Directional statistics for spatiotemporal wave analysis

I've been searching for a good distribution that can be used to summarize how the distribution of phases and amplitudes evolves during transient synchronization events of beta (~20 Hz) local field potentials (LFP) in motor cortex. So far, it seems difficult to find a single distribution family that works in all cases. 

Spatiotemporal wave activity in beta oscillations in motor cortex can be described in terms of the beta-band analytic LFP signal, which has both a magnitude and a phase, and who's real part is equal to the time-domain value of the beta-filtered signal. 

\begin{equation}z_k(t) = \beta_k (t) + i\cdot  \operatorname{Hilbert}(\beta_k)(t) = r_k(t) e^{i \theta_k(t)},\\\textrm{ where $k$ indexes over channels}\end{equation}

Circular statistics can be used to summarize the distribution of analytic signal phase, in order to detect synchrony and wave events.

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Figure 2: Neither the complex Gaussian nor log-polar statistics perfectly describe the distributions of analytic signal. In these plots, the black ellipse represents a complex Gaussian model of the data, with the ellipse boundary at one standard deviation, and the ellipse axes representing the eigenvectors of the covariance matrix $\Sigma$. The cyan contours represent a log-polar model of the data, which uses the mean and standard deviation of the log-amplitude, as well as the circular mean and standard deviation of the phases, to model the data in log-polar space. (a) When phase is concentrated, and not correlated with amplitude, both the log-polar statistics and the complex Gaussian distribution describe the data well. (b) When phase and amplitude are correlated, the log-polar model cannot capture the phase-amplitude dependence. (c) During traveling wave events, signal amplitude is high, and there is dispersion in phase. In these cases, the log-polar statistics are more appropriate than the complex Gaussian. (d) Traveling wave events appear to often evolve from states that show a mixture of synchrony and standing wave dynamics. The log-polar statistics break down when the phase distribution is bimodal, but the complex Gaussian can describe these states well. (e) At low signal amplitudes, the system is often asynchronous, and the phase and amplitude of the log-polar model are poorly defined. (f) Although rare or absent in our data, a hypothetical distribution with uniform phase and concentrated amplitude could occur, say, during traveling wave events with short wavelength. In this case, the complex Gaussian model is especially bad.

Three unbiased estimators of spike-field coherence

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In these notes we show a simplified expression for the Pairwise Phase Consistency (PPC) measure of Vinck et al. (2010). We illustrate it's relation to a bias-corrected spike-field coherence measure from Grasse et al. (2010), and discuss an third notion of spike-field coherence that is intermediate between the two. 

Here, we use the term "event-triggered" rather than "spike-triggered", because we want to apply these measures to neural events other than spikes (e.g. beta-frequency transients in motor cortex).

Pairwise Phase Consistency

Vinck et al. (2010) define Pairwise Phase Consistency (PPC) as the expected dot product between all pairs of (spike-triggered) phase measurements. 

\[\hat\Upsilon = \frac 2 {N(N-1)} \sum_{j=1..N-1} \sum_{k=j+1..N} \cos(\theta_j - \theta_k)\]

There is an alternative way to express PPC that is faster to calculate, and also reveals a relationship between PPC and similar alternatives.