# -*- coding: utf-8 -*-
"""
Statistical measures of spike trains (e.g., Fano factor) and functions to
estimate firing rates.
Tutorial
--------
:doc:`View tutorial <../tutorials/statistics>`
Run tutorial interactively:
.. image:: https://mybinder.org/badge.svg
:target: https://mybinder.org/v2/gh/NeuralEnsemble/elephant/master
?filepath=doc/tutorials/statistics.ipynb
.. current_module elephant.statistics
Functions overview
------------------
Rate estimation
~~~~~~~~~~~~~~~
.. autosummary::
:toctree: statistics/
mean_firing_rate
instantaneous_rate
time_histogram
sskernel
Spike interval statistics
~~~~~~~~~~~~~~~~~~~~~~~~~
.. autosummary::
:toctree: statistics/
isi
cv
lv
cv2
Statistics across spike trains
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. autosummary::
:toctree: statistics/
fanofactor
complexity_pdf
:copyright: Copyright 2014-2020 by the Elephant team, see `doc/authors.rst`.
:license: Modified BSD, see LICENSE.txt for details.
"""
from __future__ import division, print_function
# do not import unicode_literals
# (quantities rescale does not work with unicodes)
import numpy as np
import quantities as pq
import scipy.stats
import scipy.signal
import neo
from neo.core import SpikeTrain
import elephant.conversion as conv
import elephant.kernels as kernels
import warnings
cv = scipy.stats.variation
[docs]def isi(spiketrain, axis=-1):
"""
Return an array containing the inter-spike intervals of the spike train.
Accepts a `neo.SpikeTrain`, a `pq.Quantity` array, or a plain
`np.ndarray`. If either a `neo.SpikeTrain` or `pq.Quantity` is provided,
the return value will be `pq.Quantity`, otherwise `np.ndarray`. The units
of `pq.Quantity` will be the same as `spiketrain`.
Parameters
----------
spiketrain : neo.SpikeTrain or pq.Quantity or np.ndarray
The spike times.
axis : int, optional
The axis along which the difference is taken.
Default: the last axis.
Returns
-------
intervals : np.ndarray or pq.Quantity
The inter-spike intervals of the `spiketrain`.
"""
if axis is None:
axis = -1
if isinstance(spiketrain, neo.SpikeTrain):
intervals = np.diff(
np.sort(spiketrain.times.view(pq.Quantity)), axis=axis)
else:
intervals = np.diff(np.sort(spiketrain), axis=axis)
return intervals
[docs]def mean_firing_rate(spiketrain, t_start=None, t_stop=None, axis=None):
"""
Return the firing rate of the spike train.
Accepts a `neo.SpikeTrain`, a `pq.Quantity` array, or a plain
`np.ndarray`. If either a `neo.SpikeTrain` or `pq.Quantity` array is
provided, the return value will be a `pq.Quantity` array, otherwise a
plain `np.ndarray`. The units of the `pq.Quantity` array will be the
inverse of the `spiketrain`.
The interval over which the firing rate is calculated can be optionally
controlled with `t_start` and `t_stop`.
Parameters
----------
spiketrain : neo.SpikeTrain or pq.Quantity or np.ndarray
The spike times.
t_start : float or pq.Quantity, optional
The start time to use for the interval.
If None, retrieved from the `t_start` attribute of `spiketrain`. If
that is not present, default to 0. Any value from `spiketrain` below
this value is ignored.
Default: None.
t_stop : float or pq.Quantity, optional
The stop time to use for the time points.
If not specified, retrieved from the `t_stop` attribute of
`spiketrain`. If that is not present, default to the maximum value of
`spiketrain`. Any value from `spiketrain` above this value is ignored.
Default: None.
axis : int, optional
The axis over which to do the calculation.
If None, do the calculation over the flattened array.
Default: None.
Returns
-------
float or pq.Quantity or np.ndarray
The firing rate of the `spiketrain`
Raises
------
TypeError
If `spiketrain` is a `np.ndarray` and `t_start` or `t_stop` is
`pq.Quantity`.
Notes
-----
If `spiketrain` is a `pq.Quantity` or `neo.SpikeTrain`, and `t_start` or
`t_stop` are not `pq.Quantity`, `t_start` and `t_stop` are assumed to have
the same units as `spiketrain`.
"""
if t_start is None:
t_start = getattr(spiketrain, 't_start', 0)
found_t_start = False
if t_stop is None:
if hasattr(spiketrain, 't_stop'):
t_stop = spiketrain.t_stop
else:
t_stop = np.max(spiketrain, axis=axis)
found_t_start = True
# figure out what units, if any, we are dealing with
if hasattr(spiketrain, 'units'):
units = spiketrain.units
else:
units = None
# convert everything to the same units
if hasattr(t_start, 'units'):
if units is None:
raise TypeError('t_start cannot be a Quantity if '
'spiketrain is not a quantity')
t_start = t_start.rescale(units)
elif units is not None:
t_start = pq.Quantity(t_start, units=units)
if hasattr(t_stop, 'units'):
if units is None:
raise TypeError('t_stop cannot be a Quantity if '
'spiketrain is not a quantity')
t_stop = t_stop.rescale(units)
elif units is not None:
t_stop = pq.Quantity(t_stop, units=units)
if not axis or not found_t_start:
return np.sum((spiketrain >= t_start) & (spiketrain <= t_stop),
axis=axis) / (t_stop - t_start)
else:
# this is needed to handle broadcasting between spiketrain and t_stop
t_stop_test = np.expand_dims(t_stop, axis)
return np.sum((spiketrain >= t_start) & (spiketrain <= t_stop_test),
axis=axis) / (t_stop - t_start)
[docs]def fanofactor(spiketrains):
r"""
Evaluates the empirical Fano factor F of the spike counts of
a list of `neo.SpikeTrain` objects.
Given the vector v containing the observed spike counts (one per
spike train) in the time window [t0, t1], F is defined as:
.. math::
F := \frac{var(v)}{mean(v)}
The Fano factor is typically computed for spike trains representing the
activity of the same neuron over different trials. The higher F, the
larger the cross-trial non-stationarity. In theory for a time-stationary
Poisson process, F=1.
Parameters
----------
spiketrains : list
List of `neo.SpikeTrain` or `pq.Quantity` or `np.ndarray` or list of
spike times for which to compute the Fano factor of spike counts.
Returns
-------
fano : float
The Fano factor of the spike counts of the input spike trains.
Returns np.NaN if an empty list is specified, or if all spike trains
are empty.
"""
# Build array of spike counts (one per spike train)
spike_counts = np.array([len(t) for t in spiketrains])
# Compute FF
if all([count == 0 for count in spike_counts]):
fano = np.nan
else:
fano = spike_counts.var() / spike_counts.mean()
return fano
[docs]def lv(v, with_nan=False):
r"""
Calculate the measure of local variation LV for a sequence of time
intervals between events.
Given a vector v containing a sequence of intervals, the LV is defined as:
.. math::
LV := \frac{1}{N} \sum_{i=1}^{N-1}
\frac{3(isi_i-isi_{i+1})^2}
{(isi_i+isi_{i+1})^2}
The LV is typically computed as a substitute for the classical coefficient
of variation for sequences of events which include some (relatively slow)
rate fluctuation. As with the CV, LV=1 for a sequence of intervals
generated by a Poisson process.
Parameters
----------
v : pq.Quantity or np.ndarray or list
Vector of consecutive time intervals.
with_nan : bool, optional
If True, `lv` of a spike train with less than two spikes results in a
np.NaN value and a warning is raised.
If False, a `ValueError` exception is raised with a spike train with
less than two spikes.
Default: True.
Returns
-------
float
The LV of the inter-spike interval of the input sequence.
Raises
------
ValueError
If an empty list is specified, or if the sequence has less than two
entries and `with_nan` is False.
If a matrix is passed to the function. Only vector inputs are
supported.
Warns
-----
UserWarning
If `with_nan` is True and the `lv` is calculated for a spike train
with less than two spikes, generating a np.NaN.
References
----------
.. [1] S. Shinomoto, K. Shima, & J. Tanji, "Differences in spiking
patterns among cortical neurons," Neural Computation, vol. 15,
pp. 2823–2842, 2003.
"""
# convert to array, cast to float
v = np.asarray(v)
# ensure the input is a vector
if len(v.shape) > 1:
raise ValueError("Input shape is larger than 1. Please provide "
"a vector as an input.")
# ensure we have enough entries
if v.size < 2:
if with_nan:
warnings.warn("Input size is too small. Please provide "
"an input with more than 1 entry. lv returns 'NaN'"
"since the argument `with_nan` is True")
return np.NaN
else:
raise ValueError("Input size is too small. Please provide "
"an input with more than 1 entry. lv returned any"
"value since the argument `with_nan` is False")
# calculate LV and return result
# raise error if input is multi-dimensional
return 3. * np.mean(np.power(np.diff(v) / (v[:-1] + v[1:]), 2))
[docs]def cv2(v, with_nan=False):
r"""
Calculate the measure of CV2 for a sequence of time intervals between
events.
Given a vector v containing a sequence of intervals, the CV2 is defined
as:
.. math::
CV2 := \frac{1}{N} \sum{i=1}^{N-1}
\frac{2|isi_{i+1}-isi_i|}
{|isi_{i+1}+isi_i|}
The CV2 is typically computed as a substitute for the classical
coefficient of variation (CV) for sequences of events which include some
(relatively slow) rate fluctuation. As with the CV, CV2=1 for a sequence
of intervals generated by a Poisson process.
Parameters
----------
v : pq.Quantity or np.ndarray or list
Vector of consecutive time intervals.
with_nan : bool, optional
If True, `cv2` of a spike train with less than two spikes results in a
np.NaN value and a warning is raised.
If False, `ValueError` exception is raised with a spike train with
less than two spikes.
Default: True.
Returns
-------
float
The CV2 of the inter-spike interval of the input sequence.
Raises
------
ValueError
If an empty list is specified, or if the sequence has less than two
entries and `with_nan` is False.
If a matrix is passed to the function. Only vector inputs are
supported.
Warns
-----
UserWarning
If `with_nan` is True and `cv2` is calculated for a sequence with less
than two entries, generating a np.NaN.
References
----------
.. [1] G. R. Holt, W. R. Softky, C. Koch, & R. J. Douglas, "Comparison of
discharge variability in vitro and in vivo in cat visual cortex
neurons," Journal of Neurophysiology, vol. 75, no. 5, pp. 1806-1814,
1996.
"""
# convert to array, cast to float
v = np.asarray(v)
# ensure the input ia a vector
if len(v.shape) > 1:
raise ValueError("Input shape is larger than 1. Please provide "
"a vector as an input.")
# ensure we have enough entries
if v.size < 2:
if with_nan:
warnings.warn("Input size is too small. Please provide"
"an input with more than 1 entry. cv2 returns `NaN`"
"since the argument `with_nan` is `True`")
return np.NaN
else:
raise ValueError("Input size is too small. Please provide "
"an input with more than 1 entry. cv2 returns any"
"value since the argument `with_nan` is `False`")
# calculate CV2 and return result
return 2. * np.mean(np.absolute(np.diff(v)) / (v[:-1] + v[1:]))
[docs]def instantaneous_rate(spiketrain, sampling_period, kernel='auto',
cutoff=5.0, t_start=None, t_stop=None, trim=False):
"""
Estimates instantaneous firing rate by kernel convolution.
Parameters
-----------
spiketrain : neo.SpikeTrain or list of neo.SpikeTrain
Neo object(s) that contains spike times, the unit of the time stamps,
and `t_start` and `t_stop` of the spike train.
sampling_period : pq.Quantity
Time stamp resolution of the spike times. The same resolution will
be assumed for the kernel.
kernel : str or `kernels.Kernel`, optional
The string 'auto' or callable object of class `kernels.Kernel`.
The kernel is used for convolution with the spike train and its
standard deviation determines the time resolution of the instantaneous
rate estimation. Currently implemented kernel forms are rectangular,
triangular, epanechnikovlike, gaussian, laplacian, exponential, and
alpha function.
If 'auto', the optimized kernel width for the rate estimation is
calculated according to [1]_ and with this width a gaussian kernel is
constructed. Automatized calculation of the kernel width is not
available for other than gaussian kernel shapes.
Default: 'auto'.
cutoff : float, optional
This factor determines the cutoff of the probability distribution of
the kernel, i.e., the considered width of the kernel in terms of
multiples of the standard deviation sigma.
Default: 5.0.
t_start : pq.Quantity, optional
Start time of the interval used to compute the firing rate.
If None, `t_start` is assumed equal to `t_start` attribute of
`spiketrain`.
Default: None.
t_stop : pq.Quantity, optional
End time of the interval used to compute the firing rate (included).
If None, `t_stop` is assumed equal to `t_stop` attribute of
`spiketrain`.
Default: None.
trim : bool, optional
If False, the output of the Fast Fourier Transformation being a longer
vector than the input vector by the size of the kernel is reduced back
to the original size of the considered time interval of the
`spiketrain` using the median of the kernel.
If True, only the region of the convolved signal is returned, where
there is complete overlap between kernel and spike train. This is
achieved by reducing the length of the output of the Fast Fourier
Transformation by a total of two times the size of the kernel, and
`t_start` and `t_stop` are adjusted.
Default: False.
Returns
-------
rate : neo.AnalogSignal
Contains the rate estimation in unit hertz (Hz). In case a list of
spike trains was given, this is the combined rate of all spike trains
(not the average rate). `rate.times` contains the time axis of the rate
estimate. The unit of this property is the same as the resolution that
is given via the argument `sampling_period` to the function.
Raises
------
TypeError:
If `spiketrain` is not an instance of `neo.SpikeTrain`.
If `sampling_period` is not a `pq.Quantity`.
If `sampling_period` is not larger than zero.
If `kernel` is neither instance of `kernels.Kernel` nor string 'auto'.
If `cutoff` is neither `float` nor `int`.
If `t_start` and `t_stop` are neither None nor a `pq.Quantity`.
If `trim` is not `bool`.
ValueError:
If `sampling_period` is smaller than zero.
If `kernel` is 'auto' and the function was unable to calculate optimal
kernel width for instantaneous rate from input data.
Warns
-----
UserWarning
If `cutoff` is less than `min_cutoff` attribute of `kernel`, the width
of the kernel is adjusted to a minimally allowed width.
If the instantaneous firing rate approximation contains negative values
with respect to a tolerance (less than -1e-8), possibly due to machine
precision errors.
References
----------
.. [1] H. Shimazaki, & S. Shinomoto, "Kernel bandwidth optimization in
spike rate estimation," J Comput Neurosci, vol. 29, pp. 171–182,
2010.
Examples
--------
>>> import quantities as pq
>>> from elephant import kernels
>>> kernel = kernels.AlphaKernel(sigma=0.05*pq.s, invert=True)
>>> rate = instantaneous_rate(spiketrain, sampling_period=2*pq.ms,
... kernel=kernel)
"""
# Merge spike trains if list of spike trains given:
if isinstance(spiketrain, list):
_check_consistency_of_spiketrainlist(
spiketrain, t_start=t_start, t_stop=t_stop)
if t_start is None:
t_start = spiketrain[0].t_start
if t_stop is None:
t_stop = spiketrain[0].t_stop
spikes = np.concatenate([st.magnitude for st in spiketrain])
merged_spiketrain = SpikeTrain(np.sort(spikes),
units=spiketrain[0].units,
t_start=t_start, t_stop=t_stop)
return instantaneous_rate(merged_spiketrain,
sampling_period=sampling_period,
kernel=kernel, cutoff=cutoff,
t_start=t_start,
t_stop=t_stop, trim=trim)
# Checks of input variables:
if not isinstance(spiketrain, SpikeTrain):
raise TypeError(
"spiketrain must be instance of :class:`SpikeTrain` of Neo!\n"
" Found: %s, value %s" % (type(spiketrain), str(spiketrain)))
if not (isinstance(sampling_period, pq.Quantity) and
sampling_period.dimensionality.simplified ==
pq.Quantity(1, "s").dimensionality):
raise TypeError(
"The sampling period must be a time quantity!\n"
" Found: %s, value %s" % (
type(sampling_period), str(sampling_period)))
if sampling_period.magnitude < 0:
raise ValueError("The sampling period must be larger than zero.")
if kernel == 'auto':
kernel_width_sigma = sskernel(
spiketrain.magnitude, tin=None, bootstrap=False)['optw']
if kernel_width_sigma is None:
raise ValueError(
"Unable to calculate optimal kernel width for "
"instantaneous rate from input data.")
kernel = kernels.GaussianKernel(kernel_width_sigma * spiketrain.units)
elif not isinstance(kernel, kernels.Kernel):
raise TypeError(
"kernel must be either instance of :class:`Kernel` "
"or the string 'auto'!\n"
" Found: %s, value %s" % (type(kernel), str(kernel)))
if not (isinstance(cutoff, float) or isinstance(cutoff, int)):
raise TypeError("cutoff must be float or integer!")
if not (t_start is None or (isinstance(t_start, pq.Quantity) and
t_start.dimensionality.simplified ==
pq.Quantity(1, "s").dimensionality)):
raise TypeError("t_start must be a time quantity!")
if not (t_stop is None or (isinstance(t_stop, pq.Quantity) and
t_stop.dimensionality.simplified ==
pq.Quantity(1, "s").dimensionality)):
raise TypeError("t_stop must be a time quantity!")
if not (isinstance(trim, bool)):
raise TypeError("trim must be bool!")
# main function:
units = pq.CompoundUnit(
"{}*s".format(sampling_period.rescale('s').item()))
spiketrain = spiketrain.rescale(units)
if t_start is None:
t_start = spiketrain.t_start
else:
t_start = t_start.rescale(spiketrain.units)
if t_stop is None:
t_stop = spiketrain.t_stop
else:
t_stop = t_stop.rescale(spiketrain.units)
time_vector = np.zeros(int((t_stop - t_start)) + 1)
spikes_slice = spiketrain.time_slice(t_start, t_stop) \
if len(spiketrain) else np.array([])
for spike in spikes_slice:
index = int((spike - t_start))
time_vector[index] += 1
if cutoff < kernel.min_cutoff:
cutoff = kernel.min_cutoff
warnings.warn("The width of the kernel was adjusted to a minimally "
"allowed width.")
t_arr = np.arange(-cutoff * kernel.sigma.rescale(units).magnitude,
cutoff * kernel.sigma.rescale(units).magnitude +
sampling_period.rescale(units).magnitude,
sampling_period.rescale(units).magnitude) * units
r = scipy.signal.fftconvolve(time_vector,
kernel(t_arr).rescale(pq.Hz).magnitude,
'full')
if np.any(r < -1e-8): # abs tolerance in np.isclose
warnings.warn("Instantaneous firing rate approximation contains "
"negative values, possibly caused due to machine "
"precision errors.")
if not trim:
r = r[kernel.median_index(t_arr):-(kernel(t_arr).size -
kernel.median_index(t_arr))]
elif trim:
r = r[2 * kernel.median_index(t_arr):-2 * (kernel(t_arr).size -
kernel.median_index(t_arr))]
t_start += kernel.median_index(t_arr) * spiketrain.units
t_stop -= (kernel(t_arr).size -
kernel.median_index(t_arr)) * spiketrain.units
rate = neo.AnalogSignal(signal=r.reshape(r.size, 1),
sampling_period=sampling_period,
units=pq.Hz, t_start=t_start, t_stop=t_stop)
return rate
[docs]def time_histogram(spiketrains, binsize, t_start=None, t_stop=None,
output='counts', binary=False):
"""
Time Histogram of a list of `neo.SpikeTrain` objects.
Parameters
----------
spiketrains : list of neo.SpikeTrain
`neo.SpikeTrain`s with a common time axis (same `t_start` and `t_stop`)
binsize : pq.Quantity
Width of the histogram's time bins.
t_start : pq.Quantity, optional
Start time of the histogram. Only events in `spiketrains` falling
between `t_start` and `t_stop` (both included) are considered in the
histogram.
If None, the maximum `t_start` of all `neo.SpikeTrain`s is used as
`t_start`.
Default: None.
t_stop : pq.Quantity, optional
Stop time of the histogram. Only events in `spiketrains` falling
between `t_start` and `t_stop` (both included) are considered in the
histogram.
If None, the minimum `t_stop` of all `neo.SpikeTrain`s is used as
`t_stop`.
Default: None.
output : {'counts', 'mean', 'rate'}, optional
Normalization of the histogram. Can be one of:
* 'counts': spike counts at each bin (as integer numbers)
* 'mean': mean spike counts per spike train
* 'rate': mean spike rate per spike train. Like 'mean', but the
counts are additionally normalized by the bin width.
Default: 'counts'.
binary : bool, optional
If True, indicates whether all `neo.SpikeTrain` objects should first
be binned to a binary representation (using the
`conversion.BinnedSpikeTrain` class) and the calculation of the
histogram is based on this representation.
Note that the output is not binary, but a histogram of the converted,
binary representation.
Default: False.
Returns
-------
neo.AnalogSignal
A `neo.AnalogSignal` object containing the histogram values.
`neo.AnalogSignal[j]` is the histogram computed between
`t_start + j * binsize` and `t_start + (j + 1) * binsize`.
Raises
------
ValueError
If `output` is not 'counts', 'mean' or 'rate'.
Warns
-----
UserWarning
If `t_start` is None and the objects in `spiketrains` have different
`t_start` values.
If `t_stop` is None and the objects in `spiketrains` have different
`t_stop` values.
See also
--------
elephant.conversion.BinnedSpikeTrain
"""
min_tstop = 0
if t_start is None:
# Find the internal range for t_start, where all spike trains are
# defined; cut all spike trains taking that time range only
max_tstart, min_tstop = conv._get_start_stop_from_input(spiketrains)
t_start = max_tstart
if not all([max_tstart == t.t_start for t in spiketrains]):
warnings.warn(
"Spiketrains have different t_start values -- "
"using maximum t_start as t_start.")
if t_stop is None:
# Find the internal range for t_stop
if min_tstop:
t_stop = min_tstop
if not all([min_tstop == t.t_stop for t in spiketrains]):
warnings.warn(
"Spiketrains have different t_stop values -- "
"using minimum t_stop as t_stop.")
else:
min_tstop = conv._get_start_stop_from_input(spiketrains)[1]
t_stop = min_tstop
if not all([min_tstop == t.t_stop for t in spiketrains]):
warnings.warn(
"Spiketrains have different t_stop values -- "
"using minimum t_stop as t_stop.")
sts_cut = [st.time_slice(t_start=t_start, t_stop=t_stop) for st in
spiketrains]
# Bin the spike trains and sum across columns
bs = conv.BinnedSpikeTrain(sts_cut, t_start=t_start, t_stop=t_stop,
binsize=binsize)
if binary:
bin_hist = bs.to_sparse_bool_array().sum(axis=0)
else:
bin_hist = bs.to_sparse_array().sum(axis=0)
# Flatten array
bin_hist = np.ravel(bin_hist)
# Renormalise the histogram
if output == 'counts':
# Raw
bin_hist = bin_hist * pq.dimensionless
elif output == 'mean':
# Divide by number of input spike trains
bin_hist = bin_hist * 1. / len(spiketrains) * pq.dimensionless
elif output == 'rate':
# Divide by number of input spike trains and bin width
bin_hist = bin_hist * 1. / len(spiketrains) / binsize
else:
raise ValueError('Parameter output is not valid.')
return neo.AnalogSignal(signal=bin_hist.reshape(bin_hist.size, 1),
sampling_period=binsize, units=bin_hist.units,
t_start=t_start)
[docs]def complexity_pdf(spiketrains, binsize):
"""
Complexity Distribution of a list of `neo.SpikeTrain` objects.
Probability density computed from the complexity histogram which is the
histogram of the entries of the population histogram of clipped (binary)
spike trains computed with a bin width of `binsize`.
It provides for each complexity (== number of active neurons per bin) the
number of occurrences. The normalization of that histogram to 1 is the
probability density.
Implementation is based on [1]_.
Parameters
----------
spiketrains : list of neo.SpikeTrain
Spike trains with a common time axis (same `t_start` and `t_stop`)
binsize : pq.Quantity
Width of the histogram's time bins.
Returns
-------
complexity_distribution : neo.AnalogSignal
A `neo.AnalogSignal` object containing the histogram values.
`neo.AnalogSignal[j]` is the histogram computed between
`t_start + j * binsize` and `t_start + (j + 1) * binsize`.
See also
--------
elephant.conversion.BinnedSpikeTrain
References
----------
.. [1] S. Gruen, M. Abeles, & M. Diesmann, "Impact of higher-order
correlations on coincidence distributions of massively parallel
data," In "Dynamic Brain - from Neural Spikes to Behaviors",
pp. 96-114, Springer Berlin Heidelberg, 2008.
"""
# Computing the population histogram with parameter binary=True to clip the
# spike trains before summing
pophist = time_histogram(spiketrains, binsize, binary=True)
# Computing the histogram of the entries of pophist (=Complexity histogram)
complexity_hist = np.histogram(
pophist.magnitude, bins=range(0, len(spiketrains) + 2))[0]
# Normalization of the Complexity Histogram to 1 (probabilty distribution)
complexity_hist = complexity_hist / complexity_hist.sum()
# Convert the Complexity pdf to an neo.AnalogSignal
complexity_distribution = neo.AnalogSignal(
np.array(complexity_hist).reshape(len(complexity_hist), 1) *
pq.dimensionless, t_start=0 * pq.dimensionless,
sampling_period=1 * pq.dimensionless)
return complexity_distribution
"""
Kernel Bandwidth Optimization.
Python implementation by Subhasis Ray.
Original matlab code (sskernel.m) here:
http://2000.jukuin.keio.ac.jp/shimazaki/res/kernel.html
This was translated into Python by Subhasis Ray, NCBS. Tue Jun 10
23:01:43 IST 2014
"""
def nextpow2(x):
"""
Return the smallest integral power of 2 that is equal or larger than `x`.
"""
n = 2
while n < x:
n = 2 * n
return n
def fftkernel(x, w):
"""
Applies the Gauss kernel smoother to an input signal using FFT algorithm.
Parameters
----------
x : np.ndarray
Vector with sample signal.
w : float
Kernel bandwidth (the standard deviation) in unit of the sampling
resolution of `x`.
Returns
-------
y : np.ndarray
The smoothed signal.
Notes
-----
1. MAY 5/23, 2012 Author Hideaki Shimazaki
RIKEN Brain Science Insitute
http://2000.jukuin.keio.ac.jp/shimazaki
2. Ported to Python: Subhasis Ray, NCBS. Tue Jun 10 10:42:38 IST 2014
"""
L = len(x)
Lmax = L + 3 * w
n = nextpow2(Lmax)
X = np.fft.fft(x, n)
f = np.arange(0, n, 1.0) / n
f = np.concatenate((-f[:int(n / 2)], f[int(n / 2):0:-1]))
K = np.exp(-0.5 * (w * 2 * np.pi * f) ** 2)
y = np.fft.ifft(X * K, n)
y = y[:L].copy()
return y
def logexp(x):
if x < 1e2:
y = np.log(1 + np.exp(x))
else:
y = x
return y
def ilogexp(x):
if x < 1e2:
y = np.log(np.exp(x) - 1)
else:
y = x
return y
def cost_function(x, N, w, dt):
"""
Computes the cost function for `sskernel`.
Cn(w) = sum_{i,j} int k(x - x_i) k(x - x_j) dx - 2 sum_{i~=j} k(x_i - x_j)
"""
yh = np.abs(fftkernel(x, w / dt)) # density
# formula for density
C = np.sum(yh ** 2) * dt - 2 * np.sum(yh * x) * \
dt + 2 / np.sqrt(2 * np.pi) / w / N
C = C * N * N
# formula for rate
# C = dt*sum( yh.^2 - 2*yh.*y_hist + 2/sqrt(2*pi)/w*y_hist )
return C, yh
[docs]def sskernel(spiketimes, tin=None, w=None, bootstrap=False):
"""
Calculates optimal fixed kernel bandwidth, given as the standard deviation
sigma.
Parameters
----------
spiketimes : np.ndarray
Sequence of spike times (sorted to be ascending).
tin : np.ndarray, optional
Time points at which the kernel bandwidth is to be estimated.
If None, `spiketimes` is used.
Default: None.
w : np.ndarray, optional
Vector of kernel bandwidths (standard deviation sigma).
If specified, optimal bandwidth is selected from this.
If None, `w` is obtained through a golden-section search on a log-exp
scale.
Default: None.
bootstrap : bool, optional
If True, calculates the 95% confidence interval using Bootstrap.
Default: False.
Returns
-------
dict
'y' : np.ndarray
Estimated density.
't' : np.ndarray
Points at which estimation was computed.
'optw' : float
Optimal kernel bandwidth given as standard deviation sigma
'w' : np.ndarray
Kernel bandwidths examined (standard deviation sigma).
'C' : np.ndarray
Cost functions of `w`.
'confb95' : tuple of np.ndarray
Bootstrap 95% confidence interval: (lower level, upper level).
If `bootstrap` is False, `confb95` is None.
'yb' : np.ndarray
Bootstrap samples.
If `bootstrap` is False, `yb` is None.
If no optimal kernel could be found, all entries of the dictionary are
set to None.
References
----------
.. [1] H. Shimazaki, & S. Shinomoto, "Kernel bandwidth optimization in
spike rate estimation," Journal of Computational Neuroscience,
vol. 29, no. 1-2, pp. 171-82, 2010. doi:10.1007/s10827-009-0180-4.
"""
if tin is None:
time = np.max(spiketimes) - np.min(spiketimes)
isi = np.diff(spiketimes)
isi = isi[isi > 0].copy()
dt = np.min(isi)
tin = np.linspace(np.min(spiketimes),
np.max(spiketimes),
min(int(time / dt + 0.5),
1000)) # The 1000 seems somewhat arbitrary
t = tin
else:
time = np.max(tin) - np.min(tin)
spiketimes = spiketimes[(spiketimes >= np.min(tin)) &
(spiketimes <= np.max(tin))].copy()
isi = np.diff(spiketimes)
isi = isi[isi > 0].copy()
dt = np.min(isi)
if dt > np.min(np.diff(tin)):
t = np.linspace(np.min(tin), np.max(tin),
min(int(time / dt + 0.5), 1000))
else:
t = tin
dt = np.min(np.diff(tin))
yhist, bins = np.histogram(spiketimes, np.r_[t - dt / 2, t[-1] + dt / 2])
N = np.sum(yhist)
yhist = yhist / (N * dt) # density
optw = None
y = None
if w is not None:
C = np.zeros(len(w))
Cmin = np.inf
for k, w_ in enumerate(w):
C[k], yh = cost_function(yhist, N, w_, dt)
if C[k] < Cmin:
Cmin = C[k]
optw = w_
y = yh
else:
# Golden section search on a log-exp scale
wmin = 2 * dt
wmax = max(spiketimes) - min(spiketimes)
imax = 20 # max iterations
w = np.zeros(imax)
C = np.zeros(imax)
tolerance = 1e-5
phi = 0.5 * (np.sqrt(5) + 1) # The Golden ratio
a = ilogexp(wmin)
b = ilogexp(wmax)
c1 = (phi - 1) * a + (2 - phi) * b
c2 = (2 - phi) * a + (phi - 1) * b
f1, y1 = cost_function(yhist, N, logexp(c1), dt)
f2, y2 = cost_function(yhist, N, logexp(c2), dt)
k = 0
while (np.abs(b - a) > (tolerance * (np.abs(c1) + np.abs(c2)))) \
and (k < imax):
if f1 < f2:
b = c2
c2 = c1
c1 = (phi - 1) * a + (2 - phi) * b
f2 = f1
f1, y1 = cost_function(yhist, N, logexp(c1), dt)
w[k] = logexp(c1)
C[k] = f1
optw = logexp(c1)
y = y1 / (np.sum(y1 * dt))
else:
a = c1
c1 = c2
c2 = (2 - phi) * a + (phi - 1) * b
f1 = f2
f2, y2 = cost_function(yhist, N, logexp(c2), dt)
w[k] = logexp(c2)
C[k] = f2
optw = logexp(c2)
y = y2 / np.sum(y2 * dt)
k = k + 1
# Bootstrap confidence intervals
confb95 = None
yb = None
# If bootstrap is requested, and an optimal kernel was found
if bootstrap and optw:
nbs = 1000
yb = np.zeros((nbs, len(tin)))
for ii in range(nbs):
idx = np.floor(np.random.rand(N) * N).astype(int)
xb = spiketimes[idx]
y_histb, bins = np.histogram(
xb, np.r_[t - dt / 2, t[-1] + dt / 2]) / dt / N
yb_buf = fftkernel(y_histb, optw / dt).real
yb_buf = yb_buf / np.sum(yb_buf * dt)
yb[ii, :] = np.interp(tin, t, yb_buf)
ybsort = np.sort(yb, axis=0)
y95b = ybsort[np.floor(0.05 * nbs).astype(int), :]
y95u = ybsort[np.floor(0.95 * nbs).astype(int), :]
confb95 = (y95b, y95u)
# Only perform interpolation if y could be calculated
if y is not None:
y = np.interp(tin, t, y)
return {'y': y,
't': tin,
'optw': optw,
'w': w,
'C': C,
'confb95': confb95,
'yb': yb}
def _check_consistency_of_spiketrainlist(spiketrainlist, t_start=None,
t_stop=None):
for spiketrain in spiketrainlist:
if not isinstance(spiketrain, SpikeTrain):
raise TypeError(
"spike train must be instance of :class:`SpikeTrain` of Neo!\n"
" Found: %s, value %s" % (
type(spiketrain), str(spiketrain)))
if t_start is None and not spiketrain.t_start == spiketrainlist[
0].t_start:
raise ValueError(
"the spike trains must have the same t_start!")
if t_stop is None and not spiketrain.t_stop == spiketrainlist[
0].t_stop:
raise ValueError(
"the spike trains must have the same t_stop!")
if not spiketrain.units == spiketrainlist[0].units:
raise ValueError(
"the spike trains must have the same units!")
