James P. Keener

Singular perturbation theory of traveling waves in excitable media (a review)

Abstract Waves of chemical or electrical activity, traveling through space, have been observed in several contexts: chemical reaction mixtures in non-convecting liquid phase, cell suspensions, nerve axons, and neuromuscular tissues. Typically, wave-supporting preparations are excitable; that is, they respond and sensitivity to perturbations are rapidly damped out, but suprathreshold disturbances trigger an abrupt and substantial response. The abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media. Singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious, but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways. We compare and contrast the approaches taken by Greenberg, Zykov, Fife, Krinskii and his collaborators, and ourselves, with particular emphasis on the case of rotating spiral waves. Our intention is to bring some order to the important but difficult theory of propagating waves in two-dimensional excitable media. In conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three-dimensional space.

Propagation and Its Failure in Coupled Systems of Discrete Excitable Cells

We study propagation and its failure in systems of discrete coupled excitable cells. It is shown that propagation fails when coupling is weak, but succeeds if coupling is strong enough. We use a theorem of Moser on maps of the plane to establish the existence of an infinite variety of stable standing solutions when coupling is small. We use upper and lower solution techniques and perturbation analysis to show that propagation is successful when the coupling strength is large enough. These results are applied to the cubic FitzHugh–Nagumo dynamics as well as to the more realistic Beeler–Reuter model of action potentials in myocardial cells.

Integrate-and-Fire Models of Nerve Membrane Response to Oscillatory Input

Nerve membranes exhibit curious responses to alternating current stimulation, among which are phase locking, as well as responses without apparent periodic pattern. We investigate these phenomena by presenting a complete analysis of the response to periodic input of an integrate-and-fire model, which is a simplified version of the Hodgkin–Huxley theory for space clamped nerves.

Spiral waves in the Belousov-Zhabotinskii reaction

Abstract The beautiful spiral waves of oxidation in the Belousov-Zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media. It has long been known that these spirals are similar to involutes of circles, at least some distance from the center, but until now, no way has been known to determine the correct wavelength and frequency. In this paper, we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution. The critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two-dimensional media, and the dispersion of plane waves Our analytical results are shown to be in good agreement with experimental data for the Belousov-Zhabotinskii reagent.

A geometrical theory for spiral waves in excitable media

In this paper, we develop a geometrical theory for waves in excitable reacting media. Using singular perturbation arguments and dispersion of traveling plane wave trains, we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems, governed by the eikonal equation. Using this theory, we study the effect of curvature on waves in excitable media, specifically, rotating spiral patterns in planar regions. From this theory we are able to determine the frequency and wavelength for spiral patterns in excitable, nonoscillatory media.

The Perron-Frobenius theorem and the ranking of football teams

The author describes four different methods to rank teams in uneven paired competition and shows how each of these methods depends in some fundamental way on the Perron–Frobenius theorem.

Chaotic behavior in piecewise continuous difference equations

A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.

The effects of discrete gap junction coupling on propagation in myocardium

A modified cable theory for a bi-domain model of myocardium that incorporates the effect of gap junctions as discrete objects coupling cardiac cells is derived. The theory is shown to be in agreement with a number of experiments that cannot be explained using standard continuous cable theory, and resolves some apparent contradictions on failure of propagation in two-dimensional anisotropic tissue. In addition, some as yet untested predictions of the theory are mentioned.

Waves in Excitable Media

A general class of two-component reaction-diffusion systems with excitable dynamics is studied by means of singular perturbation theory. It is shown how stable traveling pulses and periodic wavetrains in one spatial dimension evolve from initial data. This information is applied to two-dimensional regions for which it is shown that steady rotating structures (spirals) exist.The perturbation results are also used to show that a one-dimensional semi-infinite medium exhibits hysteresis when used as a periodic signaling device. Finally, other nonexcitable dynamics are analyzed, and their stable one-dimensional structures listed.

A numerical method for the solution of the bidomain equations in cardiac tissue.

A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.