Henry S. Greenside

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.

Calculation of three-dimensional MHD equilibria with islands and stochastic regions

Abstract A three-dimensional MHD equilibrium code is described that does not assume the existence of good flux surfaces. Given an initial guess for the magnetic field, the code proceeds by calculating the pressure driven current and then by updating the field using Amperes law. The numerical algorithm to solve the magnetic differential equation for the pressure driven current is described, and demonstrated for model fields having islands and stochastic regions. The numerical algorithm which solves Amperes law in three dimensions is also described. Finally, the convergence of the code is illustrated for a particular stellarator equilibrium with no large islands.

Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method

A recently developed space-time adaptive mesh refinement algorithm (AMRA) for simulating isotropic one- and two-dimensional excitable media is generalized to simulate three-dimensional anisotropic media. The accuracy and efficiency of the algorithm is investigated for anisotropic and inhomogeneous 2D and 3D domains using the Luo-Rudy 1 (LR1) and FitzHugh-Nagumo models. For a propagating wave in a 3D slab of tissue with LR1 membrane kinetics and rotational anisotropy comparable to that found in the human heart, factors of 50 and 30 are found, respectively, for the speedup and for the savings in memory compared to an algorithm using a uniform space-time mesh at the finest resolution of the AMRA method. For anisotropic 2D and 3D media, we find no reduction in accuracy compared to a uniform space-time mesh. These results suggest that the AMRA will be able to simulate the 3D electrical dynamics of canine ventricles quantitatively for 1 s using 32 1-GHz Alpha processors in approximately 9 h.

Stable propagation of a burst through a one-dimensional homogeneous excitatory chain model of songbird nucleus HVC.

We demonstrate numerically that a brief burst consisting of two to six spikes can propagate in a stable manner through a one-dimensional homogeneous feedforward chain of nonbursting neurons with excitatory synaptic connections. Our results are obtained for two kinds of neuronal models: leaky integrate-and-fire neurons and Hodgkin-Huxley neurons with five conductances. Over a range of parameters such as the maximum synaptic conductance, both kinds of chains are found to have multiple attractors of propagating bursts, with each attractor being distinguished by the number of spikes and total duration of the propagating burst. These results make plausible the hypothesis that sparse, precisely timed sequential bursts observed in projection neurons of nucleus HVC of a singing zebra finch are intrinsic and causally related.

Fractal Basins of Attraction Associated with a Damped Newton's Method

An intriguing and unexpected result for students learning numerical analysis is that Newtons method, applied to the simple polynomial z 3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newtons method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting.

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 simple stochastic model for the onset of turbulence in Rayleigh-Bénard convection

Abstract A simple stochastic model of a particle diffusing randomly in an external two-well potential is proposed to stimulate the onset of turbulence in a medium aspect ratio cylindrical Rayleigh-Benard cell. The model is studied numerically and both time series and their power spectra are obtained. The results are compared with experimental data as well as with the results of deterministic models consisting of a finite number of interacting modes.

Pattern formation and dynamics in Rayleigh-Benard convection : numerical simulations of experimentally realistic geometries.

Abstract Rayleigh–Benard convection is studied and quantitative comparisons are made, where possible, between theory and experiment by performing numerical simulations of the Boussinesq equations for a variety of experimentally realistic situations. Rectangular and cylindrical geometries of varying aspect ratios for experimental boundary conditions, including fins and spatial ramps in plate separation, are examined with particular attention paid to the role of the mean flow. A small cylindrical convection layer bounded laterally either by a rigid wall, fin, or a ramp is investigated and our results suggest that the mean flow plays an important role in the observed wavenumber. Analytical results are developed quantifying the mean flow sources, generated by amplitude gradients, and its effect on the pattern wavenumber for a large-aspect-ratio cylinder with a ramped boundary. Numerical results are found to agree well with these analytical predictions. We gain further insight into the role of mean flow in pattern dynamics by employing a novel method of quenching the mean flow numerically. Simulations of a spiral defect chaos state where the mean flow is suddenly quenched is found to remove the time dependence, increase the wavenumber and make the pattern more angular in nature.

Lyapunov spectral analysis of a nonequilibrium Ising-like transition.

By simulating a nonequilibrium coupled map lattice that undergoes an Ising-like phase transition, we show that the Lyapunov spectrum and related dynamical quantities, such as the dimension correlation length

Edge turbulence measurements in TFTR

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