Spiral waves: linear and nonlinear theory

(1)

Spiral waves: linear and nonlinear theory

Björn Sandstede

Toan Nguyen

Arnd Scheel Kevin Zumbrun Margaret Beck Stephanie Dodson

(2)

Spiral waves

CO oxidation on platinum [Nettesheim, von Oertzen,

Rotermund, Ertl]

cAMP signalling of amoebae [Newell]

Calcium waves in Xenopus oocytes [Clapham et al.]

Belousov-Zhabotinsky reaction [Swinney et al.]

(3)

Spiral waves

Dynamics of core / spiral tip

Modulations of wave trains in far field

[Li, Ouyang, Petrov, Swinney]

Meander (far field)

V

^OLUME

77, N

^UMBER

10 P H Y S I C A L R E V I E W L E T T E R S 2 S

^EPTEMBER

1996

Transition from Simple Rotating Chemical Spirals to Meandering and Traveling Spirals

Ge Li,* Qi Ouyang, Valery Petrov, and Harry L. Swinney

Center for Nonlinear Dynamics and Department of Physics, The University of Texas at Austin, Austin, Texas 78712

(

Received 8 May 1996

)

Experiments on the Belousov-Zhabotinksy reaction unfold the bifurcation from simple (temporally periodic) rotating spirals to meandering (quasiperiodic) spirals in the neighborhood of a codimension-2 point. There are two types of meandering spirals, inward-petal (epicycloid) spirals and outward-petal (hypocycloid) spirals. These two types of meandering regimes are separated in the phase diagram by a line of traveling spirals that terminates at the codimension-2 point. The observations are in good accord with theory. [S0031-9007(96)01014-9]

PACS numbers: 82.40.Bj, 82.20.Mj, 87.90. + y

Rotating spiral waves are ubiquitous in systems rang- ing from excitable reaction-diffusion media [1,2] to ag- gregating slime-mold cells [3] to cardiac muscle tissue [4]. Winfree discovered that under certain conditions a spiral tip meanders rather than follows a periodic circu- lar orbit [5]. Meandering spirals have been subsequently extensively studied experimentally [6–9] and theoreti- cally [10 –16]. Experiments [8,9] and theoretical analyses [12,13,16] have shown that the meandering is often not an erratic motion; rather, the spiral tip moves in epicycloid- like [17] orbits (flowerlike orbits with inward petals) or hypocycloidlike orbits [17] (with outward petals) that are B quasiperiodic in time [8]. Meandering is of interest in part because of its predicted relation to defect mediated turbulence [18–20]. It may also provide a clue to the cause of cardiac arrythmias, which can lead to ventricular fibrillation [21].

There have been few definitive experimental results on meandering other than the observation that the on- set of meandering is a periodic-quasiperiodic transition [8,22]. We present here experiments on the Belousov- Zhabotinsky (BZ) reaction that reveal, as predicted by Barkley [16], an unfolding of the bifurcation to meander- ing about a codimension-2 point that is the terminus of a line of traveling spirals.

Figure 1 shows the orbit of a spiral tip for the two types of meandering motion [23]: (a) an outward-petal meandering spiral and (b) an inward-petal meandering spiral. We take the spiral tip to be the point with maximum local curvature on the wave front. Hypocycloid motion with outward petals is illustrated in Fig. 1(c), where the primary circle (radius r

₁

) orbits the secondary circle (radius r

₂

) in one direction with frequency f

₂

and spins about its center in the opposite direction with frequency f

₁

; epicycloid motion with both rotations in the same sense and inward petals is illustrated in Fig. 1(d).

Figure 2 illustrates all four types of spiral motion that we have observed. Figure 2(a) is a simple periodic rotat- ing spiral, which becomes unstable as a parameter is var- ied. Depending on the control parameter, the system then chooses outward-petal meandering, as in Fig. 2(b), or

inward-petal meandering, as in Fig. 2(d). The tip of a me- andering spiral emits waves that are compressed in front of the tip and dilated behind the tip. This produces super- spirals, as can be seen in Fig. 2(b), which has a retrograde superspiral, and in Fig. 2(d), which has a prograde super- spiral. At the transition from outward-petal to inward- petal spirals, the spiral tip travels in a straight line; see Fig. 2(c).

FIG. 1. Meandering spiral with (a) outward and (b) inward petals. The white lines in the images show the trajectories of spiral tip. (c) and (d) illustrate, respectively, a hypocycloid and an epicycloid, analogous to the motion in (a) and (b). The only parameter different in (a) and ( b) is the concentration of sulfuric acid in reservoir B: (a) 0.46M, (b) 0.40M. The other control parameter in the present experiments is the concentration of malonic acid in reservoir B, which is fixed in this figure and Figs. 2 and 3 at 0.8M but varied in Fig. 4. Other conditions are fixed in our experiments at the values given in Ref. [23]. The pictures in (a) and (b) are 1.7 3 1.7 mm².

0031-9007 y96y77(10)y2105(4)$10.00 © 1996 The American Physical Society 2105

Meander (core) Drift

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Spiral waves in cardiac tissue

Cardiac arrhythmias can be caused by spiral waves pinned to inhomogeneities

Low-energy excitation can remove spiral waves [Luther et al.]

Proposed strategies are sensitive to the relative phase difference between excitation waves and externally induced stimuli

Heart model [Fenton et al.]

(P , 0.002). Furthermore, LEAP effectiveness was demonstrated for ventricular fibrillation in vitro (n 5 7 canine preparations, 12 defibril- lation episodes and 28 LEAP episodes). In these experiments (Fig. 1d), the average energy reduction of LEAP versus a single shock was 85%

(P , 10 ²⁵ ). The spatio-temporal excitation dynamics of the right atrium in vitro before, during and after control are shown in Fig. 1d (see also Supplementary Movie 1). During fibrillation, waves of tur- bulent electric activation propagate across the atria. At t 5 0, a sequence of five electric pulses is applied at the coil electrodes, followed by a transient, spatio-temporal reorganization of the activation waves. After each pulse, the area that is activated increases, indicating progressive synchronization of the myocardium; fibrillation then

terminates and normal sinus rhythm (Supplementary Movie 2) can resume.

To elucidate the mechanism of defibrillation by LEAP, we studied the response of quiescent atrial and ventricular tissue to a homogeneous, pulsed electric field (Fig. 2a, b). In Fig. 2c, images taken at 1.5 ms, 3 ms and 6 ms after the pulse (0.22 V cm ² ¹ ) show depolarization induced by a single source. However, with increasing electric field strengths of 0.22 V cm ²¹ , 0.39 V cm ²¹ and 0.50 V cm ²¹ , the number of sources increases to several dozen over the entire tissue. The locations of these sources and the wave propagation patterns are summarized in the iso- chronal maps shown in Fig. 2d. The density of sources, shown in Fig. 2c and d, increased with increasing field strength for both the ventricle and the atrium, thereby decreasing the activation time (Fig. 2b).

The results can be explained in the context of virtual electrodes

^15–20

. In the bidomain representation, the voltage in cardiac tissue is the potential drop between the intracellular and extracellular medium. Theory pre- dicts

¹⁶

that, in the presence of an electric field, discontinuities in tissue conductivity, such as blood vessels, changes in fibre direction, fatty tissue and intercellular clefts, induce a redistribution of intracellular and extra- cellular currents that can locally hyperpolarize or depolarize the cells. At the depolarization threshold, an excitation wave is emitted

^15–17,21

.

The electric field that is necessary to produce an activation, as a function of the size of the conduction discontinuity in quiescent tissue,

–4 –3 –2 –1 0 1 2 3 4

–0.5 0 0.5 1 1.5

–4 –3 –2 –1 0 1 2 3 4

–0.5 0 0.5 1

Time (s)

MAP (a.u.)

Atrial fibrillation (f = 6.8 Hz) LEAP Sinus rhythm

o 2

0 1 1.5

0.5 Defibrillation LEAP LEAP

0.76 J

0.08 J 0.14 J

1.05 J

Pulse ener gy (J)

In vivo

In vitro

Defibrillation

b

c

d

Optical mapping (a.u.)

In vivo In vitro

Time (s)

e ^{–2.366 s} –2.332 s –2.198 s –2.164 s

Atrial fibrillation

2.258 s 2.272 s 2.284 s 2.298 s

Sinus rhythm LEAP

0.020 s 0.118 s 0.218 s 0.316 s

–2.130 s

2.320 s 0.416 s

a

–80 20 Membrane potential

(mV)

0 1 2 3

0.14 J 1.51 J

In vitro

Atrial fibrillation Ventricular fibrillation

Pulse ener gy (J)

LEAP Defibrillation

2 cm

RA LA

CE

RV LV

1 cm

Figure 1 | Low-energy termination of cardiac electrical turbulence in vivo and in vitro . a, Schematic of the anatomy of the heart. LA, left atrium; LV, left ventricle; RA, right atrium; RV, right ventricle. A pulsed electric field was

applied with standard cardioversion coiled wire electrodes (CE) inserted into the left and right atria by catheters (see Supplementary Information).

b, Monophasic action potential (MAP) recording of termination of AF using LEAP in vivo. a.u., arbitrary units. Dominant frequency f

_v

5 6.8 6 0.1 Hz, n 5 5 pulses, pulse duration Dt 5 8 ms, pacing cycle length T

_p

5 99 ms, pulse energy W 5 0.074 6 0.012 J. c, Termination of AF in vitro, measured from the atrial epicardium of the same heart as in b, by optical mapping (see e). The signal from a 0.3 3 0.3 mm

²

region is shown (f

_v

5 6.8 6 0.1 Hz, n 5 5, Dt 5 8 ms, T

_p

5 90 ms, W5 0.066 6 0.017 J). d, Reduction in pulse energy using LEAP versus standard defibrillation. In vivo AF (n 5 7): LEAP (56 episodes, mean energy W _

5 0.14 6 0.08 J); defibrillation (22 episodes, W _

5 0.89 6 0.56 J). In vitro AF (n 5 5): LEAP (46 episodes, W _

5 0.10 6 0.07 J); defibrillation (39 episodes, W _

5 1.15 6 0.58 J). In vitro ventricular fibrillation (n 5 7): LEAP (28 episodes, W _

5 0.17 6 0.16 J); defibrillation (12 episodes, W _

5 1.34 6 0.89 J;

see Supplementary Information). Box plots show the median and the 25th and 75th percentiles. Whiskers indicate the statistically significant data range and red crosses mark outliers. e, Optical mapping of the AF termination that is also shown in c. During AF, complex spatio-temporal propagation of electrical

excitation waves was observed (white line indicates boundary of atrium). LEAP (n 5 5, Dt 5 90 ms) progressively synchronized the tissue (see Supplementary Movies 1 and 2). Data are given as mean 6 s.d. unless stated otherwise.

E

Activation sites

Right atrium

1 cm

Right ventricle Apex

0 0 1 2 3

5 10 15 20 25 30

Activation time (ms)

Field strength (V cm

^–1

)

d c

a b

Atrium Ventricle

Time (ms)

0 73

25

0 (ms)

20

0 (ms)

18

0 (ms) Time

0.22 V cm

^–1

0.39 V cm

^–1

0.50 V cm

^–1

1.5 ms

3.0 ms

6.0 ms

Figure 2 | Sites of activation in a cardiac preparation. a, Canine wedge preparation (7.5 3 5.6 cm

²

) consisting of right atrium and right ventricle. At

t 5 0 s, an electric field pulse of strength E 5 0.34 V cm

²¹

was applied for 5 ms.

The colour indicates the time of local activation observed with fluorescence

imaging on the endocardium; the greyscale trans-illumination image shows the complex anatomy of the endocardium. The white square marks the area shown in panel c. b, Mean activation times t(E) for atria (blue circles, n 5 3 preparations, 17 measurements of t(E)) and ventricles (red circles, n 5 6 preparations, 24

measurements of t(E)) in response to an electric field pulse of strength E and duration 5 ms. Error bars indicate s.d. c, Activation of the atrium (in the region indicated by the white square in panel a) after an electric field pulse at t 5 0. With increasing field strength, the number of activation sites increased and the time interval for total activation decreased. The colour code for each row is shown in d (see Supplementary Movie 3). For E , 0.2 V cm

²¹

, no waves were observed.

d, Isochronal maps of the activation sequences shown in c.

RESEARCH LETTER

2 3 6 | N A T U R E | V O L 4 7 5 | 1 4 J U L Y 2 0 1 1

Macmillan Publishers Limited. All rights reserved

©2011

Canine heart (in vitro) [Luther et al.]

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Spiral waves in cardiac tissue

Ventricular Tachycardia (VT) Abnormally fast heart rate

Ventricular Fibrillation (VF) Fatal unless treated immediately

Arrhythmic pattern

Alternans rhythm

(6)

Spatio-temporal period-doubling of spiral waves

Line defects in light- sensitive BZ-reaction [Yoneyama, Fujii, Maeda]

(supercritical?) Alternans in

cardiac tissue (subcritical?)

2 S. DODSON, B. SANDSTEDE

0 1 2 3 4

(a) (b) Time

Figure 1. (a) Stationary line defect in the w-component of the R¨ ossler system. (b) Time evolution of alternans instability in the u-component of the Karma model on a square of side length 16cm with homogeneous Neumann boundary conditions. System parameters as defined in Section 2.

wave.

37 Nonlinear reaction-di↵usion systems qualitatively capture transitions to complex mean-

38 dering, drifting, and the period-doubled line defects and alternans patterns. In these systems,

39 planar spiral waves are stationary solutions in a rotating polar coordinate frame and converge

40 to one-dimensional periodic travelling waves away from the core. Stability and bifurcations

41 can be studied by considering the spectra of the operator obtained by linearizing the nonlinear

42 system about the spiral wave solution. The spectrum consists of isolated eigenvalues and a

43 set determined by the operator in the far-field limit.

44 Bounded domains are of interest in applications to cardiac dynamics and laboratory ex-

45 periments. Neumann boundary conditions naturally represent lower conductance tissue sep-

46 arating regions of the heart or the physical walls of containers. Mathematically, on finite

47 domains planar spiral waves are truncated and matched with a boundary sink, which adds

48 extra structure to the spiral wave and alters the spectrum of the linear operator [13, 33, 32].

49 The boundary sink itself directly contributes an additional set of eigenvalues, and the finite

50 domain modifies the spectrum associated with the far-field dynamics. Furthermore, radial

51 growth in the eigenfunctions is permitted, and those that would not be integrable on the full

52 plane now emerge as true eigenfunctions on the bounded domain [13, 33, 32]. These eigenval-

53 ues are associated with intrinsic properties of the spiral wave and are attributed to the spiral

54 core. The spectrum of the operator on a bounded domain is therefore a union of three disjoint

55 sets that are associated with instabilities from the far-field, boundary conditions, and core.

56 Knowing which set unstable eigenvalues belong to provides information about how instabilities

57 will manifest themselves on unbounded or bounded domains.

58 Meander and drift instabilities are the result of a Hopf bifurcation originating from the

59 core: the emerging dynamics is understood through actions of the symmetry group of trans-

60 lations and rotations on the plane and a center manifold reduction [5, 39]. However, previous

61 studies investigating alternans and line defects provide inconsistent and incomplete evidence

62 for which spectral set the unstable eigenvalues belong to.

63 Due to the clinical significance, the alternans instability has been a recent area of focus in

64 the cardiac dynamics community. In single cells, alternans are widely attributed to a period-

65 doubling instability observed in simple 1D maps [18]. However, this condition, known as the

66 restitution hypothesis, has received contradictory evidence [10, 7] and does not appear to

67 be relevant for excitable tissues that support traveling waves. The formation and stability

68 This manuscript is for review purposes only.

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Outline

Wave trains: spectral stability and modulation equations

Planar spiral waves: spiral spectra on the plane and on bounded disks

One-dimensional defects: anticipated dynamics, and nonlinear stability

results

Re(λ)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Im(λ)

-4 -3 -2 -1 0 1 2 3

4 R 20 Psuedospectra and R 30 Spiral Spectra

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Figure 4: Pseudospectra of radius 20 spiral. Black circles are eigenvalues calculated with eig of radius 30 spiral.

2 Di↵usion Added to the v-equation

Essential and absolute spectra calculated from wave trains. The spiral spectra is calculated with direct methods (eig) on a spiral of radius 20 discretized with 64 Fourier spectra grid points in the angular direction and 100 4th order centered finite di↵erence in the radial direction. The short grid approach is used, so there is only one grid point at the origin with a 9-point laplacian there.

Re(λ)

-6 -5 -4 -3 -2 -1 0 1

Im(λ)

-4 -3 -2 -1 0 1 2 3 4

δ = 0.2

Essential Spectra Absolute Spectra Spiral Spectra

Figure 5: Essential, absolute and spiral spectra with = 0.2.

3

(8)

Slowly varying modulations

k spatial wavenumber ω=ω*(k) temporal frequency cp=ω*/k phase velocity

cg=dω*/dk group velocity u*(φ)

cp

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λ*(iγ) = -icgγ - dγ² + Ο(|γ|³)

λ*(iγ) Im λ

Re λ Spectrum of wave trains

e^λt cos(γx) = e^−ic^g^γte^iγx = e^iγ(x^−c^g^t) = cos(γ(x − c^gt))

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e^λt cos(γx) ≈ e^−ic^g^γte^iγx = e^iγ(x^−c^g^t) = cos(γ(x − c^gt))

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Harmonic modulations

Linear temporal response to spatial modulation with wavenumber γ is

given by λ=λ*(iγ)

(9)

Slowly varying modulations

Formal derivation: [Howard & Kopell], [Kuramoto]

Validity over natural time scale 1/ε²: [Doelman, S., Scheel, Schneider]

Stability of wave trains: [S., Scheel, Schneider, Uecker], [Johnson, Zumbrun], [Iyer, S.]

Viscous Burgers equation:

for slowly varying wavenumber modulations q(X,T) on scale X=ε(x-cgt) and T=ε²t/2 with 0<ε≪1

q_T = λ′′_*(0)q_XX − ω′′_*(k)(q²)_X

Re λ Im λ

λ*(iγ) = -icgγ - dγ² + Ο(|γ|³) q(X,T)

local wavenumber cg

q_t + c_gq_x = 0

(10)

Outline

Wave trains: spectral stability and modulation equations

Planar spiral waves: spiral spectra on the plane and on bounded disks

One-dimensional defects: anticipated dynamics, and nonlinear stability

results

Re(λ)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Im(λ)

-4 -3 -2 -1 0 1 2 3

4 R 20 Psuedospectra and R 30 Spiral Spectra

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Figure 4: Pseudospectra of radius 20 spiral. Black circles are eigenvalues calculated with eig of radius 30 spiral.

2 Di↵usion Added to the v-equation

Essential and absolute spectra calculated from wave trains. The spiral spectra is calculated with direct methods (eig) on a spiral of radius 20 discretized with 64 Fourier spectra grid points in the angular direction and 100 4th order centered finite di↵erence in the radial direction. The short grid approach is used, so there is only one grid point at the origin with a 9-point laplacian there.

Re(λ)

-6 -5 -4 -3 -2 -1 0 1

Im(λ)

-4 -3 -2 -1 0 1 2 3 4

δ = 0.2

Essential Spectra Absolute Spectra Spiral Spectra

Figure 5: Essential, absolute and spiral spectra with = 0.2.

3

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Planar spiral waves

Goals:

Understand the structure of the spectrum of the linearization about a spiral wave Relate the spectra of asymptotic wave train and spiral wave

Reaction-diffusion systems:

Rotating spiral waves:

Linearization in co-rotating frame:

Archimedean spiral waves:

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u_⇤(r, ) ⇡ u^wt(r + ), r 1

L⇤ ⇡ D@^rr + f_u(u_wt(r + )) + !_⇤@

<latexit sha1_base64="jAoXx+6o0j7XT81H65nWmWvDmTw=">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</latexit>

Theorem [S., Scheel] Assume that is a generic spiral wave so that the asymptotic wave train has , then there is an such

that as .

u_*(r, ϕ)

c_g > 0 a ≠ 0

|u_*(r, ϕ) − u_wt(κr + ϕ − a log r)| → 0 r → ∞

(12)

Spectra of planar spiral waves

Linear dispersion relation of asymptotic wave train in laboratory frame: λ(iγ)

Far field eigenfunctions: v(r, ϕ) ≈ e

^iγr

e

^iℓϕ

v

_∞

(κr + ϕ)

essential

spectrum discrete

eigenvalues 2iω*

iω*

0

Re λ Im λ

λ(iγ)

(13)

Spectra of planar spiral waves

Spectrum on spaces with exponentially decaying weights e

^−r

Adjoint eigenfunctions associated with eigenvalues are exponentially localized

:= point spectrum in weighted spaces λ = 0, ± iω

_*

Σ

_ext

( ^core )

essential

spectrum discrete

eigenvalues 2iω*

iω*

0

Re λ Im λ

(14)

Spiral waves on large disks

Spiral wave on large disk:

Glue planar spiral wave and boundary sink together

Time (sink) = angle (spiral)

Boundary sink

1d space

time Neumann

boundary conditions

SOURCE OF PERIOD-DOUBLING INSTABILITIES IN SPIRAL WAVES 5

Wave trains are solutions to (3.1) of the form U (x, t) = U ₁ (x !t) where U ₁ is 2⇡-periodic

149 in its argument, so that  is the spatial wave number, and ! is the temporal frequency. In

150 the traveling coordinate ⇠ = x !t, wave trains are stationary solutions of

151 U _t =  ² DU _⇠⇠ + !U _⇠ + F (U ), ⇠ 2 R.

(3.2)

152 153

Generically, wave trains arise as one-parameter families for which ! and  are connected by

154 the nonlinear dispersion relation ! = ! _⇤ () and the profile U ₁ (⇠; ) depends smoothly on

155 . The group velocity of the wave train is defined from the nonlinear dispersion relation as

156 c _g = ^d! _d : it is equal to the speed with which perturbations are transported along the wave

157 train in the original laboratory frame.

158 To prepare for our discussion of spiral waves on bounded domains, we introduce the concept

159 of boundary sinks which connect wave trains of (3.1) at x = 1 with a Neumann boundary

160 condition at x = 0. We say that U (x, t) = U _bdy (x, !t) where U _bdy (x, ⌧ ) is 2⇡-periodic in ⌧

161 and satisfies the following one-dimensional equation on the half line in the laboratory frame

162 !U _⌧ = DU _xx + F (U ), (x, t) 2 ( 1, 0) ⇥ S ¹ , (3.3)

163 U _x (0, ⌧ ) = 0, ⌧ 2 S ¹

164 165

and converges to a wave train U ₁ (x !t) with c _g > 0 as x ! 1 such that

166 U _bdy (x, ·) U ₁ (x ·) _C

¹

_(S

¹

₎ ! 0.

167 168

An example of a boundary sink is shown in Figure 2.

169 (b)

Neumann Boundary

t = 0

t = T Space

Time

x = 0

4

0 1 2 3 Space x = 0

(a)

Figure 2. (a) Illustration of a boundary sink. The e↵ect of Neumann boundary condition is highlighted by the temporal cross section on the right. (b) Comparison between planar spiral wave (left) and spiral wave on a bounded disk with Neumann boundary conditions (right). Both spirals shown on disks for comparison purposes.

3.2. Planar spiral waves and truncation to bounded disks. We say that the reaction-

170 di↵usion system (2.1) has a planar spiral wave solution of the form U (x, t) = U _⇤ (r, !t)

171 This manuscript is for review purposes only.

Planar spiral wave

SOURCE OF PERIOD-DOUBLING INSTABILITIES IN SPIRAL WAVES 5

Wave trains are solutions to (3.1) of the form U (x, t) = U₁(x !t) where U₁ is 2⇡-periodic

149

in its argument, so that  is the spatial wave number, and ! is the temporal frequency. In

150

the traveling coordinate ⇠ = x !t, wave trains are stationary solutions of

151

U_t = ²DU_⇠⇠ + !U_⇠ + F (U ), ⇠ 2 R.

(3.2)

152153

Generically, wave trains arise as one-parameter families for which ! and  are connected by

154

the nonlinear dispersion relation ! = !_⇤() and the profile U₁(⇠; ) depends smoothly on

155

. The group velocity of the wave train is defined from the nonlinear dispersion relation as

156

c_g = ^d!_d: it is equal to the speed with which perturbations are transported along the wave

157

train in the original laboratory frame.

158

To prepare for our discussion of spiral waves on bounded domains, we introduce the concept

159

of boundary sinks which connect wave trains of (3.1) at x = 1 with a Neumann boundary

160

condition at x = 0. We say that U (x, t) = U_bdy(x, !t) where U_bdy(x, ⌧ ) is 2⇡-periodic in ⌧

161

and satisfies the following one-dimensional equation on the half line in the laboratory frame

162

!U_⌧ = DU_xx + F (U ), (x, t) 2 ( 1, 0) ⇥ S¹, (3.3)

163

U_x(0, ⌧ ) = 0, ⌧ 2 S¹

164165

and converges to a wave train U₁(x !t) with c_g > 0 as x ! 1 such that

166

U_bdy(x, ·) U₁(x ·) _C¹_(S¹₎ ! 0.

167168

An example of a boundary sink is shown in Figure 2.

169

(b)

Neumann Boundary

t = 0

t = T Space

Time

x = 0

4

0 1 2 3 Space x = 0

(a)

Figure 2. (a) Illustration of a boundary sink. The e↵ect of Neumann boundary condition is highlighted by the temporal cross section on the right. (b) Comparison between planar spiral wave (left) and spiral wave on a bounded disk with Neumann boundary conditions (right). Both spirals shown on disks for comparison purposes.

3.2. Planar spiral waves and truncation to bounded disks. We say that the reaction-

170

di↵usion system (2.1) has a planar spiral wave solution of the form U (x, t) = U_⇤(r, !t)

171

This manuscript is for review purposes only.

Neumann boundary conditions

at r=R

Wave trains are solutions to (3.1) of the form U (x, t) = U

₁

(x !t) where U

₁

is 2⇡-periodic

149

in its argument, so that  is the spatial wave number, and ! is the temporal frequency. In

150

the traveling coordinate ⇠ = x !t, wave trains are stationary solutions of

151

U

_t

= 

²

DU

_⇠⇠

+ !U

_⇠

+ F (U ), ⇠ 2 R.

(3.2)

152153

Generically, wave trains arise as one-parameter families for which ! and  are connected by

154

the nonlinear dispersion relation ! = !

_⇤

() and the profile U

₁

(⇠; ) depends smoothly on

155

. The group velocity of the wave train is defined from the nonlinear dispersion relation as

156

c

_g

=

^d!_d

: it is equal to the speed with which perturbations are transported along the wave

157

train in the original laboratory frame.

158

To prepare for our discussion of spiral waves on bounded domains, we introduce the concept

159

of boundary sinks which connect wave trains of (3.1) at x = 1 with a Neumann boundary

160

condition at x = 0. We say that U (x, t) = U

_bdy

(x, !t) where U

_bdy

(x, ⌧ ) is 2⇡-periodic in ⌧

161

and satisfies the following one-dimensional equation on the half line in the laboratory frame

162

!U

_⌧

= DU

_xx

+ F (U ), (x, t) 2 ( 1, 0) ⇥ S

¹

, (3.3)

163

U

_x

(0, ⌧ ) = 0, ⌧ 2 S

¹

164165

and converges to a wave train U

₁

(x !t) with c

_g

> 0 as x ! 1 such that

166

U

_bdy

(x, ·) U

₁

(x ·)

_C¹_(S¹₎

! 0.

167168

An example of a boundary sink is shown in Figure 2.

169

(b)

Neumann Boundary

t = 0

t = T Space

Time

x = 0

4

0 1 2 3 Space x = 0

(a)

Figure 2. (a) Illustration of a boundary sink. The e↵ect of Neumann boundary condition is highlighted by the temporal cross section on the right. (b) Comparison between planar spiral wave (left) and spiral wave on a bounded disk with Neumann boundary conditions (right). Both spirals shown on disks for comparison purposes.

3.2. Planar spiral waves and truncation to bounded disks. We say that the reaction-

170

di↵usion system (2.1) has a planar spiral wave solution of the form U (x, t) = U

_⇤

(r, !t)

171

This manuscript is for review purposes only.

(15)

Spectra of spiral waves on disks of radius R

Absolute spectrum: depends on wave train only (1/R convergence) Discrete eigenvalues: depend on spiral wave (exp(-R) convergence)

absolute

spectrum discrete

eigenvalues iω*

0

Re λ Im λ

Theorem [S., Scheel]

as . Σ(bounded disk) → Σ_abs ∪ Σ_ext(^core) ∪ Σ_ext(boundary sink) R → ∞

(16)

Case study: Period doubling of spiral waves

Goal:

What causes these instabilities?

Core, boundary, or absolute spectrum?

Line defects in Rössler model

2 S. DODSON, B. SANDSTEDE

0 1 2 3 4

(a) (b) Time

Figure 1. (a) Stationary line defect in the w-component of the R¨ossler system. (b) Time evolution of alternans instability in the u-component of the Karma model on a square of side length 16cm with homogeneous Neumann boundary conditions. System parameters as defined in Section 2.

wave.

37

Nonlinear reaction-di↵usion systems qualitatively capture transitions to complex mean-

38

dering, drifting, and the period-doubled line defects and alternans patterns. In these systems,

39

planar spiral waves are stationary solutions in a rotating polar coordinate frame and converge

40

to one-dimensional periodic travelling waves away from the core. Stability and bifurcations

41

can be studied by considering the spectra of the operator obtained by linearizing the nonlinear

42

system about the spiral wave solution. The spectrum consists of isolated eigenvalues and a

43

set determined by the operator in the far-field limit.

44

Bounded domains are of interest in applications to cardiac dynamics and laboratory ex-

45

periments. Neumann boundary conditions naturally represent lower conductance tissue sep-

46

arating regions of the heart or the physical walls of containers. Mathematically, on finite

47

domains planar spiral waves are truncated and matched with a boundary sink, which adds

48

extra structure to the spiral wave and alters the spectrum of the linear operator [13, 33, 32].

49

The boundary sink itself directly contributes an additional set of eigenvalues, and the finite

50

domain modifies the spectrum associated with the far-field dynamics. Furthermore, radial

51

growth in the eigenfunctions is permitted, and those that would not be integrable on the full

52

plane now emerge as true eigenfunctions on the bounded domain [13, 33, 32]. These eigenval-

53

ues are associated with intrinsic properties of the spiral wave and are attributed to the spiral

54

core. The spectrum of the operator on a bounded domain is therefore a union of three disjoint

55

sets that are associated with instabilities from the far-field, boundary conditions, and core.

56

Knowing which set unstable eigenvalues belong to provides information about how instabilities

57

will manifest themselves on unbounded or bounded domains.

58

Meander and drift instabilities are the result of a Hopf bifurcation originating from the

59

core: the emerging dynamics is understood through actions of the symmetry group of trans-

60

lations and rotations on the plane and a center manifold reduction [5, 39]. However, previous

61

studies investigating alternans and line defects provide inconsistent and incomplete evidence

62

for which spectral set the unstable eigenvalues belong to.

63

Due to the clinical significance, the alternans instability has been a recent area of focus in

64

the cardiac dynamics community. In single cells, alternans are widely attributed to a period-

65

doubling instability observed in simple 1D maps [18]. However, this condition, known as the

66

restitution hypothesis, has received contradictory evidence [10, 7] and does not appear to

67

be relevant for excitable tissues that support traveling waves. The formation and stability

68

This manuscript is for review purposes only.

Alternans in Karma model

2 S. DODSON, B. SANDSTEDE

0 1 2 3 4

(a) (b) Time

Figure 1. (a) Stationary line defect in the w-component of the R¨ ossler system. (b) Time evolution of alternans instability in the u-component of the Karma model on a square of side length 16cm with homogeneous Neumann boundary conditions. System parameters as defined in Section 2.

wave.

37 Nonlinear reaction-di↵usion systems qualitatively capture transitions to complex mean-

38 dering, drifting, and the period-doubled line defects and alternans patterns. In these systems,

39 planar spiral waves are stationary solutions in a rotating polar coordinate frame and converge

40 to one-dimensional periodic travelling waves away from the core. Stability and bifurcations

41 can be studied by considering the spectra of the operator obtained by linearizing the nonlinear

42 system about the spiral wave solution. The spectrum consists of isolated eigenvalues and a

43 set determined by the operator in the far-field limit.

44 Bounded domains are of interest in applications to cardiac dynamics and laboratory ex-

45 periments. Neumann boundary conditions naturally represent lower conductance tissue sep-

46 arating regions of the heart or the physical walls of containers. Mathematically, on finite

47 domains planar spiral waves are truncated and matched with a boundary sink, which adds

48 extra structure to the spiral wave and alters the spectrum of the linear operator [13, 33, 32].

49 The boundary sink itself directly contributes an additional set of eigenvalues, and the finite

50 domain modifies the spectrum associated with the far-field dynamics. Furthermore, radial

51 growth in the eigenfunctions is permitted, and those that would not be integrable on the full

52 plane now emerge as true eigenfunctions on the bounded domain [13, 33, 32]. These eigenval-

53 ues are associated with intrinsic properties of the spiral wave and are attributed to the spiral

54 core. The spectrum of the operator on a bounded domain is therefore a union of three disjoint

55 sets that are associated with instabilities from the far-field, boundary conditions, and core.

56 Knowing which set unstable eigenvalues belong to provides information about how instabilities

57 will manifest themselves on unbounded or bounded domains.

58 Meander and drift instabilities are the result of a Hopf bifurcation originating from the

59 core: the emerging dynamics is understood through actions of the symmetry group of trans-

60 lations and rotations on the plane and a center manifold reduction [5, 39]. However, previous

61 studies investigating alternans and line defects provide inconsistent and incomplete evidence

62 for which spectral set the unstable eigenvalues belong to.

63 Due to the clinical significance, the alternans instability has been a recent area of focus in

64 the cardiac dynamics community. In single cells, alternans are widely attributed to a period-

65 doubling instability observed in simple 1D maps [18]. However, this condition, known as the

66 restitution hypothesis, has received contradictory evidence [10, 7] and does not appear to

67 be relevant for excitable tissues that support traveling waves. The formation and stability

68 This manuscript is for review purposes only.

(17)

Case study: Period doubling of spiral waves

Line defects in Rössler model

λ~iω*/2

for W (x, ⌧ ) where the temporal direction is scaled to be 2⇡-periodic in ⌧ .

478

Solutions in the far-field are obtained by translating asymptotic wave trains in time and

479

space using the angular frequency and spatial wave number from the spiral and imposing

480

the cuto↵ function (x). Applying (1 (x)) to the translation yields an initial condition

481

for W (x, ⌧ ). Domain sizes were selected to fit 6 periods of the wave train, which accurately

482

captured both the Neumann boundary conditions and convergence to the far-field dynamics.

483

Asymptotic wave trains were computed from the one-dimensional problem (3.2) using Fourier

484

spectral methods on a periodic grid of N_t points. The translation of wave train to boundary

485

sink resulted in N_s = 6N_s. To take spatial derivatives, the pattern was initially posed on a

486

larger spatial grid of 8 periods with Neumann boundary conditions on each end. When solving

487

for the final pattern, the left two periods were removed to eliminate left-hand side boundary

488

e↵ects and simulate a half-infinite line.

489

5. Results. Spirals on each domain are numerically calculated for the Karma and R¨ossler

490

models, and the influence of the spiral regions is determined by comparing the spectra of the

491

three operators L⇤,R, L^R,nr, and L^bdy.

492

-0.05 0 0.05

Re( )

0 0.5 1 1.5

Im()

-0.4 -0.3 -0.2 -0.1 0 0.1

Re( )

0 0.5 1 1.5

Im()

(a)

(c)

(b)

(d)

Absolute Point Essential

3!/2

!

!/2

3!/2

!

!/2

Figure 4. R¨ossler Model: (a) Spectra for L⇤,R representing a stable spiral on a disk with parameter µR = 2 and radius R = 125. Labels on right side of imaginary axis indicate half-multiples of angular frequency. (b) Spectra of unstable spiral, µR = 3.4, (c) Spiral on bounded disk of radius R = 125 exhibiting a single stationary line defect. Parameter µR = 3.4. (d) Unstable point eigenfunction responsible for line defects with µR = 3.4.

Corresponds to eigenvalue = 0.043 + 0.54i = 0.043 + !/2i.

5.1. R¨ossler Model: Line defects are driven by the boundary. At the onset of period

493

doubling, point eigenvalues with imaginary parts approximately equal to ^!₂ + `!, ` 2 Z desta-

494

bilize, followed by branches of essential and then absolute spectra upon increasing µ_R further.

495

The unstable eigenfunctions are localized at the boundary (Figure 4), indicating that line

496

defects are a result of instabilities of the boundary conditions. The spectra of L_⇤,R, L^R,nr,

497

for W (x, ⌧ ) where the temporal direction is scaled to be 2⇡-periodic in ⌧ .

478

Solutions in the far-field are obtained by translating asymptotic wave trains in time and

479

space using the angular frequency and spatial wave number from the spiral and imposing

480

the cuto↵ function (x). Applying (1 (x)) to the translation yields an initial condition

481

for W (x, ⌧ ). Domain sizes were selected to fit 6 periods of the wave train, which accurately

482

captured both the Neumann boundary conditions and convergence to the far-field dynamics.

483

Asymptotic wave trains were computed from the one-dimensional problem (3.2) using Fourier

484

spectral methods on a periodic grid of N_t points. The translation of wave train to boundary

485

sink resulted in N_s = 6N_s. To take spatial derivatives, the pattern was initially posed on a

486

larger spatial grid of 8 periods with Neumann boundary conditions on each end. When solving

487

for the final pattern, the left two periods were removed to eliminate left-hand side boundary

488

e↵ects and simulate a half-infinite line.

489

5. Results. Spirals on each domain are numerically calculated for the Karma and R¨ossler

490

models, and the influence of the spiral regions is determined by comparing the spectra of the

491

three operators L_⇤,R, L^R,nr, and L^bdy.

492

-0.05 0 0.05

Re( ) 0

0.5 1 1.5

Im()

-0.4 -0.3 -0.2 -0.1 0 0.1

Re( ) 0

0.5 1 1.5

Im()

(a)

(c)

(b)

(d)

Absolute Point Essential

3!/2

!

!/2

3!/2

!

!/2

Figure 4. R¨ossler Model: (a) Spectra for L⇤,R representing a stable spiral on a disk with parameter µ_R = 2 and radius R = 125. Labels on right side of imaginary axis indicate half-multiples of angular frequency. (b) Spectra of unstable spiral, µ_R = 3.4, (c) Spiral on bounded disk of radius R = 125 exhibiting a single stationary line defect. Parameter µR = 3.4. (d) Unstable point eigenfunction responsible for line defects with µR = 3.4.

Corresponds to eigenvalue = 0.043 + 0.54i = 0.043 + !/2i.

5.1. R¨ossler Model: Line defects are driven by the boundary. At the onset of period

493

doubling, point eigenvalues with imaginary parts approximately equal to ^!₂ + `!, ` 2 Z desta-

494

bilize, followed by branches of essential and then absolute spectra upon increasing µ_R further.

495

The unstable eigenfunctions are localized at the boundary (Figure 4), indicating that line

496

defects are a result of instabilities of the boundary conditions. The spectra of L_⇤,R, L^R,nr,

497

λ~3iω*/2

Alternans in Karma model

λ~3iω*/2

(a)

(d)

3!/2

!

!/2

(c)

0 1 2 3 4

Time

3!/2

!

!/2

Absolute Point (b)

Essential

Figure 7. Karma Model: (a) Spectra for stable spiral, µK = 0.6. Labels on right side of imaginary axis indicate half-multiples of angular frequency. (b) Spectra of unstable spiral, µK = 1.4, (c) Development of alternans in time evolution of spiral on bounded square with homogeneous-Neumann boundary conditions.

Parameter µK = 1.4. Square of side length 16cm. (d) Unstable point eigenfunction responsible for alternans.

Corresponds to eigenvalue = 2.6 + 75.9i ⇡ 2.6 + 3!/2i, µ^K = 1.4. Domain radius R = 5 with homogeneous- Neumann boundary conditions.

small when the eigenvalue is near the essential spectrum, with Re ⌫ positive (negative) for

542

the eigenvalue to the left (right) of the continuous spectrum curve. Small Re ⌫ indicates

543

little radial growth, and Im ⌫ = is set by the essential spectrum point. The wave train

544

eigenfunction is determine by the eigenfunction on the essential spectrum, V_ess. Using these

545

observations, the spiral eigenfunction to leading order in the radius is given by

546

V (r, ) = e^{i r}e^`V_ess(r + ).

(5.1)

547548

A numerically constructed eigenfunction is shown at the top of Figure 9a. Note that the

549

derivative of the wave train @_⇠U₁(⇠) is the eigenfunction on the essential spectrum with

550

eigenvalue = i`!, ` 2 Z. The alternans eigenfunction crosses ⌃^ess near one of these points,

551

and therefore V_ess(⇠) is close to U₁⁰ (⇠) (Figure 9b). The derivative of the wave train is the

552

highest at the wave fronts and backs and it is this shape that leads to the changing width of the

553

spiral bands and form of alternans. Moreover, the structure of the constructed eigenfunction

554

is in good agreement with the alternans eigenfunction from L_⇤,R which is reproduced on the

555

bottom of Figure 9a.

556

When the alternans eigenvalue is to the left of the essential spectrum, approximately

557

µ_K < 1.4, the overall shape of the eigenfunction is comparable to those shown in Figure 9a,

558

but there is slight radial growth toward the boundary, corresponding to a spatial eigenvalue,

559

⌫, with a small positive real part. Radial growth of the alternans eigenfunction for several

560

values of bifurcation parameter µ_K is visible in Figure 9c. As the eigenvalue approaches the

561

(a)

(d)

3!/2

!

!/2

(c)

0 1 2 3 4

Time

3!/2

!

!/2

Absolute Point (b)

Essential

Figure 7. Karma Model: (a) Spectra for stable spiral, µK = 0.6. Labels on right side of imaginary axis indicate half-multiples of angular frequency. (b) Spectra of unstable spiral, µK = 1.4, (c) Development of alternans in time evolution of spiral on bounded square with homogeneous-Neumann boundary conditions.

Parameter µK = 1.4. Square of side length 16cm. (d) Unstable point eigenfunction responsible for alternans.

Corresponds to eigenvalue = 2.6 + 75.9i ⇡ 2.6 + 3!/2i, µ^K = 1.4. Domain radius R = 5 with homogeneous- Neumann boundary conditions.

small when the eigenvalue is near the essential spectrum, with Re ⌫ positive (negative) for

542

the eigenvalue to the left (right) of the continuous spectrum curve. Small Re ⌫ indicates

543

little radial growth, and Im ⌫ = is set by the essential spectrum point. The wave train

544

eigenfunction is determine by the eigenfunction on the essential spectrum, V_ess. Using these

545

observations, the spiral eigenfunction to leading order in the radius is given by

546

V (r, ) = e^{i r}e^`V_ess(r + ).

(5.1)

547548

A numerically constructed eigenfunction is shown at the top of Figure 9a. Note that the

549

derivative of the wave train @_⇠U₁(⇠) is the eigenfunction on the essential spectrum with

550

eigenvalue = i`!, ` 2 Z. The alternans eigenfunction crosses ⌃^ess near one of these points,

551

and therefore V_ess(⇠) is close to U₁⁰ (⇠) (Figure 9b). The derivative of the wave train is the

552

highest at the wave fronts and backs and it is this shape that leads to the changing width of the

553

spiral bands and form of alternans. Moreover, the structure of the constructed eigenfunction

554

is in good agreement with the alternans eigenfunction from L⇤,R which is reproduced on the

555

bottom of Figure 9a.

556

When the alternans eigenvalue is to the left of the essential spectrum, approximately

557

µ_K < 1.4, the overall shape of the eigenfunction is comparable to those shown in Figure 9a,

558

but there is slight radial growth toward the boundary, corresponding to a spatial eigenvalue,

559

⌫, with a small positive real part. Radial growth of the alternans eigenfunction for several

560

values of bifurcation parameter µ_K is visible in Figure 9c. As the eigenvalue approaches the

561

λ~iω*/2

Spiral waves: linear and nonlinear theory