Improved measures of phase-coupling between spikes and the Local Field Potential

Abstract

An important tool to study rhythmic neuronal synchronization is provided by relating spiking activity to the Local Field Potential (LFP). Two types of interdependent spike-LFP measures exist. The first approach is to directly quantify the consistency of single spike-LFP phases across spikes, referred to here as point-field phase synchronization measures. We show that conventional point-field phase synchronization measures are sensitive not only to the consistency of spike-LFP phases, but are also affected by statistical dependencies between spike-LFP phases, caused by e.g. non-Poissonian history-effects within spike trains such as bursting and refractoriness. To solve this problem, we develop a new pairwise measure that is not biased by the number of spikes and not affected by statistical dependencies between spike-LFP phases. The second approach is to quantify, similar to EEG-EEG coherence, the consistency of the relative phase between spike train and LFP signals across trials instead of across spikes, referred to here as spike train to field phase synchronization measures. We demonstrate an analytical relationship between point-field and spike train to field phase synchronization measures. Based on this relationship, we prove that the spike train to field pairwise phase consistency (PPC), a quantity closely related to the squared spike-field coherence, is a monotonically increasing function of the number of spikes per trial. This derived relationship is exact and analytic, and takes a linear form for weak phase-coupling. To solve this problem, we introduce a corrected version of the spike train to field PPC that is independent of the number of spikes per trial. Finally, we address the problem that dependencies between spike-LFP phase and the number of spikes per trial can cause spike-LFP phase synchronization measures to be biased by the number of trials. We show how to modify the developed point-field and spike train to field phase synchronization measures in order to make them unbiased by the number of trials.

Fig. 1

Illustration of point-field phase synchronization measures. Spikes (vertical, solid lines) are related to LFPs by determining the instantaneous LFP phases at the times of spiking by centering a window (horizontal, solid lines) around every spike. Thus, the phase for each spike is determined from the LFP snippet that is centered around it. The instantaneous LFP phase at the time of spiking can be directly gauged by using the dotted lines that connect the spikes to the LFP. The frequency resolution, which is inversely proportional to time resolution, is proportional to the window length, which needs to be sufficiently large. However, if the windows are too long, then the loss of time-resolution may lead to a decrease in spike-LFP phase consistency. The instantaneous spike-LFP phases are depicted by the unit arrows that originate from the spikes. The LFP was modeled as band-stop filtered Gaussian white noise, as in Zeitler et al. (2006). Here, we show only two (m = 1 and m = 2) out of M available trials. The measures of point-field phase consistency are computed over the n relative spike-LFP phases. By vector addition of the n spike-LFP phase vectors, a resultant vector is obtained, whose length is conventionally taken as a measure of point-field phase consistency

Fig. 2

Illustration of spike train to field phase synchronization measures. Spikes (vertical, solid lines) are related to LFPs by computing, using the Fast Fourier Transform, the respective phase offsets of the spike train and the LFP at a particular frequency (grey arrows) and subsequently determining the relative phase between these phase offsets (black arrows). Thus, the relative phases are computed for a time-period, not for an individual spike. The LFP was generated using the same model as in Fig. 1. Shown here are two trials (m = 1 and m = 2). The measures of spike train to field phase consistency are computed over the M relative phases. By vector addition of the M relative phase vectors, a resultant vector is obtained, whose length is conventionally taken as a measure of the consistency of the relative phases between LFPs and spike trains

Fig. 3

Illustration of the point-field PPC and the point-field PPC . (a) Nine spike-LFP phases from four different trials are recorded. These spike-LFP phases are depicted by the arrows, which are duplicated on the x- and y-axis. This gives 9 ·9 = 81 potential 2-combinations of spike-LFP phases across trials. The extent to which two spike-LFP phase vectors coincide is measured by the dot product. The point-field PPC equals the average dot product of spike-LFP phase vectors across all 2-combinations of different spikes. The nine 2-combinations of spikes with themselves are not taken into account, which is indicated by the diagonal patterns. (b) Same as in (a), but the point-field PPC equals the average dot product of spike-LFP phase vectors across all 2-combinations of spikes from different trials.The 2-combinations of spikes from the same trial are not taken into account, which is indicated by the diagonal patterns

Fig. 4

Influence of dependence between spike-LFP phases on point-field phase synchronization measures. (a) Estimated, expected point-field PPC (solid), and point-field PPC values (dotted) (y-axis) as a function of the number of trials (x-axis), in the presence of an absolute refractory period of 8 ms. The grey portion of the solid curve highlights that no estimate exists when only one trial is available. (b) Similar to (a), but now in the presence of an absolute refractory period of 40 ms. (c) Influence of burstiness on point-field phase synchronization measures. Estimated expected point-field PPC (solid), and point-field PPC value (dotted) (y-axis) as a function of the number of trials (x-axis). For every spike fired, we added one extra spike with the same phase. (d) Influence of incomplete coverage of LFP phases on estimated, expected point-field PPC (solid), and point-field PPC values (dotted) (y-axis) as a function of number of trials (x-axis), with the neuron firing only during 20% of the cycle duration

Fig. 5

Influence of dependence of spike-LFP phase consistency and spike count on point-field phase synchronization measures. Estimated, expected point-field PPC (solid), and point-field PPC values (black, dotted) (y-axis) as a function of the number of trials (x-axis), where trials with a large number of spikes have a noisier phase distribution

Fig. 6

Comparison of uncorrected and corrected spike train to field PPC . (a) Y-axis: corrected spike train to field PPC (dotted) and uncorrected spike train to field PPC (solid). X-axis: number of spikes per trial. Different lines represent different coupling strength, with the von Mises dispersion parameter κ ∈ { 0.1, 0.5, 1, 20 }. (b) Empirically observed and analytically predicted correction factor for spike train to field PPC (y-axis) as a function of the number of spikes per trial (x-axis), again for κ ∈ { 0.1, 0.5, 1, 20 }

Fig. 7

Influence of firing rate variance on spike train to field phase synchronization measures. Dashed: spike train to field PPC for Poisson distribution minus spike train to field PPC for zero-variance distribution, as a function of the number of spikes per trial. Positive values indicate a larger value for Poisson distribution. Solid: as for dashed, but now for spike train to field PPC . Dashed-dotted: as solid, but now for spike train to field PPC