The projected isotropic normal distribution with applications in neuroscience

TL;DR

This paper introduces the projected isotropic normal distribution, explores its properties, and develops practical approximations for phase analysis in EEG signals under flash stimulation.

Contribution

It revisits the projected isotropic normal distribution, derives new properties, and provides approximations to facilitate phase-based EEG data analysis.

Findings

Derived closed-form trigonometric moments.

Developed two practical approximations for the distribution of the resultant.

Applied the methodology successfully to EEG data from flash-stimulation experiments.

Abstract

This paper is motivated by a cutting-edge application in neuroscience: the analysis of electroencephalogram (EEG) signals recorded under flash stimulation. Under commonly used signal-processing assumptions, only the phase angle of the EEG is required for the analysis of such applications. We demonstrate that these assumptions imply that the phase has a projected isotropic normal distribution. We revisit this distribution and derive several new properties, including closed-form expressions for its trigonometric moments. We then examine the distribution of the mean resultant and its square -- a statistic of central importance in phase-based EEG studies. The distribution of the resultant is analytically intricate; to make it practically useful, we develop two approximations based on the well-known resultant distribution for the von Mises distribution. We then study inference problems for this projected isotropic normal distribution. The method is illustrated with an application to EEG data from flash-stimulation experiments.