elephant.spike_train_correlation.covariance

elephant.spike_train_correlation.covariance(binned_spiketrain, binary=False, fast=True)

Calculate the NxN matrix of pairwise covariances between all combinations of N binned spike trains.

For each pair of spike trains \((i,j)\), the covariance \(C[i,j]\) is obtained by binning \(i\) and \(j\) at the desired bin size. Let \(b_i\) and \(b_j\) denote the binned spike trains and \(\mu_i\) and \(\mu_j\) their respective averages. Then

\[C[i,j] = <b_i-\mu_i, b_j-\mu_j> / (L-1)\]

where <., .> is the scalar product of two vectors, and \(L\) is the number of bins.

For an input of N spike trains, an N x N matrix is returned containing the covariances for each combination of input spike trains.

If binary is True, the binned spike trains are clipped to 0 or 1 before computing the covariance, so that the binned vectors \(b_i\) and \(b_j\) are binary.

Parameters:

binned_spiketrain(N, ) elephant.conversion.BinnedSpikeTrain

A binned spike train containing the spike trains to be evaluated.

binarybool, optional

If True, the spikes of a particular spike train falling in the same bin are counted as 1, resulting in binary binned vectors \(b_i\). If False, the binned vectors \(b_i\) contain the spike counts per bin. Default: False

fastbool, optional

If fast=True and the sparsity of binned_spiketrain is > 0.1, use np.cov(). Otherwise, use memory efficient implementation. See Notes [2]. Default: True

Returns:

C(N, N) np.ndarray

The square matrix of covariances. The element \(C[i,j]=C[j,i]\) is the covariance between binned_spiketrain[i] and binned_spiketrain[j].

Raises:

MemoryError

When using fast=True and binned_spiketrain shape is large.

Warns:

UserWarning

If at least one row in binned_spiketrain is empty (has no spikes).

See also

correlation_coefficient

Pearson correlation coefficient

Notes

1. The spike trains in the binned structure are assumed to cover the complete time span [t_start, t_stop) of binned_spiketrain.

2. Using fast=True might lead to MemoryError. If it’s the case, switch to fast=False.

Examples

Covariance matrix of two Poisson spike train processes.

>>> import neo
>>> import numpy as np
>>> import quantities as pq
>>> from elephant.spike_train_generation import StationaryPoissonProcess
>>> from elephant.conversion import BinnedSpikeTrain
>>> from elephant.spike_train_correlation import covariance

>>> np.random.seed(1)
>>> st1 = StationaryPoissonProcess(rate=10*pq.Hz,
... t_stop=10.0*pq.s).generate_spiketrain()
>>> st2 = StationaryPoissonProcess(rate=10*pq.Hz,
... t_stop=10.0*pq.s).generate_spiketrain()
>>> cov_matrix = covariance(BinnedSpikeTrain([st1, st2], bin_size=5*pq.ms))
>>> cov_matrix 
array([[ 0.05432316, -0.00152276],
   [-0.00152276,  0.04917234]])