This is habilitation dissertation thesis on the importance of EEE and LPF phase for understanding of brain state dynamics of possible quantum coherent macroscopic phase

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"Studies of the dynamics of changes in brain functional states by methods of
digital phase analysis of field potentials"
Thesis · August 2023
DOI: 10.13140/RG.2.2.33695.12962
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Jerzy Zbigniew Achimowicz
Warsaw Medical Academy
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2. 1
"Studies of the dynamics of changes in brain functional
states by methods of digital phase analysis of field
potentials".
Jerzy Z. Achimowicz
Department of Flight Safety,
Military Institute of Aviation Medicine, 54
Krasinskiego St. , 01-755 Warsaw
jachimow@wiml.waw.pl
Table of Contents:
1. Introduction.
1.1. Methodological aspects of brain research at the mesoscopic level.
1.2. Current views on the role of field potentials in brain processes.
2. Research objective and study material.
3. Methods for analyzing the phase features of brain electrical activity signals.
3.1. Linear methods - phase synchronization (coherence)
3.2. Nonlinear methods - quadrature phase synchronization (bicoherence).
4. Studies of the spatio-temporal dynamics of the processes of synchronization of
the activity of local populations of neurons of the cortex and limbic system
5. Investigating the dynamics of nonlinear interactions between neuronal
populations in the limbic system and cortex.
6. Implications for modeling the dynamics of changes in brain action states and
the role of field potentials in modulating the excitability of neuronal networks.
8. Conclusion.
9. Literature,
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3. 2
1. Introduction.
1.1. Methodological aspects of brain research at the mesoscopic level.
The research presented in this paper deals with methods for analyzing signals of electrical
brain activity, recorded extracortically, to assess the dynamics of functional states of the central
nervous system.
Measurement and analysis of electrical signals in neural tissue, as a method of assessing the activity
of the brain's neuronal networks, are particularly useful for these purposes. Compared to other
indicators of brain activity, electrical potentiates have high temporal resolution and potentially high
spatial resolution. Visualization of the brain's electrical activity using, for example, microelectrode
arrays , provides information unavailable with very expensive methods such as positron emission
tomography - PET or magnetic resonance imaging - MRI.
Despite this, the prevailing view in electrophysiology is that the main method of studying
central nervous system (CNS) function, is the study of neuronal action potentials. Even in the
current NIMH publication, in the chapter on "Recommendations for Future Research" (Koslow et
al. , 1995), one can only find the statement;
"... there is a need to develop new electrode arrays and data analysis techniques to record and
analyze spike train activity in real time from large ensembles of simultaneously recorded
neurons in behaving animals...."
This demonstrates an underestimation of the role of field potentials as synthetic indicators
of the spatially and temporally synchronized activity of a pool of neurons. Field potentials are
generally associated with EEG signals, recorded in clinics for diagnostic purposes such as epilepsy
or in evoked potential studies as a tool for localizing receptor-specific areas of the cortex. For many
years, there was also the view that field potentials are perceived through volume conduction and are
therefore nonspecific and do not provide information about the location of electrically active
neuronal ensembles. The absence or variability of the relationship between Local Field Potentials
(LPF) and cellular activity, was regarded as irrelevant to the former.
My research shows the falsity of these colloquial opinions. LFP field potentials can be an
indicator of the dynamics of the functional states of the central nervous system and, in
particular, the degree of synchronization of the excitation and inhibition processes of the brain's
neuronal networks. They contain a large amount of information due to their local nature and
high temporal dynamics, which changes depending on the functional state of the CNS.
Unlike classical EEG signals and neuronal activity recorded intracellularly, which give
information at the macro- or micro-scale level of the brain, local field potentials provide
information at the intermediate (mesoscopic) level. They give information about the level of
synchrony in a population of neurons from the immediate vicinity of the recording electrodes. The
lack of constant LFP-neuronal activity correlations, indicates that LFPs contain new information,
with a different level of CNS organization.
Moreover, the results of these studies indicate that by using appropriate methods of
digital analysis of signals, such as phase feature analysis, especially with the help of nonlinear
spectral analysis, it is possible to draw conclusions relevant to modeling the dynamics of brain
processes at the level of neuronal ensembles. This includes the hypothesis of a possible active
role of field potentials as a factor modulating the excitability of neuronal networks and the
discrete nature of changes in the level of cooperativity (synergism) of neuronal pools. It may
point to the important role of mechanisms involving
"competition" of different neuronal pathways (pathway switching) in the processes of conscious
stimulus perception.

4. 3
Only recently has more attention begun to be paid to oscillatory field potentials and, in
particular, to cortical rhythms and those observed in the hippocampus.
In recent years, the electrophysiology of the nervous system, thanks to the development of
the technique of receiving these signals by means of subdurally or intracerebrally implanted
multielectrodes and the use of computer-based modern methods of analyzing multidimensional
signals, has again experienced rapid development. Particular attention is being paid to the
functional role of rhythmic and spatially synchronized activity of local neuronal populations, the
very sensitive measures of which are frequency domain correlation coefficients. They can refer to
measures of linear correlations like the coherence function or nonlinear ones like the bicoherence
function. They describe the phasic features of two or more field potential signals, which, due to
their local character, can serve as indicators of the coordination (spatio-temporal dependence) of
the excitation and inhibition processes of neurons of different brain structures underlying brain
functioning (Konorski, 1964).
Understanding the neuronal mechanisms of changes in these indicators studied by me,
accompanying behaviorally controlled functional states of the body, can be, in my opinion, a
path to a better understanding of brain function in normal and pathological states.
Within the recently established field of so-called "Cognitive Neuroscience" - cognitive
neurophysiology, the common view is that information processing by the brain takes place in vast
areas of the cortex, in such an integrated way as to realize coherent (coherent) functions of
perception and behavior. The problem of integrating information processing on a brain-wide scale
is attempted to be solved within the framework of the model of temporal-spatial organization of
functional connections of small populations of neurons (local networks). It is believed that the
temporal synchronization of the activity of neuronal ensembles, manifested in the phase
synchronization (coherence) of local cortical field potentials, is the physiological basis of these
integration functions. Their realization allows flexible adaptation to the changing environment and
enables the realization of goal-oriented behavior (Bressler, 1996).
Before discussing the results of my work, conceptually close to the research paradigm
discussed above , I will briefly outline current views on the functional significance of so-called
"bioelectric brain rhythms."
Subsequently, methods of digital description of their interrelationships will also be
discussed. The methods used by the author focus not on the amplitude characteristics of the LFP
signal, which are an indicator of the intensity of the processes of excitation or inhibition of neurons
or the number of neurons participating in these processes, but on their temporal interrelationships
expressed by their phase characteristics. This ignores an additional uncontrolled variable in the
form of the amplitude of the LFP signal. Its big disadvantage is the generally unknown variance,
related to the difficulty of obtaining unambiguous (repeatable) localization of electrodes in the
studied anatomical structures. Uncontrolled variance also arises, related to the method of
measurement; including, among other things, the geometry and resistance of the electrodes. This
indeterminacy makes it difficult to compare the results of similar tests performed in different
laboratories.

5. 4
1.2. Current views on the role of rhythmic field potentials in brain processes.
In studies of the nervous system, researchers have long encountered a range (hierarchy) of
rhythms with different frequencies (see review paper, Bullock, Achimovich 1994).
From the point of view of the theory of system dynamics, the occurrence of rhythms, so frequent in
biology, is most often a manifestation of the activity of regulatory systems with so-called feedback
loops. In contrast to control systems , which do not exhibit oscillations, such systems and especially
systems in which non-lyrical oscillations are observed (e.g., so-called chaotic oscillations) have
certain advantages. Their advantage over other systems lies in their ability to respond very quickly
to signals from the environment, as well as their great ability to coordinate (synchronize) vital
electrical or chemical processes observed in nervous tissue.
At present, there is no doubt that rhythmic changes in potentials recorded in neural tissue
extra-cellularly, the so-called field potentials, are a manifestation of either synchronized activity
(action potentials) of a certain population of neurons or synchronized but subthreshold fluctuations
of neuronal potentials of cellular membranes. Synchrony is achieved either by excitation of one
network by another (driving) or mutual coupling.
Often this type of coupling occurs between cortical and subcortical structures. Currently,
however, the rhythms observed in the cerebral cortex of mammals, particularly primates, which are
associated with integrative brain functions, are of greatest interest to researchers. It should be
remembered, however, that rhythms are already observed at a lower level of brain organization, e.g.
in receptors such as the retina or in subcortical structures , where they have been observed for
years, and not only in mammalian brains.
Some overview of the types of these rhythms and their division in terms of their temporal
relationship to stimuli (so-called evoked rhythms) is contained in a comparative work (Bullock and
Achimovich 1995).
Interactions between neurons, or groups of neurons, can also be modulatory (excitatory or
inhibitory) and the physical carrier can be the connection of neurons by chemical or electrical
synapses or even field (extrasynaptic) effects (Achimovich, Bullock, 1994).
Traditionally, it was thought that the rhythms observed in the cerebral cortex, such as the
alpha rhythm characteristic of the visual cortex, are indicative only of the passive state of the CNS
(idle state); for the opening of the eyes or the application of a stimulus such as visual stimulus
causes their disappearance (desynchronization of EEG activity). This interpretation is related to the
term "activation" (arousal), introduced by Moruzzi and Magoun (1949), in their study of the effects
of brainstem (reticular formation) stimulation on cortical electrical activity. Current studies of
cortical rhythms (Steriade et.al. 1996) indicate that there are two types of cortical rhythms:
• - Slow rhythms, which also include components of evoked potentials, characterized by polarity
reversal (phase reversal) as a function of depth in the cortex in the range of 0.25 to 0.5 mm and
are characterized by synchronization over large distances, such as the order of centimeters
between electrodes on the scalp ( in the EEG) and whose amplitude actually decreases during
the stimulus response (arousal).
• - Fast rhythms with frequencies of 30-40 Hz, (the so-called gamma rhythms), which do not
disappear, but are actually induced by activation of ARAS (Ascending Reticular Activating
System), and thus also occur in the awake state (awake) and in the REM phase of sleep, and also
appear during the depolarizing phases of slow-wave sleep. They are not observed in the white
matter and

6. 5
They do not exhibit polarity reversal. Their coherence (synchronization) is spatially restricted
generally to the cortical and near-neighbor columns.
However, the two types of rhythms are not independent, since slow rhythms indicate a
periodic predominance of either inhibitory or excitatory processes , and are probably an indicator of
subliminal fluctuations of inhibitory and excitatory influences, which may, for example, modulate
interactions between cortical and thalamic neurons, facilitating or inhibiting the appearance of fast
oscillations evoked by external or internal stimuli, for example, in the REM phase of sleep.
Currently, the occurrence of fast (in the range of 20-60 Hz) spontaneous rhythms in various
functional states associated with enhanced attention has been found experimentally (Lopes da
Silva, 1970 , Bouyer et.al. 1981, Freeman and Van Dijk, 1987, Murthy and Fetz, 1992) or evoked
rhythms associated with stimulus perception in humans and animals (Eckhorn et. Al. 1988, Gray
et.al, 1989, Pantev et.al 1991, Jagadeesh et.al, 1992, Llinas and Ribary, 1993, Desmet
and Tomberg, 1994, Gray 1994).
Since then, a number of papers have been published explaining the mechanisms of
rhythmic bioelectrical activity of the subcortical structures and cortex of the brain, as well as the
mechanisms of its synchronization and desynchronization (see, for example, Steriade et al. 1990).
Particular importance is attributed to gamma rhythms (30-50 Hz) in the mechanisms of perception
of moving visual stimuli ("feature binding") not only in anesthetized cats but also in monkeys
under natural conditions (see in a collection of papers edited by Buzsaki et al. , 1994). Reports of
the role of these rhythms in attentional processes (Murthy and Fetz, 1989), (Niebur et al., 1993)
have also been confirmed in work conducted in Poland (Wróbel et al. 1994).
• More and more experimental data support the hypothesis of modulatory effects of slow
oscillations on the activity of neuronal networks. This hypothesis was initiated by works that
demonstrated the existence of electric field effects on the activity of many neurons, ganglia or
cortex ( Terzulo I Bullock 1956, Watanabe and Bullock 1960, Korn and Faber 1980, Faber and
Korn 1983) and the relationship between slow field potential changes and changes in EEG and
unit activity ( Laming 1983, Laming and Ewert 1984, Laming et al. 1984, 1991). Some
researchers claim that slow potentials can modulate the excitability of cortical neuronal
networks (Muller et.al. 1994, Elbert, 1992). Potentials with negative polarity such as CNV
(Contigent Negative Variation) and the so-called "readiness potential" (Bereitschaftpotential)
indicate the predominance of excitatory processes . In contrast, potentials of positive polarity ,
such as the P300 component of visual evoked potentials, indicate the predominance of
inhibitory processes of "disfacilitation" or even induce it. There are more and more supporters
of the long-known hypothesis that, synchronous oscillations are not just an indicator of
synchronized neuronal activity, but that they play an active role in higher brain functions.
(Bullock, 1993), (Elbert, 1992).
• Currently, the view that the experimentally observed high dynamics of field potentials is a direct
manifestation of the processes of dynamically synchronized activity of neuronal ensembles,
which can actively influence the resultant activity of the cortex (cortical output), is becoming
increasingly popular. In Bressler's review paper (Bressler 1995), the author states that the
synchronization of the activity of neuronal ensembles between, often very distant, areas of the
cortex can be a mechanism for selectively choosing the active neuronal ensembles necessary for
processing information arriving from the environment. This simultaneous activation of multiple
neurons may be necessary to activate, through projection pathways, the target output neurons,
due to the synchronous excitation of multiple synapses on their dendritic trees. Such processes
may underlie the coordinated transmission of multimodal cortical information, the existence of
which has been postulated both in the hippocampal area (Eichbaum, Otto I Cohen, 1994 ) and in
the basal nuclei (basal ganglia), (Graybiel, Aosaki, Flahert I Kimura, 1994).
2. Research objective and study material.

7. 6
2.1. Purpose of the study
• The main research problem I have been concerned with is the role of field potentials in the
functioning of the central nervous system and, in particular, the answer to the question :
Are the rhythms of bioelectrical activity of the brain only indicators of brain processes or do
they play an active e.g. modulatory role in the processes of stimulus perception ?
• The purpose of my work, was to study the dynamics of changes in CNS field potentials in time
and space and, in particular, electrical activity of an oscillatory nature (rhythms) and their role in
the regulation of functional states (functional) of the brain. I attempted to solve this problem
within the framework of the traditional phenomenological approach, by studying their
relationship to CNS functional states, such as wakefulness, sleep or seizure activity in clinical
cases. Patients diagnosed with epilepsy, in whom, due to the ineffectiveness of drug treatment, a
decision was made to treat them surgically, constitute a unique group of patients who have been
implanted with either subdural or intracerebral electrodes to localize the epileptic focus.
• For ethical reasons, I did not use planned experiential manipulation of the behavioral state of the
aforementioned patients, but limited myself to selecting data from continuous 24-hour
recordings, defined by an experienced clinician neurologist as different phases of sleep, seizure
activity or wakefulness.
• I was also interested in comparing the spatial structure of field potentials recorded by classical
EEG and micro-EEG from subdural and intracerebral electrodes to determine the role of volume
conduction in the generation of these local field potentials.
• In order to verify the hypothesis of the active role of field potentials in brain function, I also
studied, using methods of nonlinear spectral analysis, the relationship between components of
signals of different frequencies. The finding of modulatory interactions of these components,
manifested in significantly different from zero in their bikohrences, may provide a rationale for
accepting the validity of this hypothesis.
• I also wanted to verify the thesis about the particular usefulness of the phasic features of the
signals of local field potentials, spontaneous activity, as indicators of the functional state of the
CNS. As part of my thesis, I demonstrated the usefulness of phase for assessing the variability
of evoked potentials received from the cranial surface by EEG. I also showed that the variability
depended on the degree of phase synchronization of spontaneous brain electrical activity and, in
particular, the alpha and theta rhythms. This fact suggested the potential usefulness of phase
analysis also of local field potentials for determining the functional state of the CNS.
The hypothesis of the unique role of phase synchronization processes of rhythmic field
potentials has long been known, but has not been fully experimentally verified,

8. 7
probably mainly for methodological reasons. Evaluating phase synchronization for such a
dynamically changing (non-stationary) signal as EEG is only possible with modern digital signal
analysis methods and adequate computer processing power.
Dr. Ross Adey of the UCLA Brain Research Institute was undoubtedly a forerunner of
research showing the importance of phase measurement in brain research. He presented, as early as
1963, the results of a study on changes in the phase synchronization of theta rhythm in limbic
structures in monkeys during the teaching of spatial orientation in a maze (Adey and Walter, 1963).
In the course of these studies, for example, reproducible differences were observed in the phase
relationships between EEG signals recorded from electrodes chronically implanted into the
posterior and ethmoral areas of the monkeys' hippocampus during correct and incorrect solutions to
a spatial discrimination task in a T maze. To investigate the sequence of synchronization processes
in the neuronal network under study, the delays of the theta rhythm were quantified by measuring
the phase shifts using digital signal analysis methods. To assess the stability over time of these
delays, so-called coherence functions were also calculated, which is the equivalent of the
correlation function in the frequency domain (Adey, 1963).
It was possible to conduct this research because UCLA had a computer with sufficient
computing power at the time, and because Dr. Tukey of Princeton University provided a program
developed for Convair in San Diego t o study the vibrations of ballistic missiles.
These studies best illustrate the fact that, the role of computers in neuroscience and
especially in electrophysiology , as a tool for the quantitative analysis of rhythms and their
interactions is very large because it allows the measurement of parameters not measurable in the
classical way.
When I came across the methodology of EEG research in 1979, my first impression was
that, the relationship between human behavior and EEG recordings was very complex and
ambiguous.I was even more intrigued by the work (Banqet, 1973), which studied the dynamics of
the amplitude of alpha and theta rhythms during autogenic training. Although these processes
showed great inter-individual variation, the repeatability of the recordings in a given individual was
astonishingly good. This was especially true for the inter-hemispheric phase synchronization of
these rhythms, as determined by the coherence function ( Levine, Hebert et. al. , 1988). Since my
own research confirmed the aforementioned facts, I became convinced that quantitative indices of
the degree of phase synchronization of collective potentials ( EEG ) could be a sensitive indicator
of the functional state of the brain. In the studies described here, this state can be described as a
prolonged state of pure auditory attention, since the subjects achieved a state of relaxation by
repeating, without vocalizing, a single sound.
The superiority of phase parameterization over amplitude parameterization can probably
be attributed to the parallel nature of sensory information processing by the brain. The
amplitude of rhythmic field potentials is determined not only by the degree to which the activity
of individual neurons is coordinated over time, but also by their spatial orientation relative to the
recording electrode and the relative number of cells with the appropriate spatial orientation, due
to the vectorial nature of the input from individual neurons. Examination of the phase of the
signal gives information mainly about the temporal relationship between the activities of various
brain structures, excluding that uncontrolled variance characteristic of the amplitude of field
signals.
In summary, we can say that,

9. 8
The main direction of my work has been the study of the temporal-spatial dynamics of
synchronization-desynchronization (S-D) processes of neuronal populations, in various cortical and
subcortical structures, including the limbic system, its usefulness for assessing the functional states
of the central nervous system and the possible active modulatory role of field potentials in brain
function.
2.2 Research material.
In my research, I recorded signals of electrical brain activity recorded extracellularly,
which are referred to in the literature by the term field potentials or EEG. In contrast, potentials
received from electrodes in contact with the surface of the cortex are often referred to as electro-
corticogram (ECoG). Potentials from electrodes inserted into the brain are referred to in the
literature as micro-EEG or local field potentials.
Field potentials were received in humans from electrodes on the scalp, placed according to
international standard 10-20. These potentials are often referred to by the term collective potentials,
as they reflect the electrical activity of large populations of cortical neurons, mainly from areas
directly under the electrode. However, they also contain components from distant brain structures,
thanks to the effect of volume conduction. I conducted my studies using the EEG method at the
Military Institute of Aviation Medicine and the Zyrardow City Hospital.
Records from electrodes that are in direct contact with nerve tissue are mainly from
epilepsy patients, especially those in whom drug treatment has failed. In cases where surgical
treatment was decided upon, either subdural or intracerebral electrodes were inserted by removing
part of the tissue to localize the area of the brain initiating seizures (epileptic focus).
During my two-year work in the laboratory of Prof. T. Bullock (Department of
Neuroscience, University of California, San Diego), I collaborated with UCSD Medical Center
(Dr. Iraqui), where subdural electrodes were used in the form of linear and rectangular arrays
containing 8 to 64 contacts, with an active area of 2 mm square and distances between them of 10
to 5 mm. In collaboration with Yale University School of Medicine (Dr. S. Spencer and Dr.
B.Ducrow), I analyzed recordings from a linear array of electrodes 5 to 7 mm apart, implanted into
limbic structures.The position of the electrodes, extending through the body of the hippocampus to
the amygdala nucleus, was controlled by nuclear magnetic resonance imaging (MRI) and X-ray
computed tomography (CAT).
Records from electrodes implanted chronically into the cortex and hippocampus in rats, I
obtained as part of a collaboration between the Neurobiologist Unit, UCSD School of Medicine in
San Diego led by Prof. T.Bullock with Dr. Jose Gaztelu of the Medical Academy in Madrid and Dr.
Buzsaki's lab at the Center for Molecular and Behavioral Neuroscience at Rutgers University in
Newark, USA.

10. 9
3. Methods for digital analysis of phase features of bioelectric signals.
In my research, I mainly used two methods (linear and nonlinear), to study the phase
characteristics of field potential signals. These consisted of evaluating the linear phase synchronism
of the rhythms by estimating the coherence function, and nonlinear phase coupling by nonlinear
spectral analysis methods, by estimating the bi-coherence function. The coherence function is well-
known and used in engineering applications for stationary signal analysis.
In my research I used it mainly to evaluate the temporal dynamics of field potentials. This required
replacing classical estimation methods with so-called parametric methods, which allow estimating
the phase parameters of short fragments of the signal, for which the assumption of its local (short-
time) stationarity can be made. On the other hand, the bi-coherence function is rarely used in
practice, due to methodological problems such as problems of normalization (ambiguity of its
definition) and evaluation of estimation errors (statistical significance of the estimate). Therefore, I
conducted my own simulation studies, on synthetic signals with known phase characteristics, to
assess the accuracy (statistical reliability), of the analysis methods I used. I wrote most of the
computer programs in Fortran, Pascal or the meta-language of the MATLAB system (MathWorks
Inc., USA).
3.1. Linear methods;
Coherence is a very important concept in digital signal analysis, not the least of which is not
often used in neuroscience for methodological reasons related t o t h e non-stationarity of
bioelectrical brain signals. Several types of coherence are distinguished, such as temporal, spectral
or spatial, each of which has its own interpretation (Gardner, 1992). In electrophysiology, spatial
coherence is most often studied as a measure of the constancy of the phase relationship
(synchronization), between signals recorded from two electrodes, located in different brain
structures. The coherence function can be interpreted as the equivalent of the correlation function in
the frequency domain. It is a measure of the ratio of the fraction of phase-synchronized signal
energy to the total signal energy, at a given frequency. In the case of rhythm, it is a measure of the
constancy of the phase relationship between two signals. In the case of a wide-band signal (such as
coherent noise), it can be taken as a measure of the degree of linearity of the dependence
relationship between the two signals under study.
Most studies of the processes of synchronization of EEG rhythms available in the
literature deal only with the spatial aspect of coherence changes, that is, the dependence of
coherence function on the location of electrodes and on the functional state of the brain.
However, the variability of phase synchronization processes over time , tacitly assuming the
stationarity of the signals under study, which simplifies the methodology of research, has
generally been neglected.
Phase relationships between signals received from different areas of the cortex or between
cortical and subcortical structures were also analyzed to determine the strength of functional
connections between different neuronal systems. A high coherence of two field potential signals,
each of which is an indicator of discharge synchrony or correlated subthreshold fluctuations of
neuronal membrane potentials of n e u r o n a l populations in the vicinity of the recording electrode,
indicates the existence of direct neuronal projections or that the two structures under study are
synchronized by another structure (common input). This method has been used to study, among
other things, the mechanisms of rhythm generation, e.g. the role of corticocortical association
connections (Thatcher et al. 1986) (Tucker et al., 1986) including their role in cognitive processes
(Rapelsberger and Petche, 1989).

11. 10
Studies available i n t h e literature have used classical methods of estimating coherence
functions based on the Fourier transform, which have numerous limitations (Blinowska et al. ,
1985). Their main drawbacks include low frequency resolution and the need for long signal
recording times (Carter, Knapp and Nuttal). Only the introduction of modern parametric methods
(Blinowska, 1994) (Wood et al. , 1992), e.g. the autoregressive model - AR, made it possible to
estimate instantaneous values of the coherence function, but this fact was not used in
electrophysiological studies of the brain.
The use of the above-mentioned parametric methods allowed me to assess the dynamics of
changes in coherence and, therefore, changes in the strength of functional connections between
populations of neurons as a function of time, which was one of the main directions of my research.
(Achimovich, 1991). When coherence fluctuations were oscillatory in nature, I statistically
evaluated the probability of these slow rhythms using the permutation method (Bullock,
Achimovich, McClune, 1992).
3.2. Non-linear methods
In my research, I was the first to apply methods of nonlinear analysis of the phase features
of EEG signals to the study of the spatio-temporal dynamics of brain processes (Achimovich and
Bullock, 1993). These methods, based on the so-called nonlinear spectral analysis (bispectral
analysis), make it possible to determine functional relationships between rhythms of different
frequencies and, in particular, to determine the strength of interactions between rhythms of
modulatory nature. For this purpose, the bicoherence function is used to measure the so-called
quadrature phase synchronism between rhythms of different frequencies. This method makes it
possible, for example, to distinguish the independent beta rhythm (20-25 Hz) from the so-called
higher harmonics of the alpha rhythm in the visual cortex (Achimovich and Bullock, 1993),
induced by visual stimuli.
4. Study of temporal-spatial dynamics of processessynchronization (coherence) of
spontaneous activity of neuronal populations of the cortex and limbic system
As mentioned earlier, there are few reports in the literature on studies of the temporal
dynamics of synchronization-desynchronization (S-D) processes of neuronal populations from
electrodes on the scalp. Such work includes a study (Schuler et al., 1988), which demonstrated the
existence of NREM sleep phase-specific, cyclic changes in the coherence of the alpha rhythm with
a period of 4 to 20 sec (i.e., a frequency range of 0.05 Hz to 0.25 Hz).
Using the parametric AR method to estimate coherence for 1-second consecutive fragments
of the EEG signal, received from the scalp electrodes, I investigated changes in the timing of
hemispheric and intrahemispheric synchronization during TM relaxation training (Achimovich,
1991). In contrast to the resting state with eyes closed, the alpha and theta rhythms did not show a
randomly appearing high interhemispheric coherence (between homologous cortical areas) of the
spindles of these rhythms, with a duration of about 3-4 sec. but the coherence reached high values
(above 0.9) and persisted for periods of the order of 30 seconds. Moreover, in contrast to literature
reports on electrophysiological correlates of the hypnotic state (Sabouri et al., 1990),
intrahemispheric coherence (e.g., between frontal and occipital leads) also reached high, sustained
values over time, despite the much greater distance between electrodes.
The high coherence of the theta rhythm of the frontal cortex region allows us to assume that
the autogenic training (TM) state is a state of enhanced and stable attention over time. Also of
interest is the fact I found that in the studied steady-state functional state of the brain, coherence
shows some variability of a specific discrete nature. The multidimensional distributions of
coherence between frontal, occipital and

12. 11
The parietal values were discontinuous in nature. Certain combinations (patterns) of spatial
coherence values were clearly favored (Achimovich, 1991). This type of temporal variability is also
observed in the NREM sleep state referred to as alpha sleep (Schuler et al.,1990).
The high coherence lasted for a period of ca. 30 seconds, to then abruptly change to a state
of low coherence, lasting for a similar length of time to resemble a rhythm with a period of 60
seconds (the so-called minute rhythm). These results inspired me to hypothesize that changes in
functional states of the brain, determined by the degree of intensity of synchronization-
desynchronization (S-D) processes of cortical rhythms, are discontinuous (discrete) in nature
(Achimovich,1991). Further own studies on the dynamics of cortical reactivity to visual stimulus,
determined by the variability of single evoked potentials, also indicate the veracity of this
hypothesis. The probable reason for the lack of literature reports on this subject is that in typical
experimental models, the dynamics of brain processes is so large that the classical (non-parametric)
methods used to measure S-D processes, in which long fragments of EEG recordings are evaluated
(averaging over time), did not allow us to capture this effect.
Current studies, conducted in the US (NIH) using intracerebral electrodes confirming a
visual stimulus-induced short-term (150 ms) increase in coherence between multiple cortical areas
in monkeys, during a visual stimulus recognition task (Bressler, Coppola, & Nakamura, 1993) and
during exposure to a moving visual stimulus in turtles (Prechtl, 1994), support the hypothesis of an
important role of S-D process dynamics in perceptual processes.
In order to clarify the mechanism of generation and reception of field potentials (local
and bulk) , and especially to determine the role of volume conduction , I decided to compare the
changes in coherence as a function of the distance between electrodes, depending on how these
signals are received.
Unlike the coherence of EEG signals received in humans from the surface of the skull,
which is high despite large distances between electrodes and often increases with distance, the
coherence between electrodes placed subdurally on the temporal cortex decreases with increasing
distance. Epileptic patients generally show a monotonic decrease in it to a value of about 0.5
already at distances of 6-8 mm (Achimowicz, Bullock and McClune, 1992).
However, half of the subjects showed an increase in coherence with distance in the 35-50
Hz band indicating the presence of spatially extensive synchronous gamma rhythms. This effect
was equally frequent in the sleep and awake states. In the lower frequency range, this decrease was
observed for all signal frequencies (Bullock, Achimovich, McClune, Iraqui, Ducrow and Spencer
1995). The fact that coherence evaluated for long time slices, the classical FFT method does not
show the existence of privileged frequency ranges (synchronous rhythms), but does not exclude
their existence. This result can only indicate the non-stationarity of these rhythms in terms of
amplitude and frequency. For some pairs of electrodes 10 mm distant, calculating coherence
functions using the AR method for successive 1-second EEG episodes, we found the existence of
synchronous rhythms with a frequency of about 5 , 20 and 40 Hz. During epileptic seizures,
coherence increased especially in the frequency range above 20 Hz.
We obtained the above results by averaging after multiple pairs of electrodes placed on the
surface of the temporal cortex. This procedure, however, is not fully justified and causes a large
loss of information, for the coherence shows considerable local spatial heterogeneity. This
inhomogeneity is due, on the one hand, to the significant functional diversity of the cortex and, on
the other hand, to the varying degree of electrical coupling between the subdural electrodes and the
cortex. This implies the need for further studies of the processes

13. 12
S-D, with greater spatial and temporal resolution. Subdural electrodes provide synchronization
information on much smaller ensembles of neurons than electrodes on the scalp. The lack of
distinct rhythms in the subdural recording may also indicate that the main contribution to the
micro-EEG signal is made not by potentials associated with discharges in neurons but rather by
subthreshold but populationally synchronized fluctuations in cell membrane potentials.
I also found it interesting to compare the temporal dynamics of S-D processes of local
potentials with the relatively well-studied dynamics of analogous changes in collective potentials,
i.e.EEG.
To determine the temporal dynamics of the degree of synchronization of the electrical
activity of cortical neuronal populations received with subdural electrodes 10 mm apart, we
estimated the coherence function for successive EEG segments, with a fixed length of 1 second.
(Bullock, Achimowicz, McClune, 1992). For each segment, we calculated the average coherence
values in selected frequency bands covering the band from 1 to 100 Hz and created a new time
series describing the fluctuations over time of the EEG synchrony level. To verify the hypothesis of
the oscillatory nature of these changes, we used the statistical method of modified periodograms
(Odell et al. 1975).
We found the occurrence of slow oscillations in the range of 0.008 to 0.5 Hz, with
oscillations of 60 and 10 seconds being the most frequently observed. The probability of
occurrence of these oscillations depended on lateralization and changed at the transition from the
wakefulness phase to the sleep phase. The most frequent minute coherence rhythm was found
during the waking phase in the left temporal region in the 25-40 Hz band. It did not occur during
the sleep phase. Similar periodic changes in coherence were also found by the author in rat cortex
and limbic structures in humans and rats (Achimovich and Bullock, 1993) (Bullock and
Achimovich, 1995) which may indicate the universal nature of this phenomenon.
Interpretation of the above results may be facilitated by recalling similar observations made
by other researchers. Slow fluctuations in the parameters of gamma band rhythms (their
amplitudes) were observed in auditory areas of the cortex in scalp electrodes (Galambos,1966),
studying rhythmic activity evoked by acoustic stimuli. Also other authors, recording EEG from
scalp electrodes in human visual cortex in the awake state (Novak, Lepicowska and Dostalek
1992), (Novak and Lepicowska 1992), observed rhythmic modulation of rhythms in the theta and
alpha wave range. The amplitude of these rhythms changed with hyperventilation.
Since the frequencies of these rhythms are the same as the frequencies of the rhythms
occurring in the autonomic blood pressure regulatory system, it is possible to interpret these
changes as the result of the modulating excitability of the cortex due to the influence of brainstem
and thalamic structures. This effect, in my opinion, may also be the result of changes in cortical
activity caused by periodic changes in cerebral flow. These effects have also been detected,
including additionally for the beta rhythm, and quantified by measuring the coherence between
heart rate and energy of EEG rhythms (Bianchi et al. 1992). If the hypothesis about the role of
neuronal mechanisms in this phenomenon (Jennings, 1992) is true, then studies of slow rhythmic
changes in the degree of synchronization of cortical rhythms may provide new information about
integrative mechanisms in the central nervous system related to interactions between cortical and
subcortical processes including the role of the central nervous system in regulating autonomic
nervous system (AUN) function.
5. Studies of the dynamics of nonlinear interactions (bicoherence)
between populations of neurons in the limbic system and cortex.

14. 13
Most of the studies available in the literature on the dynamics of the degree of
synchronization of the rhythmic activity of neuronal populations have been carried out using
methods of linear correlation analysis, the equivalent of which in the frequency domain is
coherence. It determines the degree of synchronization of the phase of a rhythm at a given
frequency between two neuronal structures.
Since my studies of the non-stationarity of the rhythms of the cortex and
hippocampus showed the existence of periodic modulation of the amplitude of these rhythms,
I became interested in the problem of the interaction and phase synchronization of rhythms
of different frequency.
A tool for studying these interactions is the nonlinear correlation function (biocorrelation),
whose counterpart in the frequency domain is bicoherence, which is a nonlinear analogue of
coherence. It determines the constancy of the phase relationship (quadrature phase coupling)
between two rhythms of different frequencies, the relationship of which consists, for example, in
mutual modulation (of amplitude or frequency) and, like coherence, takes values in the range from
0 to 1. A bicoherence equal to zero indicates the complete independence of rhythms of different
frequencies. Just as high coherence indicates a linear relationship between the studied signals, bio-
coherence indicates their non-linearity (reciprocal or so-called quadrature coherence) or non-linear
mechanism of generation of a signal (auto-bicoherence).
Among the best-studied rhythms, in addition to cortical rhythms, is the theta rhythm of the
hippocampus, known in the literature as RSA (Rhythmic Slow Activity). It is characteristic of
exploratory activity in rats, disappears in the slow-wave sleep phase and is observed in the REM
phase. Since a number of higher-frequency (e.g., gamma) rhythms have now also been found to
occur in limbic structures (Buzsaki, Bragin et al., 1994), I decided to investigate possible
interrelationships between these different forms of activity.
I found significant differences between the interactions of theta and gamma rhythms
depending on the animal's behavioral state (Achimovich and Bullock 1993). During exploratory
activity, the theta rhythm is observed synchronized with the gamma frequency rhythm (in the 60-80
Hz band) . This effect disappears during slow-wave sleep, while in the REM phase a phase-
synchronized rhythm of 25 Hz frequency appears in addition to the theta rhythm. These phase
relationships cannot be discerned using classical linear spectral analysis.
The significance of these interactions between theta and gamma rhythms is not known at
this time, no less, however, they are also observed in limbic structures in humans, particularly in
the amygdala area (Achimovich and Bullock 1993). It appears mainly in the period immediately
preceding seizure activity, observed in the anterior part of the hippocampus (pes hippocampi). The
mechanism of seizure initiation in limbic structures is not fully understood, and perhaps a
systematic analysis of the nonlinear phase synchronicities I observed will provide a better
understanding of them.
The mechanism of phase synchronization of rhythms of different frequencies, probably
generated generally by different neuronal populations, may involve mutual modulation of neuronal
excitability either through synaptic connections or directly through the field effect. The results of
my work on the dependence of visual evoked potentials on the coherence of spontaneous EEG
activity indirectly confirm the hypothesis, which has been known for many years, that slow cortical
rhythms can modify (Abraham et al. 1973) the reactivity (excitability) of cortical neuronal pools
and thus also the way in which stimuli are perceived and received.
6. Implications of the results of our own research for modeling the dynamics of brain action
states and the role of field potentials in modulating the excitability of neuronal networks

15. 14
The mechanisms of generation of field potentials, received extracellularly, are not fully
investigated, although it seems that their main source is the spatially and temporally coordinated
postsynaptic potentials of active neurons (Bullock, 1993). Based on studies of the spatial structure
of potentials received from subdural and intracerebral electrodes, however, the contribution of
time-synchronized but subthreshold fluctuations of neuronal membrane potentials cannot be
excluded either (Achimowicz and Bullock 1993).
The fact that there are modulatory interactions between rhythmic electrical activity of the
cortex at different frequencies can be explained by the existence of modulatory interactions of
disconnected populations of neurons. These interactions can be realized not only through synaptic
connections (Aerstsen and Preisl 1991) but also through field (electrotonic) interactions
(Achimovich, Bullock 1993).
The results of my study, indicating the presence of significant phase correlations between
different EEG rhythms from electrodes on the scalp as well as from electrodes implanted into the
brain, seem to support the hypothesis that free brain potentials play an active role in regulating the
intensity of excitation and inhibition processes in neuronal networks (Abraham et al. 1973),
(Landfield 1972), (Elbert 1992) (Emad, Eskander et al., 1992). Nonlinear properties of neurons
may play a large role in these interactions. Many experimental data indicate (Rall ,1970) that,
already at the level of a single neuron, nonlinear effects can play a very large role. The random
potential of the discharge triggering zone in a neuron (the axonal hillock), is not a simple sum of
postsynaptic potentials from the body areas and dendrites of the neuron. Nonlinear interactions
(e.g., multiplicative) are also possible, and may well explain the modulatory effects of the
convergence of impulses from different neuronal pathways (Achimovich and Bullock, 1993).
However, these mechanisms have not been accounted for in biological models of neuronal
networks to date.It is also likely that the oscillatory activity of neuronal populations is of a
nonlinear (chaotic) oscillation nature. While the mechanisms of mutual synchronization of rhythms
may play an important role in brain processes, for example, in the regulation of attention (Krukov
et al. 1992), the nonlinearity of oscillations, generally characterized by significant bicoherence,
may provide a more accurate and rapid mutual synchronization of different neuronal populations
(Ditto and Pecora 1994).
The results obtained in my research on the dynamics of synchronization-desynchronization
processes and the dynamics of cortical evoked potentials inspired me to propose a new model of the
functional states of the central nervous system (Achimovich, 1991). In my opinion, collective
potentials provide information about the functional state of the CNS mainly in the temporal
sequence of bioelectrical events, that is, in the phase of the EEG signal. The amplitudes of
spontaneous and evoked EEG activity show very large fluctuations, which may be due to the fact of
parallel information processing in the central nervous system. The number of neurons carrying out
a given function can be subject to large fluctuations, which, although manifested in the appearance
of a random component of the amplitude of field potentials, does not cause disruption of the
functions carried out by neuronal networks of the brain (functional redundancy). Phase changes in
EEG signals and evoked potentials tend to assume discrete (discontinuous) values, which indicates
the deterministic nature of brain processes (Achimovich 1991). Functional (functional) states of the
nervous system can be well-defined over short stretches of time, while transitions between different
states prvably take place in leaps and bounds as a result of stimulus reception, or spontaneously, for
example, as a result of changes in the degree of arousal due to internal stimulation.
The high dynamics of brain processes makes that, achieving a stationary state of the
CNS in an experimental model is very difficult and requires precise planning of the experiment
(Achimovich 1991), in terms of controlling the behavioral state of the biological object under
study.

16. 15
7. Conclusion
Research conducted by me showed that:
• The coherence of the main EEG rhythms (from electrodes on the scalp), in some functional
states, is very high, even for electrodes several centimeters apart, and can increase with
increasing distance (Achimovich 1991), indicating the important role of cortical-subcortical
projections and volume conduction.
• Field potentials recorded directly in neural tissue reflect the electrical activity of a population of
neurons from the immediate vicinity of the recording electrode and the contribution of distant
brain structures, e.g. through volume conduction, is small (Bullock, Achimovich , 1992).
• The local nature of the field potentials is evidenced by the fact that the coherence of the signals
decreases with the distance of the intracerebral electrodes, to a value of 0.5 already at a distance
of about 5 mm. This measure of the collective behavior of neurons in time and space, which can
be called the
"coherence length", is slightly greater in the human brain than in more primitive organisms
(Bullock, Achimovich et al. 1995). This means that studies of field potentials provide
information about the behavior of neuronal populations at the mesoscopic level , which is not
redundant with the results of studies conducted at the level of single neurons or by the EEG
method.
• The rhythms observed in local field potentials show high temporal dynamics and exhibit
fluctuations of a periodic nature. The period of these oscillations falls mainly in the range of 1 to
100 seconds (0.01 to 1 Hz), which coincides with the frequency range of slow rhythms observed
in the autonomic nervous system (AUN). The amplitude and frequency of these oscillations also
depend on the functional state of the central nervous system (CNS) and the location of the
electrodes. This fact indicates the existence of interactions of a modulatory nature between the
AUN and CNS, whose potential integrative role, probably important for optimal control of the
living organism, should be the subject of further research. (Bullock, Achimowicz et al. 1995).
• The existence of phase relationships (quadrature phase coupling), between rhythms in the CNS
at different frequencies, as indicated by the high value of bicoherence, indicates the importance
of nonlinear mechanisms in the generation of field potentials. Learning about the neuronal
mechanisms, dependent on the functional state of the central nervous system, of the processes of
linear and nonlinear synchronization of rhythms, can shed new light on understanding the
functional role of synchronous activity of neuronal ensembles. The detection of modulatory
interactions of rhythms provides new grounds for the hypothesis of the active role of brain field
potentials in the regulation of excitation and inhibition processes and the influence of
subcortical structures on cortical processes. (Achimovich, Bullock, 1993).
• The deterministic model I proposed for the dynamics of changes in the brain's functional states,
as manifested in spikes or periodic changes in the intensity of synchronization and
desynchronization processes, can help interpret research on the electrophysiological correlates
of arousal and attention in terms of "competition" (Desimone, 1995), of different neuronal
pathways. This hypothesis suggests that a change in the functional state of the CNS takes place
not through a gradual (smooth) change in the degree of excitation of different neuronal
pathways, but rather through their "switching" (Achimovich, 1991).
• Previous studies of CNS rhythms have used methods of linear spectral analysis and implicitly
assumed the underlying assumption of the independence of rhythms o

17. 16
different frequencies. In view of the current development of research attributing a special role to
rhythms of 20 and 40 Hz in conscious stimulus perception, the nonlinear methods introduced by
the author can be very useful in determining their functional relationships. They may allow
interpreting their neuronal mechanisms in terms of changes in the level of attention and
perception of stimuli as a whole (feature binding), or only in the form of simple harmonic
relations of these rhythms, indicating the nonlinear mechanism of their generation.
• The results of my work also testify to the need to further develop methods for the digital
analysis of the phasic features of signals of integrated electrical activity of the brain, especially
those
, which take into account the nonlinear nature and dynamics (nonstationarity) of these signals.
Simultaneous analysis of field potentials in the time and frequency domains (Bullock and
Achimovich, 1995), can provide new data relevant to understanding the mechanisms of the
central nervous system at the mesoscopic level of organization of the brain's neuronal networks,
which is currently very poorly understood.
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