Detecting pairwise correlations in spike trains: an objective comparison of methods and application to the study of retinal waves

Abstract

Correlations in neuronal spike times are thought to be key to processing in many neural systems. Many measures have been proposed to summarize these correlations and of these the correlation index is widely used and is the standard in studies of spontaneous retinal activity. We show that this measure has two undesirable properties: it is unbounded above and confounded by firing rate. We list properties needed for a measure to fairly quantify and compare correlations and we propose a novel measure of correlation-the spike time tiling coefficient. This coefficient, the correlation index, and 33 other measures of correlation of spike times are blindly tested for the required properties on synthetic and experimental data. Based on this, we propose a measure (the spike time tiling coefficient) to replace the correlation index. To demonstrate the benefits of this measure, we reanalyze data from seven key studies, which previously used the correlation index to investigate the nature of spontaneous activity. We reanalyze data from β2(KO) and β2(TG) mutants, mutants lacking connexin isoforms, and also the age-dependent changes in wild-type and β2(KO) correlations. Reanalysis of the data using the proposed measure can significantly change the conclusions. It leads to better quantification of correlations and therefore better inference from the data. We hope that the proposed measure will have wide applications, and will help clarify the role of activity in retinotopic map formation.

Keywords: activity; correlations; development; retina; retinotopic map; spike times.

Figure 1.

Diagram to demonstrate the calculation of the spike time tiling coefficient. The four quantities required to calculate the spike time tiling coefficient are P_A, P_B, T_A, T_B. The only free parameter is Δt. Values and scales are for demonstration only.

Figure 2.

Examples of simulated data used to test measures. Data generated from Model 1 used a Poisson spiking model where both neurons fire at 1.5 Hz with increasing percentage of spike times which are shared with a spike in the other train: 0% (A), 87% (B), and 99% (C). Recording duration T = 300 s. Data generated from Model 2 used a Poisson burst model with a burst rate of λ = 0.05 Hz, where the number of spikes in each burst is drawn from a Poisson distribution with mean N = 8. The positions of the spikes relative to the center of the burst (indicated by a red arrow) are drawn from a uniform distribution on [−1, 1] s (σ = 2). The center of the burst of the second train is offset from the center of the first by a fixed amount: O = 0 s (A), 1 s (B), and 2 s (C), T = 3600 s. Data generated from Model 3 shows regular out-of-synchrony firing with increasing firing rate generated using an integrate-and-fire model (see Materials and Methods for details). The firing rates are 0.76 Hz (A), 1.27 Hz (B), and 2.5 Hz (C), T = 3000 s.

Figure 3.

The correlation index is dependent on firing rate. The correlation index of two identical Poisson spike trains is plotted for varying firing rates. Simulation values were generated by simulating one Poisson spike train and then calculating the correlation index comparing this train with itself. Mean with error bars of ±1SD are plotted for 10 trials, each of duration 300 s. The theoretical expected value of the correlation index under this model (red line) is given by Equation 3.

Figure 4.

Twenty-one measures are rejected because they are dependent on firing rate (lack property N2). All measures which showed rate-dependence when tested on the autocorrelation of Poisson spike trains are plotted. Three measures which did not show rate-dependence are also shown in the bottom row for comparison (green). One Poisson spike train was simulated for 300 s for varying rates (0.05–5 Hz) and the measures were calculated comparing this spike train to itself. The mean of 10 repeats are plotted and ±1 SD is shown by gray shading. The identity of each measure appears in Table 3. Note that the correlation index is not presented here, but in Figure 3 and that measure 12 has two versions; one appears here and the other in Figure 5A; see Materials and Methods for details.

Figure 5.

Nine measures are rejected using remaining necessary properties. A, Measures which are dependent on firing rate (lack property N2) where dependency is not obvious from autocorrelation are applied to data generated from the following Poisson spiking model: one train has rate 3 Hz and the other's rate varies (0.1–5 Hz). There are no shared spike times (T = 300 s). The second version of Measure 12 is shown (the first version is in Fig. 4); see Materials and Methods for details. B, Measures which are dependent on recording time (lack property N3) are applied to data generated from the following Poisson spiking model: both neurons fire at rate 1 Hz and 10% of spike times are shared. The recording duration varies from 50 to 300 s and Δt = 0.6 s (higher than usual because for these measures smaller values cause issues with computational precision). C, Measures which cannot distinguish anticorrelation from no correlation (lack property N6) are applied to regular out-of-synchrony spikes of varying rate (0.25–4.2 Hz) generated using an integrate-and-fire model described by Dayan and Abbott, 2001 (their Fig. 5.20). The parameters are as in their figure with the following exceptions: P_max = 0.5, R_mI_e = 18 mV. τ_s varied from 0.05–1.5 s and E_s from 0 mV to −70 mV, T = 3000 s. In all panels, Measure 35 (which possesses the necessary properties) is shown for comparison (green), the mean of 10 repeats are plotted and error bars are omitted for visual clarity.

Figure 6.

Detailed examination of the four measures which possess all necessary properties eliminates three on the basis of the desirable properties. A, The spike count correlation with different bin widths (d; see Materials and Methods) is applied to data from the following Poisson burst model which has increasing range of spike offsets within a burst: both neurons have a burst rate of 0.05 Hz, burst centers have 0 s offset, each burst contains eight spikes whose positions are drawn from a uniform distribution of varying width (0.1–4 s) centered on the burst center (T = 3600 s). B, The spike time tiling coefficient (STTC) applied to identical data to that in A. The spike count correlation with d = 1 ms is plotted (black) for comparison. C, The Kerschensteiner and Wong correlation measure with different lengths of the averaging window (w; see Materials and Methods) is applied to data from the following Poisson burst model with increasing number of spikes per burst: both neurons have a burst rate of 0.05 Hz, burst centers have 0 s offset, the number of spikes in each burst is drawn from a Poisson distribution with increasing mean (from 1 to 15). Spike positions are drawn from a uniform distribution of width 2 s centered on the center of the burst (T = 3600 s). D, The Kruskal measure is applied to spike times from the following model: two independent Poisson neurons are simulated each with rate 0.1 Hz. For each spike (in either train) a burst is generated in the other train with 0 s offset of the burst center and one to six spikes whose positions are drawn from a uniform distribution of width 2Δt around the burst center (T = 2000 s, Δt = 0.1 s). The spike time tiling coefficient (green) is plotted in C and D for comparison. For all panels the mean of 10 repeats is plotted, error bars are omitted for visual clarity.

Figure 7.

Evaluating correlations in retinal waves recorded from connexin mutant mice shows that the spike time tiling coefficient can significantly alter conclusions. A, Raster plots of 10 spike trains over a 10 min interval, recorded from retinas isolated from P12 wild-type mouse and two mutant mice (lacking either one or two connexin isoforms- Cx45 and Cx36/Cx45), P11 Cx45ko and P10 Cx36/Cx45dko. Data are from Blankenship et al. (2011) and raster plots follow the presentation of their Figure 2A. The mean firing rate and number of animals (n) from each genotype is recorded in the legend. B, Pairwise correlation index as a function of intercellular distance for each genotype. Data points are medians over all recordings and error bars indicate the interquartile range (IQR). Inset, The same data normalized (multiplicatively) by genotype so that the correlation indices are identical at zero distance, following Figure 2B in the original publication. C, Same as B, using the STTC in place of the correlation index. Compare with both B and B, inset. In both B and C, Δt = 100 ms as in the original publication. The distances at which correlations are measured are the discrete set of separations possible on the MEA grid.

Figure 8.

Reanalysis using the spike time tiling coefficient supports the conclusion that correlations in spontaneous activity in the developing ferret and mouse retina decreases with age. A, The correlation index is calculated pairwise and shown as a function of electrode separation for spontaneous retinal activity in developing ferret for four different ages (data from Wong et al., 1993). The distances at which correlations are measured were binned (bin width 20 μm) due to high density. B, Same as A using the STTC in place of the correlation index. C, The correlation index is calculated pairwise and shown as a function of electrode separation for spontaneous retinal activity in developing mouse for four different ages (data from Demas et al., 2003). The distances at which correlations are measured are the discrete set of separations possible on the MEA grid. D, Same as C using the STTC. In all panels, median values are plotted and IQRs are only shown at the smallest separation distance. Other IQRs are omitted for visual clarity. Mean firing rates and number of animals (n) for each age are recorded in the legend.

Figure 9.

Reanalysis of experimental data using the spike time tiling coefficient supports the conclusions that the β2(KO) and β2(TG) mouse phenotypes show lower correlations in spontaneous retinal activity than those of wild-type. A, The correlation index (left) or STTC (right) is plotted pairwise against electrode separation for recordings of spontaneous retinal activity for P6 wild-type and β2(KO) phenotypes (data from Stafford et al., 2009). B, The correlation index (left) or spike time tiling coefficient (right) is plotted pairwise against electrode separation for recordings of spontaneous retinal activity for P5 wild-type and two β2(KO) phenotypes: Xu and Picciotto (Pic) (data from Sun et al., 2008). C, The correlation index (left) or spike time tiling coefficient (right) is plotted pairwise against electrode separation for recordings of spontaneous retinal activity for P4 wild-type and β2(TG) phenotypes (data from Xu et al., 2011). In all panels, Δt = 100 ms, as in original publications, medians are plotted and the error bars show the IQR. Mean firing rates and number of animals (n) for each phenotype are recorded in the legend. All recordings at 37°C. The distances at which correlations are measured are the discrete set of separations possible on the MEA grid.

Figure 10.

Reanalysis of age related changes in β2(KO) mutants shows that the spike time tiling coefficient is able to more accurately quantify correlations which the correlation index ascribes as extremely correlated. A, Raster plots of 10 spike trains over a 10 min interval, recorded from retinas isolated from P5 wild-type mouse and P4 β2(KO) mouse. All data are controls from Kirkby and Feller (2013). B, Pairwise correlation index as a function of intercellular distance for P5 wild-type mouse and β2(KO) mouse of different ages (P4–P7). C, Same as B but using STTC. The mean firing rate and number of animals (n) from each genotype is recorded in the legend. Data points are medians over all recordings and error bars indicate the IQR. For visual clarity, IQRs are only shown at the smallest separation distance. The distances at which correlations are measured are the discrete set of separations possible on the MEA grid. Δt = 100 ms, as in original publication and recordings were performed at 33–35°C.

Figure 11.

Varying the window of synchrony Δt can be informative about correlational timescales inherent in data. A, The STTC of spike trains from Stafford et al. (2009) was calculated pairwise (as in Fig. 9B) and the median value at the smallest electrode separation is plotted for varying Δt. Error bars show the IQR. The genotypes shown are P6 wild-type and β2(KO). B, Same as A, but data from Sun et al., 2008; Fig. 9D). The genotypes shown are P5 wild-type and two β2(KO) phenotypes: Xu and Picciotto (Pic). C, Same as A, but data from Xu et al., 2011 (Fig. 9F). The genotypes shown are P4 wild-type and β2(TG) mouse. Vertical lines at Δt = 1 s indicate separation between region with strong Δt dependency (Δt ≤ 1) and weaker dependency (note x-axis has a log-scale and that the limit of the spike time tiling coefficient as Δt tends to infinity is one).