Measuring phase-amplitude coupling between neural oscillations of different frequencies via the Wasserstein distance

Phase-amplitude coupling (PAC) is a key neuronal mechanism. Here, a novel method for quantifying PAC via the Wasserstein distance is presented.

The Wasserstein distance is an optimization algorithm for minimizing transportation cost and distance. For the first time, the author has applied this distance function to quantify PAC and named the Wasserstein Modulation Index (wMI). As the wMI accommodates the product of the amplitude value in each phase position and the coupling phase position, it allows for extraction of more detailed PAC features from the data.

The validity of the wMI calculations was examined using various simulation data, including sinusoidal and non-sinusoidal waves and empirical data sets. The current findings showed that the wMI is a more robust and stable index for quantifying PAC under various measuring conditions. Specifically, it can better reflect the timing of coupling and distinguish the shape of the coupling distribution than other measurements, both of which are the most significant parameters related to the functionality of PAC. Furthermore, the wMI is also suitable for many applications, such as more data-driven approaches and direct comparisons.

Compared with Euler-based PAC methods and the Modulation Index (MI), the wMI is not easily affected by the non-sinusoidal nature of neural oscillation and the short data length and enables better reflection of the natures of PAC, such as the timing of coupling and the amplitude distribution in the phase plane, than the MI.

The wMI is expected to extract more detailed PAC characteristics, which could considerably contribute to the neuroscience field.