Rate-adjusted spike-LFP coherence comparisons from spike-train statistics

Abstract

Coherence is a fundamental tool in the analysis of neuronal data and for studying multiscale interactions of single and multiunit spikes with local field potentials. However, when the coherence is used to estimate rhythmic synchrony between spiking and any other time series, the magnitude of the coherence is dependent upon the spike rate. This property is not a statistical bias, but a feature of the coherence function. This dependence confounds cross-condition comparisons of spike-field and spike-spike coherence in electrophysiological experiments. Taking inspiration from correction methods that adjust the spike rate of a recording with bootstrapping ('thinning'), we propose a method of estimating a correction factor for the spike-field and spike-spike coherence that adjusts the coherence to account for this rate dependence. We demonstrate that the proposed rate adjustment is accurate under standard assumptions and derive distributional properties of the estimator. The reduced estimation variance serves to provide a more powerful test of cross-condition differences in spike-LFP coherence than the thinning method and does not require repeated Monte Carlo trials. We also demonstrate some of the negative consequences of failing to account for rate dependence. The proposed spike-field coherence estimator accurately adjusts the spike-field coherence with respect to rate and has well-defined distributional properties that endow the estimator with lower estimation variance than the existing adjustment method.

Keywords: Coherence; Point processes; Rhythms; Spike–field; Synchrony.

Figure 1

A: Representative realization of AR(2) LFP and corresponding spike times at μ = 80 spikes per second. B: Sample multitaper LFP spectral density for NW = 5, 9 tapers, and 100 trials. C: Corresponding multitaper estimate of the spike-LFP coherence.

Figure 2

Mean (−) and jackknife 95% confidence interval (- -) for the standard (blue) and adjusted (red) spike-field coherence for simulated spike trains at different target rates μ*, but constant C_λy. The unadjusted coherence increases as the rate increases through the parameter α. Each panel displays the adjustment to a different μ* (indicated by vertical lines). Horizontal lines give the mean for the unadjusted coherence at μ = μ*.

Figure 3

A: Sample histograms for 1000 realizations of $z={\tanh }^{-1}({\widehat{C}}_{\mathit{ny}})$, the Fisher-transformed spike field coherence, at μ₁ = 40 (blue) and μ₂ = 60 (red) with N = 100. Normal distributions are shown corresponding to the ensemble sample (−) and theoretical (- -) variance (based on (12)). B: Sample histogram and normal distributions of the higher rate condition, adjusted to the spike rate of the low-rate condition. Corresponding Gaussian distributions for the sample (− ·) and theoretical (×) distributions with adjusted mean and variance given by (13) and (14), are similar.

Figure 4

Comparison of the proposed rate adjustment with the thinning method using simulated data. Both conditions had the same C_λy, but different spike rates. Sample sizes for both conditions was N = 100. Shaded areas are jackknife 95% confidence intervals. Difference in C_ny between adjusted, thinned, and low-rate conditions are non-significant (p > .05).

Figure 5

Discrepancy between real and assumed type-I error rate for the test of cross-condition comparisons in spike-field coherence. When C_ny is used as a proxy for C_λy, then the null distribution should have non-zero mean, even while the standard practice is to assume zero mean. The x-axis indicates the type-I error rate that the experimenter thinks she is using when assuming zero-mean. The y-axis indicates the type-I error that is actually achieved for the given threshold. Solid line indicates the error rate for the adjusted test.

Figure 6

Receiver operating characteristics for the test for cross-condition differences in spike-field coherence under various conditions. Smaller rate is μ = 10 sps. Larger intensity-field coherence is C_λy = .3. Smaller intensity-field coherence is C_λy = .1. Because the proposed estimator has smaller variance than the unadjusted, or thinned estimator, the test using the proposed rate-adjusted estimator (×) always has larger AUC than the test using the thinned estimator, or the unadjusted condition with equal spike rates. Red markers are covered by green markers, since thinning and adjustment result is consistent testing properties across experimental outcomes.