1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

A simply-connected region of the Belousov–Zhabotinskii (B–Z) reaction medium may produce a variety of three-dimensional wave forms of unusual geometry. The reaction–diffusion hypothesis even in its linear formulation supports iso-concentration surfaces which include helical surfaces and toroidal scroll waves with twists1. A variety of other wave forms has been proposed by Winfree and Strogatz2. Experimental difficulties associated with observations in three dimensions have precluded confirmation of these waves in situ. We now present direct experimental evidence of evolving three-dimensional chemical waves. Our design is easy to implement, lends itself to time-delay photography and is essentially free of difficulties pointed out earlier3. The dominant wave forms are toroidal scrolls1, 3–5; other wave forms observed include the sphere-like structure predicted recently2.

Three-dimensional chemical waves in the Belousov-Zhabotinskii reaction

So-called “excitable” media support nonlinear waves that propagate as pulses, e.g., the action potential in nerve fibers and heart muscle, and the oxidation pulse in certain chemical reactions. In two-dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction. Their investigation in diverse excitable media has dramatically expanded in the USA and USSR since their discovery in both countries in 1970. In three-dimensional media the pivot of the rotating structure is not a point but a curved line, generically a closed ring. Rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium. It is of interest to understand how stimuli create organizing centers, and how organizing centers persist or decay. The numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention. This paper illustrates organizing centers by computer g...

Stable particle-like solutions to the nonlinear wave equations of three-dimensional excitable media

Two regimes of the scroll ring drift in the three-dimensional active media

Singular filaments organize chemical waves in three dimensions: III. Knotted waves

Applying the Debye limit to a suitable superposition of singular solutions of the linear two-component reaction-diffusion equations in three dimensions leads to helical and twisted toroidal scroll waves of constant concentration. An integral representation of the solutions of the Helmholtz equation in toroidal coordinates establishes a connection between helical and toroidal waves.

J Gomatam

Papers

Pattern synthesis from singular solutions in the Debye limit: helical waves and twisted toroidal scroll structures