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Method for Evaluating the Number of Signal Sources and Application to Non-invasive Brain-computer Interface

Alexandra Bernadotte, Victor Buchstaber

TL;DR

A mathematical model based on polyharmonic signals is implemented to interpret the data from brain-computer interface sensors based on a mathematical model of the signal in the form of a polyharmonic signal for evaluating the number of sources, or active brain oscillators.

Abstract

This paper provides a brief introduction of the mathematical theory behind the time series unfolding method. The algorithms presented serve as a valuable mathematical and analytical tool for analyzing data collected from brain-computer interfaces. In our study, we implement a mathematical model based on polyharmonic signals to interpret the data from brain-computer interface sensors. The analysis of data coming to the brain-computer interface sensors is based on a mathematical model of the signal in the form of a polyharmonic signal. Our main focus is on addressing the problem of evaluating the number of sources, or active brain oscillators. The efficiency of our approach is demonstrated through analysis of data recorded from a non-invasive brain-computer interface developed by the author.

Method for Evaluating the Number of Signal Sources and Application to Non-invasive Brain-computer Interface

TL;DR

A mathematical model based on polyharmonic signals is implemented to interpret the data from brain-computer interface sensors based on a mathematical model of the signal in the form of a polyharmonic signal for evaluating the number of sources, or active brain oscillators.

Abstract

This paper provides a brief introduction of the mathematical theory behind the time series unfolding method. The algorithms presented serve as a valuable mathematical and analytical tool for analyzing data collected from brain-computer interfaces. In our study, we implement a mathematical model based on polyharmonic signals to interpret the data from brain-computer interface sensors. The analysis of data coming to the brain-computer interface sensors is based on a mathematical model of the signal in the form of a polyharmonic signal. Our main focus is on addressing the problem of evaluating the number of sources, or active brain oscillators. The efficiency of our approach is demonstrated through analysis of data recorded from a non-invasive brain-computer interface developed by the author.

Paper Structure

This paper contains 12 sections, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Brain Signal Model
  3. Time series multidimensional unfolding method
  4. Basic concepts and definitions of TSMU method
  5. Ranks of Time Series' Unfoldings
  6. Time Series Reconstruction
  7. Evaluating the Number of Sources (Oscillators)
  8. Algorithm for Evaluating the Number of Sources (Oscillators)
  9. Results of applying the TSMU algorithm to the model and BCI data
  10. Comparative analysis of the TSMU algorithm on model and BCI data
  11. Evaluating the Number of Sources (Oscillators) of BCI data
  12. CONCLUSIONS

Figures (4)

  • Figure 1: Schematic representation of brain oscillators and BCI sensors.
  • Figure 2: The model functions projections onto the planes of the principal components. $T = 16, T = 16, \Delta t = 1, t_1 = 0, N = 256$.
  • Figure 3: The model function $g(t) = \sin(\frac{2\pi}{16}t) + \frac{1}{2}\sin(\frac{2\pi}{8}t) + \frac{1}{3}\sin(\frac{2\pi}{6}t) + \frac{1}{4}\sin(\frac{2\pi}{5}t) + \varepsilon (t)$, where $\varepsilon (t)$ is set by the Gaussian noise generator ($D = 1, E = 0$) and the BCI signal. $n = 16, \Delta t = 1, t_1 = 0, N = 256.$
  • Figure 4: Unfoldings of the model function $g(t) = \sin(\frac{2\pi}{16}t) + \frac{1}{2}\sin(\frac{2\pi}{8}t) + \frac{1}{3}\sin(\frac{2\pi}{6}t) + \frac{1}{4}\sin(\frac{2\pi}{5}t) + \varepsilon (t)$, where $\varepsilon (t)$ is set by the Gaussian noise generator ($D = 1, E = 0$) and the BCI signal. $n = 16, \Delta t = 1, t_1 = 0, N = 256.$ Projections of time series unfoldings onto 1-2, 3-4, 5-6 and 7-8 eigenvectors' plans.