Computational NeuroscienceUnivariate normalization of bispectrum using Hölder's inequality

- We suggest a new normalization factor for a bispectrum which is a product of univariate quantities. The absolute value of the normalized bispectrum is proved to be bounded by zero and one using Hölders inequality.
- The new normalization is compared to three widely used normalizations, specifically showing that the proposed normalization factor, in contrast to other factors leading to bounded measures, is itself not affected by coupling between the signals.
- The significance of the bicoherence values of different normalization methods are compared to each other using statistical tests. We did not find evidence to conclude that the normalization has a relevant effect on the statistical power of the bicoherence values.

Considering that many biological systems including the brain are complex non-linear systems, suitable methods capable of detecting these non-linearities are required to study the dynamical properties of these systems. One of these tools is the third order cummulant or cross-bispectrum, which is a measure of interfrequency interactions between three signals. For convenient interpretation, interaction measures are most commonly normalized to be independent of constant scales of the signals such that its absolute values are bounded by one, with this limit reflecting perfect coupling. Although many different normalization factors for cross-bispectra were suggested in the literature these either do not lead to bounded measures or are themselves dependent on the coupling and not only on the scale of the signals. In this paper we suggest a normalization factor which is univariate, i.e., dependent only on the amplitude of each signal and not on the interactions between signals. Using a generalization of Hölder's inequality it is proven that the absolute value of this univariate bicoherence is bounded by zero and one. We compared three widely used normalizations to the univariate normalization concerning the significance of bicoherence values gained from resampling tests. Bicoherence values are calculated from real EEG data recorded in an eyes closed experiment from 10 subjects. The results show slightly more significant values for the univariate normalization but in general, the differences are very small or even vanishing in some subjects. Therefore, we conclude that the normalization factor does not play an important role in the bicoherence values with regard to statistical power, although a univariate normalization is the only normalization factor which fulfills all the required conditions of a proper normalization.