# -*- 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
import numpy as np
from quantities import ms, mV, Hz, Quantity, dimensionless
from neo import SpikeTrain
import random
from elephant.spike_train_surrogates import dither_spike_train
import warnings
[docs]def threshold_detection(signal, threshold=0.0 * 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 object
'signal' is an analog signal.
threshold : A quantity, e.g. in mV
'threshold' contains a value that must be reached
for an event to be detected. Default: 0.0 * mV.
sign : 'above' or 'below'
'sign' determines whether to count thresholding crossings
that cross above or below the threshold.
format : None or 'raw'
Whether to return as SpikeTrain (None)
or as a plain array of times ('raw').
Returns
-------
result_st : neo SpikeTrain object
'result_st' contains the spike times of each of the events (spikes)
extracted from the signal.
"""
assert threshold is not None, "A threshold must be provided"
if sign is 'above':
cutout = np.where(signal > threshold)[0]
elif sign in '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 = SpikeTrain(events_base, units=signal.times.units,
t_start=signal.t_start, t_stop=signal.t_stop)
return result_st
[docs]def peak_detection(signal, threshold=0.0 * mV, sign='above', format=None):
"""
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 object
'signal' is an analog signal.
threshold : A quantity, e.g. in mV
'threshold' contains a value that must be reached
for an event to be detected.
sign : 'above' or 'below'
'sign' determines whether to count thresholding crossings that
cross above or below the threshold. Default: 'above'.
format : None or 'raw'
Whether to return as SpikeTrain (None) or as a plain array
of times ('raw'). Default: None.
Returns
-------
result_st : neo SpikeTrain object
'result_st' contains the spike times of each of the events
(spikes) extracted from the signal.
"""
assert threshold is not None, "A threshold must be provided"
if sign is 'above':
cutout = np.where(signal > threshold)[0]
peak_func = np.argmax
elif sign in 'below':
cutout = np.where(signal < threshold)[0]
peak_func = np.argmin
else:
raise ValueError("sign must be 'above' or 'below'")
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 = np.concatenate((border_start, border_end))
borders = np.append(0, borders)
borders = np.append(borders, len(cutout)-1)
borders = np.sort(borders)
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 empy 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
if format is None:
result_st = SpikeTrain(events_base, units=signal.times.units,
t_start=signal.t_start, t_stop=signal.t_stop)
elif 'raw':
result_st = events_base
else:
raise ValueError("Format argument must be None or 'raw'")
return result_st
def _homogeneous_process(interval_generator, args, 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`.
"""
def rescale(x):
return (x / mean_rate.units).rescale(t_stop.units)
n = int(((t_stop - t_start) * mean_rate).simplified)
number = np.ceil(n + 3 * np.sqrt(n))
if number < 100:
number = min(5 + np.ceil(2 * n), 100)
assert number > 4 # if positive, number cannot be less than 5
isi = rescale(interval_generator(*args, size=int(number)))
spikes = np.cumsum(isi)
spikes += t_start
i = spikes.searchsorted(t_stop)
if i == len(spikes):
# ISI buffer overrun
extra_spikes = []
t_last = spikes[-1] + rescale(interval_generator(*args, size=1))[0]
while t_last < t_stop:
extra_spikes.append(t_last)
t_last = t_last + rescale(interval_generator(*args, size=1))[0]
# np.concatenate does not conserve units
spikes = Quantity(
np.concatenate(
(spikes, extra_spikes)).magnitude, units=spikes.units)
else:
spikes = spikes[:i]
if as_array:
spikes = spikes.magnitude
else:
spikes = SpikeTrain(
spikes, t_start=t_start, t_stop=t_stop, units=spikes.units)
return spikes
[docs]def homogeneous_poisson_process(rate, t_start=0.0 * ms, t_stop=1000.0 * ms,
as_array=False):
"""
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*Hz.
Parameters
----------
rate : Quantity scalar with dimension 1/time
The rate of the discharge.
t_start : Quantity scalar with dimension time
The beginning of the spike train.
t_stop : Quantity scalar with dimension time
The end of the spike train.
as_array : bool
If True, a NumPy array of sorted spikes is returned,
rather than a SpikeTrain object.
Raises
------
ValueError : If `t_start` and `t_stop` are not of type `pq.Quantity`.
Examples
--------
>>> from quantities import Hz, ms
>>> spikes = homogeneous_poisson_process(50*Hz, 0*ms, 1000*ms)
>>> spikes = homogeneous_poisson_process(
20*Hz, 5000*ms, 10000*ms, as_array=True)
"""
if not isinstance(t_start, Quantity) or not isinstance(t_stop, Quantity):
raise ValueError("t_start and t_stop must be of type pq.Quantity")
rate = rate.rescale(1 / t_start.units)
mean_interval = 1 / rate.magnitude
return _homogeneous_process(
np.random.exponential, (mean_interval,), rate, t_start, t_stop,
as_array)
[docs]def inhomogeneous_poisson_process(rate, as_array=False):
"""
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
If True, a NumPy array of sorted spikes is returned,
rather than a SpikeTrain object.
Raises
------
ValueError : If `rate` contains any negative value.
"""
# Check rate contains only positive values
if any(rate < 0) or not rate.size:
raise ValueError(
'rate must be a positive non empty signal, representing the'
'rate at time t')
else:
#Generate n hidden Poisson SpikeTrains with rate equal to the peak rate
max_rate = np.max(rate)
homogeneous_poiss = homogeneous_poisson_process(
rate=max_rate, 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.magnitude *
homogeneous_poiss.units)
# Accept each spike at time t with probability rate(t)/max_rate
u = np.random.uniform(size=len(homogeneous_poiss)) * max_rate
spikes = homogeneous_poiss[u < rate_interpolated.flatten()]
if as_array:
spikes = spikes.magnitude
return spikes
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
----------
times : Quantity vector(time)
The time points for which the interpolation is computed
signal : neo.core.AnalogSignal
The analog signal containing the discretization of the function to
interpolate
Returns
------
out: Quantity array representing the values of the interpolated signal at the
times given by times
Notes
-----
If `signal` has sampling period `dt=signal.sampling_period`, its values
are defined at `t=signal.times`, such that `t[i] = signal.t_start + i * dt`
The last of such times is lower than
signal.t_stop`:t[-1] = signal.t_stop - dt`.
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]`.
'''
dt = 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 = np.floor(((times - t_start) / dt).rescale(
dimensionless).magnitude).astype('i')
# Compute the slope m of the signal at each time in times
y1 = signal_extended[time_ids]
y2 = signal_extended[time_ids + 1]
m = (y2 - y1) / dt
# Interpolate the signal at each time in times by linear interpolation
out = (y1 + m * (times - times_extended[time_ids])) * signal.units
return out.rescale(signal.units)
[docs]def homogeneous_gamma_process(a, b, t_start=0.0 * ms, t_stop=1000.0 * 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*Hz.
Parameters
----------
a : int or float
The shape parameter of the gamma distribution.
b : Quantity scalar with dimension 1/time
The rate parameter of the gamma distribution.
t_start : Quantity scalar with dimension time
The beginning of the spike train.
t_stop : Quantity scalar with dimension time
The end of the spike train.
as_array : bool
If True, a NumPy array of sorted spikes is returned,
rather than a SpikeTrain object.
Raises
------
ValueError : If `t_start` and `t_stop` are not of type `pq.Quantity`.
Examples
--------
>>> from quantities import Hz, ms
>>> spikes = homogeneous_gamma_process(2.0, 50*Hz, 0*ms, 1000*ms)
>>> spikes = homogeneous_gamma_process(
5.0, 20*Hz, 5000*ms, 10000*ms, as_array=True)
"""
if not isinstance(t_start, Quantity) or not isinstance(t_stop, Quantity):
raise ValueError("t_start and t_stop must be of type pq.Quantity")
b = b.rescale(1 / t_start.units).simplified
rate = b / a
k, theta = a, (1 / b.magnitude)
return _homogeneous_process(np.random.gamma, (k, theta), rate, t_start, t_stop, as_array)
def _n_poisson(rate, t_stop, t_start=0.0 * ms, n=1):
"""
Generates one or more independent Poisson spike trains.
Parameters
----------
rate : Quantity or Quantity array
Expected firing rate (frequency) of each output SpikeTrain.
Can be one of:
* a single Quantity value: expected firing rate of each output
SpikeTrain
* a Quantity array: rate[i] is the expected firing rate of the i-th
output SpikeTrain
t_stop : Quantity
Single common stop time of each output SpikeTrain. Must be > t_start.
t_start : Quantity (optional)
Single common start time of each output SpikeTrain. Must be < t_stop.
Default: 0 s.
n: int (optional)
If rate is a single 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 of return error
try:
for r in rate.reshape(-1, 1):
r.rescale('Hz')
except AttributeError:
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 (=%s) must be < t_stop (=%s)' % (t_start, t_stop))
# Set number n of output spike trains (specified or set to len(rate))
if not (type(n) == int and n > 0):
raise ValueError('n (=%s) must be a positive integer' % str(n))
rate_dl = rate.simplified.magnitude.flatten()
# Check rate input parameter
if len(rate_dl) == 1:
if rate_dl < 0:
raise ValueError('rate (=%s) must be non-negative.' % rate)
rates = np.array([rate_dl] * n)
else:
rates = rate_dl.flatten()
if any(rates < 0):
raise ValueError('rate must have non-negative elements.')
sts = []
for r in rates:
sts.append(homogeneous_poisson_process(r * Hz, t_start, t_stop))
return sts
[docs]def single_interaction_process(
rate, rate_c, t_stop, n=2, jitter=0 * ms, coincidences='deterministic',
t_start=0 * ms, min_delay=0 * ms, return_coinc=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: quantities.Quantity
Total time of the simulated processes. The events are drawn between
0 and `t_stop`.
rate: quantities.Quantity
Overall mean rate of the time series to be generated (coincidence
rate `rate_c` 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 `rate_c`.
* 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 `rate_c`.
rate_c: quantities.Quantity
Coincidence rate (rate of coincidences for the n-dimensional SIP).
The SIP spike trains will have coincident events with rate `rate_c`
plus independent 'background' events with rate `rate-rate_c`.
n: int, optional
If `rate` is a single 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
jitter: quantities.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`.
coincidences: string, optional
Whether the total number of injected coincidences must be determin-
istic (i.e. rate_c is the actual rate with which coincidences are
generated) or stochastic (i.e. rate_c is the mean rate of coincid-
ences):
* 'deterministic': deterministic rate
* 'stochastic': stochastic rate
Default: 'deterministic'
t_start: quantities.Quantity, optional
Starting time of the series. If specified, it must be lower than
t_stop
Default: 0 * ms
min_delay: quantities.Quantity, optional
Minimum delay between consecutive coincidence times.
Default: 0 * ms
return_coinc: 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
EXAMPLE:
>>> import quantities as qt
>>> import jelephant.core.stocmod as sm
>>> sip, coinc = sm.sip_poisson(n=10, n=0, t_stop=1*qt.sec, \
rate=20*qt.Hz, rate_c=4, return_coinc = True)
*************************************************************************
"""
# Check if n is a positive integer
if not (isinstance(n, int) and n > 0):
raise ValueError('n (=%s) must be a positive integer' % str(n))
# 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 * Hz:
raise ValueError(
'rate (=%s) must be non-negative.' % str(rate))
rates_b = np.array(
[rate.magnitude for _ in range(n)]) * rate.units
else:
rates_b = np.array(rate).flatten() * rate.units
if not all(rates_b >= 0. * Hz):
raise ValueError('*rate* must have non-negative elements')
# Check: rate>=rate_c
if np.any(rates_b < rate_c):
raise ValueError('all elements of *rate* must be >= *rate_c*')
# Check min_delay < 1./rate_c
if not (rate_c == 0 * Hz or min_delay < 1. / rate_c):
raise ValueError(
"'*min_delay* (%s) must be lower than 1/*rate_c* (%s)." %
(str(min_delay), str((1. / rate_c).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 - rate_c, t_stop=t_stop, t_start=t_start)
# Convert the trains from neo SpikeTrain objects to simpler Quantity
# objects
embedded_poisson_trains = [
emb.view(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 quantity from the Quantity class
# in the quantities module
if coincidences == 'deterministic':
Nr_coinc = int(((t_stop - t_start) * rate_c).rescale(dimensionless))
while True:
coinc_times = t_start + \
np.sort(np.random.random(Nr_coinc)) * (t_stop - t_start)
if len(coinc_times) < 2 or min(np.diff(coinc_times)) >= min_delay:
break
elif coincidences == 'stochastic':
while True:
coinc_times = homogeneous_poisson_process(
rate=rate_c, t_stop=t_stop, t_start=t_start)
if len(coinc_times) < 2 or min(np.diff(coinc_times)) >= min_delay:
break
# Convert coinc_times from a neo SpikeTrain object to a Quantity object
# pq.Quantity(coinc_times.base)*coinc_times.units
coinc_times = coinc_times.view(Quantity)
# Set the coincidence times to T-jitter if larger. This ensures that
# the last jittered spike time is <T
for i in range(len(coinc_times)):
if coinc_times[i] > t_stop - jitter:
coinc_times[i] = t_stop - jitter
# 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 Quantity objects to
# neo.SpikeTrain objects
sip_process = [
SpikeTrain(t, t_start=t_start, t_stop=t_stop).rescale(t_stop.units)
for t in sip_process]
coinc_times = [
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_coinc:
output = sip_process
else:
output = sip_process, coinc_times
return output
def _pool_two_spiketrains(a, b, extremes='inner'):
"""
Pool the spikes of two spike trains a and b into a unique spike train.
Parameters
----------
a, b : neo.SpikeTrains
Spike trains to be pooled
extremes: str, 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'
Output
------
neo.SpikeTrain containing all spikes in a and b falling in the
specified extremes
"""
unit = a.units
times_a_dimless = list(a.view(Quantity).magnitude)
times_b_dimless = list(b.rescale(unit).view(Quantity).magnitude)
times = (times_a_dimless + times_b_dimless) * unit
if extremes == 'outer':
t_start = min(a.t_start, b.t_start)
t_stop = max(a.t_stop, b.t_stop)
elif extremes == 'inner':
t_start = max(a.t_start, b.t_start)
t_stop = min(a.t_stop, b.t_stop)
times = times[times > t_start]
times = times[times < t_stop]
else:
raise ValueError(
'extremes (%s) can only be "inner" or "outer"' % extremes)
pooled_train = SpikeTrain(
times=sorted(times.magnitude), units=unit, t_start=t_start,
t_stop=t_stop)
return pooled_train
def _pool_spiketrains(trains, extremes='inner'):
"""
Pool spikes from any number of spike trains into a unique spike train.
Parameters
----------
trains: list
list of spike trains 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'
Output
------
neo.SpikeTrain containing all spikes in trains falling in the
specified extremes
"""
merge_trains = trains[0]
for t in trains[1:]:
merge_trains = _pool_two_spiketrains(
merge_trains, t, 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 = SpikeTrain(
merge_trains, t_stop=t_stop, t_start=t_start, units=trains[0].units)
return merge_trains
def _sample_int_from_pdf(a, n):
"""
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
-----
a: numpy.array
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: int
Number of samples generated with the function
Output
-------
array of n samples taking values between 0 and n=len(a)-1.
"""
A = np.cumsum(a) # cumulative distribution of a
u = np.random.uniform(0, 1, size=n)
U = np.array([u for i in a]).T # copy u (as column vector) len(a) times
return (A < U).sum(axis=1)
def _mother_proc_cpp_stat(A, t_stop, rate, t_start=0 * ms):
"""
Generate the hidden ("mother") Poisson process for a Compound Poisson
Process (CPP).
Parameters
----------
A : numpy.array
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 : quantities.Quantity
The stopping time of the mother process
rate : quantities.Quantity
Homogeneous rate of the n spike trains that will be genereted by the
CPP function
t_start : quantities.Quantity, optional
The starting time of the mother process
Default: 0 ms
Output
------
Poisson spike train representing the mother process generating the CPP
"""
N = len(A) - 1
exp_A = np.dot(A, range(N + 1)) # expected value of a
exp_mother = (N * rate) / float(exp_A) # rate of the mother process
return homogeneous_poisson_process(
rate=exp_mother, t_stop=t_stop, t_start=t_start)
def _cpp_hom_stat(A, t_stop, rate, t_start=0 * ms):
"""
Generate a Compound Poisson Process (CPP) with amplitude distribution
A and heterogeneous firing rates r=r[0], r[1], ..., r[-1].
Parameters
----------
A : numpy.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 : quantities.Quantity
The end time of the output spike trains
rate : quantities.Quantity
Average rate of each spike train generated
t_start : quantities.Quantity, optional
The start time of the output spike trains
Default: 0 ms
Output
------
List of n neo.SpikeTrains, 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 = len(A) - 1 # Number of trains in output
try: # Faster but more memory-consuming approach
M = len(mother) # number of spikes in the mother process
spike_matrix = np.zeros((N, M), dtype=bool)
# for each spike, take its label l
for spike_id, l in enumerate(labels):
# choose l random trains
train_ids = random.sample(range(N), l)
# 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
times = [[] for i in range(N)]
for train_id, row in enumerate(spike_matrix):
times[train_id] = mother[row].view(Quantity)
except MemoryError: # Slower (~2x) but less memory-consuming approach
print('memory case')
times = [[] for i in range(N)]
for t, l in zip(mother, labels):
train_ids = random.sample(range(N), l)
for train_id in train_ids:
times[train_id].append(t)
trains = [SpikeTrain(
times=t, t_start=t_start, t_stop=t_stop) for t in times]
return trains
def _cpp_het_stat(A, t_stop, rate, t_start=0. * ms):
"""
Generate a Compound Poisson Process (CPP) with amplitude distribution
A and heterogeneous firing rates r=r[0], r[1], ..., r[-1].
Parameters
----------
A : array
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 : Quantity (time)
The end time of the output spike trains
rate : Quantity (1/time)
Average rate of each spike train generated
t_start : quantities.Quantity, optional
The start time of the output spike trains
Default: 0 ms
Output
------
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 = len(rate) # number of output spike trains
A_exp = np.dot(A, range(N + 1)) # expectation of A
r_sum = np.sum(rate) # sum of all output firing rates
r_min = np.min(rate) # minimum of the firing rates
r1 = r_sum - N * r_min # rate of the uncorrelated CPP
r2 = r_sum / float(A_exp) - r1 # rate of the correlated CPP
r_mother = r1 + r2 # rate of the hidden mother process
# Check the analytical constraint for the amplitude distribution
if A[1] < (r1 / r_mother).rescale(dimensionless).magnitude:
raise ValueError('A[1] too small / A[i], i>1 too high')
# Compute the amplitude distrib of the correlated CPP, and generate it
a = [(r_mother * i) / float(r2) for i in A]
a[1] = a[1] - r1 / float(r2)
CPP = _cpp_hom_stat(a, t_stop, r_min, t_start)
# Generate the independent heterogeneous Poisson processes
POISS = [
homogeneous_poisson_process(i - r_min, t_start, t_stop) for i in rate]
# Pool the correlated CPP and the corresponding Poisson processes
out = [_pool_two_spiketrains(CPP[i], POISS[i]) for i in range(N)]
return out
[docs]def compound_poisson_process(rate, A, t_stop, shift=None, t_start=0 * ms):
"""
Generate a Compound Poisson Process (CPP; see [1]) with a given amplitude
distribution A and stationary marginal rates r.
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 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 : quantities.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
A : array
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 : quantities.Quantity
The end time of the output spike trains.
shift : None or quantities.Quantity, optional
If None, the injected synchrony is exact. If shift is a Quantity, all
the spike trains are shifted independently by a random amount in
the interval [-shift, +shift].
Default: None
t_start : quantities.Quantity, optional
The t_start time of the output spike trains.
Default: 0 s
Returns
-------
List of neo.SpikeTrains
SpikeTrains with specified firing rates forming the CPP with amplitude
distribution A.
References
----------
[1] Staude, Rotter, Gruen (2010) J Comput Neurosci 29:327-350.
"""
# Check A is a probability distribution (it sums to 1 and is positive)
if abs(sum(A) - 1) > np.finfo('float').eps:
raise ValueError(
'A must be a probability vector, sum(A)= %f !=1' % (sum(A)))
if any([a < 0 for a in A]):
raise ValueError(
'A must be a probability vector, all the elements of must be >0')
# Check that the rate is not an empty Quantity
if rate.ndim == 1 and len(rate.magnitude) == 0:
raise ValueError('Rate is an empty Quantity array')
# Return empty spike trains for specific parameters
elif A[0] == 1 or np.sum(np.abs(rate.magnitude)) == 0:
return [
SpikeTrain([] * t_stop.units, t_stop=t_stop,
t_start=t_start) for i in range(len(A) - 1)]
else:
# Homogeneous rates
if rate.ndim == 0:
cpp = _cpp_hom_stat(A=A, t_stop=t_stop, rate=rate, t_start=t_start)
# Heterogeneous rates
else:
cpp = _cpp_het_stat(A=A, t_stop=t_stop, rate=rate, t_start=t_start)
if shift is None:
return cpp
# Dither the output spiketrains
else:
cpp = [
dither_spike_train(cp, shift=shift, edges=True)[0]
for cp in cpp]
return cpp
# Alias for the compound poisson process
cpp = compound_poisson_process
