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The method of electroencephalography, or registering the brain electrical activity from the head skin surface, is now the most feasible way to non-invasive study of the brain processes underlying mental activity. The EEG is also an effective tool for brain disease and traumatic lesion diagnostics.
Since the EEG is a very complex signal, a great variety of mathematical methods was applied to it in attempt to extract more information concerning the brain functioning. Most of those methods, however, can provide reliable results only if the inherent "biological" properties of this signal, and especially its high nonstationarity, are taken into account.
The critical review in the beginning of this Chapter shows that even the methods suggested by various authors for treating the EEG as a nonstationary process, especially the parametric methods, are often not well suited for this signal, because they possess an internal contradiction of constructing models without considering the nonstationarity of the intervals to which the model is fitted.
This is why the nonparametric approach to the nonstationary processes suggested and developed by Brodsky and Darkhovsky appeared to be a valuable tool for the investigation of brain functioning on the basis of EEG analysis. The results of the application of change-point analysis to the human EEG described in this Chapter demonstrated the relationship between the piecewise stationary structure of this signal and the brain functioning. The segmentary structure of the electrical activity from different brain sites was shown to be considerably synchronized, and the degree of synchronization appeared to be dependent on various factors of brain activity. Thus, a new phenomenon of the "operational synchrony" was described (Kaplan et al. 1995; Kaplan et al. 1997a; Kaplan et al. 1997c).
From the methodological point of view, effective detection of the change-points on the basis of nonparametric approach by Brodsky and Darkhovsky opened a way for the development of a specific analysis of synchronization between signals, based on the methods of estimation the coupling between point processes (Kaplan et al. 1995).
The experimental evidence described in this Chapter also makes possible a novel approach to the understanding of the basic problems of brain multivariability.
Even now, in the end of the XXth century, when the artificial computing systems are just about to win man's chess crown, the natural brain keeps on surprising us by its unimaginable potential multivariability. More than 100 milliards of neurons, multiplied by 5--10 thousands of interneuronal contacts (synapses) at each neuron, and once again multiplied by tens of operational structures for each of these contacts, this is the basis which underlies the superastronomical variety of possible brain state combinations (Arbib 1976; Berns 1968).
It is evident, however, that effective control of body functioning requires strict constraining of the number of degrees of freedom at all levels of the brain structural and functional hierarchy. This is the most apparent in the control of movement. "To conquer the superfluous degrees of freedom of the moving organ, i.e., to transform it into a controllable system, just this is the main task of the co-ordination of movements" (Berns 1968). Reducing the superfluous degrees of freedom probably becomes most crucial while the cognitive activity invisible from outside is going on, when the greatest number of the neuronal systems should be involved and the lowest number of control parameters is available.
This problem of constraining the number of degrees of freedom is, most probably, very difficult to solve in a framework of brain states continuum. The abrupt reduction of the number of degrees of freedom during cognitive operations could be easily achieved if the dynamical organization of the system is constrained by a finite number of its metastable states. It also can make much more simple the interactions between elementary neuronal systems: by synchronizing the short-term periods of stabilization of "macroscopic" system variables, these elementary systems acquire the ability of interactive information exchange necessary for making an "agreed" decision.
Taking also into account the hypothesis of the hierarchy of segmental description on different time scales (Kaplan 1998) (see above in subsection 7.3.2), it can be suggested that the piecewise stationary structure of brain activity corresponding the EEG piecewise stationary structure is the framework in which a variety of fast "microscopic" variables of a large system can obey the "macroscopic" operational structure of brain activity.
This idea is based on the "slaving principle" by Haken (1977) which probably enables a potentially multivariative system to reduce greatly its number of degrees of freedom at local time intervals in accordance with order parameters.
Thus, the spatial and temporal hierarchy of discrete metastable states of
neuronal assemblies can serve as a basis of functioning of such a potentially
multivariable system like the brain. These metastable states, in their turn,
must appear in the EEG in the form of its piecewise stationary organization
which can be studied by means of the change-point analysis.
We thanks Dr. J.Roeschke (University of Mainz) for supplying the sleep