A novel nonlinear bispectrum analysis for dynamical complex oscillations.

Unlabelled: In this study, we proposed a novel set of bispectrum in constructing both frequency power and complexity spectrum. The uniform phase empirical mode decomposition (UPEMD) was implemented to obtain nonlinear extraction while guaranteeing explicit frequencies. Lepel-Ziv complexity (LZC) and frequency power per mode were used for comprehensive frequency spectra. To examine the performances of the proposed method and meanwhile optimize routine methodological parameters, either chaotic logistic maps or a default non-stationary simulation in 40 ~ 60 Hz along with several challenges were designed. The simulation results showed the UPEMD-based LZC spectrum distinguishes the degree of complexity, reflecting the bandwidth and noise level of the inputs. The UPEMD-based power spectrum on the other side presents power distribution of nonlinear and nonstationary oscillation across multiple frequencies. In addition, given gait disturbance is an unsolved symptom in adaptive deep brain stimulation (DBS) for Parkinson's disease (PD), meanwhile considering the representative of deep brain activities to the complex oscillations, such data were analyzed further. Our results showed the high-frequency band (45 ~ 80 Hz) of the UPEMD-based LZC spectrum reflects the impact of auditory cues in modulating the complexity of DBS recording. Such an increase in complexity (45 ~ 60 Hz) reduces shortly after the cue was removed. As for the UPEMD-based power spectrum, decreasing power over the higher frequency region (> 30 Hz) was shown with auditory cues. These results manifest the potential of the proposed methods in reflecting gait improvement for PD. The proposed bispectrum reflected both the nonlinear complexity and power spectrum analyses, enabling examining targeted frequencies with refined resolution.

Supplementary Information: The online version contains supplementary material available at 10.1007/s11571-023-09953-z.