Using a time series obtained from the electroencephalogram recording of a human epileptic seizure, we show the existence of a chaotic attractor, the latter being the direct consequence of the deterministic nature of brain activity. This result is compared with other attractors seen in normal human brain dynamics. A sudden jump is observed between the dimensionalities of these brain attractors (4.05 ± 0.05 for deep sleep) and the very low dimensionality of the epileptic state (2.05 ± 0.09). The evaluation of the autocorrelation function and of the largest Lyapunov exponent allows us to sharpen further the main features of underlying dynamics. Possible implications in biological and medical research are briefly discussed.Recent progress in the theory of nonlinear dynamical systems has provided new methods for the study of time series in such fields as hydrodynamics (1), chemistry (2), climatic variability (3, 4), biochemistry (5, 6), and human brain activity (7). The study of such complex systems may be performed by analyzing experimental data recorded as a series of measurements in time of a pertinent and easily accessible variable of the system. In most cases, such variables describe a global or averaged property of the system. For example, a time series may be obtained by recording at regular time intervals the mean electrical activity of a portion of the mammalian cortex. Although it may seem that such data offer only a one-dimensional view of the activity of the brain, this is not the case: it can be shown that a time series may provide information about a large number ofpertinent variables, which may subsequently be used to explore and characterize the system's dynamics (8).More specifically, by using a time series one can determine the possibility of constructing an attractor and thereby establishing the deterministic character of the dynamics of the underlying system. This topological entity portrays the essential features of the system's dynamics and may be characterized by the numerical value of its Hausdorff dimension D. A steady state is represented by a point attractor D = 0 and a time periodic regime exhibits a line attractor with D = 1. In general, ifD is a noninteger-that is, a fractal dimension-we may be in the presence of a chaotic attractor. The main feature of chaotic attractors is their sensitivity to the initial conditions. After a lapse of time, it is increasingly difficult to predict the future evolution of the system from a given initial state.In this paper, we analyze the electrical activity of the human cortex by means of the electroencephalogram (EEG) recorded from an epileptic human patient and also from normal persons during sleep cycles. First, on the basis of analyses of the time series of EEG data, we show the existence of an epileptic attractor and determine its correlation dimension I, which is easily accessible from experimental data. Then, we analyze other dynamical properties of the time series, such as Lyapunov exponents and time autocorrelation function. The next sec...

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With the help of several independent methods of nonlinear dynamics, the electrocardiograms (ECG) of four normal human hearts are studied qualitatively and quantitatively. A total of 36 leads were tested. The power spectrum, the autocorrelation function, the phase portrait, the Poincaré section, the correlation dimension, the Lyapunov exponent and the Kolmogorov entropy all point to the fact that the normal heart is not a perfect oscillator. The cardiac activity stems from deterministic dynamics of chaotic nature characterized by correlation dimensions D2 ranging from 3.6 to 5.2. Two different phase spaces are constructed for the evaluation of D2: the introduction of time lags and the direct use of space vectors give similar results. It is shown that the variabilities in interbeat intervals are not random but exhibit short range correlations governed by deterministic laws. These correlations may be related to the accelerating and decelerating physiological processes. This new approach to the cardiac activity may be used in clinical diagnosis. Also they are valuable tools for the evaluation of mathematical models which describe cardiac activity in terms of evolution equations.

The oscillatory properties of single thalamocortical neurons were investigated by using a Hodgkin-Huxley-like model that included Ca2+ diffusion, the low-threshold Ca2+ current (lT) and the hyperpolarization-activated inward current (lh). lh was modeled by double activation kinetics regulated by intracellular Ca2+. The model exhibited waxing and waning oscillations consisting of 1-25-s bursts of slow oscillations (3.5-4 Hz) separated by long silent periods (4-20 s). During the oscillatory phase, the entry of Ca2+ progressively shifted the activation function of lh, terminating the oscillations. A similar type of waxing and waning oscillation was also observed, in the absence of Ca2+ regulation of lh, from the combination of lT, lh, and a slow K+ current. Singular approximation showed that for both models, the activation variables of lh controlled the dynamics of thalamocortical cells. Dynamical analysis of the system in a phase plane diagram showed that waxing and waning oscillations arose when lh entrained the system alternately between stationary and oscillating branches.

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