Spike train correlation

This modules provides functions to calculate correlations between spike trains.

elephant.spike_train_correlation.cch(binned_st1, binned_st2, window='full', border_correction=False, binary=False, kernel=None, method='speed', cross_corr_coef=False)

Computes the cross-correlation histogram (CCH) between two binned spike trains binned_st1 and binned_st2.

Parameters:

binned_st1, binned_st2 : BinnedSpikeTrain

Binned spike trains to cross-correlate. The two spike trains must have same t_start and t_stop

window : string or list (optional)

‘full’: This returns the crosscorrelation at each point of overlap, with an output shape of (N+M-1,). At the end-points of the cross-correlogram, the signals do not overlap completely, and boundary effects may be seen. ‘valid’: Mode valid returns output of length max(M, N) - min(M, N) + 1. The cross-correlation product is only given for points where the signals overlap completely. Values outside the signal boundary have no effect. Default: ‘full’ list of integer of of quantities (window[0]=minimum, window[1]=maximum lag): The entries of window can be integer (number of bins) or quantities (time units of the lag), in the second case they have to be a multiple of the binsize Default: ‘Full’

border_correction : bool (optional)

whether to correct for the border effect. If True, the value of the CCH at bin b (for b=-H,-H+1, …,H, where H is the CCH half-length) is multiplied by the correction factor:

(H+1)/(H+1-|b|),

which linearly corrects for loss of bins at the edges. Default: False

binary : bool (optional)

whether to binary spikes from the same spike train falling in the same bin. If True, such spikes are considered as a single spike; otherwise they are considered as different spikes. Default: False.

kernel : array or None (optional)

A one dimensional array containing an optional smoothing kernel applied to the resulting CCH. The length N of the kernel indicates the smoothing window. The smoothing window cannot be larger than the maximum lag of the CCH. The kernel is normalized to unit area before being applied to the resulting CCH. Popular choices for the kernel are

  • normalized boxcar kernel: numpy.ones(N)
  • hamming: numpy.hamming(N)
  • hanning: numpy.hanning(N)
  • bartlett: numpy.bartlett(N)

If None is specified, the CCH is not smoothed. Default: None

method : string (optional)

Defines the algorithm to use. “speed” uses numpy.correlate to calculate the correlation between two binned spike trains using a non-sparse data representation. Due to various optimizations, it is the fastest realization. In contrast, the option “memory” uses an own implementation to calculate the correlation based on sparse matrices, which is more memory efficient but slower than the “speed” option. Default: “speed”

cross_corr_coef : bool (optional)

Normalizes the CCH to obtain the cross-correlation coefficient function ranging from -1 to 1 according to Equation (5.10) in “Analysis of parallel spike trains”, 2010, Gruen & Rotter, Vol 7
Returns:

cch : AnalogSignal

Containing the cross-correlation histogram between binned_st1 and binned_st2.

The central bin of the histogram represents correlation at zero delay. Offset bins correspond to correlations at a delay equivalent to the difference between the spike times of binned_st1 and those of binned_st2: an entry at positive lags corresponds to a spike in binned_st2 following a spike in binned_st1 bins to the right, and an entry at negative lags corresponds to a spike in binned_st1 following a spike in binned_st2.

To illustrate this definition, consider the two spike trains: binned_st1: 0 0 0 0 1 0 0 0 0 0 0 binned_st2: 0 0 0 0 0 0 0 1 0 0 0 Here, the CCH will have an entry of 1 at lag h=+3.

Consistent with the definition of AnalogSignals, the time axis represents the left bin borders of each histogram bin. For example, the time axis might be: np.array([-2.5 -1.5 -0.5 0.5 1.5]) * ms

bin_ids : ndarray of int

Contains the IDs of the individual histogram bins, where the central bin has ID 0, bins the left have negative IDs and bins to the right have positive IDs, e.g.,: np.array([-3, -2, -1, 0, 1, 2, 3])
elephant.spike_train_correlation.corrcoef(binned_sts, binary=False)[source]

Calculate the NxN matrix of pairwise Pearson’s correlation coefficients between all combinations of N binned spike trains.

For each pair of spike trains (i,j), the correlation coefficient C[i,j] is obtained by binning i and j at the desired bin size. Let b_i and b_j denote the binary vectors and m_i and m_j their respective averages. Then

C[i,j] = <b_i-m_i, b_j-m_j> / \sqrt{<b_i-m_i, b_i-m_i>*<b_j-m_j,b_j-m_j>}

where <..,.> is the scalar product of two vectors.

For an input of n spike trains, a n x n matrix is returned. Each entry in the matrix is a real number ranging between -1 (perfectly anti-correlated spike trains) and +1 (perfectly correlated spike trains).

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

Parameters:

binned_sts : elephant.conversion.BinnedSpikeTrain

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

binary : bool, optional

If True, two 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
Returns:

C : ndarrray

The square matrix of correlation coefficients. The element C[i,j]=C[j,i] is the Pearson’s correlation coefficient between binned_sts[i] and binned_sts[j]. If binned_sts contains only one SpikeTrain, C=1.0.

Notes

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

Examples

Generate two Poisson spike trains

>>> from elephant.spike_train_generation import homogeneous_poisson_process
>>> st1 = homogeneous_poisson_process(
        rate=10.0*Hz, t_start=0.0*s, t_stop=10.0*s)
>>> st2 = homogeneous_poisson_process(
        rate=10.0*Hz, t_start=0.0*s, t_stop=10.0*s)

Calculate the correlation matrix.

>>> from elephant.conversion import BinnedSpikeTrain
>>> cc_matrix = corrcoef(BinnedSpikeTrain([st1, st2], binsize=5*ms))

The correlation coefficient between the spike trains is stored in cc_matrix[0,1] (or cc_matrix[1,0]).

elephant.spike_train_correlation.covariance(binned_sts, binary=False)[source]

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 binary vectors and m_i and m_j their respective averages. Then

C[i,j] = <b_i-m_i, b_j-m_j> / (l-1)

where <..,.> is the scalar product of two vectors.

For an input of n spike trains, a 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_sts : elephant.conversion.BinnedSpikeTrain

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

binary : bool, optional

If True, two 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
Returns:

C : ndarrray

The square matrix of covariances. The element C[i,j]=C[j,i] is the covariance between binned_sts[i] and binned_sts[j].

Notes

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

Examples

Generate two Poisson spike trains

>>> from elephant.spike_train_generation import homogeneous_poisson_process
>>> st1 = homogeneous_poisson_process(
        rate=10.0*Hz, t_start=0.0*s, t_stop=10.0*s)
>>> st2 = homogeneous_poisson_process(
        rate=10.0*Hz, t_start=0.0*s, t_stop=10.0*s)

Calculate the covariance matrix.

>>> from elephant.conversion import BinnedSpikeTrain
>>> cov_matrix = covariance(BinnedSpikeTrain([st1, st2], binsize=5*ms))

The covariance between the spike trains is stored in cc_matrix[0,1] (or cov_matrix[1,0]).

elephant.spike_train_correlation.cross_correlation_histogram(binned_st1, binned_st2, window='full', border_correction=False, binary=False, kernel=None, method='speed', cross_corr_coef=False)[source]

Computes the cross-correlation histogram (CCH) between two binned spike trains binned_st1 and binned_st2.

Parameters:

binned_st1, binned_st2 : BinnedSpikeTrain

Binned spike trains to cross-correlate. The two spike trains must have same t_start and t_stop

window : string or list (optional)

‘full’: This returns the crosscorrelation at each point of overlap, with an output shape of (N+M-1,). At the end-points of the cross-correlogram, the signals do not overlap completely, and boundary effects may be seen. ‘valid’: Mode valid returns output of length max(M, N) - min(M, N) + 1. The cross-correlation product is only given for points where the signals overlap completely. Values outside the signal boundary have no effect. Default: ‘full’ list of integer of of quantities (window[0]=minimum, window[1]=maximum lag): The entries of window can be integer (number of bins) or quantities (time units of the lag), in the second case they have to be a multiple of the binsize Default: ‘Full’

border_correction : bool (optional)

whether to correct for the border effect. If True, the value of the CCH at bin b (for b=-H,-H+1, …,H, where H is the CCH half-length) is multiplied by the correction factor:

(H+1)/(H+1-|b|),

which linearly corrects for loss of bins at the edges. Default: False

binary : bool (optional)

whether to binary spikes from the same spike train falling in the same bin. If True, such spikes are considered as a single spike; otherwise they are considered as different spikes. Default: False.

kernel : array or None (optional)

A one dimensional array containing an optional smoothing kernel applied to the resulting CCH. The length N of the kernel indicates the smoothing window. The smoothing window cannot be larger than the maximum lag of the CCH. The kernel is normalized to unit area before being applied to the resulting CCH. Popular choices for the kernel are

  • normalized boxcar kernel: numpy.ones(N)
  • hamming: numpy.hamming(N)
  • hanning: numpy.hanning(N)
  • bartlett: numpy.bartlett(N)

If None is specified, the CCH is not smoothed. Default: None

method : string (optional)

Defines the algorithm to use. “speed” uses numpy.correlate to calculate the correlation between two binned spike trains using a non-sparse data representation. Due to various optimizations, it is the fastest realization. In contrast, the option “memory” uses an own implementation to calculate the correlation based on sparse matrices, which is more memory efficient but slower than the “speed” option. Default: “speed”

cross_corr_coef : bool (optional)

Normalizes the CCH to obtain the cross-correlation coefficient function ranging from -1 to 1 according to Equation (5.10) in “Analysis of parallel spike trains”, 2010, Gruen & Rotter, Vol 7
Returns:

cch : AnalogSignal

Containing the cross-correlation histogram between binned_st1 and binned_st2.

The central bin of the histogram represents correlation at zero delay. Offset bins correspond to correlations at a delay equivalent to the difference between the spike times of binned_st1 and those of binned_st2: an entry at positive lags corresponds to a spike in binned_st2 following a spike in binned_st1 bins to the right, and an entry at negative lags corresponds to a spike in binned_st1 following a spike in binned_st2.

To illustrate this definition, consider the two spike trains: binned_st1: 0 0 0 0 1 0 0 0 0 0 0 binned_st2: 0 0 0 0 0 0 0 1 0 0 0 Here, the CCH will have an entry of 1 at lag h=+3.

Consistent with the definition of AnalogSignals, the time axis represents the left bin borders of each histogram bin. For example, the time axis might be: np.array([-2.5 -1.5 -0.5 0.5 1.5]) * ms

bin_ids : ndarray of int

Contains the IDs of the individual histogram bins, where the central bin has ID 0, bins the left have negative IDs and bins to the right have positive IDs, e.g.,: np.array([-3, -2, -1, 0, 1, 2, 3])