Spike Train Analysis -- 125 Functions (23 Modules)¶

SC-NeuroCore includes a complete spike train analysis toolkit covering every metric from Elephant, PySpike, SpikeInterface, and NeuroTools. 125 functions across 23 modules (22 spike_stats + 1 explainability). Pure NumPy, zero external dependencies.

Quick Start¶

from sc_neurocore.analysis import (
    firing_rate, cv_isi, fano_factor, cross_correlation,
    power_spectrum, burst_detection, victor_purpura_distance,
    phase_locking_value, spade_detect, gpfa,
)
import numpy as np

# Generate a Poisson spike train
from sc_neurocore.analysis import homogeneous_poisson
train = homogeneous_poisson(rate_hz=50.0, duration_s=10.0)

# Basic statistics
print(f"Rate: {firing_rate(train):.1f} Hz")
print(f"CV(ISI): {cv_isi(train):.2f}")
print(f"Fano factor: {fano_factor(train):.2f}")

Module Reference¶

Basic (`spike_stats.basic`)¶

Spike extraction, ISI computation, rate, count, binning.

Function	Description
`spike_times(train, dt)`	Extract spike times (seconds) from a binary 0/1 array
`isi(train, dt)`	Inter-spike intervals (seconds)
`firing_rate(train, dt)`	Mean firing rate (Hz)
`spike_count(train)`	Total spike count
`bin_spike_train(train, bin_size)`	Bin a binary spike train into spike counts per bin

Variability (`spike_stats.variability`)¶

Spike train regularity, complexity, and fractal measures.

Function	Description
`cv_isi(train, dt)`	Coefficient of variation of ISI (CV=1 for Poisson, <1 regular)
`cv2(train, dt)`	Local CV2, less sensitive to rate changes (Holt et al. 1996)
`local_variation(train, dt)`	Local variation LV (Shinomoto et al. 2003)
`lvr(train, dt, refractoriness_ms)`	Revised local variation LvR corrected for refractoriness (Shinomoto et al. 2009)
`fano_factor(train, window_ms, dt)`	Fano factor: variance/mean of windowed spike counts
`isi_entropy(train, dt, bins)`	Shannon entropy of ISI distribution (bits)
`lempel_ziv_complexity(train)`	Lempel-Ziv 1976 complexity, normalized by N/log2(N)
`approximate_entropy(train, m, r_factor)`	Approximate entropy ApEn (Pincus 1991)
`sample_entropy(train, m, r_factor)`	Sample entropy SampEn, lower bias (Richman & Moorman 2000)
`permutation_entropy(train, order, delay)`	Bandt-Pompe permutation entropy, normalized to [0,1]
`hurst_exponent(train, min_window)`	Hurst exponent via detrended fluctuation analysis (Peng et al. 1994)
`allan_factor(train, dt, n_scales)`	Allan factor vs window size for fractal clustering detection
`rescaled_range(train, min_window)`	Hurst exponent via R/S analysis (Hurst 1951)
`complexity_pdf(train, dt, bins)`	ISI probability density function (Abeles 1982)
`optimal_bin_width(train, dt)`	Shimazaki-Shinomoto 2007 optimal histogram bin width
`optimal_kernel_bandwidth(train, dt)`	Silverman's rule-of-thumb bandwidth for ISI KDE

Rate Estimation (`spike_stats.rate`)¶

Instantaneous and population firing rate estimation.

Function	Description
`instantaneous_rate(train, dt, kernel, sigma_ms)`	Kernel-smoothed instantaneous rate (gaussian/exponential/rectangular)
`population_rate(trains, dt, sigma_ms)`	Population-level instantaneous rate across multiple neurons
`psth(trials, bin_ms, dt)`	Peri-stimulus time histogram across trials

Distance Metrics (`spike_stats.distance`)¶

Spike train distance and similarity measures.

Function	Description
`van_rossum_distance(a, b, dt, tau_ms)`	Van Rossum 2001 exponential-kernel distance
`victor_purpura_distance(times_a, times_b, cost_per_s)`	Victor-Purpura 1996 edit distance
`isi_distance(a, b, dt)`	ISI-distance ratio comparison (Kreuz et al. 2007)
`spike_distance(times_a, times_b, t_start, t_end)`	SPIKE-distance (Kreuz et al. 2013)
`spike_sync(times_a, times_b, t_start, t_end)`	SPIKE-synchronization, normalized to [0,1] (Kreuz et al. 2015)
`spike_sync_profile(times_a, times_b, n_bins, ...)`	Binned SPIKE-synchronization profile
`spike_profile(times_a, times_b, n_bins, ...)`	Binned SPIKE-distance profile
`isi_profile(a, b, dt, n_bins)`	Binned ISI-distance profile
`adaptive_spike_distance(times_a, times_b, ...)`	Adaptive SPIKE-distance with cost interpolation (Kreuz et al. 2013)
`schreiber_similarity(a, b, dt, sigma_ms)`	Smoothed Pearson correlation similarity (Schreiber et al. 2003)
`hunter_milton_similarity(times_a, times_b, dt_max)`	Nearest-neighbour coincidence fraction (Hunter-Milton 2003)
`earth_movers_distance(times_a, times_b, ...)`	EMD between spike time distributions (Rubner et al. 1998)
`multi_neuron_victor_purpura(times_list, cost)`	All-pairs Victor-Purpura distance matrix
`generalized_victor_purpura(times_a, times_b, cost_func)`	Victor-Purpura with arbitrary cost function
`spike_distance_matrix(times_list, metric, ...)`	All-pairs distance matrix (spike_distance/spike_sync/victor_purpura)

Correlation (`spike_stats.correlation`)¶

Cross-correlation, synchrony, and covariance measures.

Function	Description
`cross_correlation(a, b, max_lag_ms, dt)`	Cross-correlogram between two binary trains
`pairwise_correlation(trains, dt)`	Pairwise Pearson correlation matrix across neurons
`event_synchronization(a, b, dt, tau_ms)`	Quian Quiroga et al. 2002 event synchronization
`spike_train_coherence(a, b, dt)`	Magnitude-squared coherence between spike trains
`spike_time_tiling_coefficient(a, b, dt, delta_ms)`	STTC, corrects for rate bias (Cutts & Eglen 2014)
`covariance_matrix(trains, bin_size)`	Spike count covariance matrix (de la Rocha et al. 2007)
`autocorrelation_time(train, dt, max_lag_ms)`	Autocorrelation time until first zero crossing
`noise_correlation(trains, bin_size)`	Trial-to-trial variability correlation (Averbeck & Lee 2006)
`signal_correlation(trains, bin_size)`	Tuning similarity via Pearson correlation of mean responses
`spike_count_covariance(trains, window)`	Windowed spike count covariance (Kohn & Smith 2005)
`joint_psth(a, b, bin_size)`	Joint PSTH matrix (Aertsen et al. 1989)
`coincidence_index(a, b, dt, delta_ms)`	Rate-corrected coincidence index kappa (Joris et al. 2006)

Spectral (`spike_stats.spectral`)¶

Spectral analysis of spike trains.

Function	Description
`power_spectrum(train, dt)`	Power spectral density of a binary spike train

Temporal Patterns (`spike_stats.temporal`)¶

Burst detection, latency, onset, and change point analysis.

Function	Description
`burst_detection(train, dt, max_isi_ms, min_spikes)`	Detect bursts as consecutive short-ISI spikes
`first_spike_latency(train, dt)`	Time to first spike (seconds)
`response_onset(train, baseline_steps, dt, threshold_sigma)`	Detect response onset exceeding baseline + threshold
`change_point_detection(train, bin_size, threshold)`	CUSUM-based firing rate change points (Page 1954)

Stimulus (`spike_stats.stimulus`)¶

Spike-triggered analysis and receptive field estimation.

Function	Description
`spike_triggered_average(stimulus, train, window_steps)`	STA: average stimulus preceding each spike
`spike_triggered_covariance(stimulus, train, window_steps)`	STC: covariance of pre-spike stimulus (Schwartz et al. 2006)
`spatial_information(train, positions, n_bins, dt)`	Spatial information in bits/spike (Skaggs et al. 1993)
`place_field_detection(train, positions, ...)`	Detect place fields (O'Keefe & Dostrovsky 1971)
`tuning_curve(train, stimulus_values, n_bins, dt)`	Firing rate vs stimulus value (Dayan & Abbott 2001)

LFP Coupling (`spike_stats.lfp`)¶

Spike-LFP phase locking, coherence, and phase histograms.

Function	Description
`phase_locking_value(train, lfp)`	PLV via Hilbert phase at spike times
`spike_field_coherence(train, lfp, dt)`	Spike-field coherence SFC in frequency domain
`spike_phase_histogram(train, lfp, n_bins)`	Histogram of LFP phase at spike times

Surrogates (`spike_stats.surrogates`)¶

Surrogate spike train generation for significance testing.

Function	Description
`surrogate_isi_shuffle(train, seed)`	ISI-shuffled surrogate preserving rate + ISI distribution
`surrogate_dither(train, dither_ms, dt, seed)`	Spike jittering surrogate (+/- dither window)
`surrogate_trial_shuffle(trains, seed)`	Trial-order shuffling destroying trial-to-trial correlation
`homogeneous_poisson(rate_hz, duration_s, dt, seed)`	Generate homogeneous Poisson spike train
`inhomogeneous_poisson(rate_func, duration_s, dt, seed)`	Time-varying Poisson via thinning (Lewis & Shedler 1979)
`gamma_process(rate_hz, shape, duration_s, dt, seed)`	Gamma-renewal process (shape=1 Poisson, >1 regular)
`compound_poisson_process(rate_hz, burst_mean, ...)`	Compound Poisson with burst events (Snyder & Miller 1991)
`surrogate_joint_isi(train, seed)`	Joint-ISI surrogate preserving serial correlations (Louis et al. 2010)
`surrogate_bin_shuffling(train, bin_size, seed)`	Within-bin spike shuffling (Hatsopoulos et al. 2003)
`surrogate_spike_train_shifting(train, max_shift, seed)`	Circular shifting surrogate

Information Theory (`spike_stats.information`)¶

Information-theoretic measures for spike trains.

Function	Description
`mutual_information(a, b, bin_size)`	Mutual information between binned spike trains (bits)
`transfer_entropy(source, target, bin_size, lag)`	Transfer entropy from source to target (bits)
`spike_train_entropy(train, bin_size, word_length)`	Binary word entropy (Strong et al. 1998)
`noise_entropy(train, n_trials, bin_size, word_length)`	Noise entropy via pseudo-trials (de Ruyter van Steveninck et al. 1997)
`stimulus_specific_information(counts, stim_ids)`	SSI in bits (Butts 2003)
`kozachenko_leonenko_mi(x, y, k)`	k-NN mutual information estimator in nats (Kraskov et al. 2004)
`time_rescaling_ks_test(times, rate_func, ...)`	KS goodness-of-fit for point processes (Brown et al. 2002)

Causality (`spike_stats.causality`)¶

Granger causality and directed connectivity.

Function	Description
`pairwise_granger_causality(source, target, ...)`	Pairwise Granger causality log-likelihood ratio (Granger 1969)
`conditional_granger_causality(source, target, cond, ...)`	Conditional GC controlling for a third signal (Geweke 1984)
`spectral_granger_causality(trains, bin_size, order, ...)`	Frequency-domain GC via VAR model (Geweke 1982)
`partial_directed_coherence(trains, ...)`	PDC: normalized directed connectivity (Baccala & Sameshima 2001)
`directed_transfer_function(trains, ...)`	DTF: normalized transfer function (Kaminski & Blinowska 1991)

Dimensionality Reduction (`spike_stats.dimensionality`)¶

Low-dimensional projections of population activity.

Function	Description
`spike_train_pca(trains, n_components, bin_size)`	PCA on binned spike count matrix
`demixed_pca(trains_by_condition, n_components, ...)`	Demixed PCA separating condition variance (Kobak et al. 2016)
`factor_analysis(trains, n_factors, bin_size, n_iter)`	Factor analysis via EM (Rubin & Thayer 1982)

Decoding (`spike_stats.decoding`)¶

Neural population decoding algorithms.

Function	Description
`population_vector_decode(trains, directions, window)`	Georgopoulos population vector decoding
`bayesian_decode(counts, tuning_rates, prior)`	Bayesian MAP decoder (Dayan & Abbott 2001)
`maximum_likelihood_decode(counts, tuning_rates)`	Maximum likelihood Poisson decoder
`linear_discriminant_decode(data, labels, test)`	Fisher LDA decoder (Fisher 1936)
`naive_bayes_decode(data, labels, test)`	Gaussian naive Bayes decoder

Network (`spike_stats.network`)¶

Network-level analysis: connectivity, assemblies, synfire chains.

Function	Description
`functional_connectivity(trains, max_lag_ms, dt)`	Infer connectivity from peak cross-correlation
`unitary_events(trains, bin_size, alpha)`	Detect significant synchronous bins (Gruen et al. 2002)
`cell_assembly_detection(trains, bin_size, threshold)`	PCA-based cell assembly detection (Lopes-dos-Santos et al. 2013)
`synfire_chain_detection(trains, dt, max_delay_ms, ...)`	Cross-correlation peak ordering (Abeles 1991)

Point Process (`spike_stats.point_process`)¶

Point process models and hazard functions.

Function	Description
`conditional_intensity(train, dt, window_ms)`	Moving-window MLE conditional intensity (Brown et al. 2004)
`isi_hazard_function(train, dt, bins)`	ISI hazard function h(t) = f(t)/S(t) (Tuckwell 1988)
`isi_survivor_function(train, dt, bins)`	ISI survivor function S(t) = P(ISI > t)
`renewal_density(train, dt, bins)`	Renewal density normalized by mean rate (Cox 1962)

Sorting Quality (`spike_stats.sorting_quality`)¶

Spike sorting quality metrics.

Function	Description
`isolation_distance(cluster, noise)`	Mahalanobis isolation distance (Harris et al. 2001)
`l_ratio(cluster, noise)`	L-ratio cluster quality (Schmitzer-Torbert et al. 2005)
`silhouette_score(features, labels)`	Mean silhouette score (Rousseeuw 1987)
`d_prime(cluster_a, cluster_b)`	Sensitivity index between two clusters (Green & Swets 1966)
`isi_violation_rate(train, dt, refractory_ms)`	Fraction of ISIs below refractory period (Hill et al. 2011)
`presence_ratio(train, n_bins)`	Fraction of time bins with at least one spike (IBL 2019)
`amplitude_cutoff(amplitudes, bins)`	Estimated missing spike fraction (Hill et al. 2011)
`snr(waveforms)`	Signal-to-noise ratio of spike waveforms (Suner et al. 2005)
`nn_hit_rate(cluster, noise, k)`	k-NN cluster purity (Chung et al. 2017)
`drift_metric(waveforms, timestamps, n_bins)`	Waveform amplitude drift over time (IBL 2019)

Waveform (`spike_stats.waveform`)¶

Spike waveform shape analysis.

Function	Description
`waveform_width(waveform, dt)`	Trough-to-peak width in seconds (Bartho et al. 2004)
`waveform_amplitude(waveform)`	Peak-to-trough amplitude
`waveform_repolarization_slope(waveform, dt)`	Max dV/dt after trough (Bean 2007)
`waveform_recovery_slope(waveform, dt)`	dV/dt during return to baseline (Bean 2007)
`waveform_halfwidth(waveform, dt)`	Duration at half-minimum amplitude (Bartho et al. 2004)
`waveform_pt_ratio(waveform)`	Peak-to-trough ratio (Bartho et al. 2004)

Statistics (`spike_stats.statistics`)¶

Significance testing.

Function	Description
`significance_bootstrap(func, a, b, n_surrogates, seed)`	Bootstrap permutation test for pairwise statistics

Patterns (`spike_stats.patterns`)¶

Spike directionality, ordering, and higher-order patterns.

Function	Description
`spike_directionality(times_a, times_b, ...)`	Asymmetry in [-1,1]: positive means A leads B (Kreuz et al. 2015)
`spike_train_order(times_list, ...)`	Pairwise directionality matrix (Kreuz et al. 2017)
`cubic_higher_order(train, dt, max_lag)`	Third-order cumulant C3(tau1, tau2) (Nikias & Petropulu 1993)

SPADE (`spike_stats.spade`)¶

Spike Pattern Detection and Evaluation (Torre et al. 2013).

Function	Description
`spade_detect(trains, bin_ms, dt, min_support, ...)`	Detect significant spatiotemporal spike patterns via frequent itemset mining + surrogate testing

GPFA (`spike_stats.gpfa`)¶

Gaussian Process Factor Analysis (Yu et al. 2009).

Function	Description
`gpfa(trains, n_latents, bin_ms, dt, max_iter, ...)`	Extract smooth latent trajectories via EM with SE-GP priors
`gpfa_transform(new_trains, params, bin_ms, dt)`	Project new data using learned GPFA parameters

Explainability¶

`sc_neurocore.analysis.explainability` ¶

`SpikeToConceptMapper` ¶

XAI Module: Maps spike patterns to semantic concepts.

Source code in src/sc_neurocore/analysis/explainability.py

class SpikeToConceptMapper:
    """
    XAI Module: Maps spike patterns to semantic concepts.
    """

    def __init__(self, concept_map: Dict[int, str]):
        self.concept_map = concept_map

    def explain(self, spikes: np.ndarray[Any, Any]) -> str:
        """
        Input: Spike vector (n_neurons,)
        Output: "The agent is thinking about [Concept1, Concept2]"
        """
        active_indices = np.where(spikes > 0)[0]

        concepts = []
        for idx in active_indices:
            if idx in self.concept_map:
                concepts.append(self.concept_map[idx])
            else:
                concepts.append(f"Unknown({idx})")

        if not concepts:
            return "The agent is idle."

        return f"The agent is active on: {', '.join(concepts)}"

`explain(spikes)` ¶

Input: Spike vector (n_neurons,) Output: "The agent is thinking about [Concept1, Concept2]"

Source code in src/sc_neurocore/analysis/explainability.py

def explain(self, spikes: np.ndarray[Any, Any]) -> str:
    """
    Input: Spike vector (n_neurons,)
    Output: "The agent is thinking about [Concept1, Concept2]"
    """
    active_indices = np.where(spikes > 0)[0]

    concepts = []
    for idx in active_indices:
        if idx in self.concept_map:
            concepts.append(self.concept_map[idx])
        else:
            concepts.append(f"Unknown({idx})")

    if not concepts:
        return "The agent is idle."

    return f"The agent is active on: {', '.join(concepts)}"

Integrated Information (Phi*)¶

Approximate Phi (Barrett & Seth 2011) measures how much a system exceeds the sum of its parts — the consciousness-relevant integrated information.

`sc_neurocore.analysis.phi_estimation.phi_star(data, tau=1)` ¶

Geometric integrated information (Barrett & Seth 2011).

Phi* = MI(past; future) - max_partition sum MI(past_k; future_k)

Approximated via Gaussian assumption: MI = 0.5 * log(det(Sigma_X) * det(Sigma_Y) / det(Sigma_XY)).

Parameters¶

data : np.ndarray Shape (n_channels, n_timesteps). Spike counts or continuous signals. tau : int Time lag for past→future mapping.

Returns¶

float Phi* estimate (bits). Non-negative; 0 = fully reducible.

Source code in src/sc_neurocore/analysis/phi_estimation.py

def phi_star(data: np.ndarray, tau: int = 1) -> float:
    """Geometric integrated information (Barrett & Seth 2011).

    Phi* = MI(past; future) - max_partition sum MI(past_k; future_k)

    Approximated via Gaussian assumption: MI = 0.5 * log(det(Sigma_X) * det(Sigma_Y) / det(Sigma_XY)).

    Parameters
    ----------
    data : np.ndarray
        Shape (n_channels, n_timesteps). Spike counts or continuous signals.
    tau : int
        Time lag for past→future mapping.

    Returns
    -------
    float
        Phi* estimate (bits). Non-negative; 0 = fully reducible.
    """
    n, T = data.shape
    if 2 * tau >= T or n < 2:
        return 0.0

    past = data[:, :-tau]
    future = data[:, tau:]

    # Joint mutual information I(past; future)
    mi_whole = _gaussian_mi(past, future)

    # Minimum information partition: try all bipartitions
    # For tractability, only try contiguous splits (first k vs rest)
    mi_parts_min = np.inf
    for k in range(1, n):
        idx_a = list(range(k))
        idx_b = list(range(k, n))
        mi_a = _gaussian_mi(past[idx_a], future[idx_a])
        mi_b = _gaussian_mi(past[idx_b], future[idx_b])
        mi_parts_min = min(mi_parts_min, mi_a + mi_b)

    phi = max(0.0, mi_whole - mi_parts_min)
    return float(phi)

`sc_neurocore.analysis.phi_estimation.phi_from_spike_trains(spikes, bin_size=10, tau=1)` ¶

Compute Phi* from binary spike trains.

Parameters¶

spikes : np.ndarray Shape (n_neurons, n_timesteps), binary {0, 1}. bin_size : int Number of timesteps per bin for spike count computation. tau : int Time lag in bins.

Returns¶

float Phi* in bits.

Source code in src/sc_neurocore/analysis/phi_estimation.py

def phi_from_spike_trains(spikes: np.ndarray, bin_size: int = 10, tau: int = 1) -> float:
    """Compute Phi* from binary spike trains.

    Parameters
    ----------
    spikes : np.ndarray
        Shape (n_neurons, n_timesteps), binary {0, 1}.
    bin_size : int
        Number of timesteps per bin for spike count computation.
    tau : int
        Time lag in bins.

    Returns
    -------
    float
        Phi* in bits.
    """
    n_neurons, n_steps = spikes.shape
    n_bins = n_steps // bin_size
    if n_bins < 2 * tau + 2:
        return 0.0

    # Bin spike trains into spike counts
    binned = np.zeros((n_neurons, n_bins))
    for b in range(n_bins):
        binned[:, b] = spikes[:, b * bin_size : (b + 1) * bin_size].sum(axis=1)

    return phi_star(binned, tau=tau)

Spike Train Analysis -- 125 Functions (23 Modules)¶

Quick Start¶

Module Reference¶

Basic (spike_stats.basic)¶

Variability (spike_stats.variability)¶

Rate Estimation (spike_stats.rate)¶

Distance Metrics (spike_stats.distance)¶

Correlation (spike_stats.correlation)¶

Spectral (spike_stats.spectral)¶

Temporal Patterns (spike_stats.temporal)¶

Stimulus (spike_stats.stimulus)¶

LFP Coupling (spike_stats.lfp)¶

Surrogates (spike_stats.surrogates)¶

Information Theory (spike_stats.information)¶

Causality (spike_stats.causality)¶

Dimensionality Reduction (spike_stats.dimensionality)¶

Decoding (spike_stats.decoding)¶

Network (spike_stats.network)¶

Point Process (spike_stats.point_process)¶

Sorting Quality (spike_stats.sorting_quality)¶

Waveform (spike_stats.waveform)¶

Statistics (spike_stats.statistics)¶

Patterns (spike_stats.patterns)¶

SPADE (spike_stats.spade)¶

GPFA (spike_stats.gpfa)¶

Explainability¶

sc_neurocore.analysis.explainability ¶

SpikeToConceptMapper ¶

explain(spikes) ¶

Integrated Information (Phi*)¶

sc_neurocore.analysis.phi_estimation.phi_star(data, tau=1) ¶

Parameters¶

Returns¶

sc_neurocore.analysis.phi_estimation.phi_from_spike_trains(spikes, bin_size=10, tau=1) ¶

Parameters¶

Returns¶

Basic (`spike_stats.basic`)¶

Variability (`spike_stats.variability`)¶

Rate Estimation (`spike_stats.rate`)¶

Distance Metrics (`spike_stats.distance`)¶

Correlation (`spike_stats.correlation`)¶

Spectral (`spike_stats.spectral`)¶

Temporal Patterns (`spike_stats.temporal`)¶

Stimulus (`spike_stats.stimulus`)¶

LFP Coupling (`spike_stats.lfp`)¶

Surrogates (`spike_stats.surrogates`)¶

Information Theory (`spike_stats.information`)¶

Causality (`spike_stats.causality`)¶

Dimensionality Reduction (`spike_stats.dimensionality`)¶

Decoding (`spike_stats.decoding`)¶

Network (`spike_stats.network`)¶

Point Process (`spike_stats.point_process`)¶

Sorting Quality (`spike_stats.sorting_quality`)¶

Waveform (`spike_stats.waveform`)¶

Statistics (`spike_stats.statistics`)¶

Patterns (`spike_stats.patterns`)¶

SPADE (`spike_stats.spade`)¶

GPFA (`spike_stats.gpfa`)¶

`sc_neurocore.analysis.explainability` ¶

`SpikeToConceptMapper` ¶

`explain(spikes)` ¶

`sc_neurocore.analysis.phi_estimation.phi_star(data, tau=1)` ¶

`sc_neurocore.analysis.phi_estimation.phi_from_spike_trains(spikes, bin_size=10, tau=1)` ¶