# Analysis of Electroencephalograms Based on the Phase Plane Method
## 3 Determination of the Signal-to-Noise Ratio Based on the Phase Portrait
The alpha rhythm is characterized by a frequency in the range of \(8\div 13\) Hz, with an amplitude up to \(100\leavevmode\
obreak\ \upmu\)V. It is registered in \(85\div 95\%\) of healthy adults examined. It is best expressed in the occipital regions; in the anterior direction, its amplitude gradually decreases. Healthy people are characterized by a relatively narrow range of the \(\alpha\) rhythm. The \(\alpha\) rhythm has the greatest amplitude in a state of calm, relaxed wakefulness. Its amplitude varies significantly over time. Spontaneous changes in amplitude, the so-called \"spindles\", are observed quite regularly, expressed in alternating increases and decreases in amplitude. With an increase in the level of the functional activity of the brain (intense attention, intense mental work, etc. ), the amplitude of the \(\alpha\) rhythm decreases, and often, it completely disappears. High-frequency irregular activity appears in the EEG. The beta rhythm is characterized by a frequency in the range of \(14\div 40\) Hz and an amplitude up to \(15\leavevmode\
obreak\ \upmu\)V. The \(\beta\) rhythm is best recorded in the area of the anterior central gyri, but also extends to the posterior central and frontal gyri. Normally, it is very weakly expressed and usually has an amplitude of \(3\div 7\leavevmode\
obreak\ \upmu\)V. If there are artifacts, they can completely camouflage themselves. The \(\beta\) rhythm is associated with motor cortical mechanisms and produces an extinction reaction in response to motor activation. When moving, the \(\beta\) rhythm disappears in the zone of the corresponding cortical projection. In Figure 6, the EEG of an healthy adult person is shown. Since the frequency band \(0.5\div 40\) Hz is informative, the signal is filtered by a bandpass filter with cutoff frequencies of \(0.4\) and \(45\) Hz. This figure clearly shows the nonstationarity of the EEG. The spectrum of this implementation is shown in Figure 7. It is incorrect to use the spectral method in the case of a nonstationary signal, but, nevertheless, it has found wide application in electroencephalography, because it allows obtaining an approximate idea of EEG rhythms. In this figure, a high level of the \(\alpha\) rhythm can be observed, which is natural in a state of quiet wakefulness. Other rhythms are weakly expressed. The EEG is essentially uniform across the entire brain and symmetrical. However, the functional heterogeneity of the cortex leads to peculiarities in the electrical activity of different areas of the brain. However, due to a fairly gradual transition from one functional zone of the cortex to another, the change in EEG types along the length of the cortex occurs gradually. Figure 6: EEG of an awake adult. One of the main criteria when analyzing the EEG is symmetry. EEG symmetry means a significant coincidence of frequencies, amplitudes, and phases of symmetrical areas of the two hemispheres of the brain. Based on the prevalence of asymmetric pathological oscillations, they predict hemispheric disorders, when the changes cover the entire hemisphere, and focal ones, which are usually expressed under one electrode. The frequency-amplitude characteristics of the EEG described above are characteristics of a healthy, awake person. Pathological manifestations in the EEG include the appearance of slow rhythms: theta (\(\theta\)) and delta (\(\delta\)). The lower their frequency and higher the amplitude, the more pronounced the pathological process is. The theta rhythm is characterized by a frequency of \(4\div 6\) Hz, and the amplitude exceeds \(40\)\(\mu\)V, reaching \(300\)\(\mu\)V or more in some pathological conditions. The \(\delta\) rhythm is characterized by a frequency of \(0.5\div 3\) Hz, and the amplitude is the same as that of the \(\theta\) rhythm. In the EEG of an awake adult person, there may also be \(\theta\) and \(\delta\) rhythms of short duration and with an amplitude not exceeding the amplitude of the \(\alpha\) rhythm.