Characterization of the domain chaos convection state by the largest Lyapunov exponent.

Using numerical integrations of the Boussinesq equations in rotating cylindrical domains with realistic boundary conditions, we have computed the value of the largest Lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths. We study in particular the domain chaos state, which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects. We compare our results with those from Egolf, [Nature 404, 733 (2000)], who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short-lived, spatially localized dislocation nucleation events. We also show that the quantity lambda1 is not intensive for aspect ratios Gamma over the range 20

Enhanced tracer transport by the spiral defect chaos state of a convecting fluid.

To understand how spatiotemporal chaos may modify material transport, we use direct numerical simulations of the three-dimensional Boussinesq equations and of an advection-diffusion equation to study the transport of a passive tracer by the spiral defect chaos state of a convecting fluid. The simulations show that the transport is diffusive and is enhanced by the spatiotemporal chaos. The enhancement in tracer diffusivity follows two regimes. For large Péclet numbers (that is, small molecular diffusivities of the tracer), we find that the enhancement is proportional to the Péclet number. For small Péclet numbers, the enhancement is proportional to the square root of the Péclet number. We explain the presence of these two regimes in terms of how the local transport depends on the local wave numbers of the convection rolls. For large Péclet numbers, we further find that defects cause the tracer diffusivity to be enhanced locally in the direction orthogonal to the local wave vector but suppressed in the direction of the local wave vector.

Mean flow and spiral defect chaos in Rayleigh-Bénard convection.

We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.

A space-time adaptive method for simulating complex cardiac dynamics.

For plane-wave and many-spiral states of the experimentally based Luo-Rudy 1 model of heart tissue in large (8 cm square) domains, we show that a space-time-adaptive time-integration algorithm can achieve a factor of 5 reduction in computational effort and memory-but without a reduction in accuracy-when compared to an algorithm using a uniform space-time mesh at the finest resolution. Our results indicate that such an algorithm can be extended straightforwardly to simulate quantitatively three-dimensional electrical dynamics over the whole human heart.

Stationarity and redundancy of multichannel EEG data recorded during generalized tonic-clonic seizures.

To improve our understanding of the physiology of generalized tonic-clonic (GTC) seizures, we have investigated the stationarity and redundancy of 21-electrode EEG data recorded from ten patients during GTC seizures elicited by electroconvulsive therapy (ECT). Stationarity was examined by calculating probability density functions (pdfs) and power spectra over small equal-length non-overlapping time windows and then by studying, visually and quantitatively, the evolution of these quantities over the duration of the seizures. Our analysis shows that some seizures had no demonstrable stationarity, that most seizures had time intervals of at least a few seconds that were statistically stationary by several criteria, and that, in some seizures, there were leads which were delayed in manifesting the statistical changes associated with seizure onset evident in other leads. The redundancy analysis demonstrated for the first time posterior-to-anterior time delays in the mid-ictal region of GTC seizures. The implications of these findings are discussed for the analysis of GTC seizure EEG data, for the physiology of GTC seizures, and for ECT research.

A comparison of EEG signal dynamics in waking, after anesthesia induction and during electroconvulsive therapy seizures.

Evidence suggests that quantitative dynamical measures of electroencephalogram (EEG) signals are more appropriate for characterizing the differences between states in an individual rather than as absolute indices. One such measure, the largest Lyapunov exponent (lambda 1), appears to have potential for identifying seizure activity and for being of clinical utility for characterizing electroconvulsive therapy (ECT) seizures. As a result, we compared lambda 1 for the EEG recorded in 8 depressed subjects in 3 states: (1) during right unilateral ECT seizures, (2) during the pre-ECT waking state, and (3) following anesthesia administration but prior to ECT. Spectral amplitude and autocorrelation were also calculated in these states, allowing a comparison of these measures with lambda 1. We hypothesized that lambda 1 would be lowest during the ECT seizures, suggestive of greater EEG signal predictability over time during the seizures. We found that during the seizures lambda 1 was smaller, while spectral amplitude was larger. Significant inter-state differences were not found for the left temporal and occipital regions suggesting that these measures might serve as markers of the degree of seizure involvement of specific brain regions. Spectral amplitude and lambda 1 were uncorrelated and varied independently in some cases. The autocorrelation time was shortest in the waking EEG, and longest for the post-anesthesia EEG, and did not account for the differences seen in lambda 1. In contrast, the persistence of oscillations in the autocorrelation functions was greater for the ictal EEG than the other two states and may relate to lambda 1.

Power-law behavior of power spectra in low Prandtl number Rayleigh-Bénard convection.

The origin of the power-law decay measured in the power spectra of low Prandtl number Rayleigh-Bénard convection near the onset of chaos is addressed using long time numerical simulations of the three-dimensional Boussinesq equations in cylindrical domains. The power law is found to arise from quasidiscontinuous changes in the slope of the time series of the heat transport associated with the nucleation of dislocation pairs and roll pinch-off events. For larger frequencies, the power spectra decay exponentially as expected for time continuous deterministic dynamics.

The largest Lyapunov exponent of the EEG during ECT seizures as a measure of ECT seizure adequacy.

Attributes of the electroencephalogram (EEG) recorded during electroconvulsive therapy (ECT) seizures appear promising for decreasing the uncertainty that exists about how to define a therapeutically adequate seizure. In the present report we study whether one promising and not yet tested ictal EEG measure, the largest Lyapunov exponent (lambda1), is useful in this regard. We calculated lambda1 from 2 channel ictal EEG data recorded in 25 depressed subjects who received right unilateral ECT. We studied the relationship of lambda1 to treatment therapeutic outcome and to an indirect measure of treatment therapeutic potency, the extent to which the stimulus intensity exceeds the seizure threshold. We found lambda1 could be reliably calculated from ictal EEG data and that the global mean, maximum, and standard deviation of lambda1 were smaller in the more therapeutically potent moderately suprathreshold ECT and in therapeutic responders. These results imply a more predictable or consistent pattern of EEG seizure activity over time in more therapeutically effective ECT seizures. These findings also suggest the promise of lambda1 as a marker of ECT seizure therapeutic adequacy and build on our previous work suggesting that lambda1 may be useful for classifying seizures and for reflecting the relative physiologic impact of seizure activity.

A quantitative measurement of spatial order in ventricular fibrillation.

INTRODUCTION: The degree of organization in ventricular fibrillation (VF) is not known. As an objective measurement of spatial order, spatial correlation functions and their characteristic lengths were estimated from epicardial electrograms of pigs in VF. METHODS AND RESULTS: VF was induced by premature stimulation in five pigs. Electrograms were simultaneously recorded with a 22 x 23 array of unipolar electrodes spaced 1.12 mm apart. Data were obtained by sampling the signals at 2000 Hz for 20 minutes immediately after the initiation of FV. Correlations between all pairs of signals were computed at various times. Correlation lengths were estimated from the decay of average correlation as a function of electrode separation. The correlation length of the VF in pigs was found to be approximately 4 to 10 mm, varying as fibrillation progressed. The degree of correlation decreased in the first 4 seconds after fibrillation then increased over the next minute. CONCLUSION: The correlation length is much smaller than the scale of the heart, suggesting that many independent regions of activity exist on the epicardium at any one time. On the other hand, the correlation length is 4 to 10 times the interelectrode spacing, indicating that some coherence is present. These results imply that the heart behaves during VF as a high dimensional, but not random, system involving many spatial degrees of freedom, which may explain the lack of convergence of fractal dimension estimates reported in the literature. Changes in the correlation length also suggest that VF reorganizes slightly in the first minute after an initial breakdown in structure.