# -*- coding: utf-8 -*-
"""
Functions to generate spike trains from analog signals,
or to generate random spike trains.
Some functions are based on the NeuroTools stgen module, which was mostly
written by Eilif Muller, or from the NeuroTools signals.analogs module.
:copyright: Copyright 2015 by the Elephant team, see `doc/authors.rst`.
:license: Modified BSD, see LICENSE.txt for details.
"""
from __future__ import division, print_function, unicode_literals
import warnings
from functools import partial
import neo
import numpy as np
import quantities as pq
from elephant.spike_train_surrogates import dither_spike_train
from elephant.utils import deprecated_alias
__all__ = [
"spike_extraction",
"threshold_detection",
"peak_detection",
"homogeneous_poisson_process",
"inhomogeneous_poisson_process",
"homogeneous_gamma_process",
"single_interaction_process",
"compound_poisson_process"
]
[docs]def threshold_detection(signal, threshold=0.0 * pq.mV, sign='above'):
"""
Returns the times when the analog signal crosses a threshold.
Usually used for extracting spike times from a membrane potential.
Adapted from version in NeuroTools.
Parameters
----------
signal : neo.AnalogSignal
An analog input signal.
threshold : pq.Quantity, optional
Contains a value that must be reached for an event to be detected.
Default: 0.0 * pq.mV.
sign : {'above', 'below'}, optional
Determines whether to count threshold crossings that cross above or
below the threshold.
Default: 'above'.
Returns
-------
result_st : neo.SpikeTrain
Contains the spike times of each of the events (spikes) extracted from
the signal.
"""
if not isinstance(threshold, pq.Quantity):
raise ValueError('threshold must be a pq.Quantity')
if sign not in ('above', 'below'):
raise ValueError("sign should be 'above' or 'below'")
if sign == 'above':
cutout = np.where(signal > threshold)[0]
else:
# sign == 'below'
cutout = np.where(signal < threshold)[0]
if len(cutout) == 0:
events_base = np.zeros(0)
else:
take = np.where(np.diff(cutout) > 1)[0] + 1
take = np.append(0, take)
time = signal.times
events = time[cutout][take]
events_base = events.magnitude
if events_base is None:
# This occurs in some Python 3 builds due to some
# bug in quantities.
events_base = np.array(
[event.magnitude for event in events]) # Workaround
result_st = neo.SpikeTrain(events_base, units=signal.times.units,
t_start=signal.t_start, t_stop=signal.t_stop)
return result_st
[docs]@deprecated_alias(format='as_array')
def peak_detection(signal, threshold=0.0 * pq.mV, sign='above',
as_array=False):
"""
Return the peak times for all events that cross threshold.
Usually used for extracting spike times from a membrane potential.
Similar to spike_train_generation.threshold_detection.
Parameters
----------
signal : neo.AnalogSignal
An analog input signal.
threshold : pq.Quantity, optional
Contains a value that must be reached for an event to be detected.
Default: 0.*pq.mV.
sign : {'above', 'below'}, optional
Determines whether to count threshold crossings that cross above or
below the threshold.
Default: 'above'.
as_array : bool, optional
If True, a NumPy array of the resulting peak times is returned instead
of a (default) `neo.SpikeTrain` object.
Default: False.
format : {None, 'raw'}, optional
.. deprecated:: 0.8.0
Whether to return as SpikeTrain (None) or as a plain array of times
('raw').
Deprecated. Use `as_array=False` for None format and `as_array=True`
otherwise.
Default: None.
Returns
-------
result_st : neo.SpikeTrain
Contains the spike times of each of the events (spikes) extracted from
the signal.
"""
if not isinstance(threshold, pq.Quantity):
raise ValueError("threshold must be a pq.Quantity")
if sign not in ('above', 'below'):
raise ValueError("sign should be 'above' or 'below'")
if as_array in (None, 'raw'):
warnings.warn("'format' is deprecated; use as_array=True",
DeprecationWarning)
as_array = bool(as_array)
if sign == 'above':
cutout = np.where(signal > threshold)[0]
peak_func = np.argmax
else:
# sign == 'below'
cutout = np.where(signal < threshold)[0]
peak_func = np.argmin
if len(cutout) == 0:
events_base = np.zeros(0)
else:
# Select thr crossings lasting at least 2 dtps, np.diff(cutout) > 2
# This avoids empty slices
border_start = np.where(np.diff(cutout) > 1)[0]
border_end = border_start + 1
borders = sorted(np.r_[0, border_start, border_end, len(cutout) - 1])
true_borders = cutout[borders]
right_borders = true_borders[1::2] + 1
true_borders = np.sort(np.append(true_borders[0::2], right_borders))
# Workaround for bug that occurs when signal goes below thr for 1 dtp,
# Workaround eliminates empty slices from np. split
backward_mask = np.absolute(np.ediff1d(true_borders, to_begin=1)) > 0
forward_mask = np.absolute(np.ediff1d(true_borders[::-1],
to_begin=1)[::-1]) > 0
true_borders = true_borders[backward_mask * forward_mask]
split_signal = np.split(np.array(signal), true_borders)[1::2]
maxima_idc_split = np.array([peak_func(x) for x in split_signal])
max_idc = maxima_idc_split + true_borders[0::2]
events = signal.times[max_idc]
events_base = events.magnitude
if events_base is None:
# This occurs in some Python 3 builds due to some
# bug in quantities.
events_base = np.array(
[event.magnitude for event in events]) # Workaround
result_st = neo.SpikeTrain(events_base, units=signal.times.units,
t_start=signal.t_start,
t_stop=signal.t_stop)
if as_array:
result_st = result_st.magnitude
return result_st
def _homogeneous_process(interval_generator, mean_rate, t_start, t_stop,
as_array):
"""
Returns a spike train whose spikes are a realization of a random process
generated by the function `interval_generator` with the given rate,
starting at time `t_start` and stopping `time t_stop`.
"""
t_start = t_start.rescale(t_stop.units)
n_spikes_expected = int(np.ceil(
((t_stop - t_start) * mean_rate).simplified))
if n_spikes_expected < 0:
raise ValueError("Expected no. of spikes: {n_spikes} < 0. The firing "
"rate ({rate}) cannot be negative and t_stop "
"({t_stop}) must be greater than t_start "
"({t_start})".format(n_spikes=n_spikes_expected,
rate=mean_rate,
t_stop=t_stop, t_start=t_start))
spikes = []
if n_spikes_expected > 0:
# 3 STDs corresponds to 99.7%
n_spikes_expected = int(np.ceil(
n_spikes_expected + 3 * np.sqrt(n_spikes_expected)))
t_last = t_start.simplified.magnitude
while True:
isi = interval_generator(size=n_spikes_expected)
spikes = np.r_[spikes, t_last + np.cumsum(isi)]
# Check if not whole time range is covered.
index_last_spike = spikes.searchsorted(t_stop.simplified.magnitude)
if index_last_spike < len(spikes):
spikes = spikes[:index_last_spike]
spikes = (spikes / mean_rate.units).rescale(t_stop.units)
break
t_last = spikes[-1]
if as_array:
spikes = spikes.magnitude
else:
spikes = neo.SpikeTrain(spikes, t_start=t_start, t_stop=t_stop,
units=t_stop.units)
return spikes
[docs]def homogeneous_poisson_process(rate, t_start=0.0 * pq.ms,
t_stop=1000.0 * pq.ms, as_array=False,
refractory_period=None):
"""
Returns a spike train whose spikes are a realization of a Poisson process
with the given rate, starting at time `t_start` and stopping time `t_stop`.
All numerical values should be given as Quantities, e.g. `100*pq.Hz`.
Parameters
----------
rate : pq.Quantity
The rate of the discharge.
t_start : pq.Quantity, optional
The beginning of the spike train.
Default: 0 * pq.ms.
t_stop : pq.Quantity, optional
The end of the spike train.
Default: 1000 * pq.ms.
as_array : bool, optional
If True, a NumPy array of sorted spikes is returned,
rather than a `neo.SpikeTrain` object.
Default: False.
refractory_period : pq.Quantity or None, optional
`pq.Quantity` scalar with dimension time. The time period after one
spike no other spike is emitted.
Default: None.
Returns
-------
spiketrain : neo.SpikeTrain or np.ndarray
Homogeneous Poisson process realization, stored in `neo.SpikeTrain`
if `as_array` is False (default) and `np.ndarray` otherwise.
Raises
------
ValueError
If one of `rate`, `t_start` and `t_stop` is not of type `pq.Quantity`.
If `refractory_period` is not None or not of type `pq.Quantity`.
If `refractory_period` is not None and the period between two
successive spikes (`1 / rate`) is smaller than the `refractory_period`.
Examples
--------
>>> import quantities as pq
>>> spikes = homogeneous_poisson_process(50*pq.Hz, t_start=0*pq.ms,
... t_stop=1000*pq.ms)
>>> spikes = homogeneous_poisson_process(
... 20*pq.Hz, t_start=5000*pq.ms, t_stop=10000*pq.ms, as_array=True)
>>> spikes = homogeneous_poisson_process(50*pq.Hz, t_start=0*pq.ms,
... t_stop=1000*pq.ms, refractory_period = 3*pq.ms)
"""
if not (isinstance(t_start, pq.Quantity) and
isinstance(t_stop, pq.Quantity)):
raise ValueError("t_start and t_stop must be of type pq.Quantity")
if not isinstance(rate, pq.Quantity):
raise ValueError("rate must be of type pq.Quantity")
if not isinstance(refractory_period, pq.Quantity) and \
refractory_period is not None:
raise ValueError("refractory_period must be of type pq.Quantity or"
"None")
rate = rate.simplified
# Case without a refractory period
if refractory_period is None:
interval_generator = partial(np.random.exponential,
scale=1. / rate.magnitude)
return _homogeneous_process(
interval_generator, rate, t_start, t_stop,
as_array)
# Case with a refractory period
refractory_period = refractory_period.simplified
if rate * refractory_period >= 1.:
raise ValueError("Period between two successive spikes must be larger "
"than the refractory period. Decrease either the "
"firing rate or the refractory period.")
effective_rate = rate / (1. - rate * refractory_period)
def interval_generator_refractory(size):
return refractory_period.magnitude + \
np.random.exponential(1. / effective_rate.magnitude, size)
# we subtract refractory_period from t_start to be added later on
# in interval_generator_refractory()
spiketrain = _homogeneous_process(interval_generator_refractory, rate,
t_start - refractory_period, t_stop,
as_array)
if not as_array:
spiketrain.t_start = t_start
return spiketrain
[docs]def inhomogeneous_poisson_process(rate, as_array=False,
refractory_period=None):
"""
Returns a spike train whose spikes are a realization of an inhomogeneous
Poisson process with the given rate profile.
Parameters
----------
rate : neo.AnalogSignal
A `neo.AnalogSignal` representing the rate profile evolving over time.
Its values have all to be `>=0`. The output spiketrain will have
`t_start = rate.t_start` and `t_stop = rate.t_stop`
as_array : bool, optional
If True, a NumPy array of sorted spikes is returned,
rather than a SpikeTrain object.
Default: False.
refractory_period : pq.Quantity or None, optional
`pq.Quantity` scalar with dimension time. The time period after one
spike no other spike is emitted.
Default: None.
Returns
-------
spiketrain : neo.SpikeTrain or np.ndarray
Inhomogeneous Poisson process realization, of type `neo.SpikeTrain`
if `as_array` is False (default) and `np.ndarray` otherwise.
Raises
------
ValueError
If `rate` contains a negative value.
If `refractory_period` is not None or not of type `pq.Quantity`.
If `refractory_period` is not None and the period between two
successive spikes (`1 / rate`) is smaller than the `refractory_period`.
"""
# Check rate contains only positive values
if np.any(rate < 0) or rate.size == 0:
raise ValueError(
'rate must be a positive non empty signal, representing the'
'rate at time t')
if not isinstance(refractory_period, pq.Quantity) and \
refractory_period is not None:
raise ValueError("refractory_period must be of type pq.Quantity or"
"None")
rate_max = np.max(rate)
if refractory_period is not None:
if (rate_max * refractory_period).simplified >= 1.:
raise ValueError(
"Period between two successive spikes must be larger "
"than the refractory period. Decrease either the "
"firing rate or the refractory period.")
# effective rate parameter for the refractory period case
rate = rate / (1. - (rate * refractory_period).simplified)
rate_max = np.max(rate)
# Generate n hidden Poisson SpikeTrains with rate equal
# to the peak rate
homogeneous_poiss = homogeneous_poisson_process(
rate=rate_max, t_stop=rate.t_stop, t_start=rate.t_start)
# Compute the rate profile at each spike time by interpolation
rate_interpolated = _analog_signal_linear_interp(
signal=rate, times=homogeneous_poiss.times)
# Accept each spike at time t with probability rate(t)/max_rate
random_uniforms = np.random.uniform(size=len(homogeneous_poiss)) * rate_max
spikes = homogeneous_poiss[random_uniforms < rate_interpolated.flatten()]
if refractory_period is not None:
refractory_period = refractory_period.rescale(
rate.t_stop.units).magnitude
# thinning in average cancels the effect of the effective firing rate
spikes = _thinning_for_refractory_period(spikes.magnitude,
refractory_period)
if not as_array:
spikes = neo.SpikeTrain(spikes * rate.t_stop.units,
t_start=rate.t_start,
t_stop=rate.t_stop)
else:
if as_array:
spikes = spikes.magnitude
return spikes
def _thinning_for_refractory_period(spiketrain, refractory_period):
"""
Function to thin out a spiketrain, that every ISI is greater than the
refractory period.
Parameters
----------
spiketrain : np.ndarray
Magnitude of a spiketrain.
refractory_period : float
Magnitude of a refractory period.
Returns
-------
thinned_spiketrain : np.ndarray
thinned out spiketrain
"""
thinned_spiketrain = []
previous_spike = -refractory_period
for spike in spiketrain:
if spike > previous_spike + refractory_period:
thinned_spiketrain.append(spike)
previous_spike = spike
return np.array(thinned_spiketrain)
def _analog_signal_linear_interp(signal, times):
"""
Compute the linear interpolation of a signal at desired times.
Given the `signal` (neo.AnalogSignal) taking value `s0` and `s1` at two
consecutive time points `t0` and `t1` `(t0 < t1)`, for every time `t` in
`times`, such that `t0<t<=t1` is returned the value of the linear
interpolation, given by:
`s = ((s1 - s0) / (t1 - t0)) * t + s0`.
Parameters
----------
signal : neo.AnalogSignal
The analog signal containing the discretization of the function to
interpolate
times : pq.Quantity
The time points for which the interpolation is computed
Returns
-------
out: pq.Quantity
representing the values of the interpolated signal at
the times given by times
Notes
-----
If `signal` has sampling period `sampling_period=signal.sampling_period`,
its values are defined at `t=signal.times`, such that
`t[i] = signal.t_start + i * sampling_period`
The last of such times is lower than
signal.t_stop`:t[-1] = signal.t_stop - sampling_period`.
For the interpolation at times t such that `t[-1] <= t <= signal.t_stop`,
the value of `signal` at `signal.t_stop` is taken to be that
at time `t[-1]`.
"""
sampling_period = signal.sampling_period
t_start = signal.t_start.rescale(signal.times.units)
t_stop = signal.t_stop.rescale(signal.times.units)
# Extend the signal (as a dimensionless array) copying the last value
# one time, and extend its times to t_stop
signal_extended = np.vstack(
[signal.magnitude, signal[-1].magnitude]).flatten()
times_extended = np.hstack([signal.times, t_stop]) * signal.times.units
time_ids = (times - t_start) / sampling_period
time_ids = np.floor(time_ids.simplified.magnitude).astype(np.int32)
# Compute the slope of the signal at each time in times
signal_1 = signal_extended[time_ids]
signal_2 = signal_extended[time_ids + 1]
slope = (signal_2 - signal_1) / sampling_period
# Interpolate the signal at each time in times by linear interpolation
return (signal_1
+ slope * (times - times_extended[time_ids])) * signal.units
[docs]def homogeneous_gamma_process(a, b, t_start=0.0 * pq.ms, t_stop=1000.0 * pq.ms,
as_array=False):
"""
Returns a spike train whose spikes are a realization of a gamma process
with the given parameters, starting at time `t_start` and stopping time
`t_stop` (average rate will be `b/a`). All numerical values should be
given as Quantities, e.g. `100*pq.Hz`.
Parameters
----------
a : int or float
The shape parameter of the gamma distribution.
b : pq.Quantity
The rate parameter of the gamma distribution.
t_start : pq.Quantity, optional
The beginning of the spike train.
Default: 0 * pq.ms.
t_stop : pq.Quantity, optional
The end of the spike train.
Default: 1000 * pq.ms.
as_array : bool, optional
If True, a NumPy array of sorted spikes is returned, rather than a
`neo.SpikeTrain` object.
Default: False.
Returns
-------
spiketrain : neo.SpikeTrain or np.ndarray
Homogeneous Gamma process realization, stored in `neo.SpikeTrain`
if `as_array` is False (default) and `np.ndarray` otherwise.
Raises
------
ValueError
If `t_start` and `t_stop` are not of type `pq.Quantity`.
Examples
--------
>>> import quantities as pq
>>> spikes = homogeneous_gamma_process(2.0, 50*pq.Hz, 0*pq.ms,
... 1000*pq.ms)
>>> spikes = homogeneous_gamma_process(
... 5.0, 20*pq.Hz, 5000*pq.ms, 10000*pq.ms, as_array=True)
"""
# note that the rate of the gamma distribution is called 'b' and not 'rate'
# to avoid false thoughts that 'rate' could be the mean firing rate, which
# equals to b / a
if not (isinstance(t_start, pq.Quantity) and
isinstance(t_stop, pq.Quantity)):
raise ValueError("t_start and t_stop must be of type pq.pq.Quantity")
b = b.rescale(1 / t_start.units).simplified
rate = b / a
theta = 1. / b.magnitude
interval_generator = partial(np.random.gamma, shape=a, scale=theta)
return _homogeneous_process(interval_generator, rate, t_start,
t_stop, as_array)
def inhomogeneous_gamma_process(rate, shape_factor, as_array=False):
"""
Returns a spike train whose spikes are a realization of an inhomogeneous
Gamma process with the given rate profile and the given shape factor.
The implementation using operational time is inspired by Nawrot et al.
(2018) [1]_.
Parameters
----------
rate : neo.AnalogSignal
A `neo.AnalogSignal` representing the rate profile evolving over time.
Its values have all to be `>=0`. The output spiketrain will have
`t_start = rate.t_start` and `t_stop = rate.t_stop`
shape_factor : float
The shape factor of the Gamma process
as_array : bool, optional
If True, a NumPy array of sorted spikes is returned,
rather than a SpikeTrain object.
Default: False.
Returns
-------
spiketrain : neo.SpikeTrain or np.ndarray
Inhomogeneous Poisson process realization, of type `neo.SpikeTrain`
if `as_array` is False (default) and `np.ndarray` otherwise.
Raises
------
ValueError
If `rate` is not a neo AnalogSignal
If `rate` contains a negative value.
References
----------
.. [1] Nawrot, M., Boucsein, C., Denker, M., Rodriguez Molina, V.,
Riehle A., Aertsen A., & Rotter, S. (2008). Measurement of
variability dynamics in cortical spike trains. Journal of
Neuroscience Methods, 169, 374–390.
"""
if not isinstance(rate, neo.AnalogSignal):
raise ValueError('rate must be a neo AnalogSignal')
# Check rate contains only positive values
if np.any(rate < 0) or rate.size == 0:
raise ValueError(
'rate must be a positive non empty signal, representing the'
'rate at time t')
# Operational time corresponds to the integral of the firing rate over time
operational_time = np.cumsum(
(rate*rate.sampling_period).simplified.magnitude)
operational_time = np.hstack((0., operational_time))
# The time points at which the firing rates are given
real_time = np.hstack((rate.times.simplified.magnitude,
rate.t_stop.simplified.magnitude))
spiketrain_operational_time = homogeneous_gamma_process(
a=shape_factor, b=shape_factor*1.*pq.Hz,
t_start=0.*pq.s, t_stop=operational_time[-1]*pq.s, as_array=True)
# indices where between which points in operational time the spikes lie
indices = np.searchsorted(operational_time, spiketrain_operational_time)
# In real time the spikes are first aligned to the left border of the bin.
# Note that indices are greater than 0 because 'operational_time' was
# padded with zeros.
spiketrain = real_time[indices - 1]
# the relative position of the spikes in the operational time bins
positions_in_bins = \
(spiketrain_operational_time - operational_time[indices-1]) / (
operational_time[indices]-operational_time[indices-1])
# add the positions in the bin times the sampling period in real time
spiketrain += rate.sampling_period.simplified.magnitude * positions_in_bins
if as_array:
return spiketrain
return neo.SpikeTrain(spiketrain, units=pq.s, t_stop=rate.t_stop)
@deprecated_alias(n='n_spiketrains')
def _n_poisson(rate, t_stop, t_start=0.0 * pq.ms, n_spiketrains=1):
"""
Generates one or more independent Poisson spike trains.
Parameters
----------
rate : pq.Quantity scalar or pq.Quantity array
Expected firing rate (frequency) of each output SpikeTrain.
Can be one of:
* a single pq.Quantity value: expected firing rate of each output
SpikeTrain
* a pq.Quantity array: rate[i] is the expected firing rate of the i-th
output SpikeTrain
t_stop : pq.Quantity
Single common stop time of each output SpikeTrain. Must be > t_start.
t_start : pq.Quantity, optional
Single common start time of each output SpikeTrain. Must be < t_stop.
Default: 0 * pq.ms
n_spiketrains: int, optional
If rate is a single pq.Quantity value, n specifies the number of
SpikeTrains to be generated. If rate is an array, n is ignored and the
number of SpikeTrains is equal to len(rate).
Default: 1
Returns
-------
list of neo.SpikeTrain
Each SpikeTrain contains one of the independent Poisson spike trains,
either n SpikeTrains of the same rate, or len(rate) SpikeTrains with
varying rates according to the rate parameter. The time unit of the
SpikeTrains is given by t_stop.
"""
# Check that the provided input is Hertz
if not isinstance(rate, pq.Quantity):
raise ValueError('rate must be a pq.Quantity')
try:
rate.rescale(pq.Hz)
except ValueError:
raise ValueError('rate argument must have rate unit (1/time)')
# Check t_start < t_stop and create their strip dimensions
if not t_start < t_stop:
raise ValueError(
't_start (={}) must be < t_stop (={})'.format(t_start, t_stop))
# Set number n of output spike trains (specified or set to len(rate))
if not (isinstance(n_spiketrains, int) and n_spiketrains > 0):
raise ValueError(
'n (={}) must be a positive integer'.format(str(n_spiketrains)))
rate_dl = rate.simplified.magnitude.flatten()
# Check rate input parameter
if len(rate_dl) == 1:
if rate_dl < 0:
raise ValueError('rate (={}) must be non-negative.'.format(rate))
rates = np.array([rate_dl] * n_spiketrains)
else:
rates = rate_dl.flatten()
if any(rates < 0):
raise ValueError('rate must have non-negative elements.')
return [homogeneous_poisson_process(rate * pq.Hz, t_start, t_stop)
for rate in rates]
[docs]@deprecated_alias(rate_c='coincidence_rate', n='n_spiketrains',
return_coinc='return_coincidences')
def single_interaction_process(
rate, coincidence_rate, t_stop, n_spiketrains=2, jitter=0 * pq.ms,
coincidences='deterministic', t_start=0 * pq.ms, min_delay=0 * pq.ms,
return_coincidences=False):
"""
Generates a multidimensional Poisson SIP (single interaction process)
plus independent Poisson processes
A Poisson SIP consists of Poisson time series which are independent
except for simultaneous events in all of them. This routine generates
a SIP plus additional parallel independent Poisson processes.
See _[1].
Parameters
----------
t_stop : pq.Quantity
Total time of the simulated processes. The events are drawn between
0 and `t_stop`.
rate : pq.Quantity
Overall mean rate of the time series to be generated (coincidence
rate `coincidence_rate` is subtracted to determine the background
rate). Can be:
* a float, representing the overall mean rate of each process. If
so, it must be higher than `coincidence_rate`.
* an iterable of floats (one float per process), each float
representing the overall mean rate of a process. If so, all the
entries must be larger than `coincidence_rate`.
coincidence_rate : pq.Quantity
Coincidence rate (rate of coincidences for the n-dimensional SIP).
The SIP spike trains will have coincident events with rate
`coincidence_rate` plus independent 'background' events with rate
`rate-rate_coincidence`.
n_spiketrains : int, optional
If `rate` is a single pq.Quantity value, `n_spiketrains` specifies the
number of SpikeTrains to be generated. If rate is an array,
`n_spiketrains` is ignored and the number of SpikeTrains is equal to
`len(rate)`.
Default: 2
jitter : pq.Quantity, optional
Jitter for the coincident events. If `jitter == 0`, the events of all
n correlated processes are exactly coincident. Otherwise, they are
jittered around a common time randomly, up to +/- `jitter`.
Default: 0 * pq.ms
coincidences : {'deterministic', 'stochastic'}, optional
Whether the total number of injected coincidences must be determin-
istic (i.e. rate_coincidence is the actual rate with which coincidences
are generated) or stochastic (i.e. rate_coincidence is the mean rate of
coincidences):
* 'deterministic': deterministic rate
* 'stochastic': stochastic rate
Default: 'deterministic'
t_start : pq.Quantity, optional
Starting time of the series. If specified, it must be lower than
`t_stop`.
Default: 0 * pq.ms
min_delay : pq.Quantity, optional
Minimum delay between consecutive coincidence times.
Default: 0 * pq.ms
return_coincidences : bool, optional
Whether to return the coincidence times for the SIP process
Default: False
Returns
-------
output: list
Realization of a SIP consisting of n Poisson processes characterized
by synchronous events (with the given jitter)
If `return_coinc` is `True`, the coincidence times are returned as a
second output argument. They also have an associated time unit (same
as `t_stop`).
References
----------
.. [1] Kuhn, Aertsen, Rotter (2003) Neural Comput 15(1):67-101
Examples
--------
>>> import quantities as pq
>>> import elephant.spike_train_generation as stg
# TODO: check if rate_coincidence=4 is correct.
>>> sip, coinc = stg.single_interaction_process(
... rate=20*pq.Hz, coincidence_rate=4,
... t_stop=1*pq.s, n_spiketrains=10, return_coincidences = True)
"""
# Check if n is a positive integer
if not (isinstance(n_spiketrains, int) and n_spiketrains > 0):
raise ValueError(
'n (={}) must be a positive integer'.format(n_spiketrains))
if coincidences not in ('deterministic', 'stochastic'):
raise ValueError(
"coincidences must be 'deterministic' or 'stochastic'")
# Assign time unit to jitter, or check that its existing unit is a time
# unit
jitter = abs(jitter)
# Define the array of rates from input argument rate. Check that its length
# matches with n
if rate.ndim == 0:
if rate < 0 * pq.Hz:
raise ValueError(
'rate (={}) must be non-negative.'.format(rate))
rates_b = np.repeat(rate, n_spiketrains)
else:
rates_b = rate.flatten()
if not all(rates_b >= 0. * pq.Hz):
raise ValueError('*rate* must have non-negative elements')
# Check: rate>=rate_coincidence
if np.any(rates_b < coincidence_rate):
raise ValueError(
'all elements of *rate* must be >= *rate_coincidence*')
# Check min_delay < 1./rate_coincidence
if not (coincidence_rate == 0 * pq.Hz
or min_delay < 1. / coincidence_rate):
raise ValueError(
"'*min_delay* (%s) must be lower than 1/*rate_coincidence* (%s)." %
(str(min_delay), str((1. / coincidence_rate).rescale(
min_delay.units))))
# Generate the n Poisson processes there are the basis for the SIP
# (coincidences still lacking)
embedded_poisson_trains = _n_poisson(
rate=rates_b - coincidence_rate, t_stop=t_stop, t_start=t_start)
# Convert the trains from neo SpikeTrain objects to simpler pq.Quantity
# objects
embedded_poisson_trains = [
emb.view(pq.Quantity) for emb in embedded_poisson_trains]
# Generate the array of times for coincident events in SIP, not closer than
# min_delay. The array is generated as a pq.Quantity.
if coincidences == 'deterministic':
# P. Bouss: we want the closest approximation to the average
# coincidence count.
n_coincidences = (t_stop - t_start) * coincidence_rate
# Conversion to integer necessary for python 2
n_coincidences = int(round(n_coincidences.simplified.item()))
while True:
coinc_times = t_start + \
np.sort(np.random.random(n_coincidences)) * (
t_stop - t_start)
if len(coinc_times) < 2 or min(np.diff(coinc_times)) >= min_delay:
break
else: # coincidences == 'stochastic'
while True:
coinc_times = homogeneous_poisson_process(
rate=coincidence_rate, t_stop=t_stop, t_start=t_start)
if len(coinc_times) < 2 or min(np.diff(coinc_times)) >= min_delay:
break
coinc_times = coinc_times.simplified
units = coinc_times.units
# Set the coincidence times to T-jitter if larger. This ensures that
# the last jittered spike time is <T
effective_t_stop = t_stop - jitter
coinc_times = np.minimum(coinc_times.magnitude,
effective_t_stop.simplified.magnitude)
coinc_times = coinc_times * units
# Replicate coinc_times n times, and jitter each event in each array by
# +/- jitter (within (t_start, t_stop))
embedded_coinc = coinc_times + \
np.random.random(
(len(rates_b), len(coinc_times))) * 2 * jitter - jitter
embedded_coinc = embedded_coinc + \
(t_start - embedded_coinc) * (embedded_coinc < t_start) - \
(t_stop - embedded_coinc) * (embedded_coinc > t_stop)
# Inject coincident events into the n SIP processes generated above, and
# merge with the n independent processes
sip_process = [
np.sort(np.concatenate((
embedded_poisson_trains[m].rescale(t_stop.units),
embedded_coinc[m].rescale(t_stop.units))) * t_stop.units)
for m in range(len(rates_b))]
# Convert back sip_process and coinc_times from pq.Quantity objects to
# neo.SpikeTrain objects
sip_process = [
neo.SpikeTrain(t, t_start=t_start, t_stop=t_stop).rescale(t_stop.units)
for t in sip_process]
coinc_times = [
neo.SpikeTrain(t, t_start=t_start, t_stop=t_stop).rescale(t_stop.units)
for t in embedded_coinc]
# Return the processes in the specified output_format
if not return_coincidences:
output = sip_process
else:
output = sip_process, coinc_times
return output
def _pool_two_spiketrains(spiketrain_1, spiketrain_2, extremes='inner'):
"""
Pool the spikes of two spike trains a and b into a unique spike train.
Parameters
----------
spiketrain_1, spiketrain_2 : neo.SpikeTrain
Spiketrains to be pooled.
extremes : {'inner', 'outer'}, optional
Only spikes of a and b in the specified extremes are considered.
* 'inner': pool all spikes from max(a.tstart_ b.t_start) to
min(a.t_stop, b.t_stop)
* 'outer': pool all spikes from min(a.tstart_ b.t_start) to
max(a.t_stop, b.t_stop)
Default: 'inner'
Returns
-------
neo.SpikeTrain
containing all spikes of the two input spiketrains falling in the
specified extremes
"""
unit = spiketrain_1.units
spiketrain_2 = spiketrain_2.rescale(unit)
times_1_dimless = spiketrain_1.magnitude
times_2_dimless = spiketrain_2.rescale(unit).magnitude
times = np.sort(np.concatenate((times_1_dimless, times_2_dimless)))
if extremes == 'outer':
t_start = min(spiketrain_1.t_start, spiketrain_2.t_start)
t_stop = max(spiketrain_1.t_stop, spiketrain_2.t_stop)
elif extremes == 'inner':
t_start = max(spiketrain_1.t_start, spiketrain_2.t_start)
t_stop = min(spiketrain_1.t_stop, spiketrain_2.t_stop)
times = times[times > t_start.magnitude]
times = times[times < t_stop.magnitude]
else:
raise ValueError(
'extremes (%s) can only be "inner" or "outer"' % extremes)
return neo.SpikeTrain(times=times, units=unit, t_start=t_start,
t_stop=t_stop)
def _pool_spiketrains(spiketrains, extremes='inner'):
"""
Pool spikes from any number of spike trains into a unique spike train.
Parameters
----------
spiketrains: list of neo.SpikeTrain
A list of spiketrains to merge.
extremes: str, optional
Only spikes of a and b in the specified extremes are considered.
* 'inner': pool all spikes from min(a.t_start b.t_start) to
max(a.t_stop, b.t_stop)
* 'outer': pool all spikes from max(a.tstart_ b.t_start) to
min(a.t_stop, b.t_stop)
Default: 'inner'
Returns
-------
neo.SpikeTrain
containing all spikes in trains falling in the specified extremes
"""
merge_trains = spiketrains[0]
for spiketrain in spiketrains[1:]:
merge_trains = _pool_two_spiketrains(
merge_trains, spiketrain, extremes=extremes)
t_start, t_stop = merge_trains.t_start, merge_trains.t_stop
merge_trains = sorted(merge_trains)
merge_trains = np.squeeze(merge_trains)
merge_trains = neo.SpikeTrain(
merge_trains, t_stop=t_stop, t_start=t_start,
units=spiketrains[0].units)
return merge_trains
def _sample_int_from_pdf(probability_density, n_samples):
"""
Draw n independent samples from the set {0,1,...,L}, where L=len(a)-1,
according to the probability distribution a.
a[j] is the probability to sample j, for each j from 0 to L.
Parameters
----------
probability_density : np.ndarray
Probability vector (i..e array of sum 1) that at each entry j carries
the probability to sample j (j=0,1,...,len(a)-1).
n_samples : int
Number of samples generated with the function
Returns
-------
np.ndarray
An array of n samples taking values between `0` and `n=len(a)-1`.
"""
cumulative_distribution = np.cumsum(probability_density)
random_uniforms = np.random.uniform(0, 1, size=n_samples)
random_uniforms = np.repeat(np.expand_dims(random_uniforms, axis=1),
repeats=len(probability_density),
axis=1)
return (cumulative_distribution < random_uniforms).sum(axis=1)
def _mother_proc_cpp_stat(A, t_stop, rate, t_start=0 * pq.ms):
"""
Generate the hidden ("mother") Poisson process for a Compound Poisson
Process (CPP).
Parameters
----------
A : np.ndarray
Amplitude distribution. A[j] represents the probability of a
synchronous event of size j.
The sum over all entries of a must be equal to one.
t_stop : pq.Quantity
The stopping time of the mother process
rate : pq.Quantity
Homogeneous rate of the n spike trains that will be generated by the
CPP function
t_start : pq.Quantity, optional
The starting time of the mother process
Default: 0 pq.ms
Returns
-------
Poisson spike train representing the mother process generating the CPP
"""
n_spiketrains = len(A) - 1
# expected amplitude
exp_amplitude = np.dot(A, np.arange(n_spiketrains + 1))
# expected rate of the mother process
exp_mother_rate = (n_spiketrains * rate) / exp_amplitude
return homogeneous_poisson_process(
rate=exp_mother_rate, t_stop=t_stop, t_start=t_start)
def _cpp_hom_stat(A, t_stop, rate, t_start=0 * pq.ms):
"""
Generate a Compound Poisson Process (CPP) with amplitude distribution
A and heterogeneous firing rates r=r[0], r[1], ..., r[-1].
Parameters
----------
A : np.ndarray
Amplitude distribution. A[j] represents the probability of a
synchronous event of size j.
The sum over all entries of A must be equal to one.
t_stop : pq.Quantity
The end time of the output spike trains
rate : pq.Quantity
Average rate of each spike train generated
t_start : pq.Quantity, optional
The start time of the output spike trains
Default: 0 pq.ms
Returns
-------
list of neo.SpikeTrain
with n elements, having average firing rate r and correlated such to
form a CPP with amplitude distribution a
"""
# Generate mother process and associated spike labels
mother = _mother_proc_cpp_stat(
A=A, t_stop=t_stop, rate=rate, t_start=t_start)
labels = _sample_int_from_pdf(A, len(mother))
n_spiketrains = len(A) - 1 # Number of trains in output
spiketrains = [[]] * n_spiketrains
try: # Faster but more memory-consuming approach
n_mother_trains = len(mother) # number of spikes in the mother process
spike_matrix = np.zeros((n_spiketrains, n_mother_trains), dtype=bool)
# for each spike, take its label
for spike_id, label in enumerate(labels):
# choose label random trains
train_ids = np.random.choice(n_spiketrains, label, replace=False)
# and set the spike matrix for that train
for train_id in train_ids:
spike_matrix[train_id, spike_id] = True # and spike to True
for train_id, row in enumerate(spike_matrix):
spiketrains[train_id] = mother[row].view(pq.Quantity)
except MemoryError: # Slower (~2x) but less memory-consuming approach
print('memory case')
for mother_spiketrain, label in zip(mother, labels):
train_ids = np.random.choice(n_spiketrains, label)
for train_id in train_ids:
spiketrains[train_id].append(mother_spiketrain)
return [neo.SpikeTrain(times=spiketrain, t_start=t_start, t_stop=t_stop)
for spiketrain in spiketrains]
def _cpp_het_stat(A, t_stop, rates, t_start=0. * pq.ms):
"""
Generate a Compound Poisson Process (CPP) with amplitude distribution
A and heterogeneous firing rates r=r[0], r[1], ..., r[-1].
Parameters
----------
A : np.ndarray
CPP's amplitude distribution. A[j] represents the probability of
a synchronous event of size j among the generated spike trains.
The sum over all entries of A must be equal to one.
t_stop : pq.Quantity
The end time of the output spike trains
rates : pq.Quantity
Array of firing rates of each spike train generated with
t_start : pq.Quantity, optional
The start time of the output spike trains
Default: 0 pq.ms
Returns
-------
list of neo.SpikeTrain
List of neo.SpikeTrains with different firing rates, forming
a CPP with amplitude distribution `A`.
"""
# Computation of Parameters of the two CPPs that will be merged
# (uncorrelated with heterog. rates + correlated with homog. rates)
n_spiketrains = len(rates) # number of output spike trains
# amplitude expectation
expected_amplitude = np.dot(A, np.arange(n_spiketrains + 1))
r_sum = np.sum(rates) # sum of all output firing rates
r_min = np.min(rates) # minimum of the firing rates
# rate of the uncorrelated CPP
r_uncorrelated = r_sum - n_spiketrains * r_min
# rate of the correlated CPP
r_correlated = r_sum / expected_amplitude - r_uncorrelated
# rate of the hidden mother process
r_mother = r_uncorrelated + r_correlated
# Check the analytical constraint for the amplitude distribution
if A[1] < (r_uncorrelated / r_mother).rescale(
pq.dimensionless).magnitude:
raise ValueError('A[1] too small / A[i], i>1 too high')
# Compute the amplitude distribution of the correlated CPP, and generate it
A = A * (r_mother / r_correlated).magnitude
A[1] = A[1] - r_uncorrelated / r_correlated
compound_poisson_spiketrains = _cpp_hom_stat(
A, t_stop, r_min, t_start)
# Generate the independent heterogeneous Poisson processes
poisson_spiketrains = \
[homogeneous_poisson_process(rate - r_min, t_start, t_stop)
for rate in rates]
# Pool the correlated CPP and the corresponding Poisson processes
return [_pool_two_spiketrains(compound_poisson_spiketrain,
poisson_spiketrain)
for compound_poisson_spiketrain, poisson_spiketrain
in zip(compound_poisson_spiketrains, poisson_spiketrains)]
[docs]@deprecated_alias(A='amplitude_distribution')
def compound_poisson_process(
rate, amplitude_distribution, t_stop, shift=None, t_start=0 * pq.ms):
"""
Generate a Compound Poisson Process (CPP; see _[1]) with a given
`amplitude_distribution` :math:`A` and stationary marginal rates `rate`.
The CPP process is a model for parallel, correlated processes with Poisson
spiking statistics at pre-defined firing rates. It is composed of
`len(A)-1` spike trains with a correlation structure determined by the
amplitude distribution :math:`A`: A[j] is the probability that a spike
occurs synchronously in any `j` spike trains.
The CPP is generated by creating a hidden mother Poisson process, and then
copying spikes of the mother process to `j` of the output spike trains with
probability `A[j]`.
Note that this function decorrelates the firing rate of each SpikeTrain
from the probability for that SpikeTrain to participate in a synchronous
event (which is uniform across SpikeTrains).
Parameters
----------
rate : pq.Quantity
Average rate of each spike train generated. Can be:
- a single value, all spike trains will have same rate rate
- an array of values (of length len(A)-1), each indicating the
firing rate of one process in output
amplitude_distribution : np.ndarray or list
CPP's amplitude distribution :math:`A`. `A[j]` represents the
probability of a synchronous event of size `j` among the generated
spike trains. The sum over all entries of :math:`A` must be equal to
one.
t_stop : pq.Quantity
The end time of the output spike trains.
shift : pq.Quantity, optional
If `None`, the injected synchrony is exact.
If shift is a `pq.Quantity`, all the spike trains are shifted
independently by a random amount in the interval `[-shift, +shift]`.
Default: None
t_start : pq.Quantity, optional
The `t_start` time of the output spike trains.
Default: 0 pq.ms
Returns
-------
list of neo.SpikeTrain
SpikeTrains with specified firing rates forming the CPP with amplitude
distribution :math:`A`.
References
----------
.. [1] Staude, Rotter, Gruen (2010) J Comput Neurosci 29:327-350.
"""
if not isinstance(amplitude_distribution, np.ndarray):
amplitude_distribution = np.array(amplitude_distribution)
# Check A is a probability distribution (it sums to 1 and is positive)
if abs(sum(amplitude_distribution) - 1) > np.finfo('float').eps:
raise ValueError(
"'amplitude_distribution' must be a probability vector: "
"sum(A) = {} != 1".format(sum(amplitude_distribution)))
if np.any(amplitude_distribution < 0):
raise ValueError("'amplitude_distribution' must be a probability "
"vector with positive entries")
# Check that the rate is not an empty pq.Quantity
if rate.ndim == 1 and len(rate) == 0:
raise ValueError('Rate is an empty pq.Quantity array')
# Return empty spike trains for specific parameters
if amplitude_distribution[0] == 1 or np.sum(np.abs(rate.magnitude)) == 0:
return [neo.SpikeTrain([] * t_stop.units,
t_stop=t_stop,
t_start=t_start)] * (
len(amplitude_distribution) - 1)
# Homogeneous rates
if rate.ndim == 0:
compound_poisson_spiketrains = _cpp_hom_stat(
A=amplitude_distribution, t_stop=t_stop, rate=rate,
t_start=t_start)
# Heterogeneous rates
else:
compound_poisson_spiketrains = _cpp_het_stat(
A=amplitude_distribution, t_stop=t_stop, rates=rate,
t_start=t_start)
if shift is not None:
# Dither the output spiketrains
compound_poisson_spiketrains = \
[dither_spike_train(spiketrain, shift=shift, edges=True)[0]
for spiketrain in compound_poisson_spiketrains]
return compound_poisson_spiketrains
# Alias for :func:`compound_poisson_process`
cpp = compound_poisson_process
