Singular filaments organize chemical waves in three dimensions II. Twisted waves

Physica 9D (1983) 65--80 North-Holland Publishing Company

SINGULAR FILAMENTS ORGANIZE CHEMICAL WAVES IN THREE DIMENSIONS

II. TWISTED WAVES

A.T. WINFREE and S.H. STROGATZ Department of Biological Sciences, Purdue University, West Lafayette, IN 47907, USA

Received 26 December 1982 Revised 3 March 1983

This is the second of a series of papers on the anatomy of three-dimensional wave sources in excitable media. We here ask whether all self-consistent structures are topologically equivalent to the experimentally verified scroll ring. As a test case we examine a scroll ring containing one full cycle of "twist". We develop methods for unravelling the anatomy of topologically diverse scroll rings, and find that this one violates the requirements of physical chemistry. However a mutually-linked pair of such waves violates no known law. We specify the initial conditions required to create one and speculate on its stability.

I. Introduction

Since 1948 two-dimensional excitable media have been known to support a rotating wave [1]. Such a wave resembles the geometric involute of the boundary of a hole around which the wave circu- lates. Since 1963 [2-8] a spiral wave has been known to circulate stably around a "core" region even in medium without holes. This core region contains an organizing center called a "rotor". In 1973 it was shown that the rotor is a tiny disc only in the two-dimensional limit: in three dimensions it is a filament surrounded by a scroll-like wave, resembling a spiral in every cross-section [9]. The core filament typically closes in a ring, resembling a long thin toroid. Paper 1 of this series [10] explored the geometry of this toroidal rotor and the "scroll ring" wave field emanating from it. In any plane perpendicular to the source filament and containing its axle (the ring's axis of rotational symmetry), we see the filament as two tiny disks, each surrounded by a spiral wave (see figs. 4 and 22 ofref. 10). The two waves collide along the axle. The three-dimensional scroll ring (in its perfectly symmetric version) is a surface of revolution of this picture. It is thus not much different from what was already familiar in the plane. The scroll is just a spiral projected by

translation into three dimensions; the scroll ring is just a spiral projected into three dimensions by revolution around a distant axle.

We now enquire whether there may exist anat- omically diverse kinds of scroll ring, all sharing the common reaction chemistry, but differing in spatial organization. Could such scroll rings even be topologically distinct from the one previously demonstrated? For example, is it conceivable that a scroll ring could be knotted? Or could it be twisted, in a sense to be clarified below? In this paper and the next we develop criteria for chemical realizability and show that neither the solitary scroll ring, twisted without knotting, nor the solitary scroll ring, knotted without twisting, can exist. But in the process, we find that closely related, topologically distinct scroll rings can exist: a knotted and twisted scroll ring, a pair of mutually linked twisted scroll rings, or a pair of linked scroll knots. These alternative spatial arrangements are compatible with the assumed reaction kinetics and with the requirements of physical chemistry. They are probably stable. They can be created by arranging appropri- ate initial conditions; our procedure gives a recipe for deriving those initial conditions.

We thus confront the specter of a discrete tax- onomy of self-maintaining wave sources in excitable

0167-2789/83/0000--0000/$03.00 © 1983 North-Holland

66 A.T. Winfree and S.H. Strogatz/Twisted chemical waves in three dimensions

media. Each species is topologically distinct from the others and is characterized by a handful of integers specifying its twistedness, knottedness, and linkage with disjoint rings. It is our objective to clarify the rules governing admission of putative structures to this "periodic table". Each admissible structure is an organizing center; it radiates waves into the surrounding medium, organizing the chemical processes of that environment in time and space. This paper describes the simplest twisted organizing centers, the "twisted scroll rings". The next paper [11] describes the simplest knotted organizing centers, the "scroll knots".

2. A twisted scroll ring

Is a iwisted scroll ring chemically realizable? This structure could be created (in a gedanken experiment) by encasing the untwisted scroll (fig. la) in a toroidal glass tube (which we suppose to be flexible and impermeable). The tube is then cut,

O. b.

Fig. 1. A scroll ring may be created by joining the ends of an imaginary cylinder containing a scroll wave. This can be done as in (a) or as in (b), impart ing a 360 ° twist to the cylinder before joining its ends.

* We need to distinguish between dynamic rotations due to time-evolution of the chemical system, and purely geometric rotations. Henceforth, to formalize this distinction, we reserve the' word "rotate" for rotations in time, at fixed location, e.g. "rotat ing spiral wave". When time is fixed but location is variable, we use the word "revolve", e.g. "we generate a sphere by revolving a circle about one o f its diameters." The more specific word "orbi t" means a revolution about a distant symmetry axis or an axle, e.g. "in a once-twisted scroll ring, paired spirals revolve around one another as they orbit the axle of the torus".

straightened into a cylinder, and gently twisted along its length through a 360 ° rotation about its axis. Finally, we curve the cylinder back into a toroid, and seal the ends together. This yields a toroidal glass tube containing a twisted scroll ring (fig. lb).

So far, there are no complications arising from the twist: the glass tube forms a protective boundary which prevents portions of the structure from encountering distant portions in 3-space. Within the glass tube, portions of wave join only their neighbors along the length of the ring. Because we began with a pre-existing scroll, all the new wavefronts, phase contours and concentration contours match perfectly along the ring and across the seal. Thus with regard to purely local considerations, the twisted scroll ring seems as plausible as the established untwisted scroll.

But now comes the crux of the problem. For the twisted scroll ring to be chemically realizable, we need to be able to remove the glass tube without mishap. A// portions of the structure must now be mutually compatible, in terms of both phase and concentration contours. Without the glass tube surrounding it, the twisted scroll ring is subjected to severe global constraints.

So stringent are these constraints that the solitary twisted scroll is not realizable. To see this, envelop the scroll ring axis with a diagnostic surface (an imaginary surface, like the Gaussian pillboxes of electrostatics, convenient for singling out some behavior on a two-dimensional locus. Here we examine phase behavior along a toroidal diagnostic surface.). Any surface of fixed phase intersects the diagnostic toroid in a closed ring (fig. 2). All such isophase rings cross the equator in the toroid's hole exactly once, because the scroll's involute cross-sections revolve* as they orbit around the hole axis. Hence, along the border of a disk plugging the hole, the isophase tings are encountered one by one. So in a full circuit of this border, phase changes through one full cycle.

This leads to the main step in our argument: because the winding number of phase along the disk's border is non-zero, phase cannot be or-

A.T. Winfree and S.H. Strogatz/Twisted chemical waves in three dimensions 67

Fig. 2. Rings of fixed phase on a diagnostic torus surrounding a twisted scroll. Only half of each ring is visible. One of the rings is an edge of the current wavefront; the others are its earlier and later positions. In a circuit around the border of any disk plugging the hole, the rings are touched in succession. Phase thus increases clockwise through a full cycle around this border.

ganized continuously throughout the disk: the

interior must contain at least one unpaired phase

singularity. (For discussion of this key point, see the appendix of paper 1 of this series [10].) More-

over, since the above argument applies to any

membrane plugging the toroid's hole, there must be a one-dimensional f i lament of phaseless points

linking the toroid. To sum up, a unconfined twisted scroll ring

cannot exist alone. It would engender a peculiar linking filament of phase singularity.

3. The axle singularity

The above arguments are somewhat abstract.

Though they demonstrate the existence of an induced singular filament, they neither locate this singularity nor show how it originates. To under-

stand its genesis, we examine the isophase curves on a different diagnostic surface: the equatorial disk plugging the toroid's hole. First consider a radial line on the disk, at fixed time. Along that radius, points closer to the center are further from the circular scroll axis, the source of waves. Thus points nearer the center are at earlier phase. Sec- ond, observe that by revolving fig. 2 we would

effectively relabel the isophase curves. So, still at

f i x e d time, it is possible to remain at fixed phase by a suitable combination of radial motion and rotation. This rule implies that the isophase curves on

the disk are spirals (fig. 3). All the isophase curves (e.g. the moving wavefront) radiate inward from

the scroll axis, colliding at the center in a peculiar

way. The collision point is a new phase singularity. It originates as a result o f wave-focusing, or- ganized by the twist of the scroll ring.

Since this argument applies to any membrane plugging the toroid 's hole, the collision locus must

actually be a one-dimensional filament coinciding with the toroid's rotational symmetry axis. Along

this vertical line (the axle) the converging wavefront resembles a scroll near the middle, and a screw surface farther out. Fig. 4 is a stereo pair taken from a videotape animation of this wave

(prepared on an APPLE microcomputer with

modified "Bill Budge 3D Graphics System" and joystick for real-time rotation, then transferred to

an Evans-Sutherland PS-300 for higher resolution photography). The movie shows the rotating

twisted scroll ring from the viewpoint of a helicop- ter news crew flying through it, then shows serial cross-sections. Snapshot Fig. 5 shows a perspective

Fig. 3. The twisted scroll ring of fig. 2 is repeated to show the progression of phase through one full cycle around an equatorial ring. Every disk bounded by that ring must therefore contain a phase singularity, here located along the hole axis (the "axle"). Near the axle the surfaces of fixed phase intersect the disk along a spiral which propagates inward, toward the axle.


Fig. 4. The twisted scroll ring, permitted to exist by the singularity threading the ring along its axle. The left image is for the right eye and vice-versa: hold the picture at arm's length and cross your eyes to see it in stereographic perspective. The gridwork of line segments is, of course, intended to represent a smooth surface. Its termination at an outer boundary is another artifact of the graphics display: unless the axial singularity closes in a ring, the wave extends to infinity.

computer animation has not done the same, only more subtly and more convincingly. However this particular pattern has been confirmed analytically as a solution to linear reaction-diffusion equations in a recent remarkable publication [12].

Fig. 5. As in fig. 4 but viewed in monocular perspective from a point far out along the axle. The singular ring conspicuously radiates one spiral outward and one inward, toward the axle.

view along the axle. Fig. 6 shows various cut-away views.

We hasten to remark that a computer-made movie of an exotic organizing center is not an existence proof. One of us (secret) is stupid enough to have hand-sketched an object (fig. 7) that cannot possibly exist. It is hard to be certain that a

4. The collision interface

In the foregoing version of the twisted scroll ring-curn-singular-axle, waves converge toward the axle. The axle behaves like a wave sink. However observations in Belousov-Zhabotinsky reagent suggest that phase singularities are wave sources, not sinks.. The pattern above could be revised accordingly by placing an outward travelling scroll wave along the axle. The outgoing wave dominates the region near the axle, but farther out it collides with the wave incoming from the scroll ring. Collisions occur along some partitioning interface. The regions on either side are governed by the waves from the associated singularities. We now seek to describe the geometry of this interface, as well as the instantaneous locus of collisions within it.

In the following description, we exploit a symmetry inherent in the overall pattern: a translation in time is equivalent to a revolution about the axle.


\

\

Fig, 7. One half of a topologically-non-sensical scroll ring. This Escher-like construction blatantly violates continuity. Our task is to ensure that more subtle violations of physical chemistry are not perpetrated in scroll rings and knots here alleged to exist in real excitable media.

Fig. 6. Thin sections through the object seen ,entire in figs. 4 and 5.

This follows because: 1) Translation in time effectively rotates each

involute cross-section within its plane; in other words, each involute undergoes an in-plane rotation about the scroll axis [8];

2) Because the involute cross-section twists through 360 ° along the circumference, this in-plane revolution amounts to an equal revolution of the scroll ring about its axle. This twist being inflicted equally on each spiral cross-section, the whole structure is still a uniformly once-twisted scroll with the same axle as the original;

3) So for the twisted scroll ring alone, time- translation = revolution about the axle;

4) Since this is also true for the remainder of the structure, i.e. the involute scroll radiating from the axle, it is true for the overall pattern.

Now we use this symmetry principle to analyze the collision locus. In any diagnostic plane perpendicular to the axle, the incoming and outgoing wavefronts collide at point P as in fig. 8. As time advances, our principle indicates that this point rotates about the axle, and thus traces out a certain circle (shown dotted in fig. 8). Slide the horizontal plane to another position along the axle. At a fixed time, the corresponding new point P is generally revolved with respect to the old P. Repeating this diagnostic procedure (always at fixed time) yields a continuous curve consisting of such points P.

70

Fig. 8. A diagnostic plane perpendicular to the axle. A wave emanating from the axle singularity collides at P with an incoming wave radiating from the ring singularity. As time advances, P traces out a circle about the axle (shown dotted).

A.T. Winfree and S.H. Strogatz/Twisted chemical waves in three dimensions

New

Scroll o ~

This curve is the instantaneous collision locus of incoming and outgoing waves. It generally resembles a coil spring. As time increases, the collision locus orbits about the axle (by the symmetry principle), thus sweeping out a surface of revolution: the collision interface.

5. Linked scroll rings

The coincidence of a singularity exactly with the whole (infinite) axle appears to depend crucially on certain symmetry assumptions made above (e.g. the scroll ring axis is perfectly circular). Also an endless line of singularity seems unsatisfactory on chemical grounds: infinity should be left quiescent, since our manipulations are performed in finite time. Fortunately, the phase singularity induced by a twisted scroll need not be a straight line. Indeed, as discussed in the first paper of this series [10], one-dimensional phase singularities are generically closed rings. The original ring being threaded by an induced arc of singularity that is generically part of a ring, we are led to consider a pair of linked scroll rings.

We may find the twist of the new scroll ring as follows. Surround it with a diagnostic torus. Note that one singular filament (the old scroll axis) threads the torus's hole. Applying the earlier winding number argument in reverse, we infer that phase changes through one full cycle around the border of a plugging equatorial disk (fig. 9). Thus every isophase curve on the torus crosses the inner equator once and so the new scroll ring is itself

Fig. 9. Scroll rings linked once and encased in diagnostic tori. Diagnostic disks plug the holes of the toil, and are each punctured by one singular axis. Around any circle threaded by an axis, phase changes through one cycle. Thus both scroll tings are once-twisted.

once-twisted, in fact in the same sense (left- or right-handed) as its mate.

Although the sense of the twist may be found from the above argument, it is more easily obtained from a gedanken experiment. As before, imagine an isolated twisted scroll ring encased by a glass torus. The entire structure is immersed in quiescent medium. Abruptly removing the glass tube exposes a wave edge (where the inner involute met the glass wall) along a closed ring which is not

parallel to the inner phase singularity. Because the inner involute spiral twists through 360 ° as it is revolved around the axle, its outer endpoint links

the torus (fig. 10). As discussed in paper 1 of this series [10], such a wave edge becomes a phase singularity upon encountering quiescent medium. This new ring of phase singularity links the first. The pair of cores and the spiral waves they emit thus revolve about one another as they orbit

Fig. 10. The twisted scroll is encased in an imaginary toroid. The scroll encounters the toroidal surface along a ring linked through the circular axis of toroid and scroll.

A.T. Winfree and S.H. Strogatz /Twisted chemical waves in three dimensions 71

Fig. 11. A pair of counter-rotating spiral waves revolving around one another as they orbit the hole axis. Thus the two phase singularities in each plane trace out a pair of linked ring singularities.

around the symmetry axis of the torus (fig. 11). The new scroll ring is therefore also once-twisted, and in the same (left-, or right-handed) sense as the first.

Note too that although the paired scroll rings have the same twist, they counter-rotate in time. Twist is a geometric property of scroll rings at a fixed time; counter-rotation of paired spirals or scrolls is a dynamic property, measured in a fixed diagnostic plane transverse to the singularities.

6. Anatomy of linked twisted scroll rings

6.1. Realizability criteria

The discussion above has established that an unconfined twisted scroll r ing , / f it exists, must be linked by either an endless axle or a like-twisted scroll ring. But we have not yet established that such structures can exist. We now need to find criteria of chemical realizability, and to show that the contemplated linked rings can satisfy these criteria. (We offer no existence proof, but the following arguments make the structure seem plausible, and may serve as guides to experimental implementation.)

We formulate the realizability criteria in terms of iso-concentration surfaces, or "isocons" [10]. While these criteria may at first seem abstract and formal, they are nothing more than standard chemical principles, phrased in the convenient lan-

guage of geometry: a) A chemical structure is composed of stacks of

isocons. To see this, consider one of the pertinent chemical species, say substance A. Each point of the structure lies at some concentration, [A]. Joining points of equal [A] yields an isocon surface. Since every point has some concentration of A, the entire space is partitioned into A-isocons. Notice that by definition, these surfaces do not intersect each other.

b) Higher concentrations lie immediately to one side of an isocon, while lower concentrations lie on the other. This admittedly mundane observation tells us something about the geometry of isocons: they are "orientable" surfaces, i.e. they have two sides, unlike a Mobius strip. Orientability ensures that concentration gradients are well-defined.

c) Isocons generally do not pass through themselves, as they fail to be orientable at loci of self-intersection. However, this prohibition and the last one (orientability) may be violated on a set "o f measure zero", e.g. at a cross point in a figure-8 contour separating two adjacent maxima (fig. 12a).

d) As previously discussed, isocons do not extend beyond the finite region of perturbation; infinity is to be left undisturbed ("quiescent") because real experiments are performed in finite time.

0.

1 b.

R

Fig. 12. Exceptional isocons: a) All the isocons are.non-self- intersecting and orientable, except the figure-8 isocon, which fails on both counts, but only at the crossing point; b) The middle isocon is not a closed ring but it is sheathed by surrounding closed isocons; otherwise point R would have contacted a discontinuously different concentration.


e) Isocons are generically closed surfaces, i.e. they could "hold water". Any exceptional isocon that is not closed must be enveloped by one that is (fig. 12b), or else finitely different isocons would pass arbitrarily close by, implying a discontinuous concentration field.

If these five constraints are satisfied by a hypo- thetical chemical structure, then there seems to be a good chance that it is realizable. (Note too that these constraints could be used equally well to rule out certain putative structures.) Our task now is to construct a family of mathematical surfaces which will serve as candidate isocons for the pair of linked twisted scroll rings. Successful completion of this task would yield a bonus: because the brilliant blue wave of Belousov-Zhabotinsky reagent corresponds to a certain group of isocons [10], we could anticipate the geometry of waves emitted by the source. We would thus know how to recognize it in the laboratory. In the case of the linked pair of once-twisted scroll rings, the wavefront proves to be a once-twisted band surrounded by spheres, as we shall see.

There is another motivation for attempting to characterize the isocons: success would not only ~onfinn the existence of the putative organizing center, but additionally show how to conjure it into existence from previously quiescent medium. The boundary conditions and initial conditions sought need to satisfy the topological invariants of the isocons. As we shall see, these conditions are unusual, but not complicated.

6.2. Constructing candidate isocons

The preceding discussion dealt with isocons in general. Candidate isocons for a configuration of scrolls must satisfy an additional constraint im- posed by scroll geometry. Specifically, nearly all isocons impinge on the scroll core tube, along its entire length. In [10, section 3.3] we discussed the internal structure of the core, and the connections of the isocons through it. However, in the present context of global anatomy, the details of the core assume a relatively minor role. We preserve the

main point (that nearly all isocons impinge upon the core) by supposing that the core tubes are thin enough to be regarded as one-dimensional rings. They essentially coincide with the phase singularities. Nearly all the isocons impinge upon them. This approximation improves as the scale of the overall structure increases, relative to the fixed thickness of the core tube (= about one-third wavelength). After using this approximation to establish the global anatomy, it may be discarded. Then we rearrange the isocons properly within the core tube, without altering the outer pattern.

Let us begin constructing the candidate isocons for a pair of linked twisted scroll rings. At this stage, we vaguely envision each isocon as some undefined closed surface. But exactly what type of closed surface is it? Applying the thin core approximation above, we see that each candidate must contain both of the linked phase singularities.

This observation points toward a two-step method for constructing candidates. First, beginning along the singularity rings, we continuously "develop", or extend, edged pieces of open surface. As a pre-candidate, the surface must be bounded, orientable, and free of self-intersections. The goal of this step is to build an open surface whose only boundary consists of the singularity rings. A sys- tematic method is given in appendix A. Second, because candidates must be closed surfaces, we must somehow seal the boundary with another surface similar to the first.

This second step requires another pre-candidate with the same boundary. This surface may be "puffed-out" from the original by the following trick: at each point, we displace the surface along its local normal vector, but by a slight amount which shrinks completely to zero along the boundary. Fig. 13a illustrates the idea in an easier example. For sufficiently slight displacements, the resulting puffed-out surface does not intersect itself; nor does it intersect the original, except along the boundary, where the two surfaces are fused (by construction). Finally, having done so, we observe that two such fused open surfaces constitute a single closed surface: a candidate isocon.


C j ~ 1

b.

Fig. 13. a) "Puffing" a surface 1 to produce a new surface 2. Points of 2 are displaced along the local normal vector in such a way that the displacement shrinks continuously to zero along the boundary C. This ensures that 1 and 2 coincide along C, and are both bounded by it; b) The circle C represents an untwisted ring singularity. It is the boundary of a pre-candidate disk 1 and a pre-candidate dome 2. Since 1 and 2 are fused along C together they form a closed surface equivalent to a sphere.

For example, in [10] we discussed the anatomy of a solitary untwisted scroll ring. In that case, we would first build an open surface whose boundary is a lone circle. Any disk or dome surface will d o - s e e fig. 13b. Next we puff-out a second dome from the first. Then these pre-candidates, being fused together along a common circle, yield a closed surface which is topologically equivalent to a sphere. So we predict spherical isocons in this case, though they may be much distorted.

For a linked pair of twisted scroll rings, the pre-candidate and candidate isocons are less apparent. As before, we first seek an open surface (the pre-candidate) bounded by the linked singularity rings. In fig. 14a, we show the interlocked rings spanned by a shaded surface. To recognize the surface, we deform the rings into the topologically equivalent arrangement of Fig. 14b. The pre-candidate is seen to be a strip with one ful l twist. As required, it is bounded and has no self-intersections. Finally, it is orientable (see fig. 14c - if the strip had an odd number of ha/f-twists, it would not be orientable.).

The resulting closed surface (the candidate isocon) is a toroid. Perhaps this is most readily perceived in fig. 14c. The pre-candidate consists of two semi-circular strips and two half-twisted strips.

8 b. o. ¢.

Fig. 14. a) A pair of linked rings form the boundary of the shaded surface, a pre-candidate isocon; b) The same rings as in a), though distorted. The shaded surface is a strip with a full twist; c) Rings and strip redrawn symmetrically. The surface is orientable (has both a " f ront" and "back" side). With an odd number of half-twists, the surface would be a non-orientable Mrbius strip.

After puffing, these pieces become two bent cylinders and two half-twisted cylinders (fig. 15). Reas- sembling, we have a toroid. Roughly speaking, we may slip the nozzle of an air pump just between the positive and negative sides of the full-twisted strip, and inflate it into an inner tube!

Although we have shown that the candidate isocons are tori, we have not demonstrated that space may be partitioned into a nested stack of them (recall 6.1, criterion a). To generate such a nest, we inflate a single toroid through a continuous series of stages. Each stage of the inflation spawns a single new toroid. Together, the tori comprise a family of candidates which coincide along the two ring singularities. Eventually the tori may become so swollen that their holes vanish through a process peculiar to chemical waves (see appendix B). Thereafter the candidate isocons are sphere-like, surrounding the entire structure in

E E

67%

Fig. 15. The full-twisted strip of fig. 14c is subdivided into half-twists and semi-circular strips. After puffing, these become bent cylinders and half-twisted cylinders, which recombine to form a toroid.


Fig. 16. Expanding toroidal waves collide internally, accom- panied by mutual annihilation of the surfaces involved, just as though the collision had been against an inert obstacle. Thus the characteristic hole of the toroid is squeezed shut, and the isocons become topologically equivalent to spheres. The linked pair of twisted scroll tings is thus enveloped in concentric spheres radiating outward at the characteristic frequency of rotors.

concentric waves propagat ing outward into quiescent medium (fig. 16). Together, these two types

of candidates may be stacked to fill space. Until now, we have concentrated on the candi-

date isocons for only a single substance, A. But wave propagat ion in excitable media depends on

the spatial configuration of two or more concentrations. (Our analysis deals exclusively with those

"excitable reactions that promptly reduce themselves f rom a higher-dimensional state-space to a 'slow manifold ' o f only two dimensions" [10].)

Outside the core tube, the candidate isocons for a second substance, B, should also be toroids and spheres; they are parallel to the A-isocons. Inside

the tubes, patches of B-toroids intersect those of A transversally; crossed concentration gradients

characterize the core [10].

hole in the other. With further inflation, the outer walls recede from the field of view, leaving something like fig. 17 in the center. A sphere about this center would cut the rubber surfaces along a single

closed ring: the boundary of the surface in fig. 17. Capping this boundary with a disk-like surface in either of two perpendicular ways would recreate

one of the two inner-tubes.

6.4. Sum m ary

Beginning along a pair of linked ring singularities, we extend an open surface. This "pre-

candidate isocon" is a once-twisted strip. It satisfies the realizability criteria of non-self- intersection, boundedness, and orientability. The linked rings constitute its only boundary. "Puffing-out" produces a family of closed surfaces,

the "candidate isocons" - these are nested toroids, the outermost and most swollen of which become

spheres as waves collide in the toroid's hole. Fi- nally, candidate isocons for A and B intermesh in a way which creates both crossed concentration

gradients inside the core tubes, and parallel concentration gradients outside the cores.

There are two versions of the linked pair: both scroll rings may be left-twisted, or both may be

right-twisted. These structures are mirror images; either is a reasonable candidate for laboratory

synthesis.

6.3. The collision interface

Each scroll ring consists o f outward-propagat ing waves. I f the least distance separating the two ring singularities is more than a few wavelengths of the involute spiral, then each may be regarded in approximat ion as a source of expanding waves. These collide along some interface midway between the two rings. The collision interface may be imagined as the surface of contact between two

linked rubber inner-tubes, each inflated to fill the

( Fig. 17. The interface between linked tori, swollen into exten- sive contact, approximates the locus of collisions between scroll waves that start along linked tings.


7. The needed experiment

In [10] we described three methods for creating rotors in two-dimensional excitable media. The essential objective of all methods is to construct transverse concentration gradients similar to those found in a rotor. The crucial similarities are the spatial scale and the topological invariants of the isocons. We now have a procedure for deriving those invariants for three-dimensional waves, given the configuration of twist and linkage in component scroll rings.

In the case of a mutually-linked pair of once- twisted scroll rings, we found that (near the singular filaments) the isocons are tori. An A* toroid and a B* toroid can intersect along two linked rings, with local concentration gradients revolving about one another as they orbit around the axle. This arrangement mimics the fundamentals of mutual linkage between once-twisted scroll rings.

Such toroidal gradients could be established quite simply, in principle. Referring to [10, section 5 ("Creating singularities")], we use method 2 ("Transverse concentration gradients"). A source of substance A is made into a ring, called Ring A. A source of B is made into a nearby linked Ring B. Now a pair of counter-rotating singularities may be expected in every plane transecting the two rings, as shown in fig. 18. These singularities lie along disjoint linked filaments that twist once as they orbit, just as in fig. 11.

We surmise that fig. 18 contains the conceptual essentials for an experiment in which this organizing center would be deliberately created for the first time. We do not yet know whether the practical difficulties can be overcome in existing versions of the Belousov-Zhabotinsky reagent.

8. Discussion and generalizations

8.1. More complex links

In retrospect, the construction of isocons for linked twisted scroll rings depended most heavily on the first step: finding an open surface with all

Fig. 18. A concentration maximum of A is arranged along a circular thread, while a B maximum is established along a linking circular thread. In each of the planes shown (and all those in between) the two concentration gradients intersect in the way believed to initiate a pair of mirror image rotors (1 and 2). These rotors lie along linked rings; each generates a twisted scroll wave. Compare to fig. 11.

the correct properties. We were lucky to find the once-twisted strip "by inspection" of fig. 13. For a more intricate configuration of linked scroll rings, this step could pose a formidable challenge. Fortu- nately, there is a simple algorithm which solves the problem for all cases of practical interest (i.e. those involving "piecewise linear" links). For this classi- cal algorithm, due to Seifert, see appendix A.

Some cases, though, really can be solved reliably by inspection. For example, consider a twice-

twisted scroll ring. Could its ring singularity exist in isolation? As before, invoke phase and winding number arguments. These establish that such a ring singularity must be linked, in fact twice, by other phase singularities. There are a few ways to manage this. One way is suggested by the physical argument of section 5. If we abruptly remove a glass tube encasing the twice-twisted scroll ring, we expose a wave-edge which is twice linked with the scroll ring axis. Upon encountering the ambient quiescent medium, the wave-edge itself becomes a ring singularity. So we are led to consider a pair of twice-linked rings (fig. 19). Then, by inspection, the desired open surface is a strip with two full twists. Upon puffing, we find toroidal candidates. With slight modifications, this argument applies to any pair of N-twisted scroll rings linked N times. The candidate isocons are always toroids.


9

O. b.

Fig. 19. A pair of twice-twisted tings, depicted symmetrically on the tight. The corresponding pre-candidate isocon (shaded surface bounded by the rings) is a strip with two twists.

However, note that there are other valid candidates. Had we chosen another view of the pair o f twice-linked rings (fig. 20a), the candidates might

not have been toroids; they could have been toroids perforated by any number of " tunnels" or, equiv-

alently, adorned with any number of additional "handles" (fig. 20b). From a chemical point of

view, this is acceptable-per iodical ly-s tacked isocons acquire a tunnel near any isolated volume of

relatively uniform concentration. But for simplicity we restrict attention to the isocons of "minimal genus" (i.e. fewest handles/tunnels), bearing in

mind that more may be trivially added. Generalizing in a different direction, we may

imagine links involving several scroll rings. In this

.. @-60o Fig. 20. a) The rings of fig. 19, redrawn to show four symmetric half-twists. Puffing the shaded surface yields a surface b) equivalent to a toroid with two .handles. This differs from the toroid that would have arisen by puffing the surface in fig. 19 (even though the rings are unchanged, topologically). This shows that a given configuration of rings may admit topologically diverse isocons.

Fig. 21. Three mutually linked ring singularities. Each disk is punctured twice, once by each of the other rings. As indicated by the arrows, any two singularities penetrate in the same direction through a disk plugging the third. Hence phase changes through two cycles in one circuit of the disk's border; thus each ring must be twisted twice.

case, tight constraints are placed on the possible values of the twist. For example, the triple link of

fig. 21 cannot be chemically realized unless each scroll ring is twisted twice. To see this, recall the

arguments of section 2. Surround each ring with a diagnostic torus. Then any plugging equatorial

disk would be punctured twice, in the same direc-

tion, so phase must change through two cycles along its border. Hence the scroll ring inside must

be twice-twisted. Three points emerge from this last example.

First, for computing winding numbers around

such disks, we need count only the algebraic

number of punctures; the twists of the puncturing axes are irrelevant. Second, several distinct linkage

configurations may be compatible with a multiply- twisted scroll ring. A twice-twisted scroll ring cannot exist alone; but it is made viable by either

of the arrangements discussed above. Third, twist' and linkage cannot both be pre-assigned arbitrarily. (As we shall see in [11], knottedness is also implicated; the present discussion excludes the case of linked knots.) The twist on a given scroll ring must equal the number of puncturing singular axes, counted according to direction of pene- tration. This rule is a special case of some more general conservation law: a topological index must be conserved in any chemically realizable configuration of scroll rings, just as in creating


plane spirals only in mirror-image pairs. This (unknown) index depends on the twistedness and the knottedness of each ring. And the collective index over many coexisting rings must depend on how the rings are linked.

8.2. Stability

It will eventually become necessary to inquire about the stability of such complex organizing centers. There is not much at present to be said and we say it here. The stability of the observed structures (the two-dimensional spiral, the scroll, the scroll ring) is limited by the chemical exhaus- tion of their medium, and by the constraint common to all reaction/diffusion structures, that they cannot be confined in too small a space [13, 14]. It is also qualified by the observed [15], but as yet not understood [16], "meandering" of the rotor's innermost parts, which compromises the rigid rotation observed farther out. However our feeling is that the admissible twisted, knotted, and linked structures will prove about as stable as those already observed (the two-dimensional spiral, the scroll, the scroll ring). The stability of the known structures is enhanced by topological constraints forbidding the local creation or destruction of an unpaired rotor [10], and by the circumstance that rotors emit waves near the shortest period sustain- able in the excitable medium, thus fending off potentially intrusive lower-frequency disturbances.

Twisted, knotted, or linked scroll rings differ only globally from the elementary scroll, which is stable in the sense qualified above. The rotation period is determined by the interplay of reaction and diffusion within the core. Twistedness, knottedness, and linkedness become apparent only on a spatial scale greater by an order of magnitude. As reaction/diffusion interplay typically has a time constant proportional to the square of the length scale, we expect any new components of rotor dynamics, whether stabilizing or de-stabilizing, to make themselves known on a time scale exceeding the rotation period by two orders of magnitude at least. On that time scale, existing versions of the

Belousov-Zhabotinsky reagent deplete their chemical sources of free energy. Heart muscle in vitro is capable of supporting rotors for hundreds, even millions, of rotations, but we do not yet know whether its rotors are more than trivially three- dimensional.

It seems worthy of note that these structures do not arise by instability of a spatially uniform steady-state: uniform quiescence is stable to small perturbations. In two-dimensional media, uniform quiescence and a pair of complementary rotors (or any number of pairs) are discrete alternative modes of the excitable medium. In the chemical reagent or in uniform nerve membrane, they are apparently alternative solutions to the corresponding para- bolic partial differential equations. They seem little affected by each other or by boundary geometry, so long as the boundary is a wavelength distant or more (ATW unpublished computations, with thanks to IBM Thomas Watson Research Labora- tory). In three-dimensional media the variety of discrete solutions is greater, as are their continuous degrees of freedom (spatial positions and orien- tations),

Quantitative theoretical analysis in this area promises to be challenging. We suspect that experimental results (in vivo, in vitro, and in numero) will come sooner.

Acknowledgements

This paper was drafted during the tenure of a Senior Visiting Fellowship (GR/B/90313) to ATW from the Science and Engineering Research Coun- cil of Great Britain, courtesy of J.D. Murray, the Mathematical Institute, University of Oxford. ATW also benefited from conversation with Drs. Andrew Presley and Graeme Segal at Oxford, and Christopher Zeeman and David Epstein at War- wick. Michael J. Bailey of the Purdue CADLAB made possible the videotape and Figures 4, 5, 6. The GC Dental and Industrial Corporation of Osaka kindly donated our principal research tool:


one kilogram of red "Utility Wax" for experiments in topology. ATW thanks the National Science Foundation for grant CHE-8103322.

Appendix A

Here is an algorithm for the construction of pre-candidate isocons. It was discovered by the topologist H. Seifert while he was founding the mathematical theory of knots and links. These surfaces are usually called "Seifert surfaces". (Con- sult [17] for a lucid introduction to this subject, and the text by Rolfsen [18] for more thorough cov- erage.)

The problem may be stated as follows: we have a certain configuration of linked ring singularities. (From now on, we will refer to this configuration as simply, "the link L" . ) We seek a surface S bounded precisely by L. In other words, S has no other boundary and all the constituent rings ("components") of L bound some part of the surface. As explained earlier, we also insist that S be orientable and non-self-intersecting. Finally, we need to assume, for technical reasons, that L is "piecewise-linear" (i.e. all its components are topologically equivalent to polygons or polygonal knots; this rules out pathological cases, e.g. an infinite number of linked components converging to a point.)

To begin the construction, assign an orientation to each component of L. Draw a picture of L in such a way that all cross-overs involve at most two segments (fig. 22a). Delete all cross-overs; in their place, substitute "short-cut" arcs that preserve orientation (fig. 22b). This results in a collection of disjoint rings, each of which bounds a disk. It may turn out that the disks are nested; if so, remedy this by lifting their interiors off the plane, working outward from the innermost disks. In other words, pull out the disks as if extending an old-fashioned telescope. This telescoping results in disjoint disks; without this step, the eventual surface S might intersect itself. Next connect the disks together at the former cross-overs with half-twisted strips (fig. 22c). The edges of each strip are the original

Q.

b.

C. ~"~_..4 "/k.x ~ : i

d. Q +

Fig. 22. a) The crossover shown is forbidden because it involves three segments. One component of the link may be displaced slightly, without changing the topology, to effect only two- segment crossovers; b) Short-cuts instead of crossovers. Note that orientation is preserved; c) The previous step produced a number of disks• They are then joined at former edge crossovers by inserting half-twisted strips• The half-twist restores the crossovers that were short-cut; d) If the surface is in pieces, join them by hollow tubes. This operation introduces no new boundary and results in a connected surface bounded by the link L.

crossing segments of L; this rule specifies the sense of the half-twist. If the resultant surface is connected, it is the desired S. Otherwise, connect the various pieces of S together with cylindrical tubes (fig. 22d).

A technical point: we have not shown that S is orientable. In fact it is. Not only that, it is "bi- collarable", which means roughly that it is sand- wiched between adjacent Seifert surfaces [18]. In conclusion, S has all the properties of an admissible pre-candidate.

Note finally that since a knot is just a link with one component, this method also shows how to construct pre-candidates for knotted scroll rings. We will use the method frequently in paper 3 of this series [11], where pre-candidate isocons usually cannot be found "by inspection".

Appendix B

We were led by topological argument to the surprising conclusion that the isocons of a pair of


C

Fig. 23. In a diagnostic plane that includes the axle, we see three isocons of the linked pair of twisted scrolls. Each is a toroid obtained by "puffing out" the twisted band that joins the two ring singularities (labeled 1,2). Note that the right side is the left side, upside-down. The interior of the A* isocon is shaded.

linked once-twisted scroll rings are tori. As sketched in fig. 23, the spacing between tori thus constructed appears to vary quite a lot. But we know that real chemical waves are locally struc- tured by the interplay of local reaction with molec- ular diffusion; they tend to maintain a characteristic arrangement of chemical gradients within the propagating wave, and a uniform propagation velocity. Moreover, rotating chemical waves are characteristically spirals of fixed pitch and wavelength; nothing similar is evident in fig. 23.

These contradictions can be reconciled, but it would have been needlessly distracting to do so in the main text. We do it here, as follows. Fig. 23 is topologically faithful, but geometrically simplified. The actual waves resemble fig. 23, but modified (fig. 24) to "reel in" the isocons as suggested in fig. 25. If the reeling is done in such a way as to form involute spirals around the nominal core, then the arrangement will be preserved under a forward displacement of every segment of every isocon

outside the core by a fixed distance in each small increment of time [1, 8]. Waves repeat at exact intervals, each wave consisting of the same isocons spaced in the same way.

One consequence is that, beyond the innermost turns of each involute, waves radiate outward. They collide midway between the two source rings, closing the hole in a former toroid. Beyond that collision, each isocon separates into an internal

1

0 2

Fig. 24. The left side of fig. 23 is simplified to present the shaded disk (cross-section through a toroid) as a long ellipse. By winding it into a spiral we satisfy the physical requirements of propagating reaction/diffusion waves, without doing violence to the topological invariants of the isocon.

Fig. 25. The shaded toroid of fig. 23 is "reeled in" about each singular ring is sketched in fig. 24. As the resulting spiral waves propagate away from each singular ring, they collide and rejoin as shown. The inner structures (resembling the numeral "3" in cross-section) still connect as a toroid, albeit peculiarly sculp- ted. But these collide in such a way that the singularities ultimately become enveloped in concentric spherical waves.


t o ro ida l pa r t and an ou te r spher ical par t . Each

ro ta t ion o f the scroll br ings a b o u t ano the r col l is ion

and creates ano the r spher ical shel l-wave b o u n d e d

by a pa i r o f concent r ic spher ica l isocons.

This spool ing o f i socons into involu te spirals ,

with the consequen t in terna l col l is ions, and bud-

ding off o f concent r ic r ing-l ike waves is ev ident in

figs. 1 and 2 o f [10] in two different two-

d imens iona l exci table media .

A litt le care is needed in discussing the t o p o l o g y

o f a " w a v e " and o f a " w a v e f r o n t " . W e use " w a v e "

here to mean a consp icuous pa r t o f a t ravel l ing

pa t te rn : the region b o u n d e d by an i socon o f some

pa r t i cu la r value. Every i socon has an assoc ia ted

or ien ta t ion , given by the concen t r a t i on gradient .

The i socon b o u n d i n g a wave moves oppos i t e to its

o r i en ta t ion where it is the " w a v e f r o n t " , bu t a long

its o r i en ta t ion where it b o u n d s the rear o f the

mov ing wave. Fig. 26 shows a spher ical wave which has been a t t enua t ed in a d isk- l ike area. The

b o u n d i n g isocon was a pa i r o f concent r ic spheres,

now jo ined in a single surface which is a t opo -

"I . " " . " ' ..:'.'I'." .... . • . :. /.'.j

"''..i"-~ : " • ." t ' J " :: .".: "" "';~::~:.'7...-... . ~ ? ~ i '

Fig. 26. A spherical wave with a disk-like puncture. The puncture has different effects on the topology of "the wave" and of "the wavefront".

logical ly equiva lent to a single sphere. The wave-

f ront was the ou te r sphere bu t is now topo log ica l ly

equiva lent to a disk b o u n d e d by the puncture .

Similar ly, the i socons o f ou r l inked pa i r o f twis ted

scrolls are to ro ids , bu t only pa r t o f one to ro id is

a wavefront : tha t pa r t is t opo log ica l ly equiva len t

to a twisted band , b o u n d e d by the two s ingular

rings. The twisted band is coi led up into a spiral

a r o u n d each s ingular i ty .

R e f e r e n c e s

[1] N. Wiener and A. Rosenblueth, Arch. Inst. Cardiol. Mex. 16 (1948) 205.

[2] R. Suzuki, S. Sato and J. Nagumo, Notes of Professional Group on Nonlinear Theory of IECE (Japan) Feb. 26, 1963.

[3] G. Gerisch, Wil. Roux Arch. Ent. Org. 156 (1965) 127. [4] S. Yoshizawa, S. Amari and J. Nagumo, Denshi Tsushin

Gakki Shi. 54 (1971) 1354 (in Japanese). [5] A.M. Zhabotinsky, Thesis, Biophysics 030391 (Puschino,

USSR, 1970). [6] A.M. Zhabotinsky, Oscillatory Processes in Biological and

Chemical Systems 2 (Puschino, USSR, 1971) (in Russian). [7] A.T. Winfree, Forskning och Framsteg 6 (1971) 9 (in

Swedish). [8] A.T. Winfree, Science 175 (1972) 634. [9] A.T. Winfree, Science 181 (1973) 937.

[10] A.T. Winfree and S.H. Strogatz, Physica 8D (1983) 35. [11] A.T. Winfree and S.H. Strogatz, Physica D, to be pub-

lished. [12] J. Gomatam, J. Phys. A 15 (1982) 1463. [13] H.G. Othmer, Lectures on Mathematics in the Life Sci-

ences 9 (1977) 57. [14] E. Conway, D. Hoffand J. Smoller, SIAM J. Apial. Math.

35 (1978) 1. [15] A.T. Winfree, The Geometry of Biological Time (Springer,

New York, 1980). [16] O.E. Rossler and C. Kahlert, Z. Naturf. 35a (1979) 565. [17] L. Neuwirth, Scientific American 235 (June 1979) 110. [18] D. Rolfsen, Knots and Links (Publish or Perish Press,

Berkeley, 1976).

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Singular filaments organize chemical waves in three dimensions II. Twisted waves