The instantaneous phase of neural rhythms is important to many neuroscience-related studies. In this letter, we show that the statistical sampling properties of three instantaneous phase estimators commonly employed to analyze neuroscience data share common features, allowing an analytical investigation into their behavior. These three phase estimators—the Hilbert, complex Morlet, and discrete Fourier transform—are each shown to maximize the likelihood of the data, assuming the observation of different neural signals. This connection, explored with the use of a geometric argument, is used to describe the bias and variance properties of each of the phase estimators, their temporal dependence, and the effect of model misspecification. This analysis suggests how prior knowledge about a rhythmic signal can be used to improve the accuracy of phase estimates.

The coherence between neural spike trains and local-field potential recordings, called spike-field coherence, is of key importance in many neuroscience studies. In this work, aside from questions of estimator performance, we demonstrate that theoretical spike-field coherence for a broad class of spiking models depends on the expected rate of spiking. This rate dependence confounds the phase locking of spike events to field-potential oscillations with overall neuron activity and is demonstrated analytically, for a large class of stochastic models, and in simulation. Finally, the relationship between the spike-field coherence and the intensity field coherence is detailed analytically. This latter quantity is independent of neuron firing rate and, under commonly found conditions, is proportional to the probability that a neuron spikes at a specific phase of field oscillation. Hence, intensity field coherence is a rate-independent measure and a candidate on which to base the appropriate statistical inference of spike field synchrony.