Exploring data-driven modeling and analysis of nonlinear pathological tremors

Abstract In this paper, we study various data-driven approaches to modeling periodic and quasiperiodic synthetic tremor signals. The investigation is a preliminary step towards modeling real pathological tremors. We first observe the experimental Parkinsonian tremor motion data in terms of periodicity and spectral properties, which suggests that tremor dynamics are nonlinear. The data-driven modeling is then carried out by embedding tremor signals in their nonlinear time-delay dimensions. The signals are first modeled with the model-free approach via long short-term memory recurrent neural network. Then we employ, for the first time, the dynamic mode decomposition (DMD) and its extended nonlinear version (EDMD) based on Koopman spectral analysis to model tremor time series. Compared to existing algorithms that are only effective in short-term tremor predictions, the proposed models excel in long-term predictions of periodic and quasiperiodic synthetic tremor signals. We quantitatively compare a variety of DMD/EDMD methods under different signal and modeling conditions. Our results show that by including nonlinear observation terms, EDMD can obtain better models with fewer delay embedding dimensions. We then investigate the optimal EDMD that promotes robustness and sparsity in the modeling process. We also demonstrate that EDMD models can provide dynamical information pertaining to the system underlying the modeled time series. Finally, we observe that while EDMD is limited in modeling experimental tremor signals with non-periodic features, it is still capable of approximating the spectral content of the modeled time series.