Wherever my web site address is mentioned, it should be changed to: http://marley.biosci.arizona.edu/~art, which also
points to a different machine for 20 PowerPoint lectures: http://eeb8.biosci.arizona.edu/art/2000_lectures
And my email address likewise changed to winfree@email.arizona.edu

See also the list of 183 Questions flagged in page margins, referred to below as Q's

page xi: Rashevsky's Mathematical Biophysics was (1948)

pages 114, 174: "1979" should be "1979a,b"

page 117 line 5, "1973" should be "1973a"

page 119, bottom equation, indices 1,2 should be reversed

page 122 and 232 remark that treatments of laser physics usually adopt the metaphor of populations of identical oscillators, thus making no use of generic findings about mutual entrainment. This is corrected by Oliva and Strogatz (2001) in Int.J.Bifur.Chaos 11, 2359-2374. [added 2 Dec 2001]

page 124 (and 235): The supercritical or inverted bifurcation to mutual synchronization was in fact confirmed in populations of simple clocks but I had not yet seen it while writing in 1999: Daido (1996) Physica D 91, 24-66, received from Daido on 16 Oct 2001. So this marginal "Q" can be removed. The one on page 235 is probably superfluous too, now.

page 126 top Qs invite attention to syntalansis threshold in various cellular and biochemical oscillators. A nice inorganic case is presented by Vanag and Epstein in this week's (26 Oct) Science 294, 835 (2001): it seems that oscillating aqueous BZ reaction droplets suspended in oil similarly interact and await analysis in these terms.

page 126, re the Question flagged in the margin as still awaiting attention: bifurcation analysis is now provided by Ariaratnam and Strogatz (2001), Phys. Rev. Lett. 86, 4278-4282, though application to any actual science remains still for the future.

page 127, top equation: (phi_j - --> (phi_i -
and next to bottom equation (copying the first equation), right hand side after "a"
was intended to use the typeset left hand side of the equality just above, but inadvertently
substituted the right hand side to no purpose (thus simply repeating the first equation).

page 131 three lines up from bottom: delete "recently" (1988!)

pages 132, 756: Stephen Yeung's name is mis-spelled "Yueng". Apologies, my fault.

page 172, below the middle "1974: should be "1973a"

page 191 bottom of shaded area: "this" case means "using cerium instead of bromide for the perturbation"

page 197 top 3 lines: Honma and Honma found Type 0 resetting in a rodent

page 223 middle: "only betrays" and "also reflects"

page 232 and 122 remark that treatments of laser physics usually adopt the metaphor of populations of identical oscillators, thus making no use of generic findings about mutual entrainment. This is corrected by Oliva and Strogatz (2001) in Int.J.Bifur.Chaos 11, 2359-2374. [added 2 Dec 2001]

page 233, re the Question flagged in the margin as still awaiting attention: bifurcation analysis is now provided by Ariaratnam and Strogatz (2001), Phys. Rev. Lett. 86, 4278-4282, though application to any actual science remains still for the future.

page 235: The supercritical or inverted bifurcation to mutual synchronization was confirmed in populations of simple clocks but I had not yet seen it while writing in 1999: Daido (1996) Physica D 91, 24-66, received from Daido 16 Oct 2001. So this marginal "Q" about clocks in general is probably superfluous too, the effect being confirmed in a special case.

page 245: Phase waves around a phase singularity are also reported in oscillatory BZ reaction by Vanag and Epstein in this week's (26 Oct, 2001) Science 294, 835. The gradients are steep enough that they seem to provoke propagating trigger waves, which in this case can form inward as well as outward spirals. This seems to answer the Question by confirming long expectation. (In 1980's GL and lambda-omega models of oscillatory reaction-diffusion fields showed phase singularities, with inward or outward spiral surrounds depending on whether low-amplitude oscillations are slower or faster than on the limit cycle ... the mystery was only that neither tidy phase singularities nor inward spirals were observed in the lab.)

page 248: Box C is the first in this Chapter 8. This is not an error. Old Boxes A and B were bibliography lists, very modern and fashionable in 1978, but tiresomely long in 1999, so I just left them out.

page 255 lines 9-12 [added 8 Nov 2001]: Rotating epidemics arising from the suggested geographical rotors indeed have been studied in numerical models, e.g., Savill et al (1997) J.Theor.Biol 188, 11 and Gurney et al. (1998) Ecology 79, 2516. No evidence yet in nature, so that much of the question still stands.

page 262 bottom Question flagged in margin: These "traffic jams" were observed in experiments with Scott Caudle for Physics 482 Senior project in Fall 1997. We put CHD agar gel in a capillary tube, made video movies always during the same 10 min interval after mixing the reagents (so that "aging" was not a factor), then image-processed the data to become a space-time plot with roughly parallel diagonal curves showing the motions of pulses along the tube. Pulses entering and exiting the "traffic jams", and the speed of the jam, were quantified in terms of phase velocities and group velocities along the observed dispersion curve. Hamik et al. (2001) J Phys Chem 105, 6144 repeated this experiment almost identically but with a CHD recipe more like Koros' original of 1994, and got the same results. See also under p381.

page 265 footnote, line 4: "D" means "of the propagator species"

page 266: this figure introduces "D" as diffusion coefficient: well, it was so used also on pages 139, 140.

page 276: at the Question flagged in margin: re Roth's 70 msec, see also similar experiment at 100 msec reported by Cheng et al. (2000) J Cardiovasc Electrophysiol 11, 998

pages 277-78 and 301-02 citing Pearlman and Ronney neglected to mention that this only works if we choose not the material diffusion coefficient but the thermal diffusion coefficient at a suitably guessed temperature. Scott et al. justify this by modeling of Pearlman's flames in terms usual for excitable media, finding that the thermal rate equation (in contrast to the chemical reaction rate equation) supports the fast excitation: J.Chem.Soc.Faraday Trans. (1997) 93, 1733-1739. The "Q" of that paper is heat energy, not the "quality factor" of this book.

page 278 line 6: "...,D, of the propagator species..."

page 286: in the middle of the shaded area all the "D"s are intended as scalar (all the same
or only one non-zero) but one got printed as bold (matrix). If the ratios of D_ij
in a matrix are changed, the period generally changes. And in that middle paragraph I
trust it is understood that abruptly changing D requires also rescaling the initial and
boundary conditions in proportion to sqrt(D); this can be neglected only if the change is
made "adiabatically" and walls are remote.

page 292 Figure 21: re departure of white tip from ideal involute curve, and the corresponding evolute circle, see p383 Figure 3.

page 296 phrase deleted by printer near end of shaded area, to replace "scrollz":
"scroll ring with a cycle of twist + writhe (see Chapter 16's Box B) locked into it as in
this diagram"

page 299 Question flag in margin of 1978 text should be removed: answers are long known, at least from computer simulations, viz., Yes, Yes, No. Next sentence ("not yet undertaken" in 1978) could now point to 3d numerical solutions of the Ginzburg-Landau oscillator field

page 301, about the demise of vortex atoms: resurrected by Berry, M V "Knotted zeros
in the quantum states of hydrogen" Fundamentals of Physics, preprint (2002)

pages 301 and 684: "Faddeev" mis-spelled with a single "d". For an update on this topic
see Battye, R. A. and Sutcliffe, P. M. (1999) Solitons, links, and knots. Phil. Trans. Roy. Soc. Lond. A 455, 4305-4331.

page 313: There has been a remarkable renaissance of the Cooke and Zeeman (1976) interpretation of development, and of somitogenesis in particular, in terms of a local cellular oscillator and a traveling wavefront or cell division boundary. See Cooke (1998) Trends Genet. 14, 85-88, Stern (1998) Bioessays 20, 528-531, Salazar-Ciudad et al. (2000) J.Theor.Biol. 205, 587-603 and (2001) Evol.Dev. 3, 84-103, Szathmary (2001) Nature 411, 143-44, and Tabin (2001) Nature 412, 780-81.

page 323 top: vault not yet opened or contents not yet divulged

page 327 middle, term "regions" should be "domains" for consistency with prior publications and here (referring to an area of parameter space, not state space or geographical space.)

It is also of interest on this page that the chemical composition and reactivity of the rotor necessarily differ from those found anywhere else during wave propagation. Though this was stressed theoretically in Winfree (1973b), its demonstration in reality waited 17 years until Agladze and Steinbock found the rotor tracing a black line on corroding steel (J.Phys.Chem.A 104, 9816-9819).

page 328 question whether a rotor or vortex filament can react to incoming periodic wavetrain by moving toward its source is answered "Yes" for marginally excitable media, as theorized in Ermakova et al (1986) Biophysics 31, 348 and Vinson (1998) Physica D 116, 313

page 331 on excitable media lacking multi-arm spirals distinct from adjacent single-arm spirals:
In the complex Ginzburg-Landau oscillator field they do exist: Aranson et al (1993), Phys. Rev. E 47, 3231. And add to end of 4th bullet: "coupled through a single species."

pages 345-46 about Norbert Wiener's spectrum for brain waves: the 1999 writing neglected to draw your attention to an excellent review by Strogatz in Lecture Notes in Biomathematics 100 (1993). And in connection with synchronous firing in epilepsy here and on p427, I neglected to mention that Durand and Warman (1994: J. Physiol. 480, 527-537) experimentally demonstrated phase scattering and attenuation of collective amplitude in response to a critically timed electrical stimulus of critical size (for follow-up: Durand,D.M., 2001: Proc IEEE 89, 1065-1082; Hahn,P.J. and Durand,D.M. 2001: J Comput NeuroSci April issue).

page 360 Figure 6 dashed curve is supposed to have an asterisk somewhere along it, anywhere.

page 366: see page 126 above about mutual synchronization of BZ oscillator droplets in oil suspension.

page 372 line 5 (first line of paragraph) 1958 --> 1959 as in the reference list.

page 381: Hamik et al. (2001) J Phys Chem 105, 6144 recently presented an "anomalous" CHD dispersion curve using the same procedure as Scott Caudle in his Senior thesis of Fall 1997 (see page 262 above). Because the CHD gel ages materially over intervals longer than about 10 min we, and they, made speed vs spacing observations always always within the same 10 min interval after the reaction starts. Rather than differentiating the life history of a single aging pulse pair we obtained a noisier dispersion curve (Figure 2.1) from the average slope of each of a large number pulses in replicate capillary tubes. Our curve is not "anomalous" perhaps because we had used a different CHD recipe.

page 391 bottom right corner is supposed to be flagged by a shaded "Q" in the margin

page 395 Figure a of Nagy-Ungvarai and Muller (1994 not 1993) Int J Bif Chaos 4, 1257.

page 397 Figure 3 ditto

page 400: The notion that experiments within a gel sandwiched between fixed-concentration reservoirs probably does not change its inner concentrations in direct proportion to reservoir concentration, and probably develops three-dimensional structures, broached in Winfree (1991) Nature 352, 568-69 in context of Turing patterns in the CDIMA reaction, seems now accepted in that case (Rudovics et al. (1999) J.Phys.Chem. A 103, 1790-1800.) However there have been no reports of its test in vycor using the other malonic acid reaction (Belousov-Zhabotinsky), perhaps because there seems to be no supplier of vycor glass as thin as used in the published experiments, and those machines have been retired. In such a situation quantitative theory about such things cannot be tested and therefore is not elaborated.

page 409 there should be two "Q" flags in the margin at lines 1-4, and at lines 14-16; at it should be noted that not only periodic illumination to entrain but also much more rapid flicker or continuous light to suppress have neither reportedly been tried in this context. And the top 5 lines should have a Question flag in the margin.

page 427: see page 345

page 435 line 5 from bottom of gray, insert "Jongsma and Tserjina (1982)"

page 455: typo, "l" left out of "Salviati"

pages 458-66 show "complex" behavior in an area to the right of a codimension 1 locus in the 2-parameter planes of 1990-91. Ashwin et al (2001) in Physica D 156, 364-382 argue that I may have been misled by transients longer than my computations: a transition to chaos may occur through a codimension 2 bifurcation starting somewhere inside that area, and the area(s) containing asymptotically chaotic hypermeander may be a very narrow wedge(s). This might help account for the peculiar apparition of Fourier spectra with abundant sharp lines to the right of my "boundary_C".

The same cautionary lesson may apply to 3d persistent organizing center computations: Paul Sutcliffe has recently extended their duration 10-fold and found that while they do indeed persist with unaltered topological invariants, they do not stay the same size: I was evidently observing a decrease to minimum size arrived at from wildly diverse initial conditions, but after passage though that common compact state, the organizing center typically expands again on a much longer time scale.

page 460 and 469: I forgot to put a big "Q" in the margin alongside mention of the one known (and confirmed in two independent publications) counterexample to the notion that there is a unique rotor period in any given excitable medium. The question is whether coexisting alternative rotor periods can be found in any other numerical medium, and similarly, why after a dozen years has it never been reported from the laboratory?

page 473 (and 485, 488, 491) mention passage on skew filaments through one another, like two consecutive hyperbolic reconnections: In Febr 2002 I learned that there is some question whether this is really possible. See Fiedler and Mantel 2000, Documenta Math 5, 695-731.

page 475 line 7 up from bottom, D³/T_o: omit typo "/" (dimensions), or better still, snopake the whole confused sentence.

page 482, sentence near middle, beginning "of additive effects" got garbled. First of all,
it hit 3 corner walls simultaneously. Secondly, words "it deformed" belong after "into a wall."

page 486 line 4 from bottom: "but with a markedly different" --> "and with a similar"
(I dont' know how I could have written such a phrase, diametrically opposite to the familiar outcome.)
Also the comments about applicability of Burgers' equation seems implicitly assuming term c₁ k to
have negligible relative size, as as indeed it would in case of equal diffusion (c₁ = 0) or k very
small (Biktashev and Holden, 1994). However neither is the case in the examples computed, so
this discussion may be nugatory.

page 488 line 3, the two cited Panfilov papers actually do not explicitly mention "tension".
page 488-90: see extensive update in new manuscript

page 489 caption Figure 7, end of line 3 "rotation" should be "precession" for consistent usage.
caption Figure 9 "I do not know why": now I do: it was a numerical artefact due to symmetry of initial conditions being identical to that of the numerical grid.

page 492: (early 2002) Extending 3d numerical experiments a decade later with faster computers: see forthcoming from Paul Sutcliffe at Univ of Kent, Canturbury. By extending the duration of computation 10-fold he found that while "persistent organizing centers" do indeed persist with unaltered topological invariants, they do not stay the same size: I was evidently observing a decrease to minimum size arrived at from wildly diverse initial conditions, but after passage though that common compact state, the organizing center typically expands again on a much longer time scale.

pages 500, 503 allusion to integrating over surfaces should be over volumes, of course

page 511, the second bulleted item is supposed to be flagged by a shaded "Q" in the margin

page 519 line 15 "At the moment..." The moment passed with August 2001 publication of Lindblom et al., J Cardiovasc Electrophysiol 12, 946-956 extending cited Lindblom et al 2000 from near field to far field (at least, 1-2 cm far, in a 3d model).

page 521 line 9: the foregoing new publication asks conversely, "Could bidomain near-field concepts be extended to make sense of far-field experiments motivated by T*S* concepts?" See also the experiments of Cheng et al., same journal 11, 998-1007 and the review/commentary by Ideker et al., pp 1008-1013, both appearing several months after this book went to press.

page 523 "This was roughly the state of affairs by 1998..." is better reviewed by Ideker et al Sept 2000, cited just above.

page 524 middle (August 2001) "It is not clear that these computations pertain to far-field experiments..": Lindblom et al. 2001 J.Cardiovasc.Electrophysiol. 12, 946-956 show they do (at least, 1-2 cm far, in a 3d model).

page 524 Febr 2002: regarding abnormaility of mocardium exposed to DAM (BDM): Nikolski et al. (2002) Am.J.Physiol 282, H565-575 find that it also changes a threshold for electrical stimulation by an order of magnitude.

page 526 near bottom "transmural activation not yet confirmed": Now it has been: Wu and Zipes (2001) Amer J Physiol 280, H2717

page 527: Question flag in the margin probably does not belong

page 528 lower Question flag belongs 2 lines higher ("this was never resolved")

page 536 seems now still further fortified by Biktashev et al. (2001) Int.J.Bif.Chaos 11, 1035-1051 and (2002) Chaos Sol.Fract. 13, 1713-1733.

Chapter 19 in general: dates from Fall 1999, when publication was expected
in early 2000, so it is uninformed by almost two year's explosive discoveries about the molecular genetics of circadian clocks. Other chapters are also somewhat out of date at publication on 5 June 2001, but this chapter dated more rapidly.

page 551 top line "Drosophila pseudoobscura" should be ital like all the rest

page 562 line 4 after "night": (i.e., sleep, presumably in the dark)

page 565 line 10: two subsequent efforts in press for Nature were unable to reproduce this

page 583, wishing for a quantitative limit-cycle model of the epigenetic mechanism of a circadain clock, checkable against experiments: now see Leloup 2000, BioEssays 22, 84-93 and 2001 et seq in press.

page 589: middle line containing new references to 1983, 1987, 1990 papers should be shaded, for consistency.

page 591 line 4: "...coupling by diffusion..." and line 5 "so" looks like a non-sequitur, but was intended to convey that because the rotor period is necessarily shorter than the spontaneous oscillation, if such shorter period is evident it betrays coupling regardless how strong or weak that may be

page 594 line 3: Drosophila pseudoobscura

page 597 middle: "suppose in the cold the developmental rate...": a testable implication of theory of rhythmic gating awaits modest laboratory effort

page 602: this "per^s" should be ital like all the others

page 641: first Question flag should be moved down a few lines

The typesetter in Manila inserted boldface page references into the reference list (pp 651-758) using an un-checked computer algorithm that proved unreliable. A few corrections noticed between 2 June and 13 September 2001:

page 666: Belmonte et al 1997 appears not on page 390 but 389, and not 398 but 399

pages 673-64: Chen (2000) appears on page 404, not 406

page 718: Nagy-Ungvarai appears on page 458, not 459

page 722: Panfilov and Pertsov, Panfilov and Rudenko do not appear on page 488

page 723: Pearlman two papers also appear on page 302

page 730: Rossler and Kahlert imputed to page 441 actually appears on p440

page 734: Schrader appears not on page 332 but on page 331

page 735: Sevcikova appears on page 191, not 192

page 750: the top 11 papers do not appear on page 253 (error in publisher's indexing program)

page 751: Winfree 1984a appears also on page 560

page 752: Winfree 1990 a,b,f do not appear on page 435

page 754: Winfree and Strogatz 1984a does not appear on page 560

page 755: Yakushevitch is not on page 468 but on page 467

On 13 Sept 2001 I desist from catching page references misplaced by Springer's automation, which I proof-read inadequately. Please just consider every page reference in the Author Index as "plus or minus 1" and remember that some few are simply missing and others imaginary.

Some corrections to the references themselves:

page 679: Dawson is 1999

page 709: on 6 Aug 2001: Lee, Cox, Goldstein became Lee Goldstein Cox in Phys.Rev.Lett. 87, 068101-1 to -4 with revised title: "Resetting Wave Forms in Dictyostelium Territories."

page 729: Rinzel and Ermentrout (1989) title is "Analysis of Neural Excitability and Oscillations"

page 731 (from Brad Roth, 29 Jan 02): Roth and Guo is "and J.P. Wikswo, Jr." Apologies, John. And I don't know where the "e" came from on the end of "Math".

page 752: Winfree (1986a) did appear in 1986, but only now I notice (Cele Abad-Zapatero points out to me) that the printed copyright page says both 1986 and 1987. Everyone but me seems has cited it as 1987. Ooops.

page 753: Winfree "2000a" (of June 1999) finally appeared in 2002, with title edited to "Are cardiac waves relevant to epileptic wave propagation?"

page 754: Winfree and Witkowski (1998) is not in "IEEE Trans Biomed Eng" but in Ann.Biomed.Eng. 26 (supplement 1), S-16

A new version of the Topics Index may appear here after I accumulate more corrections.
A few that I have noticed:

page 761 "alternative rotor periods": I forgot to place Q's in the margins of these pages noting that nothing of the sort has yet been reported from the laboratory.

page 763: Circadian Rhythms subtopic Evolution: pages 549, 568, 570, 575-582

page 764: Discoveries Overlooked: add lower page 385

page 768: Hyper-meander: not pages 462, 467 but 462-67

page 774: scroll ring/shrinking: pages 167-169 not 189

page 775: Singular filament: pass through or reconnect: pages 473, 485, 488, 491

page 776: Parkinsonian tremor: add page 115

page 777: web site: also page 471, and phase wave: also page 379

Overall: the shaded Question flags in left margin of even pages were supposed to be in right margin of odd pages, not hidden in the crack: I overlooked the mutation in all page proofs!

And finally, although NSF support is gratefully acknowledged throughout the book, I neglected to state up front that

"This material is based upon work supported by the National Science Foundation under Grant No. 9974334. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation."