeeze(spike_times[0][0])
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time, conv_rate = get_convolved_rate(times, bin_width/np.sqrt(12.))
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time, inst_rate = get_instantaneous_rate(times)
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time, binn_rate = get_binned_rate(times, bin_width)
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fig = plt.figure()
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ax1 = fig.add_subplot(411)
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ax2 = fig.add_subplot(412)
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ax3 = fig.add_subplot(413)
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ax4 = fig.add_subplot(414)
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ax1.spines["right"].set_visible(False)
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ax1.spines["top"].set_visible(False)
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ax1.yaxis.set_ticks_position('left')
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ax1.xaxis.set_ticks_position('bottom')
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ax2.spines["right"].set_visible(False)
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ax2.spines["top"].set_visible(False)
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ax2.yaxis.set_ticks_position('left')
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ax2.xaxis.set_ticks_position('bottom')
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ax3.spines["right"].set_visible(False)
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ax3.spines["top"].set_visible(False)
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ax3.yaxis.set_ticks_position('left')
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ax3.xaxis.set_ticks_position('bottom')
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ax4.spines["right"].set_visible(False)
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ax4.spines["top"].set_visible(False)
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ax4.yaxis.set_ticks_position('left')
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ax4.xaxis.set_ticks_position('bottom')
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ax1.vlines(times[times < (100000*dt)], ymin=0, ymax=1, color="dodgerblue", lw=1.5)
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ax1.set_ylabel("spikes", fontsize=10)
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ax1.set_xlim([1.5, 3.5])
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ax1.set_ylim([0, 1])
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ax1.set_yticks([0, 1])
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set_axis_fontsize(ax1, 10)
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ax1.set_xticklabels([])
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ax2.plot(time, inst_rate, lw=1.5, label="instantaneous rate")
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ax2.set_ylabel("firing rate [Hz]", fontsize=10)
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ax2.legend(fontsize=10)
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ax2.set_xlim([1.5, 3.5])
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ax2.set_ylim([0, 300])
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set_axis_fontsize(ax2, 10)
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ax2.set_xticklabels([])
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ax3.plot(time, binn_rate, lw=1.5, label="binned rate")
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ax3.set_ylabel("firing rate [Hz]", fontsize=10)
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ax3.legend(fontsize=10)
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ax3.set_xlim([1.5, 3.5])
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ax3.set_ylim([0, 300])
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set_axis_fontsize(ax3, 10)
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ax3.set_xticklabels([])
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ax4.plot(time, conv_rate, lw=1.5, label="convolved rate")
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ax4.set_xlabel("time [s]", fontsize=10)
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ax4.set_ylabel("firing rate [Hz]", fontsize=10)
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ax4.legend(fontsize=10)
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ax4.set_xlim([1.5, 3.5])
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ax4.set_ylim([0, 300])
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set_axis_fontsize(ax4, 10)
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fig.set_size_inches(7.5, 5)
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fig.subplots_adjust(left=0.1, bottom=0.125, top=0.95, right=0.95, )
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fig.set_facecolor("white")
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fig.savefig("firingrates.pdf")
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plt.close()
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if __name__ == "__main__":
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spike_times = spio.loadmat('lifoustim.mat')["spikes"]
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# plot_isi_rate(spike_times)
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# plot_bin_rate(spike_times, 0.05)
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# plot_conv_rate(spike_times, 0.025)
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plot_comparison(spike_times, 0.05, 0.025)
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158
pointprocesses/lecture/isimethod.py
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158
pointprocesses/lecture/isimethod.py
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import matplotlib.pyplot as plt
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import numpy as np
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from IPython import embed
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def set_rc():
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plt.rcParams['xtick.labelsize'] = 8
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plt.rcParams['ytick.labelsize'] = 8
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plt.rcParams['xtick.major.size'] = 5
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plt.rcParams['xtick.minor.size'] = 5
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plt.rcParams['xtick.major.width'] = 2
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plt.rcParams['xtick.minor.width'] = 2
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plt.rcParams['ytick.major.size'] = 5
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plt.rcParams['ytick.minor.size'] = 5
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plt.rcParams['ytick.major.width'] = 2
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plt.rcParams['ytick.minor.width'] = 2
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plt.rcParams['xtick.direction'] = "out"
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plt.rcParams['ytick.direction'] = "out"
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def create_spikes(isi=0.08, duration=0.5):
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times = np.arange(0., duration, isi)
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times += np.random.randn(len(times)) * (isi / 2.5)
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times = np.delete(times, np.nonzero(times < 0))
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times = np.delete(times, np.nonzero(times > duration))
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times = np.sort(times)
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return times
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def gaussian(sigma, dt):
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x = np.arange(-4*sigma, 4*sigma, dt)
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y = np.exp(-0.5 * (x / sigma)**2)/np.sqrt(2*np.pi)/sigma;
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return x, y
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def setup_axis(spikes_ax, rate_ax):
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spikes_ax.spines["right"].set_visible(False)
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spikes_ax.spines["top"].set_visible(False)
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spikes_ax.yaxis.set_ticks_position('left')
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spikes_ax.xaxis.set_ticks_position('bottom')
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spikes_ax.set_yticks([0, 1.0])
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spikes_ax.set_ylim([0, 1.05])
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spikes_ax.set_ylabel("spikes", fontsize=9)
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spikes_ax.text(-0.125, 1.2, "A", transform=spikes_ax.transAxes, size=10)
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rate_ax.spines["right"].set_visible(False)
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rate_ax.spines["top"].set_visible(False)
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rate_ax.yaxis.set_ticks_position('left')
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rate_ax.xaxis.set_ticks_position('bottom')
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rate_ax.set_xlabel('time[s]', fontsize=9)
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rate_ax.set_ylabel('firing rate [Hz]', fontsize=9)
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rate_ax.text(-0.125, 1.15, "B", transform=rate_ax.transAxes, size=10)
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def plot_bin_method():
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dt = 1e-5
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duration = 0.5
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spike_times = create_spikes(0.018, duration)
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t = np.arange(0., duration, dt)
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bins = np.arange(0, 0.55, 0.05)
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count, _ = np.histogram(spike_times, bins)
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plt.xkcd()
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set_rc()
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fig = plt.figure()
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||||
fig.set_size_inches(5., 2.5)
|
||||
fig.set_facecolor('white')
|
||||
spikes = plt.subplot2grid((7,1), (0,0), rowspan=3, colspan=1)
|
||||
rate_ax = plt.subplot2grid((7,1), (3,0), rowspan=4, colspan=1)
|
||||
setup_axis(spikes, rate_ax)
|
||||
spikes.set_ylim([0., 1.25])
|
||||
|
||||
spikes.vlines(spike_times, 0., 1., color="darkblue", lw=1.25)
|
||||
spikes.vlines(np.hstack((0,bins)), 0., 1.25, color="red", lw=1.5, linestyles='--')
|
||||
for i,c in enumerate(count):
|
||||
spikes.text(bins[i] + bins[1]/2, 1.05, str(c), fontdict={'color':'red'})
|
||||
spikes.set_xlim([0, duration])
|
||||
|
||||
rate = count / 0.05
|
||||
rate_ax.step(bins, np.hstack((rate, rate[-1])), where='post')
|
||||
rate_ax.set_xlim([0., duration])
|
||||
rate_ax.set_ylim([0., 100.])
|
||||
rate_ax.set_yticks(np.arange(0,105,25))
|
||||
fig.tight_layout()
|
||||
fig.savefig("binmethod.pdf")
|
||||
plt.close()
|
||||
|
||||
|
||||
def plot_conv_method():
|
||||
dt = 1e-5
|
||||
duration = 0.5
|
||||
spike_times = create_spikes(0.05, duration)
|
||||
kernel_time, kernel = gaussian(0.02, dt)
|
||||
|
||||
t = np.arange(0., duration, dt)
|
||||
rate = np.zeros(t.shape)
|
||||
rate[np.asarray(np.round(spike_times/dt), dtype=int)] = 1
|
||||
rate = np.convolve(rate, kernel, mode='same')
|
||||
rate = np.roll(rate, -1)
|
||||
|
||||
plt.xkcd()
|
||||
set_rc()
|
||||
fig = plt.figure()
|
||||
fig.set_size_inches(5., 2.5)
|
||||
fig.set_facecolor('white')
|
||||
spikes = plt.subplot2grid((7,1), (0,0), rowspan=3, colspan=1)
|
||||
rate_ax = plt.subplot2grid((7,1), (3,0), rowspan=4, colspan=1)
|
||||
setup_axis(spikes, rate_ax)
|
||||
|
||||
spikes.vlines(spike_times, 0., 1., color="darkblue", lw=1.5, zorder=2)
|
||||
for i in spike_times:
|
||||
spikes.plot(kernel_time + i, kernel/np.max(kernel), color="orange", lw=0.75, zorder=1)
|
||||
spikes.set_xlim([0, duration])
|
||||
|
||||
rate_ax.plot(t, rate, color="darkblue", lw=1, zorder=2)
|
||||
rate_ax.fill_between(t, rate, np.zeros(len(rate)), color="red", alpha=0.5)
|
||||
rate_ax.set_xlim([0, duration])
|
||||
rate_ax.set_ylim([0, 50])
|
||||
rate_ax.set_yticks(np.arange(0,75,25))
|
||||
fig.tight_layout()
|
||||
fig.savefig("convmethod.pdf")
|
||||
|
||||
|
||||
def plot_isi_method():
|
||||
spike_times = create_spikes(0.09, 0.5)
|
||||
|
||||
plt.xkcd()
|
||||
set_rc()
|
||||
fig = plt.figure()
|
||||
fig.set_size_inches(5., 2.5)
|
||||
fig.set_facecolor('white')
|
||||
|
||||
spikes = plt.subplot2grid((7,1), (0,0), rowspan=3, colspan=1)
|
||||
rate = plt.subplot2grid((7,1), (3,0), rowspan=4, colspan=1)
|
||||
setup_axis(spikes, rate)
|
||||
|
||||
spikes.vlines(spike_times, 0., 1., color="darkblue", lw=1.25)
|
||||
spike_times = np.hstack((0, spike_times))
|
||||
for i in range(1, len(spike_times)):
|
||||
t_start = spike_times[i-1]
|
||||
t = spike_times[i]
|
||||
spikes.annotate(s='', xy=(t_start, 0.5), xytext=(t,0.5), arrowprops=dict(arrowstyle='<->'), color='red')
|
||||
|
||||
i_rate = 1./np.diff(spike_times)
|
||||
|
||||
rate.step(spike_times, np.hstack((i_rate, i_rate[-1])),color="darkblue", lw=1.25, where="post")
|
||||
rate.set_ylim([0, 75])
|
||||
rate.set_yticks(np.arange(0,100,25))
|
||||
|
||||
fig.tight_layout()
|
||||
fig.savefig("isimethod.pdf")
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
plot_isi_method()
|
||||
plot_conv_method()
|
||||
plot_bin_method()
|
||||
BIN
pointprocesses/lecture/lifoustim.mat
Normal file
BIN
pointprocesses/lecture/lifoustim.mat
Normal file
Binary file not shown.
BIN
pointprocesses/lecture/p-unit_spike_times.mat
Normal file
BIN
pointprocesses/lecture/p-unit_spike_times.mat
Normal file
Binary file not shown.
BIN
pointprocesses/lecture/p-unit_stimulus.mat
Normal file
BIN
pointprocesses/lecture/p-unit_stimulus.mat
Normal file
Binary file not shown.
@ -1,6 +1,35 @@
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\chapter{\tr{Point processes}{Punktprozesse}}
|
||||
\chapter{Analyse von Spiketrains}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{rasterexamples}
|
||||
\caption{\label{rasterexamplesfig}Raster-Plot von jeweils 10
|
||||
Realisierungen eines station\"arenen Punktprozesses (homogener
|
||||
Poisson Prozess mit Rate $\lambda=20$\;Hz, links) und eines
|
||||
nicht-station\"aren Punktprozesses (perfect integrate-and-fire
|
||||
Neuron getrieben mit Ohrnstein-Uhlenbeck Rauschen mit
|
||||
Zeitkonstante $\tau=100$\,ms, rechts).}
|
||||
\end{figure}
|
||||
|
||||
Aktionspotentiale (``Spikes'') sind die Tr\"ager der Information in
|
||||
Nervensystemen. Dabei ist in erster Linie nur der Zeitpunkt des
|
||||
Auftretens eines Aktionspotentials von Bedeutung. Die genaue Form
|
||||
spielt keine oder nur eine untergeordnete Rolle.
|
||||
|
||||
Nach etwas Vorverarbeitung haben elektrophysiologischer Messungen
|
||||
deshalb Listen von Spikezeitpunkten als Ergebniss - sogenannte
|
||||
``Spiketrains''. Diese Messungen k\"onnen wiederholt werden und es
|
||||
ergeben sich mehrere ``trials'' von Spiketrains
|
||||
(\figref{rasterexamples}).
|
||||
|
||||
Spiketrains sind Zeitpunkte von Ereignissen --- den Aktionspotentialen
|
||||
--- und deren Analyse f\"allt daher in das Gebiet der Statistik von
|
||||
sogenannten Punktprozessen.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Punktprozesse}
|
||||
|
||||
\begin{figure}[t]
|
||||
\texpicture{pointprocessscetchB}
|
||||
@ -28,16 +57,6 @@ erzeugt. Zum Beispiel:
|
||||
Dynamik des Nervensystems und des Muskelapparates erzeugt.
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{rasterexamples}
|
||||
\caption{\label{rasterexamplesfig}Raster-Plot von jeweils 10
|
||||
Realisierungen eines station\"arenen Punktprozesses (homogener
|
||||
Poisson Prozess mit Rate $\lambda=20$\;Hz, links) und eines
|
||||
nicht-station\"aren Punktprozesses (perfect integrate-and-fire
|
||||
Neuron getrieben mit Ohrnstein-Uhlenbeck Rauschen mit
|
||||
Zeitkonstante $\tau=100$\,ms, rechts).}
|
||||
\end{figure}
|
||||
|
||||
F\"ur die Neurowissenschaften ist die Statistik der Punktprozesse
|
||||
besonders wichtig, da die Zeitpunkte der Aktionspotentiale als
|
||||
zeitlicher Punktprozess betrachtet werden k\"onnen und entscheidend
|
||||
@ -116,9 +135,9 @@ Intervalls mit sich selber).
|
||||
% \caption{\label{countstatsfig}Count Statistik.}
|
||||
% \end{figure}
|
||||
|
||||
Die Anzahl der Ereignisse $n_i$ in Zeifenstern $i$ der
|
||||
L\"ange $W$ ergeben ganzzahlige, positive Zufallsvariablen die meist durch folgende
|
||||
Sch\"atzer charakterisiert werden:
|
||||
Die Anzahl der Ereignisse (Spikes) $n_i$ in Zeifenstern $i$ der
|
||||
L\"ange $W$ ergeben ganzzahlige, positive Zufallsvariablen, die meist
|
||||
durch folgende Sch\"atzer charakterisiert werden:
|
||||
\begin{itemize}
|
||||
\item Histogramm der counts $n_i$.
|
||||
\item Mittlere Anzahl von Ereignissen: $\mu_N = \langle n \rangle$.
|
||||
@ -146,7 +165,7 @@ Insbesondere ist die mittlere Rate der Ereignisse $r$ (``Spikes pro Zeit'', Feue
|
||||
\section{Homogener Poisson Prozess}
|
||||
F\"ur kontinuierliche Me{\ss}gr\"o{\ss}en ist die Normalverteilung
|
||||
u.a. wegen dem Zentralen Grenzwertsatz die Standardverteilung. Eine
|
||||
\"ahnliche Rolle spilet bei Punktprozessen der ``Poisson Prozess''.
|
||||
\"ahnliche Rolle spielt bei Punktprozessen der ``Poisson Prozess''.
|
||||
|
||||
Beim homogenen Poisson Prozess treten Ereignisse mit einer festen Rate
|
||||
$\lambda=\text{const.}$ auf und sind unabh\"angig von der Zeit $t$ und
|
||||
@ -192,3 +211,176 @@ Der homogne Poissonprozess hat folgende Eigenschaften:
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
|
||||
\caption{\label{hompoissoncountfig}Z\"ahlstatistik von Poisson Spikes.}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Zeitabh\"angiger Feuerraten}
|
||||
|
||||
Bisher haben wir station\"are Spiketrains betrachtet, deren Statistik
|
||||
innerhlab der Analysezeit nicht ver\"andert. Meistens jedoch \"andert
|
||||
sich die Statistik der Spiketrains eines Neurons mit der
|
||||
Zeit. Z.B. kann ein sensorisches Neuron auf einen Reiz hin mit einer
|
||||
erh\"ohten Feuerrate antworten. Im folgenden Stellen wir drei Methoden
|
||||
vor, mit denen eine sich zeitlich ver\"andernde Rate des Punktprozesses
|
||||
abgesch\"atzt werden kann.
|
||||
|
||||
Eine klassische Darstellung zeitabh\"angiger neuronaler Aktivit\"at
|
||||
ist das sogenannte Peri Stimulus Zeithistogramm (peri stimulus time
|
||||
histogram, PSTH). Es wird der zeitliche Verlauf der Feuerrate $r(t)$
|
||||
dargestellt. Die Einheit der Feuerrate ist Hertz, das heisst, die
|
||||
Anzahl Aktionspotentiale pro Sekunde. Es verschiedene Methoden diese
|
||||
zu bestimmen. Drei solcher Methoden sind in Abbildung \ref{psthfig}
|
||||
dargestellt. Alle Methoden haben ihre Berechtigung und ihre Vor- und
|
||||
Nachteile. Im folgenden werden die drei Methoden aus Abbildung
|
||||
\ref{psthfig} n\"aher erl\"autert.
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=\columnwidth]{firingrates}
|
||||
\caption{\textbf{Verschiedene Methoden die zeitabh\"angige Feuerrate
|
||||
zu bestimmen. A)} Rasterplot einer einzelnen neuronalen
|
||||
Antwort. Jeder vertikale Strich notiert den Zeitpunkt eines
|
||||
Aktionspotentials. \textbf{B)} Feurerrate aus der instantanen
|
||||
Feuerrate bestimmt. \textbf{C)} klassisches PSTH mit der Binning
|
||||
Methode. \textbf{D)} Feuerrate durch Faltung mit einem Gauss Kern
|
||||
bestimmt.}\label{psthfig}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\paragraph{Instantane Feuerrate}
|
||||
Ein sehr einfacher Weg, die zeitabh\"angige Feuerrate zu bestimmen ist
|
||||
die sogenannte \textit{instantane Feuerrate}. Dabei wird die Feuerrate
|
||||
aus dem Kehrwert des \textit{Interspike Intervalls}, der Zeit zwischen
|
||||
zwei aufeinander folgenden Aktionspotentialen (Abbildung \ref{instrate}
|
||||
A), bestimmt. Die abgesch\"atzte Feuerrate (Abbildung \ref{instrate} B)
|
||||
ist g\"ultig f\"ur das gesammte Interspike Intervall
|
||||
\ref{instrate}. Diese Methode hat den Vorteil, dass sie sehr einfach zu
|
||||
berechnen ist und keine Annahme \"uber eine relevante Zeitskala (der
|
||||
Kodierung oder des Auslesemechanismus der postsynaptischen Zelle)
|
||||
macht. $r(t)$ ist allerdings keine kontinuierliche Funktion, die
|
||||
Spr\"unge in der Feuerrate k\"onnen f\"ur manche Analysen nachteilig
|
||||
sein. Des Weiteren ist die Feuerrate nie null, auch wenn lange keine
|
||||
Aktionspotentiale generiert wurden.
|
||||
|
||||
\begin{figure}[!htb]
|
||||
\includegraphics[width=\columnwidth]{isimethod}
|
||||
\caption{\textbf{Bestimmung des zeitabh\"angigen Feuerrate aus dem
|
||||
Interspike Interval. A)} Skizze eines Rasterplots einer
|
||||
einzelnen neuronalen Antwort. Jeder vertikale Strich notiert den
|
||||
Zeitpunkt eines Aktionspotentials. Die Pfeile zwischen
|
||||
aufeinanderfolgenden Aktionspotentialen illustrieren das
|
||||
Interspike Interval. \textbf{B)} Der Kehrwert des Interspike
|
||||
Intervalls ist die Feuerrate.}\label{instrate}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\paragraph{Binning Methode}
|
||||
Bei der Binning Methode wird die Zeitachse in gleichm\"aßige
|
||||
Abschnitte (Bins) eingeteilt und die Anzahl Aktionspotentiale, die in
|
||||
die jeweiligen Bins fallen, gez\"ahlt (Abbildung \ref{binpsth} A). Um
|
||||
diese Z\"ahlungen in die Feuerrate umzurechnen muss noch mit der
|
||||
Binweite normiert werden. Die bestimmte Feuerrate gilt f\"ur das
|
||||
gesamte Bin (Abbildung \ref{binpsth} B). \textbf{Tipp:} Um die Anzahl
|
||||
Spikes pro Bin zu berechnen kann die \code{hist} Funktion benutzt
|
||||
werden. Das so berechnete PSTH hat wiederum eine stufige Form, die von
|
||||
der Wahl der Binweite anh\"angt. Die Binweite bestimmt die zeitliche
|
||||
Aufl\"osung der Darstellung. \"Anderungen in der Feuerrate, die
|
||||
innerhalb eines Bins vorkommen k\"onnen nicht aufgl\"ost werden. Die
|
||||
Wahl der Binweite stellt somit eine Annahme \"uber die relevante
|
||||
Zeitskala der Verarbeitung dar. Auch hier ist $r(t)$ keine
|
||||
koninuierliche Funktion.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=\columnwidth]{binmethod}
|
||||
\caption{\textbf{Bestimmung des PSTH mit der Binning Methode. A)}
|
||||
Skizze eines Rasterplots einer einzelnen neuronalen Antwort. Jeder
|
||||
vertikale Strich notiert den Zeitpunkt eines
|
||||
Aktionspotentials. Die roten gestrichelten Linien stellen die
|
||||
Grenzen der Bins dar und die Zahlen geben den Spike Count pro Bin
|
||||
an. \textbf{B)} Die Feuerrate erh\"alt man indem das
|
||||
Zeithistogramm mit der Binweite normiert.}\label{binpsth}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\paragraph{Faltungsmethode}
|
||||
Bei der Faltungsmethode geht wird etwas anders vorgegangen um die
|
||||
Feuerrate zeitaufgel\"ost zu berechnen. Die Aktionspotentialfolge wird
|
||||
zun\"achst ``bin\"ar'' dargestellt. Das heisst, dass eine Antwort als
|
||||
(Zeit-)Vektor dargestellt wird, in welchem die Zeitpunkte der
|
||||
Aktionspotentiale als 1 notiert werden. Alle anderen Elemente des
|
||||
Vektors sind 0. Anschlie{\ss}end wir dieser bin\"are Spiketrain mit
|
||||
einem Gauss Kern bestimmter Breite gefaltet.
|
||||
|
||||
\[r(t) = \int_{-\infty}^{\infty}d\tau \omega(\tau)\rho(t-\tau) \],
|
||||
wobei $\omega(\tau)$ der Filterkern und $\rho(t)$ die bin\"are Antwort
|
||||
ist. Bildlich geprochen wird jede 1 in $rho(t)$ durch den Filterkern
|
||||
ersetzt (Abbildung \ref{convrate} A). Wenn der Kern richtig normiert
|
||||
wurde (Integral 1), ergibt sich die Feuerrate direkt aus der
|
||||
\"Uberlagerung der Kerne (Abb. \ref{convrate} B). Die Faltungsmethode
|
||||
f\"uhrt, anders als die anderen Methoden, zu einer kontinuierlichen
|
||||
Funktion was f\"ur spektrale Analysen von Vorteil sein kann. Die Wahl
|
||||
der Kernbreite bestimmt, \"ahnlich zur Binweite, die zeitliche
|
||||
Aufl\"osung von $r(t)$. Man macht also eine Annahme \"uber die
|
||||
relevante Zeitskala.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=\columnwidth]{convmethod}
|
||||
\caption{\textbf{Schematische Darstellung der Faltungsmethode. A)}
|
||||
Rasterplot einer einzelnen neuronalen Antwort. Jeder vertikale
|
||||
Strich notiert den Zeitpunkt eines Aktionspotentials. In der
|
||||
Faltung werden die mit einer 1 notierten Aktionspotential durch
|
||||
den Faltungskern ersetzt. \textbf{B)} Bei korrekter Normierung des
|
||||
Kerns ergibt sich die Feuerrate direkt aus der \"Uberlagerung der
|
||||
Kerne.}\label{convrate}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\section{Spike triggered Average}
|
||||
Die graphischer Darstellung der Feuerrate allein reicht nicht aus um
|
||||
den Zusammenhang zwischen neuronaler Antwort und einem Stimulus zu
|
||||
analysieren. Eine Methode mehr \"uber diesen Zusammenhang zu erfahren
|
||||
ist der Spike triggered average (STA). Der STA ist der mittlere
|
||||
Stimulus, der zu einem Aktionspotential in der neuronalen Antwort
|
||||
f\"uhrt.
|
||||
|
||||
\begin{equation}
|
||||
STA(\tau) = \frac{1}{\langle n \rangle} \left\langle \displaystyle\sum_{i=1}^{n}{s(t_i - \tau)} \right\rangle
|
||||
\end{equation}
|
||||
|
||||
Der STA l\"a{\ss}t sich relativ einfach berechnen, indem aus dem
|
||||
Stimulus f\"ur jeden beobachteten Spike ein entsprechender Abschnitt
|
||||
ausgeschnitten wird und diese dann mittelt (Abbildung
|
||||
\ref{stafig}).
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.5\columnwidth]{sta}
|
||||
\caption{\textbf{Spike Triggered Average eines P-Typ
|
||||
Elektrorezeptors.} Der Rezeptor wurde mit einem ``white-noise''
|
||||
Stimulus getrieben. Zeitpunkt 0 ist der Zeitpunkt des beobachteten
|
||||
Aktionspotentials.}\label{stafig}
|
||||
\end{figure}
|
||||
|
||||
Aus dem STA k\"onnen verschiedene Informationen \"uber den
|
||||
Zusammenhang zwischen Stimulus und neuronaler Antwort gewonnen
|
||||
werden. Die Breite des STA repr\"asentiert die zeitliche Pr\"azision,
|
||||
mit der Stimulus und Antwort zusammenh\"angen und wie weit das neuron
|
||||
zeitlich integriert. Die Amplitude des STA (gegeben in der gleichen
|
||||
Einheit, wie der Stimulus) deutet auf die Empfindlichkeit des Neurons
|
||||
bez\"uglich des Stimulus hin. Eine hohe Amplitude besagt, dass es
|
||||
einen starken Stimulus ben\"otigt um ein Aktionspotential
|
||||
hervorzurufen. Aus dem zeitlichen Versatz des STA kann die Zeit
|
||||
abgelesen werden, die das System braucht um auf den Stimulus zu
|
||||
antworten.
|
||||
|
||||
Der STA kann auch dazu benutzt werden, aus den Antworten der Zelle den
|
||||
Stimulus zu rekonstruieren (Abbildung
|
||||
\ref{reverse_reconstruct_fig}). Bei der \textit{invertierten
|
||||
Rekonstruktion} wird die Zellantwort mit dem STA gefaltet.
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=\columnwidth]{reconstruction}
|
||||
\caption{\textbf{Rekonstruktion des Stimulus mittels STA.} Die
|
||||
Zellantwort wird mit dem STA gefaltet um eine Rekonstruktion des
|
||||
Stimulus zu erhalten.}\label{reverse_reconstruct_fig}
|
||||
\end{figure}
|
||||
|
||||
88
pointprocesses/lecture/sta.py
Normal file
88
pointprocesses/lecture/sta.py
Normal file
@ -0,0 +1,88 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import scipy.io as scio
|
||||
from IPython import embed
|
||||
|
||||
|
||||
def plot_sta(times, stim, dt, t_min=-0.1, t_max=.1):
|
||||
count = 0
|
||||
sta = np.zeros((abs(t_min) + abs(t_max))/dt)
|
||||
time = np.arange(t_min, t_max, dt)
|
||||
if len(stim.shape) > 1 and stim.shape[1] > 1:
|
||||
stim = stim[:,1]
|
||||
for i in range(len(times[0])):
|
||||
times = np.squeeze(spike_times[0][i])
|
||||
for t in times:
|
||||
if (int((t + t_min)/dt) < 0) or ((t + t_max)/dt > len(stim)):
|
||||
continue;
|
||||
|
||||
min_index = int(np.round((t+t_min)/dt))
|
||||
max_index = int(np.round((t+t_max)/dt))
|
||||
snippet = np.squeeze(stim[ min_index : max_index])
|
||||
sta += snippet
|
||||
count += 1
|
||||
sta /= count
|
||||
|
||||
fig = plt.figure()
|
||||
fig.set_size_inches(5, 5)
|
||||
fig.subplots_adjust(left=0.15, bottom=0.125, top=0.95, right=0.95, )
|
||||
fig.set_facecolor("white")
|
||||
|
||||
ax = fig.add_subplot(111)
|
||||
ax.plot(time, sta, color="darkblue", lw=1)
|
||||
ax.set_xlabel("time [s]")
|
||||
ax.set_ylabel("stimulus")
|
||||
ax.xaxis.grid('off')
|
||||
ax.spines["right"].set_visible(False)
|
||||
ax.spines["top"].set_visible(False)
|
||||
ax.yaxis.set_ticks_position('left')
|
||||
ax.xaxis.set_ticks_position('bottom')
|
||||
|
||||
ylim = ax.get_ylim()
|
||||
xlim = ax.get_xlim()
|
||||
ax.plot(list(xlim), [0., 0.], zorder=1, color='darkgray', ls='--')
|
||||
ax.plot([0., 0.], list(ylim), zorder=1, color='darkgray', ls='--')
|
||||
ax.set_xlim(list(xlim))
|
||||
ax.set_ylim(list(ylim))
|
||||
fig.savefig("sta.pdf")
|
||||
plt.close()
|
||||
return sta
|
||||
|
||||
|
||||
def reconstruct_stimulus(spike_times, sta, stimulus, t_max=30., dt=1e-4):
|
||||
s_est = np.zeros((spike_times.shape[1], len(stimulus)))
|
||||
for i in range(10):
|
||||
times = np.squeeze(spike_times[0][i])
|
||||
indices = np.asarray((np.round(times/dt)), dtype=int)
|
||||
y = np.zeros(len(stimulus))
|
||||
y[indices] = 1
|
||||
s_est[i, :] = np.convolve(y, sta, mode='same')
|
||||
|
||||
plt.plot(np.arange(0, t_max, dt), stimulus[:,1], label='stimulus', color='darkblue', lw=2.)
|
||||
plt.plot(np.arange(0, t_max, dt), np.mean(s_est, axis=0), label='reconstruction', color='gray', lw=1.5)
|
||||
plt.xlabel('time[s]')
|
||||
plt.ylabel('stimulus')
|
||||
plt.xlim([0.0, 0.25])
|
||||
plt.ylim([-1., 1.])
|
||||
plt.legend()
|
||||
plt.plot([0.0, 0.25], [0., 0.], color="darkgray", lw=1.5, ls='--', zorder=1)
|
||||
plt.gca().spines["right"].set_visible(False)
|
||||
plt.gca().spines["top"].set_visible(False)
|
||||
plt.gca().yaxis.set_ticks_position('left')
|
||||
plt.gca().xaxis.set_ticks_position('bottom')
|
||||
|
||||
fig = plt.gcf()
|
||||
fig.set_size_inches(7.5, 5)
|
||||
fig.subplots_adjust(left=0.15, bottom=0.125, top=0.95, right=0.95, )
|
||||
fig.set_facecolor("white")
|
||||
fig.savefig('reconstruction.pdf')
|
||||
plt.close()
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
punit_data = scio.loadmat('p-unit_spike_times.mat')
|
||||
punit_stim = scio.loadmat('p-unit_stimulus.mat')
|
||||
spike_times = punit_data["spike_times"]
|
||||
stimulus = punit_stim["stimulus"]
|
||||
sta = plot_sta(spike_times, stimulus, 5e-5, -0.05, 0.05)
|
||||
reconstruct_stimulus(spike_times, sta, stimulus, 10, 5e-5)
|
||||
@ -1,22 +1,29 @@
|
||||
BASENAME=regression
|
||||
|
||||
PYFILES=$(wildcard *.py)
|
||||
PYPDFFILES=$(PYFILES:.py=.pdf)
|
||||
|
||||
pdf : $(BASENAME).pdf $(PYPDFFILES)
|
||||
all : pdf
|
||||
|
||||
# script:
|
||||
pdf : $(BASENAME)-chapter.pdf
|
||||
|
||||
$(BASENAME).pdf : $(BASENAME)-chapter.tex
|
||||
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex $(PYPDFFILES)
|
||||
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
|
||||
|
||||
$(PYPDFFILES) : %.pdf : %.py
|
||||
python $<
|
||||
|
||||
clean :
|
||||
rm -f *~ $(BASENAME).aux $(BASENAME).log $(BASENAME).out $(BASENAME).toc $(BASENAME).nav $(BASENAME).snm $(BASENAME).vrb
|
||||
rm -f *~
|
||||
rm -f $(BASENAME).aux $(BASENAME).log
|
||||
rm -f $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out
|
||||
rm -f $(PYPDFFILES) $(GPTTEXFILES)
|
||||
|
||||
cleanall : clean
|
||||
rm -f $(BASENAME).pdf
|
||||
rm -f $(BASENAME)-chapter.pdf
|
||||
|
||||
watch :
|
||||
watchpdf :
|
||||
while true; do ! make -q pdf && make pdf; sleep 0.5; done
|
||||
|
||||
|
||||
|
||||
@ -19,10 +19,13 @@
|
||||
\lstset{inputpath=programming/code}
|
||||
\include{programming/lectures/programming}
|
||||
|
||||
\chapter{Graphische Darstellung}
|
||||
|
||||
\graphicspath{{designpattern/lecture}{designpattern/lecture/figures}}
|
||||
\lstset{inputpath=designpattern/code/}
|
||||
\graphicspath{{plotting/lecture/}{plotting/lecture/images/}}
|
||||
\lstset{inputpath=plotting/code/}
|
||||
\include{plotting/lecture/plotting}
|
||||
|
||||
\graphicspath{{designpattern/lecture/}{designpattern/lecture/figures/}}
|
||||
\lstset{inputpath=designpattern/code}
|
||||
\include{designpattern/lecture/designpattern}
|
||||
|
||||
\chapter{Cheat-Sheet}
|
||||
@ -52,8 +55,4 @@
|
||||
\renewcommand{\texinputpath}{pointprocesses/lecture/}
|
||||
\include{pointprocesses/lecture/pointprocesses}
|
||||
|
||||
\graphicspath{{spike_trains/lecture/images/}}
|
||||
\lstset{inputpath=spike_trains/code/}
|
||||
\include{spike_trains/lecture/psth_sta}
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -20,6 +20,8 @@ werden. Dar\"uber hinaus wird versucht die Beziehung zwischen der
|
||||
zeitabh\"angigen neuronalen Antwort und dem zugrundeliegenden
|
||||
Stimulus zu analysieren.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%% The following moved to the pointprocesses chapter!
|
||||
\section{Darstellung der zeitabh\"angigen Feuerrate}
|
||||
|
||||
Eine klassische Darstellung zeitabh\"angiger neuronaler Aktivit\"at
|
||||
|
||||
Reference in New Issue
Block a user