Agnessa Babloyantz

Evidence of Chaotic Dynamics of Brain Activity During the Sleep Cycle

The study of complex systems may be performed by analysing 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.

Is the normal heart a periodic oscillator

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.

Thermodynamics of evolution

The physicochemical basis of biological order is a puzzling problem that has occupied whole generations of biologists and physicists and has given rise, in the it, to passionate discussions. Biological systems are highly complex and ordered objects. It is generally accepted that the present order reflects structures acquired during a long evolution. Moreover, the maintenance of order in actual living systems requires a great number of metabolic and synthetic reactions as well as the existence of complex mechanisms controlling the rate and the timing of the various processes. All these features bring the scientist a wealth of new problems. In the first place one has systems that have evolved spontaneously to extremely organized and complex forms. On the other hand metabolism, synthesis and regulation imply a highly heterogeneous distribution of matter inside the cell through chemical reactions and active transport. Coherent behavior is really the characteristic feature of biological systems (see the box on...

Predictability of human EEG: a dynamical approach

The electroencephalogram recordings from human scalp are analysed in the framework of recent methods of nonlinear dynamics. Three stages of brain activity are considered: the alpha waves (eyes closed), the deep sleep (stage four) and the Creutzfeld-Jakob coma. Two dynamical parameters of the attractors are evaluated. These are the Lyapunov exponents, which measure the divergence or convergence of trajectories in phase space and the Kolmogorov or metric entropy, whose inverse gives the mean predicting time of a given EEG signal. In all the stages considered, the results reveal the presence of at least two positive Lyapunov exponents, which are the footprints of chaos. This number increases to three positive exponents in the case of alpha waves, indicating that although for very short episodes the alpha waves seem extremely coherent, the variability of the brain increases markedly over larger periods of activity. The degree of entropy/chaos increases from coma to deep sleep and then to alpha waves. The large predicting time observed for deep sleep suggests that these waves are related to a slow rate of information processing. The predicting time of the alpha waves is much smaller, indicating a rapid loss of information. Finally, with the help of the Lyapunov exponents, the attractors dimensions are evaluated using two different conjectures and compared to values obtained previously by the Grassberger-Procaccia algorithm.

A comparative study of the experimental quantification of deterministic chaos

Abstract Topological and correlation dimensions of physiological chaotic attractors are evaluated. The latter are embedded in phase spaces reconstructed using the laging method, multi-channel recordings and the singular value decomposition technique. A comparative study shows that comparable results are obtained only if the correlation dimension is less than four.

Spatiotemporal patterns in epileptic seizures

A neuronal network model of epilepsy is investigated. The network is described in terms of differential delay equations in which strong depolarization of any unit in the ensemble results in spike inactivation and the attenuation of that cells output. It can be shown that homogeneous oscillations with the qualitative features of epileptic seizures, including the progression from tonic to clonic firing patterns, appear when a highly depolarized homogeneous steady state becomes unstable. Stability calculations and the study of a simplified model that is solved analytically point to hyperexcitation as a critical determinant of epileptic activity. Spatially inhomogeneous solutions were studied in three types of connective topologies, i) uniformly densely connected networks, ii) densely connected networks containing a number of cells (microfoci) with pathologically strong connections to each other and to other normal cells, and iii) sparsely connected networks in which the strength of connections falls off as a function of the physical distance separating the cells. Homogeneous epileptic solutions remain stable to spatial perturbations in the first two types of topology. Type iii) may however give rise to a variety of spatiotemporal patterns, including travelling waves and “chaotic” behaviour. It is suggested that such inhomogeneous patterns may occur in the early stages of a seizure.

Models for cell differentiation and generation of polarity in diffusion-governed morphogenetic fields

Models based on molecular mechanisms are presented for pattern formation in developing organisms. It is assumed that there exists a diffusion governed gradient in the morphogenetic field. It is shown that cellular differentiation and the subsequent pattern formation result from the interaction of the diffusing morphogen with the genetic regulatory mechanism of cells. In a second stage it is shown that starting from a homogeneous distribution of morphogen, polarity can be generated spontaneously in the morphogenetic field giving rise to the establishment of a gradient. The stability of these gradients is demonstrated. The onset of a morphogenetic gradient and pattern formation are combined in a single coherent model. Size invariance and its biological implications are discussed.

Fluctuations in Open Systems

The probability of occurrence of fluctuations around nonequilibrium steady states or states slowly varying in time is discussed from a kinetic viewpoint. The limitations of the thermodynamic theory of fluctuations are investigated. It is shown that in a large class of far‐from‐equilibrium systems, described by a discrete set of variables, it is possible to extend the classical Einstein fluctuation formula, provided one uses suitable steady‐state parameters rather than equilibrium quantities. This conclusion is shown to be in agreement with the results of recent work by Prigogine and Glansdorff.

Chemical instabilities and multiple steady state transitions in Monod-Jacob type models.

Abstract A number of Monod-Jacob type models for induction or repression are studied analytically. It is shown that for well defined ranges of values of certain parameters such systems may exhibit transitions between several stable steady states. The thermodynamic aspects of such transitions are discussed within the framework of the recent theory of dissipative instabilities.

A model of the inward current Ih and its possible role in thalamocortical oscillations.

We investigated the kinetic properties of the hyperpolarization-activated inward current (Ih) of thalamocortical (TC) neurons. Recently, it was shown that this current is characterized by different time constants of activation and inactivation, which was in apparent conflict with the single-exponential time course of the current. We introduce here a model of Ih based on the cooperation of a slow and a fast activation variable and show that this kinetic scheme accounts for these apparently conflicting experimental data. We also report that following the combination of such a current with other currents seen in TC cells, one observes several types of oscillating behavior, similar to the slow oscillations and the spindle-like oscillations seen in vitro.