Walter J. Karplus

The simplified-FitzHugh-Nagumo model with action potential duration restitution: effects on 2D wave propagation

Abstract A modification of the simplified FitzHugh-Nagumo (FN) equations is proposed for introducing a residual component of the slow variable, which determines the restitution of action potential duration (APD) also known as the interval-excitation duration relationship. The three-step-wise approximation of ϵ( E ) which is widely used in current publications is replaced in a new model by a four-step approximation. This change is used for studying by computer simulation the effects of APD restitution properties independently of the APD and refractory period on 2D wave propagation in an isotropic matrix (made by 128 × 128 nodes). The method for fitting the model to the given experimental restitution data (obtained from myocardial cells) is presented. The computer simulations implemented on a massively parallel computer (Connection Machine) showed at least three important qualitative distinctions in behavior which demonstrate the effect of APD restitution: changes in the speed and wavelength of propagated waves with the period of stimulation, non-stationary propagation of spiral waves, and site-specific induction of spiral waves with premature stimulation not on the tail of the previous wave. Quantitative effects of differing restitution properties are expressed in the size and location of a window of vulnerability in 2D excitable media. These windows are characterized by the appearance of single and double spiral waves in response to premature stimulation applied inside the window. Thus the APD restitution incorporated in the FN model produces a significant effect on the formation and propagation of spiral waves.

PDEL—a language for partial differential equations

Conventional computer methods available to solve continuous system problems characterized by partial differential equations are very time-consuming and cumbersome. A convenient, easy to learn and to use, high level problem oriented language to solve and study partial differential equation problems has been designed; a practical translator for the language has also been designed, and a working version of it has been constructed for a significant portion of the language. This <italic>P</italic>artial <italic>D</italic>ifferential <italic>E</italic>quation <italic>L</italic>anguage, PDEL, is outlined, and the highlights of the translator are briefly summarized.

The spectrum of mathematical modeling and systems simulation

The methodology involved in the modeling and simulation of physical, life and social science systems is viewed in perspective. A critical factor determining the validity of a model is the extent to which it can be derived from basic laws and insights into the internal structure of the system using deductive methods, rather than relying upon observations and measurements of the system input and outputs. Accordingly, the mathematical models as they arise in various application disciplines are arranged along a spectrum according to the relative amount of deduction and induction involved in their construction. This provides an insight into the ultimate validity of simulations and to what use they can properly be put.

Excitation wave propagation within narrow pathways: geometric configurations facilitating unidirectional block and reentry

Abstract The propagation of excitation waves in narrow pathways of 2D excitable media through regions of impermeable media and decreased excitability media is investigated as a model of reentrant arrhythmias in the heart. Arrhythmia occurrences are common in the presence of infarct scars where regions of normal tissue are interspersed with inexcitable ones. Propagation through narrow pathways with three idealized geometric configurations are considered: propagation through narrow paths with either parallel borders, tapered borders, or a combination of these. The focus of this study is on the conditions for unidirectional block and reentry appearance. It is shown that in narrow paths with only parallel borders unidirectional block is impossible, regardless of the properties of the border media. In this case only bidirectional propagation or block is possible, depending on the pathway width. Unidirectional block can occur, however, when the narrow path has a tapered shape. Waves propagating from the wide end of the pathway die out at the narrow end, while waves propagating in the opposite direction are able to pass through. An approximate relationship between the geometry of the pathway and wave front curvature is obtained. This provides a simple approach to estimate the critical curvature value in computer simulations of various excitable media or in experiments with real media. A new computer simulation approach is proposed to closely approximate the dependence of stationary speed of propagation on wave front curvature. In computer simulations of a two dimensional grid of 128 × 128 membrane segments, using the modified FitzHugh-Nagumo equations, the arrangement of two or more pathways in parallel permits reentry if at least one pathway has a configuration causing unidirectional block, and the medium at the site of unidirectional block has sufficient time to recover from the previous excitation. The latter is facilitated by anisotropic conduction properties of the media especially in the case with impermeable borders. Thus, there exist specific geometric configurations of two dimensional narrow pathways, which allow reentry even when the media properties are uniform.

Architectural and Software Issues in the Design and Application of Peripheral Array Processors

How can we overcome the limits of von Neumann architecture and avoid supercomputer costs? This overview suggests that peripheral array processors are the answer in many applications.

Wave propagation in cardiac tissue and effects of intracellular calcium dynamics (computer simulation study)

Computer simulation using Luo-Rudy I1 model of ventricular myocyte showed that intracellular calcium dynamics become irregular in case of high rate stimulation. This causes the transition from stationary to nonstationary spiral wave and its breakup in 2D model of cardiac tissue. Obtained results suggest how ventricular fibrillation may occur due to the abnormalities of intracellular calcium dynamics. The short review of existing cardiac cell models with calcium dynamics is presented.

A Hybrid Computer Technique for Treating Nonlinear Partial Differential Equations

When digital computers are employed to solve linear and nonlinear partial differential equations using implicit finite-difference approximations, the most time-consuming portion of the calculation is the solution of the large system of simultaneous difference equations which must be carried out for each time increment. Recognizing that an analog computer system, by virtue of its parallel mode of operation, generates solutions of simultaneous algebraic equations virtually instantaneously regardless of the number of equations involved, a combined analog and digital computer system has been developed. The system includes a small digital computer connected in a closed loop with a network of analog elements, consisting of an array of inexpensive node modules, one for each grid point in space. The purpose of this network is to solve the system of difference equations whenever this solution is required by the digital program. The analog unit, therefore, serves as a subroutine. The resulting system has a number of important advantages over pure digital computer techniques.

INTELLIGIBILITY OF SPEECH IN A VIRTUAL 3-D ENVIRONMENT

In a simulated air traffic control task, improvement in the detection of auditory warnings when using virtual 3-D audio depended on the spatial configuration of the sounds. Performance improved substantially when two of four sources were placed to the left and the remaining two were placed to the right of the participant. Surprisingly, little or no benefits were observed for configurations involving the elevation or transverse (front/back) dimensions of virtual space, suggesting that position on the interaural (left/right) axis is the crucial factor to consider in auditory display design. The relative importance of interaural spacing effects was corroborated in a second, free-field (real space) experiment. Two additional experiments showed that (a) positioning signals to the side of the listener is superior to placing them in front even when two sounds are presented in the same location, and (b) the optimal distance on the interaural axis varies with the amplitude of the sounds. These results are well predicted by the behavior of an ideal observer under the different display conditions. This suggests that guidelines for auditory display design that allow for effective perception of speech information can be developed from an analysis of the physical sound patterns.

Peripheral processors for high - speed simulation:

This paper describes some of the results of a study directed to the specification and procurement of a new cockpit simulator for an advanced class of heli copters. A part of the study was the definition of a challenging benchmark problem, and detailed analy ses of it were made to assess the suitability of a variety of simulation techniques. The analyses showed that a particularly cost-effective approach to the attainment of adequate speed for this extremely de manding application is to employ a large minicomputer acting as host and controller for a special-purpose digital peripheral processor. Various realizations of such peripheral processors, all employing state- of-the-art electronic circuitry and a high degree of parallelism and pipezining., are available or under development. The types of peripheral processors- array processors, simulation-oriented processors, and arrays of processing elements - are analyzed and com pared. They are particularly promising approaches which should be suitable for high-speed simulations of all kinds, the cockpit simulator being a case in point.

Error analysis of hybrid computer systems

were only partially known. In summary, the analyst was not aware, a priori, that two solutions existed in the neighborhood in which the initial investigation was made. The first set of computations gave a clue that this might be the case. Because of convenient computer access, the analyst could swiftly pursue his &dquo;solution search&dquo; by easily making modifications to his input data, and rapidly obtaining the new solution information. This led to the final two solutions in relatively little total elapsed time. in this example, all graphs except those in figure 10 were automatically generated by EASL on the typewriter/plotter console.