The pairwise phase consistency: a bias-free measure of rhythmic neuronal synchronization

Oscillatory activity is a widespread phenomenon in nervous systems and has been implicated in numerous functions. Signals that are generated by two separate neuronal sources often demonstrate a consistent phase-relationship in a particular frequency-band, i.e., they demonstrate rhythmic neuronal synchronization. This consistency is conventionally measured by the PLV (phase-locking value) or the spectral coherence measure. Both statistical measures suffer from significant bias, in that their sample estimates overestimate the population statistics for finite sample sizes. This is a significant problem in the neurosciences where statistical comparisons are often made between conditions with a different number of trials or between neurons with a different number of spikes. We introduce a new circular statistic, the PPC (pairwise phase consistency). We demonstrate that the sample estimate of the PPC is a bias-free and consistent estimator of its corresponding population parameter. We show, both analytically and by means of numerical simulations, that the population statistic of the PPC is equivalent to the population statistic of the squared PLV. The variance and mean squared error of the PPC and PLV are compared. Finally, we demonstrate the practical relevance of the method in actual neuronal data recorded from the orbitofrontal cortex of rats that engage in a two-odour discrimination task. We find a strong increase in rhythmic synchronization of spikes relative to the local field potential (as measured by the PPC) for a wide range of low frequencies (including the theta-band) during the anticipation of sucrose delivery in comparison to the anticipation of quinine delivery.