Temporal order of nonlinear dynamics in human brain

Brain Research Reviews 45 (2004) 79 – 95 www.elsevier.com/locate/brainresrev

Review

Temporal order of nonlinear dynamics in human brain W.S. Tirsch a,*, Ph. Stude b, H. Scherb a, M. Keidel c a

GSF-Institut fu¨r Medizinische Informatik und Institut fu¨r Biomathematik, D-85764, Neuherberg, Germany b Neurologische Klinik und Poliklinik der Universita¨t, D-45122, Essen, Germany c Kliniken fu¨r Neurologie und neurologische Rehabilitation am Bezirkskrankenhaus, D-95445, Bayreuth, Germany Accepted 21 January 2004 Available online 5 March 2004

Abstract In previous spectral analysis investigations, we demonstrated that the spontaneous activity of the alpha EEG is not stationary but rather shows cyclic alterations with a circa 1-min periodicity. Following the conclusion that a power increase in the alpha band implies a neuronal synchronization, and vice versa, an associated decrease of the EEG complexity was postulated. Accordingly, a rhythmic variation, i.e., a temporal order of the nonlinear dynamics with similar period length, was expected. Bipolar 4-min EEG recordings were obtained from 20 awake subjects (mean age: 23.5 F 2.5 years) with eyes closed for the EEG leads C3, C4, Oz, and Fz according to the 10 – 20 system. For the automatic evaluation of spontaneous alterations of complexity, a sliding computation of the so-called correlation dimension, using an analysis window length of 20 s continuously shifted by 1 s, was performed. The time series of complexity exhibited an oscillatory behavior with a mean period length of 58.7 s; the Friedman test statistic revealed no significant topological differences. For the rejection of the null hypothesis that the observed periodicity is a random one, two-group t-tests and ANOVA with repeated measures were performed, comparing the corresponding amplitudes and period lengths with those derived from 20 pseudo-random signals (taken from a multivariate Gaussian normal distribution). The mean relative change of EEG complexity was highly significantly increased ( P < 0.0001) compared to that of random data. Likewise, the difference of mean period lengths was also significant ( P < 0.01). The results indicate that the coupling strength of the neural network of the brain changes periodically, with a cyclic alteration from a central to a parallel processing mode of information, reflecting state transitions from synchronized, low-complex EEG activity to desynchronized high-complex activity, and vice versa. Various neuronal control mechanisms that may be acting as pacemakers responsible for the temporal order of such transients are discussed. A disturbance of the temporal order may be of pathophysiological significance. D 2004 Elsevier B.V. All rights reserved. Theme: Neural basis of behavior Topic: Aging Biological rhythms Keywords: EEG; Nonlinear dynamics; Correlation dimension; Complexity; Periodicity; Temporal order

Contents 1.

2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Rhythmic processes in the CNS . . . . . . . . . . . . 1.2. EEG rhythms . . . . . . . . . . . . . . . . . . . . . . 1.3. Nonlinear dynamics of the EEG . . . . . . . . . . . . Materials and methods . . . . . . . . . . . . . . . . . . . . . 2.1. Subjects and data acquisition . . . . . . . . . . . . . . 2.2. Nonlinear dynamical analysis . . . . . . . . . . . . . . 2.2.1. Geometrical reconstruction of the ‘attractor’ of 2.2.2. Estimation of dimensional complexity. . . . . 2.2.3. Sliding analysis . . . . . . . . . . . . . . . .

. . . . . . . a . .

* Corresponding author. Tel.: +49-89-3187-4186; fax: +49-89-3187-4243. E-mail address: [email protected] (W.S. Tirsch). 0165-0173/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.brainresrev.2004.01.002

. . . . . . . . . . . . . . . . . . . . . EEG . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . time series . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

80 80 80 81 81 81 82 82 82 83

80

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

2.2.4. Quantification of the cyclic structure of time series of complexity 2.2.5. Analysis of variance with random data . . . . . . . . . . . . . . 3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Oscillatory behavior of the time courses of dimensional complexity . . . . 3.1.1. Spectral analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Sine wave fitting technique . . . . . . . . . . . . . . . . . . . . 3.2. Statistical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Channel differences . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Comparison with random data . . . . . . . . . . . . . . . . . . 4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Brain state transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Circa 1-min rhythms of dimensional complexity . . . . . . . . . . . . . . 4.3. Pacemakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Pathophysiological aspects . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction 1.1. Rhythmic processes in the CNS Functions of the central nervous system (CNS) are not maintained in a steady state. Rather, they exhibit a dynamic, oscillatory behavior with a well-ordered change of activity and synergy over time. Thus, rhythmicity, considered as a paradigmatic illustration of self-organized temporal behavior, is characteristic for the CNS [36]. Self-generated, cyclic alterations of the spontaneously active ‘neuronal network’ of the CNS acting as pacemakers are responsible for these neuronal rhythms and oscillations of various period lengths. Models of neuronal populations describing the basic mechanisms for the generation of these rhythms were presented by Winfree [137], Lopes da Silva et al. [89,91], Basar [14], and Erdi [36]. Spontaneous, ‘infraslow’ alterations of cortical DC-levels [8,9] and slow-wave oscillations in the cerebral subcortical structures [100] were revealed. Periodic variations of systemic arterial pressure [94], cardiovascular and respiratory rhythms [32,76,79], or oscillations of the tremor and muscle vibrations [67,69,128] were found. A survey of ultra- and circadian rhythms was issued by Hildebrandt [55] and Hildebrandt et al. [56] describing the autonomous temporal order of biological systems. 1.2. EEG rhythms Since Berger’s [21] discovery of the EEG waves in 1929, numerous studies have been published on the oscillatory changes of the spontaneous bioelectrical activity measured from the scalp. A first review of cerebral rhythms in the waking EEG was given by Katz and Cracco [65]. Rhythms and oscillations with period lengths ranging from milliseconds to hours were observed. For example, Molnar and Weiss [101] reported on (damped) oscillations in the amplitude of acoustically evoked potentials, with a period

by means of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

sine wave fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

84 84 86 86 86 86 87 87 87 89 89 90 90 91 92 92

length of circa 100 ms observed in animal experiments. These findings are obviously in line with the alpha rhythm, the most prominent periodicity of the human brain, which was described first by Berger and later by Andersen and Andersson [10], Lippold [83], Lopes da Silva et al. [88], Lopes da Silva and Storm van Leeuwen [87], and Treisman [132]. Much slower periodicities, in the range of seconds and minutes, were observed by Keidel [66], Scheuler et al. [117], and Brenner et al. [23]. Infra- or ultradian periodicities in the order of minutes and hours occurring during human sleep were identified by Burns et al. [26] and Lacroix et al. [77]. Circa 90-min periodicities in the EEG, known as REM/NREM cycles or ‘basic rest – activity cycles’, were described by Dement and Kleitman [31], and Kleitman [75]. With respect to the waking EEG, ultradian rhythms were found by Tsuji et al. [133], Meneses and Corsi [99], and Hayashi et al. [54]. Accordingly, we have demonstrated, based on spectral analysis investigations, that the spontaneous EEG activity of awake normal persons is not maintained at a static level but rather shows cyclic, circa 1-min alterations of spectral power and coherence in the main frequency range, i.e., the alpha band, with period lengths ranging from 20 to 70 s [67,68,72,128]. Likewise, Kaplan [63] described ‘‘the phenomenon of the EEG spectral variability’’ which reflects the temporal structure of the signal. Our findings were facilitated by the introduction of a sliding or ‘running’ spectral analysis with overlapping segments. From our results, we conclude that a power increase in the alpha band may be due to an increase in synchronization or coupling strength (i.e., the degree of cooperation) between various neuronal elements within cortical networks generating the EEG signals. This assumption is in line with the considerations of Linkenkaer-Hansen et al. [82] who suggested that ‘‘EEG oscillations arise from correlated activity of a large number of neurons whose interactions are generally nonlinear’’ [86,122]. Furthermore, this assumption also agrees with Winfree’s [137] concept of coupled oscillators (i.e., the

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

phenomenon of collective synchronization of phases and frequencies), Haken’s [50] interpretation of the brain as a ‘synergetic system‘, the model of synchronization and desychronization of coupled self-oscillatory networks [14], and the concept of synchronizing oscillations at the cellular and neuronal network level [86]. Thus, we deduced that the spontaneously active network of the CNS, interpreted as the underlying dynamical system, may be more or less ordered. This should be associated by a simultaneous decrease or increase of the EEG complexity. Accordingly, Strogatz [123] has shown that the coupling architecture (i.e., the design of neuronal coupling) of interacting dynamical systems, which may behave collectively, is reflected in the complexity of the entire network. Bearing in mind the former described circa 1-min periodicity of the alpha power, a corresponding rhythmic alteration of the complexity with similar periods is postulated for long-term EEG recordings following Basar’s [15] concept of continuously spontaneous changing state of the brain. 1.3. Nonlinear dynamics of the EEG Over the past two decades, numerous mathematical models have been developed to explain the rhythmical behavior of neurobiological systems using nonlinear approaches [28,45,59,74,95]. A detailed review of the mathematical concepts in nonlinear brain dynamics is given by Dvorˇa´k and Holden [33]. Methods of nonlinear dynamical system and ‘chaos’ theory [119] may yield information about the dynamical system underlying experimentally recorded time series such as the EEG. This is based on the assumption that the investigated time series is derived from synergetic, self-organized neural systems showing a deterministic, chaotic behavior. Mathematically, ‘chaos’ can be defined as the exponential sensitive dependence on initial conditions, and can be applied to stationary evolutions of time series. To consider a system chaotic implies that there is a fundamental limit on the predictability of such a system; that is, very small changes of the input or control parameters can lead to an extensive variation of the entire system. The unpredictability of a system can also be quantified by its entropy. The entropy reflects the rate at which information about the initial condition is lost, and is equal to the sum of all positive Lyapunov exponents. The positive value of the principal Lyapunov exponent k1 indicates that the trajectories of the system (i.e., the set of points in state space, which were sequentially visited) exponentially diverge from one another in the state space. Thus, the positive value of k1 yields evidence for the chaotic behavior of the system. The information resulting from the nonlinear dynamical system analysis is mainly reflected by the complexity parameter, which is associated to the dimensionality of the underlying dynamical system. Thereby, this parameter denotes the minimum number of essential variables needed to model the system dynamics [37]. The dimensional complexity of EEG time series has been estimated by means

81

of the widely used measure of the correlation dimension. First reports were published by Babloyantz and Destexhe [11], Mayer-Kress and Holzfuss [97], Mayer-Kress and Layne [98], and Ro¨schke and Basar [115], indicating that the underlying neuronal networks may possess multiple types of attractors. These may reflect the chaotic properties of a deterministic system [43,93]. Tutorial reviews on nonlinear dynamical EEG analysis were given by Basar [17] and Pritchard and Duke [110]. Nonlinear dynamical analysis has found an important application in investigating the bioelectrical activity of the brain. This has given rise to statements such as ‘‘the EEG is not simple noise’’ and ‘‘the EEG reflects deterministic chaos’’ [17], and has led to speculations like ‘‘chaos underlies the ability of the brain to respond flexibly to the outside world and to generate novel activity patterns’’ [42]. Reviews on chaotic brain dynamics were issued by Basar [16] and Lehnertz et al. [80]. Besides the work of Linkenkaer-Hansen et al. [82] who found long-range temporal correlations and nonlinear power –law scaling behavior in human brain oscillations, systematic studies on the temporal order of dimensional complexity of long-term EEG recordings have not been performed. At best, some papers were published providing only rough indications of the temporal structure of the correlation dimension using approaches with consecutive segments of analysis [19,98]. Thus, the introduction of the sliding analysis of EEG complexity by our group has provided a better insight into the temporal fine structure of such recordings. Thereby, the sliding analysis enabled to disclose cyclic, temporal alterations (nonstationarities) in the signal’s complexity, i.e., cyclic transitions from low- to high-dimensional ‘brain chaos’ as first suggested by Keidel et al. [67,72]. Initial reports from Keidel et al. [73] and Tirsch et al. [129] illustrate these points. In the present study, we go a step further. Due to Basar’s [17] statement about noise, the amount of the relative change of EEG complexity within the oscillating time series of complexity will be subsequently related to the results of corresponding random processes. The aims of the present paper are (1) to prove the postulated periodicity of the EEG complexity with circa 1min cycles, (2) to reject the null hypothesis that the observed periodicity is a random one, and (3) to propose neuronal approaches to explain the observed cyclic transients of complexity.

2. Materials and methods 2.1. Subjects and data acquisition Twenty healthy young volunteers (mean age: 26.8 years) were examined. During the examination, the subjects were awake, sitting in a supine position. They were instructed to relax with eyes closed, corresponding to the vigilance-state ‘relaxed wakefulness’. Data acquisition was performed in a

82

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

sound-absorbed and electrically shielded room. Unipolar scalp recordings of the spontaneous EEG activity were obtained from left (C3: channel 1) and right motor cortex (C4: channel 2), midoccipital (Oz: channel 3), and midfrontal region (Fz: channel 4) corresponding to the 10 – 20 system and referred to linked earlobes. The electrode positions were chosen in order to cover the bihemispherical convexity (C3 –C4) and to record the electrical activity from a region with a preponderance of the alpha (Oz) and beta rhythm (Fz). Signals were recorded for 4 min continuously and A/D converted using a sampling rate of 500 Hz. Fig. 1 shows an example of an EEG recording (JL14) with a duration of 4 min. Visual inspection of the temporal structure of the EEG curve revealed only the well-known alpha spindles. No information about long-term fluctuations of the signal’s activity are evident. Underlying systematic, oscillatory changes of the neuronal activity in the 1-min range can only be detected by appropriate computer-assisted procedures such as the previously mentioned sliding analysis. 2.2. Nonlinear dynamical analysis 2.2.1. Geometrical reconstruction of the ‘attractor’ of a EEG time series The dynamical approach of nonlinear modeling described here is based upon the interpretation of a single scalar time series recorded from one electrode position on the scalp as the one-dimensional projection of a multidimensional phase –space trajectory of an underlying nonlinear dynamical system, which is unknown. Thus, by using samples obtained from the single time series, one can reconstruct the dynamics of the underlying system. Based on a geometrical view of the underlying process, following the delay scheme of Packardt et al. [108] and the reconstruction theorem of Takens [125], the reconstruction of the phase – space trajectory can be realized by means of m-dimensional delay vectors. This procedure embeds the one-dimensional

scalar time series into a higher-dimensional space, with embedding dimension m, using time lags [108]. This technique, which allows EEG time series to be reinterpreted as multidimensional geometrical objects [98], is a basic tool for the characterization of a dynamical system such as the human brain. It enables the reconstruction of the system’s trajectories, which may fill out the entire phase space, or which may converge to a limited subset that is called the attractor derived from the time series. The reconstruction of the attractor in the m-dimensional pseudo-state subspace is based on the assumption that this subspace is a sufficient reconstruction of the original state space. Then, the reconstructed attractor is diffeomorphic to the original attractor; both have the same metric properties and nearly the same dimension, entropy, and Lyapunov exponents. With respect to brain dynamics, ‘‘these attractors do not directly provide an adequate model, rather they may be interpreted as the result of self-organizing processes within the brain, which produce coordinated states or pattern activity’’ [47]. 2.2.2. Estimation of dimensional complexity The basic idea for the characterization of chaotic dynamical systems is to calculate the dimension of the appropriate attractors—a quantity which provides information about their topology and self-similar structure. The dimension is a statistical measure of the space filling out the attractor. The attractor of a periodic system is a closed curve, which has the dimension 1 and entropy 0. Chaotic systems have a fractal dimension and a positive entropy. Initial reports on this subject were published by Farmer et al. [37]. The dimension has been widely used in the field of nonlinear dynamical analysis to estimate the number of independent variables needed to model the dynamical process and to discriminate between a deterministic and a random activity. The dimensionality is also a measure of the complexity related to the number of independent oscillators or modes that modulate the process [34], and also approximates to the effective number of active degrees of freedom of the system,

Fig. 1. Four-minute record of EEG activity filtered by a 1 – 40 Hz band-pass and derived from the right motor cortex (C4) of subject JL14.

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

which is reflected by its entropy [127]. According to the entropy concept of Basar [14], the entropy is also a measure of the order of the system, i.e., an increased order involves a lower entropy, and vice versa. Thus, it can be deduced that the order of the system is associated with its dimensional complexity. In practice, one of the measures most commonly used to characterize the system’s attractor is the two-point correlation dimension D2, as proposed by Grassberger and Procaccia [46], utilizing the scaling structure of the attractor. This absolute measure can be obtained by quantifying the spatial correlation between pairs of randomly chosen points on the attractor. This requires the introduction of an additional concept: the correlation integral providing a measure for the probability (cumulative percentage) that pairs of phase – space vectors are separated by a distance less or equal to a prior defined value. It has been shown that the exponent of the correlation integral, known as the correlation dimension, as used in this paper is a useful measure of the local structure of an attractor [46], estimating the dimensional complexity of the dynamical process. The estimate of D2 is by no means trivial and has serious methodological limitations. Many factors may have a large effect on the estimated values of this measure, such as the finite length of the signal, noise, and choice of the reconstruction parameters. Particularly relevant is the effect of the nonstationarity of the EEG signal because the attractor concept includes the idea of an autonomous (stationary) system [38]. We emphasize that a fractal value of D2, as used in the following statistical evaluation, does not stringently imply the existence of chaotic behavior [107] and should be interpreted with care. This is due to the fact that this value represents an approximation and only rather short EEG time series were used with respect to the nonstationary character of the signals [87]. The estimation of the correlation dimension as carried out in this study is based on the scaling law of the correlation integral C(r) that is due to the exponential divergence of the trajectories: CðrÞfr D2

or

logðCðrÞÞ ¼ D2 logðrÞ þ c

in which r is the distance, and the slope of the exponent D2 is the correlation dimension. Considering the above equation, the exponent D2 can be obtained as the slope of the straight line figuring the logarithmic plot of C vs. r. However, in dealing with physiological data sets derived from living systems, this ideal function may not be apparent, and graphs in which different regions are nearly linear are usually evident [47]. The problem at this juncture is to determine the interval of the most linear scaling region, which is essential to fulfill the condition that the reconstructed attractor be self-similar and fractal. To meet this criterion, a procedure was developed which automatically localizes the corresponding optimal fitting center within the graph of the correlation integral by means of the minimum residual variance of several

83

consecutive regression lines. The slope of the final regression line fitted through the optimal fitting center is an estimate of the correlation dimension D2. This procedure is performed stepwise by considering successively higher values of the embedding dimension. For a sufficiently large value, the dimension of the attractor will be obtained as the saturation value of this procedure. This is guaranteed even if model attractors are considered. However, for physiological time series such as the EEG, the slope values of the correlation dimension do not saturate in all cases [12] due to the finite length of the time series and superimposed noise. This problem was pointed out by Mayer-Kress and Layne [98], and Pritchard and Duke [110]. A detailed description of this procedure applied to EEG data is given in Tirsch et al. [130]. Following the theoretical considerations of Eckmann and Ruelle [35], the values of the correlation dimension D2 should be limited to 8, corresponding to an EEG epoch length of N = 10 000 data points ( = 20 s) as used in our study. On the other hand, with respect to the theorem of Takens, the embedding dimension should be at least j2D2 + 1j where D2 is the true correlation dimension of the attractor. This results in a minimum embedding dimension of m = 17, which was used for our subsequent analyses. Only time series of complexity based on this value were investigated. 2.2.3. Sliding analysis In contrast to conventional analysis techniques, based on the consecutive evaluation of defined periods, we used the sliding technique for the estimation of the dimensional complexity of long-term EEG signals. This technique provides the detection of systematic oscillatory changes in the signal’s activity more clearly. A sliding shift of a defined period is performed like a ’running’ analysis window over the whole record length using a time shift Dt. From the results of the sliding spectral analysis applied to human neurobiological signals [128] and of the sliding dimensional analysis applied to EEG signals [73], the period length was selected for 20 s (i.e., 10 000 data points) and Dt for 1 s in the present study. These values have yielded the best results in disclosing temporal patterns from the signals. The problem of stationarity when applying the classical algorithm of Grassberger and Procaccia [46] to biological time series such as the EEG is recognized. To be accurate, this algorithm requires both data stationarity and long data sets [112]. However, the latter is in conflict with the highly nonstationary structure of the EEG signal [39,40,63, 90,104], requiring that a compromise be found. Recently, Fell et al. [38], and Skinner and Molnar [121] have pointed out this dilemma proposing ‘‘as a safeguard for biological data with uncertain stationarity short data epochs (running windows) in which stationarity can be assumed’’. The concept of the sliding calculation of the correlation dimension applied here is supported by the reports of Havstad and Ehlers [53]. Following the ideas already presented by Keidel et al. [68], they suggested that, for nonstationary dynamical

84

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

systems, small successive but overlapping data segments in which the system is approximately stationary should be preferred. They demonstrated the ability of such data sets to follow fluctuations in dimension with sufficient accuracy that enables studies of dimension as a function of time in nonstationary dynamical systems. In this sense, we follow the reports of Dahlhaus [30], Adak [4], Brodsky and Darkhovsky [24], Barlow [13], Fingelkurts and Fingelkurts [39], and Renshaw [113] in considering the EEG data of each analysis window to be piecewise or quasi-stationary. Within each window, the data were submitted to the following procedures: (1) elimination of linear trend and centering to the zero-line; (2) filtering by means of a 1 – 40 Hz band-pass filter to suppress artifacts using a Butterworth filter of tenth order in a parallel configuration, and elimination of phase shifts by forward and backward filtering; (3) scaling between 0 and 1; (4) calculation of the optimal time delay by means of the first minimum of the mutual information content; (5) dimensional analysis. Using an EEG recording of 4 min, this procedure produces a time series with 220 values for the correlation dimension. 2.2.4. Quantification of the cyclic structure of time series of complexity by means of sine wave fitting technique The principle of this method, which is based on Fourier decomposition, is to look for the endogenous autorhythm within a given time series approximating the time series by sinusoid functions with defined periods. Amplitude and phase were calculated by means of least square fitting—a technique which was taken from the so-called cosinor procedure by Halberg et al. [52]. Accordingly, various sine functions with stepwise increasing period lengths, ranging from 5 to 100 s with increments of 1 s, were fitted in the search for an optimal fit. This was facilitated by taking into account the minimum residual variance, i.e., the mean quadratic deviation between the time series and the approximat-

ing sine wave. By means of a ranking of several local minima, three optimal fits of first to third order were proposed by the algorithm. For the detection of the endogenous rhythm of the time series, the final optimal fit was visually selected. This was supported by the corresponding peak frequency of the concomitant power spectra. In most of the cases, the first fit was selected; the second and third fits were rarely used. The resulting period length and peak-to-peak amplitude of this fit, reflecting the mean relative change of the time series, were used as features for further statistical analyses. In contrast to the spectral analysis, this approach enables a more precise disclosure of the endogenous rhythm of a time series. This is due to the power spectrum providing periodicities with a temporal resolution, which is rather low and strongly restricted in the case of small data sets comprising only 220 values. Because the period lengths are the reciprocal values of the multiples of the spectral resolution Df = 1/T , where T denotes the length of time series (in our case T = 220 s), only period lengths of T, T/2, T/3 . . . are possible. 2.2.5. Analysis of variance with random data With respect to a statistical corroboration of the revealed periodicity of dimensional complexity in the sense that this rhythm is not a random one, the following heuristic null hypothesis was formulated: ‘‘the disclosed rhythmicity is not EEG-specific and can also be generated by a random process’’. For this purpose, a sample set of 20 random signals of 4 min duration taken from a multivariate normal Gaussian distribution with n = 4 dimensions (simulating the 4 EEG channels) was generated by means of a standard pseudorandom number generator. Mean vector and covariance matrix as the input of the random procedure were pooled from the sample set of 20 EEG recordings. In Fig. 2, an example of such a random signal is presented. In analogy to

Fig. 2. Epoch (30 s) of a multivariate, 4-channel normal Gaussian random process. Mean vector and covariance matrix required by the random procedure were derived from the EEG data of 20 subjects. The random signals were smoothed by the same band-pass filter as used for the EEG data (1 – 40 Hz).

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

85

Fig. 3. Smoothed time courses of dimensional complexity over 220 analysis windows derived from the same EEG epoch as shown on Fig. 1. Time delays used for the calculation of complexity ranged from 26 to 40 ms. Embedding dimension is running from 11 up to 20. The length of analysis window is 20 s (i.e., 10 000 data points), which is shifted by 1 s. Note the cyclic structure of the graph indicating a 50 – 60 s periodicity.

Fig. 4. Grand average over 20 smoothed power spectra derived from the time series of dimensional complexity with embedding dimension d = 17. The solid line represents the mean spectrum based on the EEG data from 20 subjects for channel 1 (C3: left motor cortex). The dashed and dotted lines correspond to the mean spectra derived from the other EEG channels, i.e., channel 2 (C4: right motor cortex), channel 3 (Oz: midoccipital region), and channel 4 (Fz: midfrontal region). On the abscissa, the frequency is drawn in steps of 0.0045 Hz as the reciprocal value of the length of time series (T = 220 s). Note the main peaks at frequency step = 3 according to a period length of 73.3 s most distinct for channels 3 and 4.

86

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

the investigated EEG signals, the random data were filtered by the same band-pass filter. Likewise, the sine wave fitting technique was applied to the random data, and corresponding period lengths and amplitudes were determined. For a statistical comparison of the EEG parameters with the random parameters, an ANOVA with repeated measures, designed with one grouping factor EEG vs. random and one within factor channels, was performed. Based on this design, the following two null hypotheses were tested: H1. There are no differences between EEG and random data in mean period length and amplitude. H2. There are no differences in period length and amplitude between the four channels.

3. Results 3.1. Oscillatory behavior of the time courses of dimensional complexity Fig. 3 shows the smoothed time courses of 220 sliding computations of the correlation dimension for the 4-min EEG

epoch illustrated in Fig. 1. Time delays for the reconstruction of phase – space vectors ranged from 26 to 40 ms. The embedding dimension increases from 11 up to 20. In contrast to consecutive analyses, pronounced rhythmic variations of dimensional complexity with periods between 50 and 60 s are clearly evident just by means of the sliding analysis. 3.1.1. Spectral analysis The purely visual evaluation of the rhythmic variations of complexity without the assistance of appropriate computer tools will be rather inaccurate. Thus, power spectra were calculated for each time series of complexity corresponding to an embedding dimension of 17, and to one channel of a 4min EEG recording. Fig. 4 shows the grand averages over 20 smoothed power spectra derived from the four EEG channels. The distinct peaks corresponding to the midoccipital (Oz) and to the midfrontal region (Fz) indicate a circa 1-min periodicity. Likewise, for the right motor cortex C4, a lower peak exhibiting the same periodicity can be discerned. 3.1.2. Sine wave fitting technique Because the temporal resolution of the period length, as revealed by spectral analysis, is rather low and strongly

Fig. 5. Results of sine wave fitting technique. Upper graph: time series of EEG complexity (embedding dimension d = 17) derived from the same EEG recording as in Fig. 1 after linear detrending and centering to the zero-line. Lower graph: optimal fits of sine waves with corresponding period lengths and amplitudes of first (T = 58 s, solid line), second (T = 40 s, dashed line), and third order (T = 33 s, dashed line). The amplitudes correspond to the relative changes of complexity related to the mean.

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95 Table 1 Results of sine wave fitting technique applied to the time series of complexity which were derived from 4-min EEG epochs of 20 subjects Time series of EEG complexity

Amplitude

Period length (s)

Channel

Mean (n = 20)

Mean (n = 20)

0.52 F 0.16 0.58 F 0.18 0.71 F 0.26 0.73 F 0.32 0.64 F 0.24 0.0049

54.1 F 15.3 60.4 F 14.2 55.6 F 16.7 64.7 F 17.8 58.7 F 16.0 0.0671

EEG lead

1 C3 2 C4 3 Oz 4 Fz ¯ (pooled) M Friedman test ( P)

Comparison of period lengths according to the four EEG channels by means of Friedman test indicated that there is no significant difference. In contrast, peak-to-peak amplitudes show a significant difference on the 1% level.

restricted, we applied the more sensitive analysis of rhythmicity like the so-called sine wave fitting technique. Fig. 5 illustrates the results of this approach when applied to the same 4-min EEG period as shown in Fig. 1. In the upper graph of Fig. 5, the time series of complexity according to an embedding dimension of 17 is drawn after linear detrending and centering to the zero-line. In the lower graph, the corrected time series is plotted together with the optimal sine waves according to the fits of first to third order. The notation contains the resulting optimal period lengths and amplitudes reflecting the relative deviations from the mean. The first fit shows a clear 1-min rhythmicity with a period length of 58 s. 3.2. Statistical evaluation 3.2.1. Channel differences Table 1 shows the results of the Friedman test, which is based on the null hypothesis of no differences between the

87

matched variables, such as period lengths or peak-to-peak amplitudes, corresponding to the four EEG channels. With respect to the period lengths, the test statistic was not significant. Thus, there were no differences between the channels. A pooled mean value of 58.7 s over the four EEG channels and over 20 cases exhibits a distinct 1-min periodicity. Fig. 6 shows the distribution of the period lengths. In contrast, for the amplitudes, there were highly significant differences ( P < 0.0049) between the channels caused by increasing amplitudes in the midoccipital (Oz) and midfrontal regions (Fz). 3.2.2. Comparison with random data Fig. 7 illustrates the grand averages of 20 smoothed power spectra derived from the time series of complexity. The solid line represents the mean spectrum, which is based on the EEG data from 20 subjects derived from channel 3 (midoccipital region Oz). The dashed line corresponds to the mean spectrum derived from the third dimension of 20 fourdimensional random signals. It is remarkable that the power of the spectrum derived from EEG data is highly increased compared to the spectrum derived from random data, indicating that fluctuations in the corresponding time series of complexity were relatively small and did not have any significance. A statistical corroboration of this finding was obtained by means of two-group t-tests and variance analysis comparing the parameters of the sine wave fitting technique derived from the time series of EEG complexity with those of random processes. Table 2 shows the results of t-tests and ANOVA with repeated measures using ‘peak-to-peak amplitude’ and ‘period length’ as variables. With respect to the amplitude parameters, the mean relative change of complexity derived

Fig. 6. Distribution of the period lengths derived from the time series of complexity over four EEG channels and 20 subjects with mean of 58.7 F 16.3 s.

88

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

Fig. 7. Grand average over 20 smoothed power spectra derived from the time series of dimensional complexity with embedding dimension d = 17. The solid line represents the mean spectrum based on the EEG data from 20 subjects for channel 3 (Oz: midoccipital region). The dashed line corresponds to the mean spectrum derived from 20 random signals. On the abscissa, the frequency is drawn in steps of 0.0045 Hz.

from the EEG data was highly significantly increased ( P < 0.0001) in all of the investigated brain regions, maximally by a factor of 3.2 compared to the corresponding random data. This increase was most distinct for the mid-

occipital and midfrontal EEG leads. Likewise, the difference of pooled means over the four data channels was highly significant ( P1 < 0.0001) according to the grouping factor EEG vs. random as revealed by the variance analysis.

Table 2 Results of ANOVA with repeated measures and of two-group t-tests based on the peak-to-peak amplitude and period length parameters calculated by means of sine wave fitting technique Time series of EEG complexity

Amplitude

Period length (s)

EEG

Random

P-value

EEG

Random

P-value

Channel

EEG lead

¯ 1 (n = 20) M

¯ 2 (n = 20) M

(t-test)

¯ 1 (n = 20) M

¯ 2 (n = 20) M

(t-test)

1 2 3 4 ¯ (pooled) M P1 (groups) P2 (channels)

C3 C4 Oz Fz

0.52 F 0.16 0.58 F 0.18 0.71 F 0.26 0.73 F 0.32 0.64 F 0.24 < 0.0001 0.0012

0.23 F 0.09 0.23 F 0.07 0.22 F 0.09 0.25 F 0.07 0.23 F 0.08

< 0.0001 < 0.0001 < 0.0001 < 0.0001

54.1 F 15.3 60.4 F 14.2 55.6 F 16.7 64.7 F 17.8 58.7 F 16.0 0.0014 0.0027

43.0 F 13.8 49.7 F 12.3 46.5 F 13.3 53.3 F 14.4 48.2 F 13.5

0.021 0.015 0.065 0.033

The parameters derived from the time series of EEG complexity of 20 subjects are compared with those of 20 Gaussian random processes. Tail probabilities correspond to the grouping factor EEG vs. random ( P1) and the repeated or within factor channels ( P2). Note that the mean relative changes in EEG complexity according to the amplitudes were highly significantly increased by the factor of about 3 (at the 0.01% level) compared to those derived from random processes.

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

According to the repeating factor channels, within differences between the channels were also significant ( P2 = 0.0012) due to the increased amplitudes of complexity derived from the EEG channels 3 and 4. Investigating the period length parameters of the time series of complexity by means of t-tests, the mean period lengths derived from the EEG data significantly differed at the 5% level from those based on random data for most of the EEG channels. The results of ANOVA revealed that the pooled mean value of 58.7 s derived from the EEG data was significantly different ( P1 = 0.0014) from the mean of 48.2 s based on random data. This value is nearly consistent with the mean value of the uniform distribution over the interval [5,100], which resulted from the search for an optimal fit using increasing period lengths from 5 to 100 s. Due to the calculated statistical significances, the null hypothesis that the EEG periodicity is a random one, that is, the corresponding features such as amplitude and period length originate from the normal distribution of the pseudo-random sample set, is rejected.

4. Discussion 4.1. Brain state transitions The results indicate that the CNS is not maintained at a static level but rather operates in an oscillatory mode with periodically changing brain states. One model, which may explain the described changes in dimensional complexity, has been reported by Basar [14]. Using the model of coupled oscillators, the brain may be regarded as a large ensemble of coupled oscillators that represents the neuronal populations of the brain. From this ensemble, a subset of N oscillators are left uncoupled. When the oscillators are coupled, the dimension will be reduced. The possible existence of ‘‘synchronized phase-locked groups of neuronal oscillators and statistically unrelated phasors’’, responsible for changes in EEG complexity, was the subject of a fundamental question previously raised by Nicolis [106]. Accordingly, Ro¨schke and Aldenhoff [114] deduced from EEG sleep data that an increased dimensionality during sleep stages I, II, and REM may be caused by a dominance of weakly coupled oscillators attributed to various neuronal networks with independent frequencies. Conversely, a decreased dimensionality observed in slow wave sleep stages, such as stage IV, may be due to an increased number of strongly coupled oscillators. Moreover, the reduced dimensionality could reflect an enhanced ‘synergetic self-organization‘ with lower entropy [49,51,127] and increased order that synchronizes some of the subsystems (oscillators) and cortical networks, respectively, and leads to state or phase transitions from high-dimensional to lowdimensional ‘brain chaos’, thereby reflecting states with high and low entropy [12,15,50,67,72]. A model of structural changes of order in chaos with chaos –chaos phase

89

transitions and dimension fluctuations was presented by Aizawa [7]. Another model for state transitions was proposed by Wright et al. [138]. A more theoretical approach from nonlinear physics dealing with these transitions was given by Gaponov-Grekhov and Rabinovich [45]. Likewise, deterministic brain states with state transitions were described by Achimowicz [2] who investigated the dynamics of the neural system. The transients in EEG complexity described above are in line with Basar’s [17] ‘operative states’, which were interpreted as functional operators of the EEG activity, thus reflecting the degree of synchronization and desynchronization in various EEG frequencies in the brain. Our findings are supported by the reports of Basar et al. [19] who observed large fluctuations in the correlation dimension ranging from 5 to 8 during 30-min EEG recordings with distinct alpha activity derived from relaxed but awake subjects, under eyes closed condition. From these results, he deduced that ‘‘the brain has two types of behavior: noisy behavior and strange attractor behavior’’. In contrast to our study, Basar used consecutive EEG epochs of 3 min duration that could not reveal the well-ordered temporal pattern of the transients in complexity. The existence of such transients obtained from experimental observations confirms the conclusions of Lopes da Silva et al. [86,92]. He suggested that the EEG signal can reflect functional states of neural networks and that, during the waking state, there are also ‘noisy’ states. A change of state may be accomplished by a transition from a random or nonoscillatory type of activity to an oscillatory mode being in line with Szentagothai and Erdi’s [124] assumption of the self-organization of neural noise. Furthermore, he concluded that the brain processes information in a parallel fashion. This makes good sense in a ‘noisy’ state where each neuron or neuronal population is firing irregularly and independently [14]. Accordingly, it may be deduced that transient periods of high complexity of the EEG, where independent areas are active, may allow a fast parallel information processing running in a distributed mode. Here, numerous processes from sensory and cognitive ‘channels’ would be executed simultaneously. This desynchronized neural state may be related to active information processing in the cortex [1,120] that is in line with Mpitsos’ [104] question of ‘‘whether an ongoing noise-like activity might enhance information-handling ability’’. This question is based on the consideration of the brain as a noisy processor [6]. The approach of parallel computing previously discussed by Haken [50] and Keidel et al. [70] is consistent with the computer metaphor of Nelson and Bower [105] including parallel distributed processing. Furthermore, this approach is consistent with the information processing paradigm of Pritchard and Duke [110] viewing the brain metaphorically working ‘like’ a digital computer. Interposed periods of serial information processing running in a central mode, with a high coupling strength between different brain structures (including hemispheres

90

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

and subcortical – cortical connections) that resembles a higher synchronization with lower complexity of the brain, may facilitate the data transfer to (and retrieval from) ’higher’ association areas, or between the hemispheres within the cortex. In this case, the brain may be maintained in a ‘resting’ state [1,18,120] (in the sense that the brain is less active) with reduced responsiveness [14,15] ‘‘carrying no new information in the future’’ [104]. This brain state is comparable to brain states encountered during epileptic seizures where depressed transfer of parallel processed information occurs [12,93]. This state was described first by Berger [22] in a rather simplified view as ‘‘a rather passive state contrasting with an active state of the brain in which alpha disappears and faster activity increases’’. It could be argued that the brain shows endogenous cyclic changes between these two processing modes. This is reflected by the changes of the chaotic dynamics of the spontaneous EEG activity, which are in line with the models of rapid information processing of Townsend [131] involving parallel and serial operations, and with the processing – load hypothesis of Pritchard and Duke [111]. The stabilization of such ordered transients in dimensional complexity seems to be an intrinsic property of the CNS with respect to the strategy of information processing. This stabilization might be responsible for higher mental functions, information storage and retrieval [2,5,67,85,106], and is consistent with Freeman’s [41] suggestions about brain state transitions: ‘‘a key step in information processing depends on an orderly transition of cortical activity from a quasi-equilibrium to a limit-cycle (synchronized oscillation state) and back again’’. Moreover, these considerations are in line with the research of Fingelkurts and Fingelkurts [39], which showed that experimentally induced or spontaneous brain activity require synchronization of large neuronal populations. Accordingly, the transient synchronization forms self-organized, unified, and metastable neuronal states, which directly underlie mental states and images either experimentally induced or internally generated. 4.2. Circa 1-min rhythms of dimensional complexity The approach of using sliding dimensional analysis of long-term recordings of human EEG signals enabled us to disclose spontaneous rhythmic variations of EEG complexity mainly in the alpha range. The numerical range of the correlation dimension was comparable with those reported in the literature [11,19,74,86,98,111]. Using a software developed by our group, we found a temporal ‘meta-order’ in the nonlinear dynamics modulating the alpha rhythm during the resting (no-task) EEG state. This finding is consistent with the suggestion of Makeig and Inlow [96] that nonlinear measures may reveal time ordering of EEG changes. Other approaches that disclose temporal dynamics in the EEG are based on a segmental description of the EEG [13,38,39,61,64]. This method uses shorter segments and is based on linear stochastic models.

We found period lengths ranging from 30 to 90 s, most frequently around 60 s with a mean of 58.7 s. Comparable results using correlation dimension have not been described before due to the lack of appropriate algorithms and lack of available computing power. We have demonstrated that these slow oscillations are not revealed by linear measures, e.g., the autocorrelation function. Furthermore, we proved that these rhythms were not caused by the so-called Slutzky effect (see, e.g., Ref. [48]) assuming additional periodicities, which were not existent in the original process and which were generated by linear smoothing of time series. Moreover, adequate statistical test procedures using pseudo-random signals were carried out to reject the null hypothesis that the observed periodicity is a random one ( P < 0.0001). The observed rhythms are in line with various reports on visual and spectral evaluations that describe similar periodicities of approximately 1 min. These earlier reports cover spontaneous alterations of cortical DC-levels [8], changes of EEG activity under anesthesia [84], EEG sleep patterns [118,126], evoked steady-state brain responses [44], changes in human vibratory output [71,72], and fluctuations in performance and EEG [3,96]. 4.3. Pacemakers The term ‘pacemaker’ is widely used in the literature and not only restricted to autonomous oscillators such as the heart or circadian clock. The first report on this subject was issued by Hoagland [57] who suggested that the ability to make temporal judgments may depend on a temporal pacemaker. The still existing hypothesis that both the alpha rhythm and the frequency of a temporal clock might be determined by a single common pacemaker is mainly derived from this work. A number of investigators examined this hypothesis. Werboff [136] found anomalous relations between EEG frequency and the duration of temporal productions. Holubar [58] concluded by means of photic driving examinations of the EEG that the alpha rhythm constitutes a temporal pacemaker. The issue of the alpha pacemaker has been controversially discussed. The classical concept of the thalamic pacemaker is based on the assumption that rhythmic activity, such as the alpha rhythm, is generated by a thalamic network through recurrent inhibition circuits. Andersen and Andersson [10] identified the thalamus nuclei as the pacemaker of basic alpha activity. Evidence that such oscillations should be caused by ‘pacemaker’ cells was given by Jahnsen and Llinas [60] who demonstrated that some types of thalamic neurons may display oscillatory behavior in vitro, even after blocking synaptic transmission. Such neurons can generate intrinsic membrane oscillations mainly in the frequency range of 6 to 10 Hz dependent on initial conditions (control parameters). According to the conclusions of Lopes da Silva et al. [93], these pacemaker neurons do not operate as real autonomous and continuous ‘pacemakers’ found in the heart or the circadian clock. They go a

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

step further when they propose an alternative stochastic model instead of the deterministic pacemaker concept. This model implies a neural network with adequate filter properties which is submitted to a random input [88]. Likewise, Treisman [132] did not support the hypothesis of a common pacemaker. On the other hand, Nicolis [106] emulated the model of the ‘‘thalamocortical pacemaker as a high amplitude dissipative self-sustained nonlinear oscillator based on the phase-locking of thalamocortical and corticothalamic feedback –feedforward loops’’. According to these models, we have used the term ‘pacemaker’ with respect to the interpretation and modeling of periodicities slower than that of the alpha rhythm. Bearing in mind the numerous reports on circa 1-min rhythms corroborating our finding on variations of EEG complexity, it seems likely that a common ‘pacemaker’, i.e., ‘‘some neuromodulation control mechanism drives the brain activity away from a simple steady state’’ [109]. Similar proposals were already made by Lugaresi et al. [94] suggesting a ‘common central underlying mechanism’, by Vassilevsky and Aleksanian [134] hypothesizing an ‘adaptogenetic brain biorhythm’, by Carpenter and Grossberg [29] establishing a small network model of the circadian pacemaker located in the suprachiasmatic nuclei (SCN) of the mammalian hypothalamus with signal patterns manifested in activity – rest cycles, and by Galambos and Makeig [44] deducing a ‘cyclic modulator’ under endogenous control with uncertain anatomical locations. A simple twomode model with coupled nonlinear oscillators, one of which is self-excited, was proposed by Gaponov-Grekhov and Rabinovich [45]. The concept of coupled oscillators was also used for the model of the so-called central pattern generator [20]. Citing the considerations of Erdi [36] and Achimowicz [2], the cyclic temporal pattern of complexity observed in our study may be interpreted as a self-organized rhythmicity that is generated at single neuron level by ‘endogenous pacemaker neurons’. This may be modeled by a set of nonlinear cortical generators driven by interacting subcortical pacemakers. As autonomous (e.g., cardiovascular or respiratory) rhythms exhibit comparable period lengths to those we show for EEG [76,78,79,94], it seems reasonable to suppose that a common neuronal oscillating structure functioning as a ‘single common pacemaker’, as previously proposed by Hoagland [57], might at least in part underlie both phenomena. Following the simplified approach of Keidel et al. [68], it may be inferred that a ‘common brainstem system’ with its ‘dynamic specificity’ and other subcortical structures may act as an oscillating network responsible for the observed rhythms in dimensional complexity. This concept is supported by the suggestions of Lenard et al. [81] who revealed a statistical relationship between respiratory cycles and EEG periodicity, and who deduced that brainstem structures act as a pacemaker. The same relationship was found by Burioka et al. [25] using the correlation dimension. Accordingly, Sandman et al. [116], and Walker and Walker

91

[135] found a coupling between the pulse pressure gradient and alpha rhythm. Likewise, Achimowicz [3] supports the idea of the common brainstem system acting as a pacemaker when he described a functional coupling between the autonomous (ANS) and central nervous systems (CNS). He suggested that slow periodic changes of rhythmic EEG activity (modulations) are related to low-frequency oscillations of cortical excitability, which may be driven by control centers at the brain stem level and attributed to brainstem cardiovascular and respiratory centers. A theoretical concept of the integration between the ANS and CNS was presented by Jennings and Coles [62]. The hypothesis of the ‘common brainstem system’ was further supported by the reports of Langhorst et al. [78,79] who found indications that reticular neurons of the lower brain stem influence not only the cardiovascular system but also the respiration, motor system, and the degree of synchronization of the EEG. Studies of Moruzzi and Magoun [103] and Moruzzi [102] have shown that the reticular formation as an arousal system has the ability to modulate the global level of cortical excitability. Following these considerations, one is tempted to speculate that the reticular formation may operate as an ascending and descending activating central controller in the sense of a central ‘master network’ or ‘central pattern generator’. Moreover, the idea of a common cortical control mechanism is supported by the results of the Friedman test, indicating that there are no differences in the mean period length of the rhythms of complexity over all four brain regions as revealed in our study. Finally, the defined central mode of generating the observed rhythmicities in the EEG complexity cannot be fully elucidated from our data. Thus, further investigations with additional recording of other physiological variables such as heart rate, EMG, and EOG are required to establish if this control mechanism may work autonomously as a ‘pacemaker’ by means of correlating the covariation of cardiovascular parameters with cyclic changes of the observed EEG complexity. 4.4. Pathophysiological aspects The nonlinear modeling of the bioelectrical activity of the brain as used in this paper and the interpretation of the resulting cyclic transients in dimensional complexity give an additional insight into the integrative functioning of the brain. Furthermore, this approach suggests a prominent role in determining nonlinear characteristics, which correspond to different physiological or pathophysiological states and which cannot be identified by conventional analyses. An additional important aspect is that the study of temporal order of nonlinear dynamics may reveal disturbances of this order which may accompany CNS dysfunctions in many diseases, e.g., in schizophrenia. Accordingly, Callaway and Naghdi [27] reported on their two-stage model of information processing that schizophrenics seem to have a defect in controlled serial information processing, but not in automatic, parallel processing.

92

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

In clinical situations, the observed long-term oscillations in EEG complexity may be of predictive value in characterizing the human brainstem function, e.g., in newborns with premature brainstem, in sudden death infant syndrome, or in monitoring of comatose patients with deteriorating brainstem function. One is tempted to predict that a dysfunction of the brainstem will be correlated with an increase of the period length of the oscillations and/or a decrease in amplitude of the described periodic changes in EEG complexity. In general, in clinical applications, it may appear that ‘lesions’ in time become more evident at an earlier stage than ‘lesions’ in structure or substrate, which, at present, form the common basis of current diagnostic and therapy concepts.

Acknowledgements We thank Michael J. Atkinson as well as the reviewers for critical and constructive suggestions on earlier drafts.

References [1] A. Accardo, M. Affinito, M. Carrozzi, F. Bouquet, Use of fractal dimension for the analysis of electroencephalographic time series, Biol. Cybern. 77 (1997) 339 – 350. [2] J.Z. Achimowicz, On the deterministic brain states revealed by VEP classification in phase domain, in: I. Dvorˇa´k, A.V. Holden (Eds.), Mathematical Approaches to Brain Functioning Diagnostics, Manchester University Press, Manchester, 1991, pp. 209 – 230. [3] J.Z. Achimowicz, Evaluation of pilot psychophysiological state in real time by analysis of spectral dynamics in EEG and ERP correlates of sensory and cognitive brain functions and its possible coupling with autonomic nervous system, Research Proposal Draft Version 10.5.1992, Human System Division, H.G. Armstrong Aero-space Medical Research Laboratory, Wright – Petterson Air Force Base, Ohio. [4] S. Adak, Time-dependent spectral analysis of nonstationarity time series, J. Am. Stat. Assoc. 93 (1998) 1488 – 1501. [5] R.W. Adey, Neurophysiological correlates of information transaction and storage of brain tissue, in: E. Steller, J.M. Sprague (Eds.), Progress in Physiological Psychology, vol. 1, Academic Press, New York, 1966, pp. 1 – 43. [6] R.W. Adey, Organization of brain tissue: is the brain a noisy processor? Int. J. Neurosci. 3 (1972) 271 – 284. [7] Y. Aizawa, Chaos – chaos phase transition and dimension fluctuation, in: G. Mayer-Kress (Ed.), Dimensions and Entropies in Chaotic Systems, Quantification of Complex Behavior, Springer Series in Synergetics, vol. 32, Springer, New York, 1986, pp. 34 – 41. [8] N.A. Aladjalova, Infraslow oscillations of the steady potential of the cerebral cortex, Nature 179 (1957) 957 – 961. [9] N.A. Aladjalova, Slow electrical processes in the brain, Progress in Brain Research, vol. 7, Elsevier, Amsterdam, 1964. [10] P. Andersen, S.A. Andersson, Physiological Basis of the Alpha Rhythm, Appleton-Century-Crofts, New York, 1968. [11] A. Babloyantz, A. Destexhe, Strange attractors in the human cortex, in: L. Rensing, U. an der Heiden, M.C. Mackey (Eds.), Temporal Disorder in Human Oscillatory Systems, Springer Series in Synergetics, vol. 36, Springer, New York, 1986, pp. 48 – 55. [12] A. Babloyantz, A. Destexhe, Low-dimensional chaos in an instance of epilepsy, Proc. Natl. Acad. Sci. U. S. A. 83 (1986) 3513 – 3517.

[13] J.S. Barlow, Methods of analysis of nonstationary EEGs, with emphasis on segmentation techniques: a comparative review, J. Clin. Neurophysiol. 2 (1985) 267 – 304. [14] E. Basar, Toward a physical approach to integrative physiology, Am. J. Physiol. 245 (1983) R510 – R533. [15] E. Basar, Synergetics of neural populations, a survey on experiments, in: E. Basar, H. Flohr, H. Haken, A.J. Mandell (Eds.), Synergetics of the Brain, Springer, New York, 1983, pp. 183 – 200. [16] E. Basar (Ed.), Chaos in Brain Function, Springer Series in Brain Dynamics, vol. 3, Springer, New York, 1990. [17] E. Basar, Chaotic dynamics and resonance phenomena in brain function: progress, perspectives and thoughts, in: E. Basar (Ed.), Chaos in Brain Function, Springer Series in Brain Dynamics, vol. 3, Springer, New York, 1990, pp. 1 – 30. [18] E. Basar, C. Basar-Eroglu, J. Ro¨schke, A. Schu¨tt, The EEG is a quasi-deterministic signal anticipating sensory-cognitive tasks, in: E. Basar, T.H. Bullock (Eds.), Springer Series in Brain Dynamics, vol. 2, Springer, New York, 1989, pp. 43 – 71. [19] E. Basar, C. Basar-Eroglu, J. Ro¨schke, J. Schult, Chaos- and alpha preparation in brain function, in: R. Cotteril (Ed.), Models of Brain Function, Cambridge Univ. Press, Cambridge, 1989, pp. 365 – 395. [20] S.J. Bay, H. Hemami, Modelling of a neural pattern generator with coupled nonlinear oscillators, IEEE Trans. Biomed. Eng. 34 (1987) 297 – 306. ¨ ber das Elektroenkephalogramm des Menschen, Arch. [21] H. Berger, U Psychiatr. Nervenkr. 87 (1929) 527 – 570. ¨ ber das Elektroenkephalogramm des Menschen, Nova [22] H. Berger, U Acta Leopold. 6 (1938) 173 – 309. [23] R.P. Brenner, N. Schaul, Periodic EEG patterns: classification, clinical correlation, and pathophysiology, J. Clin. Neurophysiol. 7 (1990) 249 – 267. [24] B.E. Brodsky, B.S. Darkhovsky, Nonparametric Statistical Diagnosis. Problems and Methods, Kluwer Academic Publishing, Dordrecht, 2000. [25] N. Burioka, G. Corne´lissen, F. Halberg, D.T. Kaplan, Relationship between correlation dimension and indices of linear analysis in both respiratory movement and electroencephalogram, Electroencephalogr. Clin. Neurophysiol. 112 (2001) 1147 – 1153. [26] S.K. Burns, A.A. Borbely, R.D. Hall, Evoked potentials: three-dimensional display, Science 157 (1967) 457 – 459. [27] E. Callaway, S. Naghdi, An information processing model for schizophrenia, Arch. Gen. Psychiatry 349 (1982) 339 – 347. [28] G.A. Carpenter, A comparative analysis of structure and chaos in models of single nerve cells and circadian rhythms, in: E. Basar, H. Flohr, H. Haken, A.J. Mandell (Eds.), Synergetics of the Brain, Springer Series in Synergetics, vol. 23, Springer, New York, 1983, pp. 311 – 329. [29] G.A. Carpenter, S. Grossberg, A neural theory of circadian rhythms: the gated pacemaker, Biol. Cybern. 48 (1983) 35 – 59. [30] R. Dahlhaus, Maximum likelihood estimation and model selection for locally stationary processes, J. Nonparametr. Stat. 6 (1996) 171 – 191. [31] W.C. Dement, N. Kleitman, Cyclic variations in the EEG during sleep and their relation to eye movements, body motility, and dreaming, Electroencephalogr. Clin. Neurophysiol. 9 (1957) 73 – 690. [32] U. Dirnagl, K.M. Einha¨upl, G. Schmieder, P. Franz, C. Garner, H.W. Pfister, Etiology and significance of 0.5 – 2/min rhythm in intracranial pressure without periodic respiration, in: G. Hildebrandt, R. Moog, F. Raschke (Eds.), Chronobiology and Chronomedecine—Basic Research and Applications, Peter Lang, Frankfurt, 1987, pp. 209 – 213. [33] I. Dvorˇa´k, A.V. Holden, Mathematical Approaches to Brain Functioning Diagnostics, Manchester University Press, Manchester, 1991. [34] J.P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (3) (1985) 617 – 656 (Part I). [35] J.P. Eckmann, D. Ruelle, Fundamental Limitations for Estimating

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95

[36]

[37] [38]

[39]

[40]

[41]

[42] [43]

[44]

[45]

[46] [47]

[48] [49] [50]

[51]

[52]

[53]

[54]

[55]

[56]

Dimensions and Liapunov Exponents in Dynamical Systems, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France, 1990, Preprint IHES/P/90/11. P. Erdi, Self-organisation in the nervous system: network structure and stability, in: I. Dvorˇa´k, A.V. Holden (Eds.), Mathematical Approaches to Brain Functioning Diagnostics, Manchester University Press, Manchester, 1991, pp. 31 – 43. J.D. Farmer, E. Ott, J.A. Yorke, The dimension of chaotic attractors, Physica, D 7 (1983) 153 – 180. J. Fell, A. Kaplan, B. Darkhovsky, J. Ro¨schke, EEG analysis with nonlinear deterministic and stochastic methods: a combined strategy, Acta Neurobiol. Exp. 60 (2000) 87 – 108. A.A. Fingelkurts, A.A. Fingelkurts, Operational architectonics of the human brain biopotential field: towards solving the mind – brain problem, Brain Mind 2 (2001) 261 – 296. A.A. Fingelkurts, A.A. Fingelkurts, A.-Ya. Kaplan, The regularities of the discrete nature of multi-variability of EEG spectral patterns, Int. J. Psychophysiol. 47 (2003) 23 – 41. W.J. Freeman, Dynamics of image formation by nerve cell assemblies, in: E. Basar, H. Flohr, H. Haken, A.J. Mandell (Eds.), Synergetics of the Brain, Springer, New York, 1983, pp. 102 – 121. W.J. Freeman, The physiology of perception, Sci. Am. 264 (1991) 78 – 85. W.J. Freeman, C.A. Skarda, Spatial EEG patterns, non-linear dynamics and perception: the neo-Sherringtonian view, Brain Res. Rev. 10 (1985) 147 – 175. R. Galambos, S. Makeig, Dynamic changes in steady-state responses, in: E. Basar (Ed.), Dynamics of Sensory and Cognitive Processing by the Brain, Springer Series in Brain Dynamics, vol. 1, Springer, New York, 1988, pp. 103 – 122. A.V. Gaponov-Grekhov, M.I. Rabinovich, Nonstationary structures—chaos and order, in: E. Basar, H. Flohr, H. Haken, A.J. Mandell (Eds.), Synergetics of the Brain, Springer Series in Synergetics, vol. 23, Springer, New York, 1983, pp. 227 – 237. P. Grassberger, I. Procaccia, Measuring the strangeness of strange attractors, Physica, D 9 (1983) 189 – 208. R.A.M. Gregson, L.A. Britton, E.A. Campell, R.G. Gates, Comparisons of the non-linear dynamics of electroencephalograms under various task loading conditions: a preliminary report, Biol. Psychol. 31 (1990) 173 – 191. U. Grenander, M. Rosenblatt, Statistical Analysis of Stationary Time Series, Wiley, New York, 1957. H. Haken, Synergetics: an introduction, in: Springer Series in Synergetics, vol. 1, Springer, New York, 1977. H. Haken, Synopsis and introduction, in: E. Basar, H. Flohr, H. Haken, A.J. Mandell (Eds.), Synergetics of the Brain, Springer Series in Synergetics, vol. 23, Springer, New York, 1983, pp. 3 – 25. H. Haken, Information and Self-Organization: A Macroscopic Approach to Complex Systems, Springer Series in Synergetics, vol. 40, Springer, New York, 1988. F. Halberg, Y.L. Tong, E.A. Johnson, Circadian system phase—an aspect of temporal morphology: procedures and illustrative examples, in: H. von Mayersbach (Ed.), The Cellular Aspects of Biorhythms, Springer, New York, 1967, pp. 20 – 48. J.W. Havstad, C. Ehlers, Attractor dimension of nonstationary dynamical systems from small data sets, Phys. Rev., A 39 (2) (1989) 845 – 853. M. Hayashi, K. Sato, T. Hori, Ultradian rhythms in task performance, self evaluation, and EEG activity, Percept. Mot. Skills 29 (1994) 791 – 800. G. Hildebrandt, The autonomous time structure and its reactive modifications in the human organism, in: L. Rensing, U. an der Heiden, M.C. Mackey (Eds.), Temporal Disorder in Human Oscillatory Systems, Springer Series in Synergetics, vol. 36, Springer, New York, 1987, pp. 69 – 78. G. Hildebrandt, R. Moog, F. Raschke (Eds.), Chronobiology and

[57] [58] [59]

[60]

[61] [62]

[63]

[64] [65] [66]

[67]

[68]

[69]

[70] [71] [72]

[73]

[74] [75]

[76]

[77]

[78]

93

Chronomedecine—Basic Research and Applications, Peter Lang, Frankfurt, 1987. H. Hoagland, The physiological control of judgements of durations: evidence for a chemical clock, J. Gen. Psych. 9 (1933) 267 – 287. J. Holubar, The Sense of Time, MIT Press, Cambridge, MA, 1969. F.C. Hoppensteadt, E.M. Izhikevich, Synaptic organizations and dynamical properties of weakly connected neural oscillators, Biol. Cybern. 75 (1996) 117 – 127. H. Jahnsen, R. Llinas, Ionic basis for the electroresponsiveness and oscillatory properties of guinea-pig thalamic neurons in vitro, J. Physiol. 349 (1984) 227 – 247. B.H. Jansen, A. Hasman, R. Lenten, Piece-wise EEG analysis: an objective evaluation, Int. J. Bio-Med. Comput. 12 (1981) 17 – 27. J.R. Jennings, M.G.H. Coles (Eds.), Handbook of Cognitive Psychophysiology, Central and Autonomic Nervous System Approaches, Wiley Psychophysiology Handbooks,Wiley, Chichester, England, 1991. A.Ya. Kaplan, Nonstationary EEG: methodological and experimental analysis, Usp. fiziol. Nauk (Success in Physiological Sciences) 29 (1998) 35 – 55. A.Ya. Kaplan, The problem of segmental description of human electroencephalogram, Hum. Physiol. 25 (1999) 107 – 114. L. Katz, R.Q. Cracco, A review of the cerebral rhythms in the waking EEG, Am. J. E.E.G. Technol. 11 (1971) 77 – 100. W.-D. Keidel, The physiological background of the electric response audiometry, in: W.-D. Keidel, W.D. Neff (Eds.), Handbook of Sensory Physiology, vol. 5, Part 3, Springer, New York, 1976, pp. 105 – 231. M. Keidel, W.-D. Keidel, W.S. Tirsch, S.J. Po¨ppl, Studying temporal order in human CNS by means of ‘running’ frequency and coherence analysis, in: L. Rensing, U. an der Heiden, M.C. Mackey (Eds.), Temporal Disorder in Human Oscillatory Systems, Springer Series in Synergetics, vol. 36, Springer, New York, 1987, pp. 69 – 78. M. Keidel, W.-D. Keidel, W.S. Tirsch, S.J. Po¨ppl, Possible central oscillators synchronizing brain waves and muscle oscillators in man, in: G. Hildebrandt, R. Moog, F. Raschke (Eds.), Chronobiology and Chronomedecine—Basic Research and Applications, Peter Lang, Frankfurt, 1987, pp. 202 – 208. M. Keidel, W.-D. Keidel, W.S. Tirsch, S.J. Po¨ppl, Amplitude cycles of tremor, muscle oscillations and EEG by means of ‘running’ frequency analysis, Electroencephalogr. Clin. Neurophysiol. 69 (1988) 14P. M. Keidel, W.S. Tirsch, S.J. Po¨ppl, Psychophysiological aspects of motor related brain activity, J. Psychophysiol. 3 (1989) 308 – 309. M. Keidel, W.S. Tirsch, S.J. Po¨ppl, Circa-minute rhythm in human vibratory output, Naturwissenschaften 76 (1989) 576 – 577. M. Keidel, W.-D. Keidel, W.S. Tirsch, S.J. Po¨ppl, Temporal pattern with circa 1-minute cycles in the running coherence function between EEG and VMG in man, J. Interdiscip. Cycle Res. 21 (1990) 33 – 50. M. Keidel, W.S. Tirsch, S.J. Po¨ppl, M. Radmacher, Circa 1-minute periodicity in brain dynamics, in: C.H.M. Brunia, A.W.K. Gaillard, A. Kok (Eds.), Psychophysiological Brain Research, Tilburg University Press, Tilburg, 1990, pp. 59 – 64. C.C. King, Fractal and chaotic dynamics in nervous systems, Prog. Neurobiol. 36 (1991) 279 – 308. N. Kleitman, Basic rest – activity cycle in relation to sleep and wakefulness, in: A. Kales (Ed.), Sleep: Physiology and Pathology, Lippincott, Philadelphia, 1969, pp. 33 – 38. H.P. Koepchen, S.M. Hilton, A. Trzebski, Central Interaction Between Respiratory and Cardiovascular Control Systems, Springer, New York, 1980. B. Lacroix, E. Stanus, G. Servoz, J. Mendlewicz, Linear and nonlinear models for sleep cycles and ultradian oscillations in EEG spectral parameters, in: W.P. Koella, E. Ru¨ther, H. Schulz (Eds.), Sleep ’84, Gustav Fischer Verlag, Stuttgart, 1985, pp. 265 – 268. P. Langhorst, M. Stroh-Werz, K. Dittmar, H. Camerer, Facultative

94

[79]

[80]

[81]

[82]

[83] [84]

[85]

[86]

[87] [88]

[89]

[90]

[91]

[92]

[93]

[94]

[95] [96]

[97]

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95 coupling of reticular neuronal activity with peripheral cardiovascular and central cortical rhythms, Brain Res. 87 (1975) 407 – 418. P. Langhorst, P. Schulz, M. Lambertz, G. Schulz, H. Camerer, Dynamic characteristics of the ‘‘ unspecific brain stem system’’, in: H.P. Koepchen, S.M. Hilton, A. Trzebski (Eds.), Central Interaction Between Respiratory and Cardiovascular Control System, Springer, New York, 1980, pp. 30 – 41. K. Lehnertz, J. Arnhold, P. Grassberger, C.E. Elger (Eds.), Proceedings of the 1999 Workshop. Chaos in Brain? World Scientific, London, 2000. H.G. Lenard, P.L. Yaneza, M. Reimer, Polygraphic recordings in subacute sclerosing panenecephalitis. A study of the pathophysiology of the periodic EEG complexes, Neuropadiatrie 7 (1976) 52 – 56. K. Linkenkaer-Hansen, V.V. Nikouline, J.M. Palva, R.J. Ilmoniemi, Long-range temporal correlations and scaling behavior in human brain oscillations, J. Neurosci. 21 (2001) 1370 – 1377. O. Lippold, The Origin of the Alpha Rhythm, Churchill Livingstone, Edinburgh, 1973. U. Lips, A. Schultz, I. Pichlmayr, Cyclic changes in absolute power of the dominant frequency band in the EEG, Electroencephalogr. Clin. Neurophysiol. 69 (1988) 18P. R.R. Llinas, The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function, Science 242 (1988) 1654 – 1664. F.H. Lopes da Silva, Neural mechanism underlying brain waves: from neural membranes to networks, Electroencephalogr. Clin. Neurophysiol. 79 (1991) 81 – 93. F.H. Lopes da Silva, W. Storm van Leeuwen, The cortical source of the alpha rhythm, Neurosci. Lett. 6 (1977) 237 – 241. F.H. Lopes da Silva, T.H.M.T. van Lierop, C.F. Schrijer, W. Storm van Leeuwen, Organization of thalamic and cortical alpha rhythms: spectra and coherences, Electroencephalogr. Clin. Neurophysiol. 35 (1973) 627 – 639. F.H. Lopes da Silva, A. Hoeks, L.H. Zetterberg, Model of brain rhythmic activity, the alpha-rhythm of the thalamus, Kybernetik 15 (1974) 27 – 37. F.H. Lopes da Silva, A. Duquesnoy, K. van Hulten, J.G. Lommen, Analysis of nonstationary electroencephalograms, in: M. Matejcek, G.K. Schenk (Eds.), Quantitative Analysis of the EEG, AEG-Telefunken, Konstanz, 1975, pp. 421 – 433. F.H. Lopes da Silva, A. van Rotterdam, P. Barts, E. van Heusden, W. Burr, Models of neuronal populations: the basic mechanisms of rhythmicity, in: M.A. Corner, D.F. Swaab (Eds.), Progress in Brain Research, vol. 45, Elsevier, Amsterdam, 1976, pp. 281 – 308. F.H. Lopes da Silva, W. Kamphuis, J.M.A.M. van Neerven, P.M. Pijn, Cellular and network mechanisms in the kindling model of epilepsy: the role of GABAergic inhibition and the emergence of strange attractors, in: E.R. John, T. Harmony, L. Prichep, M. ValdesSosa, P. Valdes-Sosa (Eds.), Machinery of the Mind, Birkhauser, Boston, MA, 1990, pp. 115 – 139. F.H. Lopes da Silva, J.P.M. Pijn, J.A. Gortier, E. van Vliet, E.W. Daalman, W. Blanes, Rhythms of the brain: between randomness and determinism, in: K. Lehnertz, J. Arnhold, P. Grassberger, C.E. Elger (Eds.), Proceedings of the 1999 Workshop. Chaos in Brain? World Scientific, London, 2000, pp. 63 – 76. E. Lugaresi, G. Coccagna, M. Mantovani, R. Lebrun, Some periodic phenomena arising during drowsiness and sleep in man, Electroencephalogr. Clin. Neurophysiol. 32 (1972) 701 – 705. M.C. Mackey, C. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977) 287 – 289. S. Makeig, M. Inlow, Lapses in alertness: coherence of fluctuations in performance and EEG spectrum, Electroencephalogr. Clin. Neurophysiol. 86 (1993) 23 – 35. G. Mayer-Kress, J. Holzfuss, Analysis of the human electroencephalogram with methods from non-linear dynamics, in: L. Rensing, U. an der Heiden, M.C. Mackey (Eds.), Temporal Disorder in Human

[98]

[99] [100]

[101] [102] [103]

[104]

[105] [106] [107]

[108] [109]

[110]

[111]

[112]

[113] [114] [115]

[116]

[117]

[118]

[119]

Oscillatory Systems, Springer Series in Synergetics, vol. 36, Springer, New York, 1987, pp. 57 – 68. G. Mayer-Kress, P.S. Layne, Dimensionality of the human electroencephalogram, in: S.H. Koslow, A.J. Mandell, M.F. Shlesinger (Eds.), Perspectives in Biological Dynamics and Theoretical Medicine, Ann. N. Y. Acad. Sci., vol. 504, The New York Academy of Sciences, New York, 1987, pp. 62 – 87. O.S. Meneses, C.M. Corsi, Ultradian rhythms in the EEG and task performance, Chronobiologia 17 (1990) 183 – 194. N.I. Moiseeva, Z.A. Aleksanian, Slow-wave oscillations or the multi-unit activity average frequency in the human brain during drowsiness and sleep, Electroencephalogr. Clin. Neurophysiol. 63 (1986) 431 – 437. C.E. Molnar, T.F. Weiss, Repetitive cortical responses to acoustic clicks, Q. Prog. Rep.-MIT Cambridge 57 (1960) 55 – 182. G. Moruzzi, Reticular influences on the EEG, Electroencephalogr. Clin. Neurophysiol. 16 (1964) 2 – 17. G. Moruzzi, H.W. Magoun, Brain stem reticular formation and activation of the EEG, Electroencephalogr. Clin. Neurophysiol. 1 (1949) 455 – 473. G.J. Mpitsos, Chaos in brain function and the problem of nonstationarity: a commentary, in: E. Basar, T.H. Bullock (Eds.), Springer Series in Brain Dynamics, vol. 2, Springer, New York, 1989, pp. 521 – 539. M.E. Nelson, J.M. Bower, Brain maps and parallel computers, Trends Neurosci. 13 (1990) 403 – 408. S.N. Nicolis, Chaotic Dynamics Applied to Biological Information Processing, Akademie-Verlag, Berlin, 1987. A.R. Osborne, A. Provenzale, Finite correlation dimension for stochastic systems with power – law spectra, Physica, D 35 (1989) 357 – 381. N.H. Packardt, J.P. Crutchfield, J.D. Farmer, R.S. Shaws, Geometry from a time series, Phys. Rev. Lett. 45 (1980) 712 – 715. L. Poupard, R. Sarte`ne, Scaling behavior in the h-wave amplitude modulation and its relationship to alertness, Biol. Cybern. 85 (2001) 19 – 26. W.S. Pritchard, D.W. Duke, Measuring chaos in the brain: a tutorial review of non-linear dynamical EEG analysis, Int. J. Neurosci. 67 (1992) 31 – 80. W.S. Pritchard, D.W. Duke, Dimensional analysis of no-task human EEG using the Grassberger – Procaccia method, Psychophysiology 29 1992, pp. 182 – 192. P.E. Rapp, E.T. Bashore, J. Mertinerie, A. Albano, A. Zimmermann, A. Meed, Dynamics of brain electrical activity, Brain Topogr. 2 (1990) 99 – 118. E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge Univ. Press, Cambridge, 1991. J. Ro¨schke, J. Aldenhoff, The dimensionality of human’s electroencephalogram during sleep, Biol. Cybern. 64 (1991) 307 – 313. J. Ro¨schke, E. Basar, Correlation dimension in various parts of the cat and human brain in different states, in: E. Basar, T.H. Bullock (Eds.), Brain Dynamics: Progress and Perspectives, Springer Series in Brain Dynamics, vol. 2, Springer, New York, 1989, pp. 131 – 148. C.A. Sandman, B.B. Walker, C. Berka, Influence of afferent cardiovascular feedback on behavior and the cortical evoked potentials, in: J.T. Cacioppo, R.E. Petty (Eds.), Perspectives in Cardiovascular Psychophysiology, Guilford Press, New York, 1982, pp. 189 – 222. W. Scheuler, P. Rappelsberger, F. Schmatz, Periodicities of the alpha sleep pattern, Electroencephalogr. Clin. Neurophysiol. 70 (1988) 70P – 71P. W. Scheuler, P. Rappelsberger, F. Schmatz, C. Pastelak-Price, H. Petsche, S. Kubicki, Periodicity analysis of sleep EEG in the second and minute ranges—example of application in different alpha activities in sleep, Electroencephalogr. Clin. Neurophysiol. 76 (1990) 222 – 234. H.G. Schuster, Deterministic Chaos: An Introduction, VCH, Weinheim, 1984.

W.S. Tirsch et al. / Brain Research Reviews 45 (2004) 79–95 [120] J.C. Shaw, K. O’Connor, C. Ongley, EEG coherence as a measure of the cerebral functional organisation, in: M.A.B. Brazier, H. Petsche (Eds.), Architectonics of the Cerebral Cortex, Raven Press, New York, 1978, pp. 245 – 255. [121] J.E. Skinner, M. Molnar, Event-related dimensional reductions in the primary auditory cortex of the conscious cat are revealed by new techniques for enhancing the non-linear dimensional algorithms, Int. J. Psychophysiol. 34 (1999) 21 – 35. [122] M. Steriade, P. Gloor, R.R. Llinas, F.H. Lopes da Silva, M.M. Mesulam, Basic report on cerebral rhythmic activities, Electroencephalogr. Clin. Neurophysiol. 76 (1990) 481 – 508. [123] S.H. Strogatz, Exploring complex networks, Nature 410 (2001) 268 – 276. [124] J. Szenta´gothai, P. Erdi, Self-organization in the nervous system, J. Soc. Biol. Struct. 12 (1989) 367 – 384. [125] F. Takens, Detetecting strange attractors in turbulence, in: D.A. Young, L.S. Young (Eds.), Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898, Springer, New York, 1981, pp. 365 – 381. [126] M.G. Terzano, L. Parrino, M.C. Spaggiari, The cyclic alternating pattern sequences in the dynamic organization of sleep, Electroencephalogr. Clin. Neurophysiol. 69 (1988) 437 – 447. [127] J. Theiler, Estimating fractal dimension, J. Opt. Soc. Am. A 7 (6) (1990) 1055 – 1073. [128] W.S. Tirsch, M. Keidel, S.J. Po¨ppl, Computer-aided detection of temporal patterns in human CNS dynamics, in: J.L. Willems, J.H. van Bemmel, J. Michel (Eds.), Progress in Computer-Assisted Function Analysis, Elsevier, Amsterdam, 1988, pp. 109 – 118.

95

[129] W.S. Tirsch, M. Keidel, M. Radmacher, S.J. Po¨ppl, Correlation between spectral density and fractal dimensionality in cyclic CNS dynamics, in: I. Dvorˇ a´ k, A.V. Holden (Eds.), Mathematical Approaches to Brain Functioning Diagnostics, Manchester University Press, Manchester, 1991, pp. 387 – 404. [130] W.S. Tirsch, M. Keidel, S. Perz, H. Scherb, G. Sommer, Inverse covariation of spectral density and correlation dimension in cyclic EEG dynamics of the human brain, Biol. Cybern. 82 (2000) 1 – 14. [131] J.T. Townsend, Serial vs. parallel processing, Psychol. Sci. 1 1990, pp. 46 – 54. [132] M. Treisman, Temporal rhythms and cerebral rhythms, in: J. Gibbon, L. Allan (Eds.), Timing and Time Perception, Ann. NY. Acad. Sci., vol. 423, (1984) 542 – 565. [133] Y. Tsuji, T. Kobayashi, Short and long ultradian EEG components in daytime arousal, Electroencephalogr. Clin. Neurophysiol. 70 (1988) 110 – 117. [134] N.N. Vassilevsky, Z.A. Aleksanian, The adaptive regulation of vegetative functions, Phiziol. Zh. SSSR 7 (1982) 948 – 952. [135] B.B. Walker, J.M. Walker, Phase relationship between cariotid pressure and ongoing electrocortical activity, Int. J. Psychophysiol. 1 (1983) 65 – 73. [136] J. Werboff, Time judgement as a function of electroencephalographic activity, Exp. Neurol. 6 (1962) 152 – 160. [137] A.T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967) 15 – 42. [138] J.J. Wright, R.R. Kydd, G.J. Lees, State-changes in the brain viewed as linear steady-states and non-linear transitions between steadystates, Biol. Cybern. 53 (1985) 11 – 17.