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.

Pacemaker-Induced Coherence in Cortical Networks.

A simple mathematical model of cortical tissue is introduced and the system's dynamics is monitored when a small subset of neurons is submitted to oscillatory inputs of various frequency and waveform. In the absence of input, the system shows desynchronized or "turbulent" behavior. The oscillatory input synchronizes the neuronal activity, which is strongest for inputs of low frequency. The increase of spatial coherence is estimated from the spatial autocorrelation function whereas the increase in temporal coherence is evaluated from correlation dimensions. The model accounts qualitatively for some of the features of the thalamocortical system.

Pattern segmentation in a binary/analog world: unsupervised learning versus memory storing.

We discuss the problem of segmentation in pattern recognition. We adopt the model and the general approach in the landmark paper by Wang, Buhmann and von der Malsburg (Neural Computation, (1990), 2, 94-106), and expand their model in a number of ways. We review their solution to the segmentation problem in associative memory, which consists in feature binding being expressed by synchrony relations between oscillators or populations of neurons. We extend the model by introducing a law of synaptic change, which allows the network to learn by structuring itself in response to stimuli with relevant features. We discuss the problem of interference between pattern completion and the learning of new memories. We also propose a form of multiplexing of input information taking advantage of the time-structure of the neurons' response. It is based on the assessment of analog as well as of binary properties of the stimuli and provides for an enhancement of the network's processing capacity. The relevance of the results for biological systems is pointed out.

A model for competing polymers leading to their spatial separation.

A model for a prebiotic polymer synthesis in a gradient of monomer is presented. In the absence of mutations the synthesis of the polymer proceeds in the region where the monomer concentration is the highest. However if a favorable mutation occurs, the latter accumulates in the high concentration zone and the initial polymer is restricted to a poorer monomer concentration region.

Mechanisms of target and spiral wave propagation in single cells.

Target and spiral wave propagation have been observed in single cells such as myocites. Moreover, in the same cells, transition from target waves to planar waves or from the latter to spiral waves was also observed. Considering an oscillatory medium described by the Ginzburg-Landau equation we suggest that such phenomena could be explained if cell nuclei and cell organelles are considered as obstacles in a small bounded medium. We discuss the role of cell geometry as well as the phenomenon of reentry at the cellular level.

Brain chaos and computation.

A model cortex comprising two interconnected spatiotemporal chaotic networks is considered. The system is able to discriminate between different patterns presented as input, and also detect motion and measure its velocity. Such cognitive processes are only possible if an "attentive" state arises in one of the networks, as a result of the stabilization of a periodic orbit out of the chaotic dynamics.

Control of spatiotemporal chaos in neuronal networks.

Different strategies for the control of chaotic neuronal networks and physical systems are presented. The variables of the dynamics span phase-spaces with very high dimensions. Novel control techniques are introduced that handle these high-dimensional cases. Depending on the system, several orbits with different spatiotemporal symmetries may be stabilized out of the chaotic attractor. The role of chaos control in information processing is pointed out.

Evidence for slow brain waves: a dynamical approach.

Various techniques of non-linear dynamics have been applied with success to EEG data and have provided a new insight into brain dynamics. Among these the 'recurrence plot' is a powerful tool for revealing the presence of drift or periodicities and is easily obtained in the framework of a non-linear dynamical analysis. When this analysis is applied to the EEG recorded from a patient suffering from Creutzfeldt-Jakob disease one observes the presence of slow periodicities of the order of 58 sec. We suggest that the recurrence plots are powerful tools for discovering hidden periodicities of EEG as well as the degree of stationarity of brain activity.

Low-dimensional chaos in an instance of epilepsy.

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.

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.

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 attractor's dimensions are evaluated using two different conjectures and compared to values obtained previously by the Grassberger-Procaccia algorithm.

Computation with chaos: a paradigm for cortical activity.

A device comprising two interconnected networks of oscillators exhibiting spatiotemporal chaos is considered. An external cue stabilizes input specific unstable periodic orbits of the first network, thus creating an "attentive" state. Only in this state is the device able to perform pattern discrimination and motion detection. We discuss the relevance of the procedure to the information processing of the brain.

Models for positional information and positional differentiation.

It is commonly thought that the formation of patterns in developing organisms is due to the existence of a gradient of a morphogen that determines the fate of cells as a function of position in the organism. A model is presented based on a molecular mechanism where the gradient is established by the active transport of a morphogen between source and sink. The cellular differentiation and the subsequent spatial pattern formation results from the interaction of this morphogen with the genetic regulatory mechanisms of cells. Some properties of the model are given and discussed in relation to grafting experiments in hydra.

Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons.

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.