# Analysis of Electroencephalograms Based on the Phase Plane Method
## 3 Determination of the Signal-to-Noise Ratio Based on the Phase Portrait
Data processing has currently become one of the most-active fields of neuroscience, in particular in the processing of electroencephalograms (EEGs). An EEG is a broadband oscillatory process in which, however, certain dominant harmonic constituents exist. The total EEG reflects the functional activity of the brain. Changes in the brain's functional activity are quite unambiguously reflected in the EEG. Therefore, in modern studies, EEG indicators are among the most-important in assessing the level of functional activity in clinical neurophysiology and psychophysiology [20,21]. The spectrum of the EEG is quite complicated. The classification of EEG rhythms by some basic ranges is introduced. The concept of \"rhythm\" in the EEG refers to a certain frequency band matching to a certain brain condition. The main rhythms are given in Table 3 [20,21]. The most-widely used method in clinical practice is the study of the signal spectrum, which is, to some extent, comparable to the manual amplitude-frequency analysis of the EEG. This method is highly informative and visual when assessing EEG components. As in manual analysis, the problem of determining the ratio of various rhythmic components in a complex EEG and determining their individual expression is solved. For this purpose, the Fourier transform (FT) is used. The FT is used when it is necessary to visually assess the holistic dynamics of the EEG during long-term observation or functional tests. In this case, the spectra for successive epochs are displayed, drawn one after the other. But, unfortunately, this method of EEG analysis does not allow diagnosing many brain diseases, for example Parkinson's and Alzheimer's diseases. Therefore, the use of other methods of EEG analysis is relevant. The ability to accurately quantify EEG parameters allows for a more-detailed and objective assessment of dynamic changes in the functional state of the brain, which turn out to be more-informative than assessing the differences between normal and pathological conditions using manual research methods. Using the power spectrum, you can easily obtain a picture of the EEG distribution over the rhythms and determine the dominant rhythm and the dominant frequency of both the entire EEG and each individual rhythm. The use of numerical methods in studying the rhythms of the biopotentials of the brains of patients with local lesions made it possible to obtain quantitative, objective assessments of EEG disorders. In addition, the use of numerical methods expands the diagnostic and, in cases of severe conditions, prognostic capabilities of EEG. The fact that EEG is a broadband oscillatory process, in which, however, certain dominant harmonics can be identified, is proof that brain neurons synchronize their work \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{EEG rhythms of an awake adult} \\ \hline rhythm & Frequency (Hz) & Amplitude (\(\mu\)V) \\ \hline \(\alpha\) & \(8\div 13\) & up to 100 \\ \hline \(\beta\) & \(14\div 40\) & up to 15 \\  & & normally \(3\div 7\) \\ \hline \multicolumn{3}{c}{Types of pathological activity for an awake adult} \\ \hline \(\delta\) & \(0.5\div 3\) & exceeds \(40\)\(\mu\)V, \\  & & reaching \(300\)\(\mu\)V or more \\  & & in some pathological \\  & & conditions \\ \hline \(\theta\) & \(4\div 6\) & \(-\ll-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: EEG rhythms. with each other. Otherwise, the sum of many thousands or even millions of electrical potentials of individual nerve cells would constitute quasi-white noise. Changes in the functional activity of the brain are reflected quite clearly in the EEG. The connection between these changes and EEG changes is so close that, in modern studies, EEG indicators are one of the most-important in assessing the level of functional activity in clinical neurophysiology and psychophysiology. The EEG rhythms of an awake adult person include the alpha, beta, and gamma rhythms [22; 23; 24; 25].