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The fundamental nature of the brain's electrical activities recorded as electroencephalogram (EEG) remains unknown. Linear stochastic models and spectral estimates are the most common methods for the analysis of EEG because of their robustness, simplicity of interpretation, and apparent association with rhythmic behavioral patterns in nature. In this paper, we extend the use of higher-order spectrum in order to indicate the hidden characteristics of EEG signals that simply do not arise from random processes. The higher-order spectrum is an extension Fourier spectrum that uses higher moments for spectral estimates. This essentially nullifies all Gaussian random effects, therefore, can reveal non-Gaussian and nonlinear characteristics in the complex patterns of EEG time series. The paper demonstrates the distinguishing features of bispectral analysis for chaotic systems, filtered noises, and normal background EEG activity. The bispectrum analysis detects nonlinear interactions; however, it does not quantify the coupling strength. The squared bicoherence in the nonredundant region has been estimated to demonstrate nonlinear coupling. The bicoherence values are minimal for white Gaussian noises (WGNs) and filtered noises. Higher bicoherence values in chaotic time series and normal background EEG activities are indicative of nonlinear coupling in these systems. The paper shows utility of bispectral methods as an analytical tool in understanding neural process underlying human EEG patterns.

Biological signals are highly complex and understandably nonlinear, may it be the firing of neurons, the beating of the heart, or breathing. The nature of the signals is dramatic and appears to be esoteric. These signals arise out of a multitude of interconnected elements comprising of the human body. These are bounded, finite, and the connections are weakly coupled across all elements. These vary in time scales ranging from nanoseconds for molecular motion to gross observable behavior in terms of days and years. This implies that biological systems are nonstationary [

Electroencephalography (EEG) is the recording of brain cortical electrical activity from electrodes placed on the scalp. The signals are also recorded subdurally or directly on the cortex and are called Electrocorticogram (ECoG). The dynamics and the structure patterns of both EEG and ECoG are similar. EEG signals are used in seizure detection, organic encephalopathies, monitoring anesthesia, and for the determination of brain death. There are perhaps 10^{5} neurons under each square millimeter of the cortical surface. Scalp EEG measures space-averaged activity of 10^{7} or more neurons implying a source area of at least a square centimeter. EEG signals are quasistationary not exceeding 2–4 seconds in eye-open state or 20–30 seconds in deep sleep. The stationary condition varies with fleeting attention, and thus analytical methods based on the assumption of the stationarity of the system are superfluous. Signal prediction, therefore, is fallacious other than the mere detection of trends. Under these circumstances, linear stochastic models remain the primary method for the analysis of biological time series due to their robustness, simplicity of interpretation, and their apparent association with rhythmic behavioral patterns in nature. The Fourier spectrum is trivial to linear methods. The linear algebraic techniques and spectral estimates over years have shown good correlation as approximation for inherently nonlinear functions that are biological data [

Over decades of research publications, the application of nonlinear time series methods involving invariant measures such as Correlation Dimension (D_{2}), Lyapunov Exponents _{1}), have not gained wide acceptance in biomedical field. The measures do not provide real insight into the biological phenomena due to their inherent high dimensionality along with long- and short-range interactions within these systems [_{2}, D_{1}, and _{2} values in the range of 2–4 in seizures and deep sleep. Similar D_{2} values are also found in normal healthy individuals in awake state [

In order to elicit better inference of EEG time series data, we have extended the Fourier transform to bispectral estimation using higher-order (third) moment [

Any process is a linear process with respect to its second-order statistics (power spectrum). The autocorrelation sequence does not give any evidence of nonlinearity. In contrast, higher-order cumulants can give evidence of nonlinearity (i.e., bispectrum for quadratic interactions). Polyspectrum estimators are the natural generalizations of the autocorrelation function, and cumulants are specific nonlinear combinations of these moments. The power spectrum does not carry information about phase which can be recovered from higher-order polyspectra. The use of higher-order moments nullifies all Gaussian random effects of the process, and the bicoherence can then quantify the degree of the remaining nonlinear coupling. The bispectrum and its normalized derivative, the bicoherence, describes the components of a time series that deviates from a Gaussian amplitude distribution. Bispectra have been used to examine various physical time series data including plasma physics and ocean waves [

Fourier transform of

Power spectrum of (

The natural estimate of the bispectrum (

The bispectrum can be also written as (

The bicoherence or the normalized bispectrum (

When the analyzed signal exhibits structure of any kind whatsoever, it might be expected that some phase coupling occurs. Bicoherence analysis is able to detect coherent signals in extremely noisy data, provided that the coherency remains constant for sufficiently long times, since the noise contribution falls off rapidly with increasing

The Lorenz and the Mackey-Glass systems are examples of classical nonlinear and chaotic systems. These are included in the study for the purpose of comparison with linear systems (WGN, filtered noises) and EEG with a view to demonstrate non-Gaussian and nonlinear characteristics of EEG. The time series of the Lorenz (

The time series of the (a) Lorenz (

The plots show time series data of (a) 1.5 Hz low-pass filtered WGN, (b) 3 Hz low-pass filtered WGN, (c) 9 Hz low-pass filtered WGN, (d) 30 Hz low-pass filtered WGN, (e) 300 low-pass filtered WGN, and (f) White Gaussian noise.

The plots show experimental time series data of various EEG activities: (a) delta EEG activity, (b) theta EEG activity, (c) alpha EEG activity, (d) beta EEG activity, and (e) indeterminate EEG activity.

The bicoherence plots of classical chaotic systems: (a) Lorenz map (

Lorenz map (

Mackey-Glass systems

Bicoherence plots of (a) 1.5 low-pass filtered WGN, (b) 3 Hz low-pass filtered WGN, (c) 9 Hz low-pass filtered WGN, (d) 30 Hz low-pass filtered WGN, (e) 300 low-pass filtered WGN, and (f) White Gaussian noise.

The bicoherence plots of (a) alpha EEG activity, (b) beta EEG activity, (c) theta EEG activity, (d) delta EEG activity, and (e) indeterminate EEG activity.

Table

Time series | |||||
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Lorenz system | 8469.61 | 0.0 | 589.87 | 46.87 | 301.35 |

Mackey-Glass system | 5765.88 | 0.00 | 408.36 | 38.49 | 203.02 |

White Gaussian noise | 2.07 | 1.00 | 0.08 | 2.25 | 0.07 |

Filtered noise 1.5 Hz | 270.08 | 0.00 | 16.31 | 8.67 | 9.93 |

Filtered noise 3 Hz | 627.06 | 0.00 | 46.60 | 12.99 | 22.74 |

Filtered noise 9 Hz | 359.46 | 0.00 | 19.83 | 9.95 | 13.20 |

Filtered noise 30.0 Hz | 120.05 | 0.00 | 5.80 | 5.89 | 4.40 |

Filtered noise 300 Hz | 55.68 | 0.21 | 0.61 | 4.03 | 1.84 |

Alpha EEG activity | 9731.75 | 0.00 | 390.91 | 35.75 | 175.05 |

Beta EEG activity | 34794.08 | 0.00 | 1407.53 | 67.65 | 628.16 |

Theta EEG activity | 4442.43 | 0.00 | 118.94 | 23.49 | 75.33 |

Delta EEG activity | 14962.96 | 0.00 | 602.83 | 44.29 | 269.02 |

Indeterminate EEG activity | 3178.88 | 0.00 | 114.14 | 20.06 | 54.83 |

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Higher-order statistics (spectra) and their application to various signal processing problems are relatively recent. There may be much more information in a stochastic non-Gaussian or deterministic signal than conveyed by its autocorrelation or power-spectral estimates. The higher-order spectra which is defined in terms of the higher-order moments or cumulants of a signal may contain this additional information. The higher-order moments are the natural generalizations of autocorrelation, and the cumulants are specific nonlinear combinations of these moments. The

For a linear process, the bicoherence is a nonzero constant. If the bispectrum is Gaussian distributed, we know that the squared bispectrum is chi-square distributed with two degrees of freedom. If the estimated bispectrum is zero, then the statistic of bicoherence is a central chi-square random variable with two degrees of freedom. The squared bicoherence is summed over m points in the nonredundant region. Then resulting statistic is chi-square distributed with 2 m degrees of freedom. The statistical test determines the consistency of bicoherence values with a central chi-square distribution. This method is suitable for extracting information about the structure of the signals as it preserves nonminimum phase information. The higher-order spectra carry phase information which is ordinarily suppressed in power spectral estimate. Extensive studies in the field of signal processing have generated information on the input-output relationships of linear systems through autocorrelation and power spectrum. However, no definite information is available on the input-output relationship of nonlinear systems to stochastic excitation and each type of nonlinearity is treated as a special case [

The present investigation uses the above concepts of higher-order spectral (bicoherence) estimates to obtain Hinich statistics for Gaussianity and linearity. The aim of the study is to contrast the Hinich statistics and bicoherence for well-understood classical chaotic system, WGN, filtered noises, and the normal background EEG activities. The results are shown in Table

The bispectral analyses of chaotic systems are found to be characteristically non-Gaussian and nonlinear. The chaos hypothesis in biological systems is based on a finite correlation dimension and a positive dominant Lyapunov exponent. A number of technical problems, however, confound the estimation of these measures. Even the instrumentation at the recording stage may render the application of nonlinear dynamics to these signals invalid. The use of analog filters during signal capture and digital filters in its analysis may account for spurious D_{2} or positive dominant Lyapunov exponent values [

The Hinich statistics (Table

The EEG time series have the lowest probability

In our detection of nonlinearity or absence of linear stochastic mechanism, we have shown that the higher-order spectra can reliably distinguish chaotic signals and EEG rhythms from the filtered noises. The filtered noises in the lower passbands show linear stochastic properties in the distribution of their bicoherence values. The results of Hinich statistics and bicoherence estimates indicate that EEG rhythms have similar properties as those of the chaotic time series. The EEG signals are unequivocally non-Gaussian and nonlinear in character. In addition, the bicoherence patterns in the nonredundant regions of the EEG time series are similar to chaotic time series, reflecting quadratic phase coupling. The bispectrum preserves nonminimum phase information of a signal and outputs zero spectrum for linear mechanism. Therefore, the bicoherence statistics of nonredundant region of the spectra (Hinich statistics) may be suitable for detecting hidden structure in signals [