A novel nonlinear bispectrum analysis for dynamical complex oscillations

Abstract

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.

Keywords: Electrophysiological oscillation; Gait; Lempel–Ziv complexity; Parkinson's disease; UPEMD-based bispectrum.

Fig. 1

UPEMD-based bispectrum algorithm

Fig. 2

Demonstration of the UPEMD-based bispectrum algorithm using a synthetic nonstationary signal

Fig. 3

UPEMD algorithm

Fig. 4

Comparison of UPEMD and EMD decompositions

Fig. 5

Effects of a-value and k-value on UPEMD-based LZC spectrum. a and b demonstrate the effects of the a-value and k-value on the LZC values across multiple frequencies, respectively. c and b are the grayscales that correspond to a and b, respectively

Fig. 6

Effect of r-value on UPEMD-based LZC spectrum. a demonstrates the effect of r-value on the LZC values across multiple frequencies and b presents its grayscale

Fig. 7

Effect of bandwidth on UPEMD-based LZC spectrum. a shows the effect of bandwidth on LZC values across multiple frequencies with a fixed center frequency. c demonstrates the effect of bandwidth with the same upper/lower bounds. b and d are the grayscales that correspond to a and c, respectively

Fig. 8

UPEMD-based LZC spectrum (abbreviated as LZC spectrum in the figure) in the logistic map. a shows the bifurcation of the logistic map. b and d demonstrate the UPEMD-based LZC spectrum of the logistic map either with a threshold restrictive condition or without. c display concept of the threshold restrictive design

Fig. 9

Effect of a-value, k-value, and r-value in the UPEMD-based power spectrum. a, b and c demonstrate the effects of a-value, k-value, and r-value on the UPEMD-based power spectrum, respectively

Fig. 10

Comparisons of linear and nonlinear signals to the UPEMD-based power spectrum (abbreviated as UPEMD spectrum in the figure) and power spectrum. a and c show the difference between the Fourier-based and the UPEMD-based power spectrum for a linear signal. b, d show the difference between the Fourier-based and the UPEMD-based power spectrum for a nonlinear signal

Fig. 11

Comparison of Wavelet scalogram and frequency spectrum between the non-best and best channels. a and b are the time–frequency scalograms for the non-best channel and best channel signal, respectively. c and d show the corresponding amplitude spectra of a and b, respectively

Fig. 12

UPEMD-based power spectrum (abbreviated as UPEMD spectrum in the figure) analysis. a and b demonstrate the UPEMD-based power spectrum of LFPs recorded from the P3 and P9 best channels, respectively

Fig. 13

Comparison of the UPEMD-based LZC spectrum between the best and non-best channels. a and c demonstrate the low-frequency range (10 ~ 45 Hz) of the UPEMD-based LZC spectrum for P1 and P5, respectively. b and d show the high-frequency range (45 ~ 80 Hz) of the UPEMD-based LZC spectrum for P1 and P5, respectively

Fig. 14

Comparison of the UPEMD-based LZC spectrum among sound states. a presents the low-frequency range (10 ~ 45 Hz) of the UPEMD-based LZC spectrum for P5 under different sound conditions. b shows the high-frequency range (45 ~ 80 Hz) of the UPEMD-based LZC spectrum for the same subject under different sound conditions. sound off 1: sound off condition before sound on; sound off 3: sound off condition after sound on