T.M.J. van Zalen

Waves in excitable media

Master thesis, defended on June 13, 2013

Thesis advisor: dr. S.C. Hille

Specialisation: Applied Mathematics

Mathematisch Instituut, Universiteit Leiden

Abstract

In this thesis we will consider waves in excitable media. These waves

can be found in dierent biological settings, from signalling in the amoeba

Dictyostelium discoideum to cardiac brillation and spiral galaxies. De-

pending on the number of spatial dimensions, the waves can be of dierent

types.

In one spatial dimension travelling waves and travelling pulses occur.

The existence and uniqueness of such waves will be proven. It follows

that there there is a unique velocity, depending on the level of a so-called

controller species, for which there exists a corresponding travelling wave.

In two spatial dimensions the spiral waves will be considered. It follows

that the normal velocity of a spiral wave front is not only determined by

the level of controller species in the front, but also by the curvature of the

front.

Contents

1 Introduction on waves in excitable media 4

2 Dictyostelium discoideum: cAMP signalling 5

2.1 The biochemistry of the Martiel-Goldbeter model . . . . . . . . . 52.2 The mathematical models . . . . . . . . . . . . . . . . . . . . . . 6

3 A reduced two-state model 12

4 Waves in one dimension 15

4.1 Travelling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.1 Transition from h− to h+ . . . . . . . . . . . . . . . . . . 154.1.2 Transition from h+ to h− . . . . . . . . . . . . . . . . . . 27

4.2 Example: the cubic case . . . . . . . . . . . . . . . . . . . . . . . 284.3 Travelling pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Spiral waves in two dimensions 31

5.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Spiral waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1 Multi-armed spiral waves . . . . . . . . . . . . . . . . . . 42

6 Discussion 44

A Stability using the Center Manifold Theorem 45

References 48

1 Introduction on waves in excitable media

Excitable media have special characteristics: there is a stable rest state andsmall perturbations from the rest state are rapidly damped out. If the per-turbation crosses a certain threshold, there will be an abrupt and substantialresponse. The medium will be refractory to further stimulation for a while untilit recovers full excitability.

Waves in excitable media are found in for example the Belousov-Zhabotinskiireaction [4] and the cAMP signalling in the cellular slime mold Dictyosteliumdiscoideum, which will be described in Chapter 2. Waves also occur on largerscale, such as the waves of infectious diseases that travel through populations,and a spiral galaxy, whose rotating arms can be treated as travelling waves ofstar formation in an excitable medium of interstellar gas and dust [9].In biology it is not always a good thing for a system that relies on a faithfulpropagation of a signal to be taken over by a self-sustained pattern: spirals onthe heart lead to cardiac brillation [8], spirals in the cortex may lead to epilepticseizures and spirals on the retina or visual cortex may cause hallucinations.

In excitable media there are dierent types of waves depending on the num-ber of spatial dimensions. In one spatial dimension, we can distinguish betweenwave fronts and wave backs and between travelling pulses, which contains onewave front and wave back, and wave trains, which are sequences of wave frontsand wave backs and can be periodic. In Chapter 4 the travelling waves andtravelling pulses in one dimension will be analysed in detail.

In two dimensions there are two characteristic patterns: the target patterns,which are created if the medium is suciently large so that more than one wavefront can exist at the same time, and the rotating spiral waves. Target patternsare expanding circular waves and require a periodic source and cannot existin a homogeneous non-oscillatory medium, while spiral waves do not require aperiodic source for their existence, since they are typically self-sustained.In Chapter 5 the spiral waves will be analysed in more detail.Experimental studies showed that spiral waves are generated by spiral breakup,which was found to occur in models where the recovery variable diuses at a highrate. In [5] is was shown that the spiral breakup also can be caused by lateralinstability of the wave front. A spiral wave can break up into a large numberof small spirals. As showed in [14], in Dictyostelium these small spirals canspontaneously form multi-armed spirals in low excitable media. These multi-armed spirals are formed due to attraction of single spirals that rotate in thesame direction.

In three dimensions the target patterns and rotating spiral waves generalizeto expanding spherical waves and rotating scroll waves respectively. In thisthesis the three dimensional cases will not be analysed. The interested readermay consult [12] and the references found there.

In this master thesis the focus is on understanding all details involved in theconstruction of travelling waves and pulses in one spatial dimension and spiralwaves in two dimensions, as presented in a series of papers, [2], [3], [4] and [12].We will not discuss the complete dynamics of the system of partial dierentialequations (PDEs) in innite dimensional state space, such as stability of thewaves, pulses and spirals. The construction of the latter 'reduces' the argumentsto a setting of ordinary dierential equations (ODEs).

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2 Dictyostelium discoideum: cAMP signalling

2.1 The biochemistry of the Martiel-Goldbeter model

In this chapter we will analyse the cyclic Adenosine 3'-5'-Monophosphate (cAMP)signalling system inDictyostelium discoideum. The biological background of thiscAMP signalling is the following, [13].

If the Dictyostelium discoideum amoebae are left to starve on an agar sur-face, they start to signal to each other by secreting cAMP. The individual cellsreceive this cAMP by binding extracellular cAMP to a membrane receptor. Thisstimulates the synthesis of cAMP from ATP by adenylate cyclase within the cell.The newly synthesized cAMP is transported to the extracellular medium so thatthe chemical signal is amplied. However, the amplication is limited since inprolonged exposure to cAMP the membrane receptors become desensitized sothat it no longer stimulates adenylate cyclase activity. In the absence of cAMPsynthesis, the concentration of cAMP decreases by the action of extracellularphosphodiesterase, which hydrolyses cAMP.

Pulses of extracellular cAMP, secreted by the signalling amoebae, travelacross the eld in the form of either expanding concentric circular waves orrotating spiral waves. As the waves pass periodically through the eld of inde-pendent amoebal cells, they stimulate a chemotactic movement of the amoebaetowards the center of the pattern, which is the origin of the circles or the pivotpoint of the spiral. Eventually all the amoebae within the domain of a singlepattern aggregate at the center to form a multicellular slug, which goes on toform a fruiting body.

The model for cAMP signalling (see Figure 2.1) in Dictyostelium correspondsto the sequence of reaction steps (a)-(e) found in (2.1), [6]. Although this isa hypothetical mechanism, there seems no other mechanism proposed for thecAMP signalling.

Figure 2.1: (a) Model for the cAMP signalling system of Dictyostelium dis-coideum. (b) The receptor box with the interconversions between the R and Dstate in the presence of P .

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(a) R k1k−1

D

(b) R + P a1d1

RP

(c) D + P a2d2

DP

(d) RP k2k−2

DP

(e) 2RP + C a3d3

E

(f) E + S d4a4 ES→

k4 E + Pi

(g) C + S d5a5 CS→

k5 C + Pi

(h) Pi →ki

(i) Pi →kt P→ke

(j) →vi S→k′ (2.1)

In reaction steps (2.1), R and D denote the active and desensitized form ofthe receptor respectively. Reactions (a)−(d) show the interconversions betweenthe R and D states in the presence of extracellular cAMP (P ). In reaction(e), two molecules of active receptor-cAMP complex (RP ) bind to less-activeadenylate cyclase (C) to form an enzyme complex of higher order activity (E).Reactions (f) and (g) describe the synthesis of intracellular cAMP (Pi) fromATP (S) by both forms of adenylate cyclase. In reaction (h) part of the Pi ishydrolysed by an intracellular phosphodiesterase, and in reaction (i) part of Piis transported across the plasma membrane into the extracellular medium whereit is hydrolysed by the membranebound and extracellular forms of this enzyme.Finally, it is assumed that the substrate ATP is synthesized at a constant rateand used in a number of reactions besides that catalysed by adenylate cyclase.

2.2 The mathematical models

Considering the time evolution of the areal densities of the species, we are ableto nd a system of dierential equations corresponding to the reactions in (2.1).First of all, we need to introduce some notations:

Denition 1 (Areal density). The areal density of a membrane bound molec-ular species is given by [·]A.

If Am is the area of the membrane, Ve the extracellular volume and Vi theintracellular volume, we can also dene, following [6]

Denition 2 (Concentration). The concentration of a membrane bound intra-cellular species is given by

[·i] :=[·i]AAmVi

, (2.2)

and the concentration of a membrane bound extracellular species is given by

[·e] :=[·e]AAmVe

. (2.3)

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Here, the membrane bound intracellular species are C, E, CS, ES, S andPi and the extracellular species are R, P , D, RP and DP (see Figure 2.1). Forsimplicity we will omit the subscripts i and e in the rest of the chapter.Further, we will write the reaction rates for the dierential equations of theareal densities as k and d for the simple rates, and a and v for the other rates.

If we look at the time evolution of the areal density of the species R, we ndthe following dierential equation:

d[R]Adt

= −k1[D]A + k−1[R]A − a1[R]A[P ]A + d1[RP ]A. (2.4)

We will make use of two conservation relations for the receptor and for adenylatecyclase:

RT = [R]A + [D]A + [RP ]A + [DP ]A + (2/h)([E]A + [ES]A) (2.5)

ET = [E]A + [ES]A + [C]A + [CS]A, (2.6)

with h a dilution factor.If we now substitute ρ = [R]/RT , δ = [D]/RT , γ = [P ]/RT and x = [RP ]/RTin (2.4), we nd the following dierential equation:

dρ

dt= −k1δ + L1k1ρ− d1

a1

a1

VeAm

ργ + d1x, (2.7)

and if we take

a1 = a1AmVe

,

we nd the equation

dρ

dt= k1(−δ + L1ρ) + d1(−ργ + x). (2.8)

The denition and the values used for the calculations of the parameters canbe found in Table 2.1, used in [11] for numerical simulations. Note that theparameters are dimensionless, except the time-scale, which is in minutes, andthe space scale, which is in mm.

The reason that [6] uses concentrations rather than surface densities for themembrane bound molecular species, may be that the authors seem to link exper-imental data concerning total concentration of these compounds in a reactionvessel.In this way we can derive the dierential equations for the time evolution of theconcentration of the other species, which can be found in Appendix A in [6].However, we had to make some adjustments and supplements to this article inorder to derive the dierential equations:

c = [C]/ET , cs = [CS]/ET , y = [DP ]/RT

β = [Pi]/KR, γ = [P ]/KR,K′m = (d5 + k5)/a5.

If we use the method described above to derive the system of equations, we alsond the reaction rates

a2 = a2AmVe

, a3 = a3A2m

V 2e

, a4 = a4AmVi

, a5 = a5AmVi

, vi = viViAm

.

7

The system of dierential equations given in Appendix A of [6] can be re-duced to a four-variable system, under the assumptions that the dierentialequations contain both fast binding and slow modication terms, so the follow-ing inequality on the rate constants holds

(k1, k−1, k2, k−2, ki, kt, ke, σ, k′) (a1, d1, a2, d2, a3, d3, a4, d4, a5, d5),

and that the dierential equations for the fastest variables (the total fractionof the receptor forms bound to cAMP (x+ y), c, e and es) reduce to algebraicequations corresponding to the quasi-steady-state hypothesis for these receptorand enzyme forms. The evolution equations for the remaining slow variablescan now be transformed, yielding the four-variable system of dierential equa-tions (2.9).

dρTdt

= −f1(γ)ρT + f2(γ)(1− ρT )

dα

dt= v − k′α− σΦ(ρT , γ, α)

dβ

dt= qσΦ(ρT , γ, α)− (ki + kt)β

dγ

dt= (ktβ/h)− keγ (2.9)

with

f1(γ) =k1 + k2γ

1 + γ, f2(γ) =

k1L1 + k2L2cγ

1 + cγ

Φ(ρT , γ, α) =α(λθ + εY 2)

1 + αθ + εY 2(1 + α), Y =

ρT γ

1 + γ.

Here, ρT denotes the total fraction of receptor in the active state, β and γdenote the intracellular and extracellular concentrations of cAMP divided bythe dissociation constant KR, and α is the intracellular ATP concentrationdivided by the Michaelis constant Km of reaction (f), so that Km = d4+k4

a4.

Experiments have indicated that the level of ATP can be kept constant,since the absence of signicant variation in ATP can be accounted for by themodel. Hence we can consider α as a parameter, and we are able to furtherreduce the number of variables. The dynamics of the cAMP signalling systemis then governed by the three dierential equations

dρTdt

= −f1(γ)ρT + f2(γ)(1− ρT )

dβ

dt= qσΦ(ρT , γ, α)− (ki + kt)β

dγ

dt= (ktβ/h)− keγ (2.10)

We can make time dimensionless by taking t = k1 · t, since we considerthe time scale in which the receptor will desensitize. Substituting this into

8

equations (2.10) results in the dierential equations

dρTdt

= −f1(γ)ρT + f2(γ)(1− ρT ) (2.11)

ε′dβ

dt= s1Φ(ρT , γ)− β (2.12)

ε′′dγ

dt= s2β − γ (2.13)

where

f1(γ) =1 + κγ

1 + γ, f2 =

L1 + κL2cγ

1 + cγ

Φ(ρT , γ) =λ1 + Y 2

λ2 + Y 2.

Name Denition Set A Set B Set C Set D Set EL1 k−1/k1 10 = = = =L2 k−2/k2 0.005 0.005 0.005 0.005 0.0005κ k2/k1 18.5 = = = =c KR/KD 10 10 10 10 45α [ATP]/Km 3 = = = =λ1

λθε 10−4 10−3 10−3 10−3 6.7 · 10−4

λ21+αθ

(1+α)ε 0.26 2.4 2.4 2.4 1.0

s1qσ

ki+ktα

1+α 690 950 950 360 80

s2 kt/keh 0.033 0.05 0.05 0.13 0.35s s1s2 23 47 47 47 28ε′ k1/(ki + kt) 0.014 0.019 0.019 0.005 0.01ε′′ k1/ke 0.0067 0.01 0.01 0.01 0.024Time-scale 1/k1 28 28 8.3 28 17

Space-scale (keD)12 /k1 10 8.2 4.5 8.2 4.1

Table 2.1: Model parameters as occuring in [6] (A-C) and [11] (D-E).

When all four parameter sets assume the same parameter value, the symbol′ =′ is used. Note that set A is used in [6] to model cAMP oscillations in awell-stirred cell suspension. Set B is used in [6] to model cAMP signal-relayingin a well-stirred cell suspension. Set C is used in the paper by Tyson et al. tocalculate spiral waves in the full three-component model. Sets D and E are usedin [11] to calculate spiral waves in the two-component model.

Equations (2.11)-(2.13) describe a homogeneous distribution of the species.However, if we allow the species to diuse freely, we must include diusion. Theequations describing the membrane-bound receptor and the intracellular cAMPare unchanged since these species are immobile, since located within cells. Theextracellular cAMP can freely diuse, so it must be described by a dimensionlessequation in the form of a reaction-diusion equation (for simplicity we will writet for the scaled time parameter)

∂γ

∂t= ε′′∆γ +

1

ε′′F (β, γ),

9

with F (β, γ) the right-hand-side of equation (2.13), and ∆ the two-dimensionalLaplacian. The spatial variables are scaled so that the diusion coecient isequal to ε′′, that is

(x, y) =kt√keD

· (x, y). (2.14)

Here x, y are the dimensional spatial variables and D is the diusion coecientof cAMP. The length scale used in (2.14) is the length scale of exponential decayof the steady-state solution of diusive species on (0,∞), with diusivityD, thatdegrade with rate ke and enter at 0 with rate kt. According to [11], we chooseD = 0.024 mm2/min. The biological reason to take this space scale is that weconsider the combination of diusion D and degradation ke of the extracellularcAMP.

Analysis of this Martiel-Goldbeter model would be simplied if we couldeliminate the equation for β. Such simplication would be possible if ε′ ε′′,so that a quasi steady-state of β can occur. This quasi-steady state will beβ = s1Φ(ρT , γ). In that case we can do a phase plane analysis of the relayand oscillations. For the parameter sets in [11] parameter sets D and E can beused, so that it is justied to take ε′ → 0 with ε′′ nite. In that case the modelequations become

∂γ

∂t= ε′′∆γ +

1

ε′′[sΦ(ρT , γ)− γ]

∂ρT∂t

= −f1(γ)ρT + f2(γ)(1− ρT ), (2.15)

where s = s1s2.Considering parameter sets D and E, we can compute the nullclines of the

ODE part of system (2.15). The γ-nullcline is

0 = sΦ(ρT , γ)− γ

0 = sλ1 +

(ρT γ1+γ

)2

λ2 +(ρT γ1+γ

)2 − γ

s

(λ1 +

(ρT γ

1 + γ

)2)

= γ

(λ2 +

(ρT γ

1 + γ

)2)

ρT (γ) = ±1 + γ

γ

√λ2γ − sλ1

s− γ. (2.16)

Since we only consider positive values of ρT , the only nullcline we will consideris the positive solution branch.From equation (2.16) we can conclude that the γ-nullcline does not have pointswith γ < sλ1

λ2or γ > s.

The ρT -nullcline is

0 = −f1(γ)ρT + f2(γ)(1− ρT )

ρT (γ) =f2(γ)

f1(γ) + f2(γ)

=(L1 + κL2cγ)(1 + γ)

(1 + κγ)(1 + cγ) + (L1 + κL2cγ)(1 + γ). (2.17)

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We can also compute the maxima of the γ-nullcline (2.16). First we haveto determine ∂ρT

∂γ

∂ρT∂γ

=

[1 + γ

γ

√λ2γ − sλ1

s− γ

]′(2.18)

=(γ + γ2)(λ2 − λ1)s− 2(λ2γ − sλ1)(s− γ)

2γ2(s− γ)2√

λ2γ−sλ1

s−γ

(2.19)

=γ2(2λ2 + s(λ2 − λ1))− γs(λ2 + 3λ1) + 2s2λ1

2γ2(s− γ)2√

λ2γ−sλ1

s−γ

(2.20)

Solving ∂ρT∂γ = 0 for γ results in

0 = γ2(2λ2 + s(λ2 − λ1)− γs(λ2 + 3λ1) + 2s2λ1 (2.21)

γ± =s[3λ1 + λ2 ±

√(λ2 − λ1)(λ2 − (8s+ 9)λ1]

2s(λ2 − λ1) + 4λ2(2.22)

Equation (2.22) is real if λ2 < λ1 or λ2 > (8s + 9)λ1. According to the modelparameters for λ1 and λ2, see Table 2.1, λ1 λ2, hence the only possibility forthe maxima of the nullcline to exists, is that λ2 > (8s+ 9)λ1. In that case, thesign of the denominator is s(λ2 − λ1) + 2λ2 > 0. Hence, we can conclude thatfor γ < γ−, ρT is increasing, for γ− < γ < γ+, ρT is decreasing and for γ > γ+,ρT is increasing again, so that this nullcline is N-shaped, which can be seen ingure 2.2.

The nullclines of system (2.15), with parameter set D, are illustrated inFigure 3.1. Note that in this gure γ is plotted against 1−ρT , since the recoveryvariable in this case is the re-sensitization of the receptor ρT .

Figure 2.2: Nullclines of system (2.15) for parameter set D.

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3 A reduced two-state model

The pair of dierential equations given in system (2.15), is of the form

ε∂u

∂t= ε2∆u+ f(u, v),

∂v

∂t= g(u, v), (3.1)

with

u = γ, v = 1−ρT , f(u, v) = sΦ(1−ρT , γ)−γ, g(u, v) = −f1(γ)(1−ρT )+f2(γ)ρT .

In the rest of this thesis we will consider a pair of reaction-diusion equationsof the form

ε∂u

∂t= ε2∆u+ f(u, v),

∂v

∂t= εδ∆v + g(u, v),

(3.2)

with u = u(x1, . . . , xn, t) and v = v(x1, . . . , xn, t), 0 < ε 1 the ratio ofthe rates of reaction of the quantities u and v, ∆ =

∑ni=1

∂∂2xi

the Laplacian

operator with n the dimensions, and δ = DvDu

the ratio of diusion coecients ofthe two components u and v. u can be seen as the trigger or excitation variableand v as the recovery variable.

We will consider in particular two cases of system (3.2): δ = 0 and δ = 1.If δ = 0 the recovery variable v does not spread in space. This is characteristic inactivity waves in neuromuscular tissue where v represents the local permeabilityof a membrane to transmembrane ionic currents. The two-dimensional casecorresponding to δ = 0 has also been considered in [3]. Moreover, Dictyosteliumcorresponds to this case.If δ = 1 both components u and v diuse equally well through space. This ischaracteristic for the Belousov-Zhabotinskii reaction. The two-dimensional casecorresponding to δ = 1 has also been considered in [4].

The functions f(u, v) and g(u, v) describe the non-linear kinetics of sys-tem (3.2). We make the following assumptions on the function f(u, v), following[2]:

(AF1) f is C1 and the nullcline f(u, v) = 0 is cubic-shaped, hence there arethree solution branches to f(u(v), v) = 0. Denote them by u = h±(v) andu = h0(v), with h−(v) ≤ h0(v) ≤ h+(v). Here h− is dened on (v−,∞),h0 on [v−, v+] and h+ on (−∞, v+). We require that v− < v+.

(AF2) ∂f∂u (h±(v), v) < 0, so these solution branches are stable, and ∂f

∂u (h0(v), v) ≥0. Therefore this solution branch is unstable.

(AF3) Above the line f(u, v) = 0, f(u, v) < 0 and below this line f(u, v) > 0.

(AF4) ∂f∂v (u, v) ≤ 0 on u ∈ [h−(v), h+(v)] for each v ∈ [v−, v+].

Example 1. 1. Let f(u, v) = P3(u)−v, with P3(u) a third order polynomialof u. Then ∂f

∂v = −1 < 0 for all v.

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2. For Dictyostelium discoideum we found the following function f(u, v):

f(u, v) = sλ1(1 + u)2 + (1− v)2u2

λ2(1 + u)2 + (1− v)2u2− u.

If we calculate ∂f∂v we obtain:

∂f

∂v(u, v) = s

−2(1− v)u2(1 + u)2(λ2 − λ1)

(λ2(1 + u)2 + (1− v)2u2)2.

Since λ2 > λ1, u > 0 and 1− v > 0, it yields that

∂f

∂v(u, v) < 0.

Because of the Implicit Function Theorem, h± are C1 functions. Dene

If (v) :=

∫ h+(v)

h−(v)

f(z, v) dz. (3.3)

Proposition 1. Assume (AF1)(AF3). If (v) has the following properties:

(i) If ∈ C1([v−, v+]).

(ii) There exists v∗± : v− < v∗− ≤ v∗+ < v+ such that If (v0) > 0 for v− ≤ v0 <v∗− and If (v0) < 0 for v∗+ < v0 ≤ v+.

(iii) If in addition (AF4) holds, then I ′f (v) < 0 on (v−, v+). Consequently thereexists a unique v∗ ∈ (v−, v+) such that If (v∗) = 0 and v∗− = v∗+ = v∗.

Proof. (i) Because f is C1, h± are C1 functions. The Fundamental Theoremof Calculus then yields that If is C1.

(ii) This follows directly from (AF3).

(iii) Because of the denition of h±, one has I′f (v) =

∫ h+(v)

h−(v)∂f∂v (z, v) dz. Since

(AF4) holds, ∂f∂v (u, v) < 0 and thus I ′f (v) < 0. From (i) and (iii) followsdirectly that there should be a v∗ ∈ (v−, v+) such that If (v∗) = 0.

Following [2], the assumptions made on g(u, v) are:

(AG1) The nullcline g(u, v) = 0 is monotone increasing.

(AG2) g(u, v) = 0 has precisely one intersection with the curve f(u, v) = 0. Thisintersection is the unique steady state solution and is located at the branchu = h−(v). It is denoted by (us, vs), such that v− < vs < v∗.

(AG3) g(u, v) < 0 above (to the left) the line g(u, v) = 0 and g(u, v) > 0 below(to the right) this line.

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The characteristic setting of these nullclines is illustrated in Figure 3.1.

f(u, v) = 0

g(u, v) = 0

u

v

f < 0g < 0

f > 0g > 0

h−(v)

h0(v)

h+(v)

us

vs

v+

v−

Figure 3.1: Typical nullclines f(u, v) = 0 and g(u, v) = 0 for the excitablesystem (3.2).

The structures for f and g as described above are typical for excitable sys-tems. A system is excitable if a perturbation over a certain threshold triggers alarge excursion from the steady state before it eventually returns to the steadystate. In other words, a system is excitable if, with a sucient initial stimulus,a pulse can be initiated which propagates throughout the medium.The excursion can be described as follows. The excitation variable u increasesrapidly until the trajectory approaches the branch f(h+(v), v). At this pointthe trajectory moves slowly up the nullcline as the recovery variable v increases.When the trajectory reaches the top of f(h+(v), v), the trajectory moves tothe the branch f(h−(v), v) as u rapidly decreases, which is followed by a slowdecrease of v back to the rest state (us, vs).

Although a pulse may be initiated, it may slow down, reverse direction, andeventually annihilate itself. In that case it will not propagate throughout themedium.

The form of f and g is such that the system of reaction-diusion equa-tions (3.2) has a so-called invariant region (cf. [10]), in particular Theorem4.3 and Example 6.C. That is, there exists a (necessarily) rectangular regionR in R2

+, containing 0, for which the vector eld associated to the reactionpart in (3.2) points inward everywhere. Then, a solution (u(t), v(t)) to (3.2)with initial condition (u0, v0) such that (u0(x), v0(x)) ∈ R for all x will have(u(t), v(t)) ∈ R for all t ≥ 0. R contains the steady state (us, vs). Thus bound-edness of solutions is assumed.

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4 Waves in one dimension

Throughout this chapter we assume that the properties of f and g, respectively(AF1)(AF4) and (AG1)(AG3), hold.

4.1 Travelling waves

First we will consider (3.2) in one spatial dimension, so that ∆ = ∂2

∂x2 . Thissystem can be treated using singular perturbation theory. Since the system hasa small parameter ε, we can consider two regions, which contains a wave front.

First of all, in the slowly varying region, when ε = 0, we nd

f(u, v) = 0,∂v

∂t= g(u, v).

Since f(u, v) = 0 has two branches of solutions that are stable and one solutionunstable to perturbations o the branch, the outer solution to (3.2) is

∂v

∂t= g(h±(v), v). (4.1)

We are looking for travelling waves so we have to nd solutions to sys-tem (3.2) of xed speed. The solutions will be of the form

u(x, t) = u(z), v(x, t) = v(z), z = x− ct,

with c the wave speed dependent on v.In that case we can write system (3.2) as

ε2uzz + εcuz + f(u, v) = 0

εδvzz + cvz + g(u, v) = 0. (4.2)

The O(1) equations, which is the lowest order in ε obtained by setting ε = 0in (4.2), are

f(u, v) = 0, cvz + g(u, v) = 0.

4.1.1 Transition from h− to h+

Consider the transition from h−(v) to h+(v) (the so-called "up-jump"). If wesuppose that for z < 0, u = h+(v) and for z > 0, u = h−(v), the solution u(z)is no longer continuous at z = 0. In that case diusion becomes important, i.e.the terms ε2uzz and εcuz, and the regions of the solution branches are patchedtogether by a boundary layer of width O(ε).Therefore we introduce a boundary layer coordinate (or a stretching transfor-mation)

ξ =z

ε.

When using this boundary layer coordinate, system (4.2) becomes

u′′ + cu′ + f(u, v) = 0

δv′′ + cv′ + εg(u, v) = 0, (4.3)

15

with u′′ = uξξ, u′ = uξ etcetera.

The boundary layer should have the correct orientation, so limξ→−∞ u(ξ) =h+(v) and limξ→∞ u(ξ) = h−(v), and c > 0, see Figure 4.1.

To lowest order in ε and requiring that v is bounded for all ξ, we have tosolve the equations

u′′ + cu′ + f(u, v) = 0

δv′′ + cv′ = 0, (4.4)

withlim

ξ→∓∞u(ξ) = h±(v), lim

ξ→±∞u′(ξ) = 0.

Solving the second equation of (4.4), we nd the general solution

v(ξ) = k1 + k2e− cδ ξ,

and since v is bounded, see Chapter 3, for all ξ, we nd that k2 = 0 andv(ξ) ≡ k1, with k1 some constant. For simplicity we will write k1 = v0, withv0 ∈ [v−, v

∗].Therefore, system (4.4) can be rewritten into

u′′ + cu′ + f(u, v) = 0, v = v0, (4.5)

withlim

ξ→∓∞u(ξ) = h±(v0) lim

ξ→±∞u′(ξ) = 0. (4.6)

This transition is sketched in Figure 4.1.

h−(v0)

h+(v0)

ξ

u

|c|

Figure 4.1: Situation sketch for the transition from h−(v0) to h+(v0).

System (4.5)- (4.6) is a non-linear eigenvalue problem for wave speed cparametrized by v0. We shall now discuss how c for which (4.5)- (4.6) hasa solution, depends on v0.

Lemma 1. Let v− ≤ v0 ≤ v+. Suppose c is such that (4.5)- (4.6) has a solutionu(ξ), then

c = cu(v0) =

∫ h+(v0)

h−(v0)f(u, v0) dz∫∞

−∞(u(ξ)′)2 dξ. (4.7)

Proof. Consideru′′ + cu′ + f(u, v0) = 0, (4.8)

16

with limξ→∓∞ u(ξ) = h±(v0) and limξ→±∞ u′(ξ) = 0. Multiplying (4.8) by u′

and integrating from ξ = −∞ to ξ =∞, we nd∫ ∞−∞

u′′(ξ)u′(ξ) + c(u′(ξ))2 + f(u(ξ), v0)u′(ξ) dξ = 0

1

2(u′(ξ))2

∣∣∣∣∞−∞

+ c

∫ ∞−∞

(u′(ξ))2 dξ +

∫ ∞−∞

f(u(ξ), v0)u′(ξ) dξ = 0. (4.9)

Since limξ→±∞ u′(ξ) = 0,1

2(u′(ξ))2

∣∣∣∣∞−∞

= 0. (4.10)

Since u = u(ξ), we have du = u′ dξ. Therefore, we can write∫ ∞−∞

f(u(ξ), v0)u′(ξ) dξ =

∫ u(∞)

u(−∞)

f(z, v0) dz

=

∫ h−(v0)

h+(v0)

f(z, v0) dz

= −∫ h+(v0)

h−(v0)

f(z, v0) dz. (4.11)

Substituting (4.10) and (4.11) into (4.9), we nd

c =

∫ h+(v0)

h−(v0)f(z, v0) dz∫∞

−∞(u′(ξ))2 dξ.

If a solution u to (4.5)- (4.6) exists for v− ≤ v0 ≤ v+, then c has the samesign as If (v0) (cf. equation (3.3). In particular

Corollary 1. If v− ≤ v0 < v∗, then c > 0. If v∗ < v0 ≤ v+, then c < 0.

Proof. From Proposition 1 we know that∫ h+(v0)

h−(v0)

f(u, v0) du > 0

for v− ≤ v0 < v∗. Combining this with (4.7) it follows that c > 0.For v∗ < v0 ≤ v+ Proposition 1 states that∫ h+(v0)

h−(v0)

f(u, v0) < 0,

and thus c < 0. This corresponds with the down-jump in Section 4.1.2.

The properties of the wave speed c as a function of v0 is given in Figure 4.2.To show that there exists a unique solution to (4.5)- (4.6), hence a unique

c(v0) > 0 for all v− < v0 < v∗, we use the approach of [1], Chapter 4.2.1 (c).Write u′ = w. Then (4.5) transforms into a rst-order system

u′ = ww′ = −cw − f(u, v0),

(4.12)

17

v0

c

cmax

cmin

v− v+v∗

Figure 4.2: Wave speed c against v0.

withlim

ξ→∓∞(u,w) = (h±(v0), 0).

Note that (u,w) = (h+(v0), 0) and (u,w) = (h−(v0), 0) are critical points forsystem (4.12). Thus we are looking for c such that there exists a heteroclinicorbit connecting (h+(v0), 0) to h−(v0), 0). First we determine the eigenvaluesof the corresponding linearisation. The Jacobian matrix corresponding to theright hand side of system (4.12) is

J(u,w) =

(0 1

−f ′(u, v0) −c

).

The linearisation around (h+(v0), 0) then equals(u′

w′

)= J(h+(v0), 0)

(u− h+(v0)

w

).

Evaluating J(u,w) in the critical point (h+(v0), 0) gives

J(u,w) =

(0 1

−f ′(h+(v0), v0) −c

). (4.13)

The eigenvalues corresponding to (4.13) are

λ±1 =−c±

√c2 − 4f ′(h+(v0), v0)

2. (4.14)

The eigenvalues for (h−(v0), 0) can be computed in a similar way, which resultsin

λ±0 =−c±

√c2 − 4f ′(h−(v0), v0)

2. (4.15)

Here the index 0 refers to the critical point x∗0 := (h−(v0), 0) and 1 to the pointx∗1 := (h+(v0), 0).

Since f ′(u, v0) < 0 for u = h±(v0), λ±0 and λ±1 are real and both havedierent signs. The critical points are therefore saddle points. Hence, theunstable manifold Wu(x∗1) of x∗1 corresponds to λ+

1 and the stable manifoldW s(x∗0) of x∗0 corresponds to λ−0 . From this we can conclude that an unstablecurve leaves x∗1 and a stable curve enters x∗0. Moreover Wu(x∗1) is tangent to

18

the line w = λ+1 (u − h+(v0)) at x∗1 and W s(x∗0) is tangent to the line w =

λ−0 (u− h−(v0)) at x∗0.In order to nd a heteroclinic orbit connecting x∗1 and x∗0, we have to nd c > 0such that Wu(x∗1) = W s(x∗0), for u > 0, w < 0.

Figure 4.3 shows a situation of the unstable manifoldWu(x∗1) and the stablemanifold W s(x∗0). Let Lε denote the vertical line through (h0(v0) + ε, 0), withε ≥ 0. For v− < v0 < v∗ it is allowed to take ε = 0. In this case we write Linstead of Lε. If v0 = v− then the critical point (h0(v−), 0) coincides with x∗0,and we must take ε > 0.

w

u

x∗0 (h0(v0), 0) x∗1

Lε

Ws(x∗0)

Wu(x∗1)

w1(c)

w0(c)

Figure 4.3: Situation sketch for Wu and W s.

Lemma 2. If c > 0, then there exist w0(c) < 0 and w1(c) < 0 such that

Wu(x∗1) ∩ Lε = (h0(v0) + ε, w1(c)), W s(x∗0) ∩ Lε = (h0(v0) + ε, w0(c)).

Proof. The unstable curve enters the region to the right of line Lε and cannotexit through the bottom, top or left hand side. Using (4.12) we deduce thatWu(x∗1) must exit this region through the line Lε at a point (h0(v0) + ε, w1(c)).Similarly, we can argue thatW s(x∗0) must cross Lε at a point (h0(v0)+ε, w0(c)).Moreover, the intersections consist of only one single point. Since u′ = w < 0Wu(x∗1) cannot re-enter Lε and W s(x∗0) cannot escape line Lε on the rightside.

Lemma 3. If c = 0, then trajectories of (4.12) are contained in level sets of

E(u,w) := w2

2 +∫ uh−(v0)

f(z, v0) dz.

Proof. If E : R2 → R is dierentiable and is a constant of motion, then

d

dtE(u(t), w(t)) = 0.

Hence∂E

∂u

du

dt+∂E

∂w

dw

dt= 0.

Consequently, since c = 0,

w∂E

∂u= f(u, v0)

∂E

∂w.

Thus we may take ∂E∂u = f(u, v0) and ∂E

∂w = w. Then

E(u,w) =w2

2+

∫ u

h−(v0)

f(z, v0) dz.

19

The level curves of E are of the form as illustrated in Figure 4.4.

u

w

w1(0)

w0(0)

Figure 4.4: Level curves of E(u,w).

At x∗1 the level curves behave as shown in Figure 4.5.

x∗1

λ+1

λ−1

Figure 4.5: Behaviour of the level curves at x∗1.

Lemma 4. Let v− ≤ v0 < v∗. If c = 0, it follows that

w0(0) > w1(0). (4.16)

Proof. If there is a heteroclinic orbit at c = 0, then x∗0 and x∗1 are on the samelevel set of E(u, v). In that case we have

E(h−(v), 0) = E(h+(v), 0)

0 =

∫ h+(v)

h−(v)

f(z, v) dz. (4.17)

From equation (4.17) follows that if there exists a heteroclinic orbit with c = 0,

then If (v) =∫ h+(v)

h−(v)f(z, v) dz = 0. From Proposition 1 follows that c(v) = 0

implies that v = v∗.

20

On one hand we have

E(L, w0(0)) = E(h−(v0), 0) = 0, (4.18)

E(L, w1(0)) = E(h+(v0), 0) =

∫ h+(v0)

h−(v0)

f(z, v0) dz, (4.19)

while on the other hand

E(L, w1(0)) =w1(0)2

2+

∫ L

h−(v0)

f(z, v0) dz, (4.20)

E(L, w0(0)) =w0(0)2

2+

∫ L

h−(v0)

f(z, v0) dz. (4.21)

Combining equations (4.18) and (4.21) gives

w0(0)2

2= −

∫ L

h−(v0)

f(z, v0) dz. (4.22)

Note that −∫ Lh−(v0)

f(z, v0) dz > 0. Combining equations (4.19) and (4.20) gives

w1(0)2

2=

∫ h+(v0)

L

f(z, v0) dz, (4.23)

with∫ h+(v0)

Lf(z, v0) dz > 0.

Since v− ≤ v0 < v∗, If (v0) > 0, see Proposition 1. Thus

w0(0)2

2= −

∫ L

h−(v0)

f(z, v0) dz <

∫ h+(v0)

L

f(z, v0) dz =w1(0)2

2.

It follows that |w0(0)| < |w1(0)|, and since wi(0) < 0, w0(0) > w1(0).

Lemma 5. Let v− ≤ v0 < v∗. Then there exists cv0 such that

w0(c) < w1(c), (4.24)

provided c > cv0 .

Proof. We will show that w0(c) < w0(0) and w1(c) > w0(0) for c sucientlylarge. Recall system (4.12).The unstable manifold Wu(x∗1) is tangent to w = λ+

1 (c)(u− h+(v0)), with

λ+1 (c) =

−c+√c2 − 4∂f∂u (h+(v0), 0)

2> 0.

For c large we have

limc→∞

λ+1 (c) = lim

c→∞

c2 −√c2 − 4∂f∂u (h+(v0), v0)

2

1

−c−√c2 − 4∂f∂u (h+(v0), v0)

= limc→∞

2∂f∂u (h+(v0), v0)

−c−√c2 − 4∂f∂u (h+(v0), v0)

= 0. (4.25)

21

Let β > 0. There exists cv0 > 0 such that |λ+1 (c)| < β for all c ≥ cv0 . If

u ∈ (h0(v0), h+(v0)), f(u, v0) > 0. Along the line segment Tβ := h0(v0) < u <h+(v0), w = β(u− h+(v0)), see Figure 4.6, we have

w′

u′=−cw − f(u, v0)

w= −c− f(u, v0)

β(u− h+(v0))= −c+

1

β

∣∣∣∣ f(u, v0)

h+(v0)− u

∣∣∣∣ .Since

∣∣∣ f(u,v0)h+(v0)−u

∣∣∣ ≤ K, with K = supu∈[h0(v0),h+(v0)]

∣∣∣∂f∂u (u, v0)∣∣∣, we see that

w′

u′≤ −c+

K

β< |β|,

on Tβ , provided c > 0 is large enough. ThereforeWu(x∗1) cannot exit S throughthe u-axis or Tβ if we choose cβ is suciently positive and we conclude that

w1(c) > β(h0(v0)− h+(v0)),

if c > cv0 .Hence for β > 0 large enough, such that |λ+

1 | < β, and c > 0 suciently positive,

w1(c) > w0(0), (4.26)

for all c > cβ .

w

u

(h0(v0), 0) x∗1

L

Tβ

S

λ+1

Figure 4.6: Situation sketch for region S.

The stable manifold W s(x∗0) is tangent to w = λ−0 (c)(u− h−(v0)), with

λ−0 (c) =−c−

√c2 − 4∂f∂u (h−(v0), v0)

2< 0,

withlimc→∞

λ−0 (c) = −∞,

so w0(c) is unbounded for large c.Now let β′ < 0. If u ∈ (h−(v0), h0(v0)) we have f(u, v0) < 0. Along the line

segment Tβ′ := h−(v0) < u < h0(v0) : w = β′(u−h−(v0)), see Figure 4.7, wehave

w′

u′= −c− f(u, v0)

w

= −c+|f(u, v0)|

w

= −c+|f(u, v0)|

β′(u− h−(v0))

22

Since∣∣∣ f(u,v0)u−h−(v0)

∣∣∣ ≤M , with M = supu∈[h−(v0),h0(v0)]

∣∣∣∂f∂u (u, v0)∣∣∣, we have

w′

u′≤ −c+

M

β′< β′,

on Tβ′ , provided c > 0 is large enough. Therefore we nd that

w0(c) < β′(h0(v0)− h−(v0)),

if cβ′ is suciently large. Hence, W s(x∗0) cannot exit S′ through Tβ′ .Hence, for all β′ < 0 there exists a cβ′ > 0 such that

w0(c) ≤ β′(h0(v0)− h−(v0)),

for all c ≥ cβ′ . Let β′ be such that w0(0) = β′(h0(v0) − h−(v0)). Then for all

c > cβ′

w0(c) < w0(0). (4.27)

w

u

x∗0 (h0(v0), 0)

L

S′

Tβ′

λ−0

Figure 4.7: Situation sketch for region S′.

Combining (4.26) and (4.27) we deduce that there exists a c > 0 such that

w0(c) < w1(c). (4.28)

Lemma 6. w0 : [0,∞)→ R is strictly decreasing and w1 : [0,∞)→ R is strictlyincreasing.

Proof. Consider equation (4.12), subject to the conditions

limξ→−∞

(u,w) = (h+(v0), 0), limξ→∞

(u,w) = (h−(v0), 0).

The eigenvalues of the linearisation around the steady states (h±(v0), 0) aregiven in (4.14) and (4.15).

We will only prove that w1 is strictly increasing. With a similar proof wecan show that c 7→ w0(c) is monotonically decreasing. For this we can rescale ξby putting τ = −ξ, so that system (4.12) transforms into

dudτ = −w,dwdτ = cw + f(u, v0),

(4.29)

withlim

τ→±∞(u,w) = (h±(v0), 0).

23

In this case, the eigenvalues for (h±(v0), 0) are

λ±0 =c±

√c2 − 4f ′(h−(v0), v0)

2, (4.30)

λ±1 =c±

√c2 − 4f ′(h+(v0), v0)

2. (4.31)

Here, the stable manifold corresponds to λ−1 and the unstable manifold corre-

sponds to λ+0 . c 7→ w0(c) will be monotonically increasing under the scaling

τ = −ξ and thus monotonically decreasing for ξ.Let c2 > c1 > 0 and u0 close to h+(v0) xed.

Remark 1. We will write Wuci for the unstable manifold Wu corresponding to

the dynamical system (4.12) with velocity c = ci, i = 1, 2.Moreover we will write wi(u0) for the solution w corresponding to a value u0

close to h+(v0) and speed ci.

If the initial condition (u0, w0) is on Wuci(h+(v0), 0), then the corresponding

solution to (4.12) satises

limξ→−∞

w′(ξ)

u′(ξ)= λ+

1 (ci).

The derivative of λ+1 (c) is

dλ+1

dc= −1

2+

1

2

c√c2 − 4f ′(h+(v0), v0)

.

Since there is no c such thatdλ+

1

dc = 0 anddλ+

1

dc < 0 for all c, λ+1 (c) is monotoni-

cally decreasing for c > 0, see (4.25).Therefore, for c2 > c1 > 0, we have the inequality

λ+1 (c2) =

−c2 +√c22 − 4f ′(h+(v0), v0))

2

<−c1 +

√c21 − 4f ′(h−(v0), v0)

2

= λ+1 (c1). (4.32)

Since w1(u0) = λ+1 (c1)(u0 − h+(v0)) and w2(u0) = λ+

1 (c2)(u0 − h+(v0)) for u0

close to h+(v0), the following inequality holds

w2(u0) < w1(u0) < 0. (4.33)

We now show that for every u ∈ (h−(v0), h+(v0)): w2(u) < w1(u) < 0. Theunstable manifold of (h+(v0), 0) is the solution to (4.5). The slope of the linetangent to the manifold at (u,wi) is

dwidu

= −ci −f(u, v0)

wi.

Assume that Wuc1(x∗1) and Wu

c2(x∗1) intersect in (u,w)|h0(v0) + ε ≤ u ≤h+(v0), w < 0. Since Wu

ci(x∗1) is a C1 submanifold, for i = 1, 2, and because

24

of equation (4.33) there exists a largest u, say u′, with u′ < u0, such thatw1(u′) = w2(u′) = w′. Consider the slopes of the tangent lines in (u′, w′):

dw1

du(u′) = −c1 −

f(u′, v0)

w′> −c2 −

f(u′, v0)

w′=dw2

du(u′).

It follows that for u < u′ close, 0 > w2(u) > w1(u) and for u > u′ close,0 > w1(u) > w2(u). Since u′ is the biggest u such that w1(u) and w2(u)intersect, there is no intersection for u′ < u < u0. Hence 0 > w1(u0) > w2(u0),which is in contradiction with equation (4.33). Therefore Wu

c1(x∗1) and Wuc2(x∗1)

do not intersect in (u,w)|h0(v0) + ε ≤ u ≤ h+(v0), w < 0.It follows that w1(u) < w2(u) < 0.

Therefore w1(c) is monotonically increasing in c.

Remark 2. Note that the argument used in the last part of the proof ofLemma 5 is valid as long as w < 0.

Proposition 2. For all v− ≤ v0 < v∗ there exists a unique c = c(v0) > 0 suchthat w0(c) = w1(c).

Proof. Since w0 and w1 depend continuously on c, we deduce from (4.16) and(4.24) that there exists a c > 0 such that

w0(c) = w1(c).

Because of Lemma 6, the solution c is unique.

Put cmax := c(v−) and cmin := c(v+).

Theorem 1. (i) If v− < v0 < v∗ there exists a unique velocity c > 0 forwhich there is a corresponding travelling wave.

(ii) If v0 = v− there exists a one-parameter family of waves with c ≥ cmax > 0.

Proof. (i) In Proposition 2 we showed that there exists a unique c > 0 suchthat w0(c) = w1(c). This means that there is a unique heteroclinic orbitconnecting x∗0 and x∗1. This yield a travelling wave of system (3.2).

(ii) Consider the following system

u′ = w

w′ = −cw − f(u, v−), (4.34)

which has as steady state solutions (u,w) = (h±(v−), 0). Note that in thiscase, the third steady state (h0(v−), 0) has coincided with (h−(v−), 0).

The eigenvalues corresponding to the linearisation of (4.34), evaluated inthe critical point (h+(v−), 0) are

λ±1 = − c2± 1

2

√c2 − ∂f

∂u(h+(v−), v−).

Since λ−1 < 0 < λ+1 , the critical point (h+(v−), 0) is a saddle point.

25

The Jacobian matrix corresponding to the linearisation of the rst-ordersystem (4.34), evaluated in the critical point (h−(v−), 0), becomes

J(h−(v−), 0) =

(0 10 −c

). (4.35)

The eigenvalues corresponding to (4.35) are

λ+0 = 0, λ−0 = −c,

with c > 0, so that λ− < 0.

We now have to determine the stability of the critical point (h−(v−), 0).Since the critical point is not hyperbolic, we need to consider the centermanifold to decide on stability. The stable manifold is tangent to thestable subspace spanned by the eigenvector

(− 1c , 1)corresponding to the

eigenvalue λ−0 = −c. The center manifold is tangent to the center subspacespanned by the eigenvector (1, 0) corresponding to the eigenvalue λ+

0 = 0.In Appendix A, Theorem 4, it is shown that (h−(v−), 0) is locally stable,at least in an open ball around x∗0 of radius δ.

With a similar proof as that of Lemma 5, we can show that Wu(x∗1)intersects the line Lε = u = h−(v−) + ε in the point w1(cmax). Thesituation sketch for v0 = v− is illustrated in Figure 4.8.

x∗1x∗0

ε

δ

Wc(x∗0)

Ws(x∗0)

Ws(x∗1)

Wu(x∗1)

Tβ

Lε

S

Figure 4.8: Situation sketch for c = cmax.

We choose ε < δ such that the intersection point W s(x∗0)∩Lε, (h−(v−) +ε, w0(cmax)), lies in B(x∗0, δ). Then also Wu(x∗1)∩Lε ∈ B(x∗0, δ). For anyinitial condition (u0, w0) ∈ B(x∗0, δ), the solution will converge to x∗0, seeFigure 4.9. Hence for c = cmax there is a heteroclinic orbit connecting x∗0and x∗1.

In Proposition 2 (ii) we showed that if c > cmax, w0(c) < w1(c) < 0.Since (h−(v−) + ε, w0(c)) ∈ B(x∗0, δ) also (h−(v−) + ε, w1(c)) ∈ B(x∗0, δ).Thus if (u0, w0) ∈ Wu

c (x∗1), then the solution starting at (u0, w0) will bein B(x∗0, δ) after same time. It will then continue by converging towards

26

ε

δ

W c(x∗0)

x∗0

W s(x∗0)

Figure 4.9: Situation sketch for the local center manifold at x∗0.

x∗0. In particular, for each c ≥ cmax the system (4.12) has a heteroclinicorbit from x∗1 to x∗0.

Therefore if v0 = v− there exists a one-parameter family of waves withc ≥ cmax > 0.

In Figure 4.2, these one-parameter families of waves are shown by the dashedlines at v0 = v− and v0 = v+.

4.1.2 Transition from h+ to h−

For the transition from h+(v0) to h−(v0), the so-called "down-jump", we haveto consider

u′′ + cu′ + f(u, v0) = 0, (4.36)

withlim

ξ→±∞u(ξ) = h±(v0).

The situation sketch for this problem is given in Figure 4.10.

h+(v0)

h−(v0)

ξ

u

|c|

Figure 4.10: Sketch of the transition from h−(v0) to h+(v0).

27

For this transition we have to show that there exists a unique c < 0 such thatthere is a heteroclinic orbit connecting h−(v0) and h+(v0). The situation forthe unstable and stable manifolds for this problem can be found in Figure 4.11.

w

ux∗0 (h0(v0), 0) x∗1

Wu Ws

L

w0(c)

w1(c)

Figure 4.11: Situation sketch for Wu and W s.

It follows that there exists a unique c(v0) < 0 for all v∗ < v0 ≤ v+. Thisproof is similar to the proof of Theorem 1.

Hence, for v− ≤ v0 < v∗ the wave moves with speed c > 0 from the lowerstate h−(v0) to the upper state h+(v0), while for v∗ < v0 ≤ v+ the waves movesback from h+(v0) to h−(v0) with speed c < 0.

4.2 Example: the cubic case

The cubic structure of f is typical of many chemical reactions involving activa-tion and inhibition. For most problems explicit calculations of many propertiesof the travelling waves are impossible and must be calculated numerically. Oneof the problems is that in general v0 ahead of the wave front is changing in time.Therefore the speed c(v0) is not constant.However, there are two cases where exact solutions are known according to [3]:if f is piecewise linear or if f is a cubic polynomial. In this section we willconsider the cubic case in detail. The piecewise linear case may be found in [7].

For the cubic case consider

f(u, v) = −A(u− u0)(u− u1)(u− u2),

with A constant, with u0, u1 and u2 depending on v. Then f(u, v) = 0 has threesolutions for u: u0, u1 and u2, such that u0 < u1 < u2.

The best known example of this cubic case is the Fitzhugh-Nagumo modelfor chemical-electrical conduction in a nerve:

f(u, v) = −u(u− β)(u− 1)− v,g(u, v) = u− dv, (4.37)

where 0 < β < 1.For the eigenvalue problem (4.5)- (4.6) we can nd c(v0) by using Equa-

tion (4.7). Another way to do this for this particular example is by making aguess: u′ = α(u− u0)(u− u2).

Lemma 7. Consider u′ = α(u− u0)(u− u2). Then α is constant.

28

Proof. Suppose that α = α(u) is not constant and α(u) 6= 0 on [u0, u2]. Substi-tuting u′ = α(u)(u− u0)(u− u2) into system (4.5) gives

u′[α′(u− u0)(u− u2) + α((u− u0) + (u− u2))]

+cα(u− u0)(u− u2)−A(u− u0)(u− u1)(u− u2) = 0

[α′(u− u0)(u− u2) + α((u− u0) + (u− u2))] + c−Au− u1

α= 0[

α′(u2 − u(u0 + u2)) + 2αu−Auα

]+ α′u0u2 − α(u0 + u2) + c+A

u1

α= 0

which is true if

α′u2 = 0 (4.38)

−u(α′(u0 + u2)− 2α+

A

α

)= 0 (4.39)

α′u0u2 − α(u0 + u2) + c+Au1

α= 0. (4.40)

From equation (4.38) it follows that α′(u) = 0, hence α is constant.

Substituting α′ = 0 into equation (4.39), we nd that −2α+ Aα = 0 hence

α = ±√A

2.

From equation (4.40) is follows that

c = −α′u0u2 + α(u0 + u2)−Au1

α

= ±√A

2(u0 + u2)∓

√2Au1

= ±√A

2(u0 + u2 − 2u1) (4.41)

Depending on the value of v, we have

c± = ±√A

2(u0 + u2 − 2u1).

4.3 Travelling pulse

In Chapter 3 it was assumed in (AG3) that there is a unique steady state solutionfor system (3.2) denoted by (us, vs), such that v− < vs < v∗. At t = 0 supposethe following pulse-shaped initial conditions (in u):

u(x, 0) =

h+(vs) for |x| < x0,h−(vs) = us for |x| ≥ x0.

(4.42)

v(x, 0) = vs, −∞ < x <∞. (4.43)

Because of symmetry of the problem, it suces to consider the case x > 0.As already mentioned in Chapter 3, f and g in system (3.2) are typical for

excitable systems. The transition from the steady state to excited state and

29

back is called a travelling pulse, and will de described in this section withoutfurther details (cf. [12]).

A wave front develops at x = x0 and propagates with speed c(vs) > 0 to theright (from h−(vs) to h+(vs)). Ahead of this front the medium is in rest untilthe front passes. Since in most of the biological settings v does not diuse inspace, we set δ = 0 in (3.2). So behind the front, v changes according to

∂v

∂t= g(u, v).

Depending on the sign of g, v increases or decreases behind the wave front.For now assume that g(h+(v), v) > 0, see Figure 3.1. In that case, v increasesbehind the wave front. Moreover, u ≈ h+(v) until v = v+. In that case thebranches h0(v) and h+(v) merge and a down-jump wave must develop. Thereare two possibilities for this down-jump:

(i) If c(vs) > |cmin| the up-jump wave moves faster than the down-jump wavewhen it has been triggered. In that case the down-jump wave travels atexactly the same speed as the up-jump wave. In other words, consideringthe one-parameter family of waves at v = v+ which travel at speeds |c| >|cmin|, see Figure 4.2, the down-jump wave travels at speed |c| = c(vs).This travelling pulse can be found in Figure 4.12(a).

(ii) If c(vs) < |cmin| the down-jump wave back moves faster when it has beentriggered than the up-jump wave. In that case the down-jump wave startso at speed |c| = |cmin| and catches up with the up-jump wave. As soonas it catches up with the up-jump wave, v doesn't have enough time toincrease for vs to v+. The down-jump wave needs to develop at somev < v+ and moves in absolute sense at a slower speed. The down-jumpwave will stabilize at a value vb such that v∗ < vb < v+ and travels atspeed |c(vb)| = c(vs). The up-jump wave and down-jump wave then moveat the same speed.This travelling pulse can be found in Figure 4.12(b).

Figure 4.12: Travelling pulses if (a) c(vs) > |cmin|, (b) c(vs) < |cmin|.

Behind the down-jump wave, u = h−(v). Assuming that g(h−(v), v) < 0, vdecreases so that the medium returns to the steady state (or rest state) (us, vs).

30

5 Spiral waves in two dimensions

In Chapter 4 we considered system (3.2) in one spatial dimension. In two spatialdimensions the dynamics of the excitable medium is again given by system (3.2),

where ∆ = ∂2

∂x2 + ∂2

∂y2 .Because we are mainly interested in the biological excitable medium setting, wewill only consider system (3.2) with δ = 0:

ε∂u

∂t= ε2∆u+ f(u, v),

∂v

∂t= g(u, v). (5.1)

Let u = u(x, y, t) be the the function that describes the spiral wave, with u = uththe location of the wave front. If u > uth, the medium is activated, while foru < uth, the medium is in rest. If one sets uth = 0 the medium would be in restfor u < 0. Since it is not possible to link u < 0 to the biology as described inChapter 3 it is more convenient to set uth > 0.

5.1 Curvature

Just as in the one dimensional case the wave front moves because of the reactionand diusion of the propagator species u. However, the normal velocity of thewave front is not only determined by the level of controller species in the front,say vf , but also by the curvature of the wave front.

Following [3], consider the stretched travelling coordinate system

x = X(εξ, η, τ), y = Y (εξ, η, τ), t = τ,

and the parametrization R = (X(εξ, η, τ), Y (εξ, η, τ)). η 7→ (X(0, η, τ), Y (0, η, τ))gives the level parametrization of the wave front, u = uth, while for ξ > 0 itparametrizes a level surface u = u with u < uth. Let N the unit normal vec-tor to that surface at some point. A sketch of this situation can be found inFigure 5.1.

ξ

η

u = uth

u = u < uth

Figure 5.1: Situation sketch for the level surfaces of ξ.

31

The coordinate system is chosen such that ξ and η are locally orthogonal

coordinates, hence ∂R∂ξ ·

∂R∂η = 0, where the subscripts ξ and η denote the partial

derivative with respect to ξ and η respectively.The orientation for the (ξ, η) coordinate axes is such that the positive ξ axis isrotated 90 degrees in the clockwise direction from the positive η axis. Thus, ifwe write

∂R

∂ξ= (εXξ, εYξ, 0)T ,

and∂R

∂η= (Xη, Yη, 0)T ,

then the orientation of the coordinate axis is such that the normal vector to thiscoordinate system is given by ∂R

∂ξ ×∂R∂η with positive z-coordinate. That is

(εXξ, εYξ, 0)T × (Xη, Yη, 0)T = (0, 0, εXξYη − εYξXη)T , (5.2)

and XξYη − YξXη > 0.One of the properties of the cross product gives∣∣∣∣∣∂R∂ξ × ∂R

∂η

∣∣∣∣∣ =

∣∣∣∣∣∂R∂ξ∣∣∣∣∣∣∣∣∣∣∂R∂η

∣∣∣∣∣ sin(θ), (5.3)

with θ the angle between ∂R∂ξ and ∂R

∂ξ , which is in this case 90 degrees. Thereforewe nd

XξYη − YξXη = |XξYη − YξXη| =√

(X2ξ + Y 2

ξ )(X2η + Y 2

η ). (5.4)

According to the chain rule,

∂

∂ξ= ε

(Xξ

∂

∂x+ Yξ

∂

∂y

), (5.5)

∂

∂η= Xη

∂

∂x+ Yη

∂

∂y, (5.6)

∂

∂τ=

∂

∂t+Xτ

∂

∂x+ Yτ

∂

∂y. (5.7)

Combining equation (5.5) and (5.6), and using (5.4), it follows that

∂

∂x=

1εYη

∂∂ξ − Yξ

∂∂η

XξYη −XηYξ,

=

1εYη

∂∂ξ − Yξ

∂∂η√

(X2ξ + Y 2

ξ )(X2η + Y 2

η ), (5.8)

∂

∂y=

Xξ∂∂η −

1εXη

∂∂ξ

XξYη −XηYξ,

=Xξ

∂∂η −

1εXη

∂∂ξ√

(X2ξ + Y 2

ξ )(X2η + Y 2

η ). (5.9)

32

Substituting these into equation (5.7) we nd

∂

∂t=

∂

∂τ−Xτ

∂

∂x− Yτ

∂

∂y

=∂

∂τ+

1√(X2

ξ + Y 2ξ )(X2

η + Y 2η )

·(

1

ε(XηYτ − YηXτ )

∂

∂ξ+ (XτYξ − YτXξ)

∂

∂η

). (5.10)

The Laplacian operator of u in the new coordinate system becomes

∆u =1

ε2

1√X2ξ + Y 2

ξ

∂

∂ξ

1√X2ξ + Y 2

ξ

∂u

∂ξ

+ εKη∂u

∂ξ

+

1√X2η + Y 2

η

∂

∂η

1√X2η + Y 2

η

∂u

∂η

−Kξ∂u

∂η

(5.11)

where Ki, i = ξ, η are the curvatures

Ki =XiYii − YiXii

(X2i + Y 2

i )32

. (5.12)

Without loss of generality we can scale the ξ coordinate so that X2ξ + Y 2

ξ = 1.Let u = u(x, y, t) be the solution to (5.1) for particular initial conditions. Weshall now derive equations satised by u when expressed as u = u(ξ, η, τ) locally,i.e. in the travelling coordinate system we seek solutions u = u(ξ), for whichthe curves ξ =constant are the level curves of u, so that ∂u

∂η = ∂u∂τ = 0. The

wave front is given by ξ = 0. For simplicity we will write u(ξ, η, τ) instead ofu(ξ, η, τ). Then system (5.1) becomes

∂2u

∂ξ2+ (Nη + aεKη)

∂u

∂ξ+ f(u, v) = 0 (5.13)

Nη∂v

∂ξ=

εNξ√X2η + Y 2

η

∂v

∂η− ε∂v

∂τ+ εg(u, v). (5.14)

with

Ni =XτYi − YτXi√

X2i + Y 2

i

, (5.15)

for i = ξ, η, as the normal velocity of the coordinate lines. In equation (5.13)a is a proportionality constant with the dimension of acceleration. To leadingorder in ε it follows from equation (5.14) that v is independent of ξ, but doesdepend on the wave front coordinate η, so that v = vf +O(ε).

Lemma 8.

Nη + aεKη = (Nη + aεKη)|ξ=0 +O(ε2). (5.16)

33

Proof. The Taylor series of Kη around ξ = 0 is given by

Kη = Kη|ξ=0 + ξ∂Kη

∂ξ

∣∣∣∣ξ=0

+O(ξ2),

with

∂Kη

∂ξ=

ε

(X2η + Y 2

η )52

[(XηξYηη +XηYηηξ − YηξXηη − YηXηηξ)(X

2η + Y 2

η )

− 3(XηYηη −XηηYη)(XηXηξ + YηYηξ)]

= O(ε)

Therefore∂Kη

∂ξ

∣∣∣∣ξ=0

= O(ε),

and thusKη = Kη|ξ=0 +O(ε),

so thatεKη = εKη|ξ=0 +O(ε2). (5.17)

Let ζ = εξ. The Taylor series of the coordinate system around ζ = 0 is givenby

X(ζ, η, τ) = X0(η, τ) + ζX1(η, τ) +O(ζ2), (5.18)

Y (ζ, η, τ) = Y0(η, τ) + ζY1(η, τ) +O(ζ2), (5.19)

where X1(η, τ) = ∂X∂ζ (0, η, τ) and Y1(η, τ) = ∂Y

∂ζ (0, η, τ).The Taylor series of Nη around ξ = 0 is

Nη = Nη|ξ=0 + ξ∂Nη∂ξ

∣∣∣∣ξ=0

+O(ξ2), (5.20)

with

∂Nη∂ξ

=ε

(X2η + Y 2

η )32

[(XξτYη +XτYξη −XξηYτ −XηYξτ )(X2

η + Y 2η )

−(XτYη −XηYτ )(XηXξη + YηYξη)] . (5.21)

Substituting the Taylor series (5.18) and (5.19) into equation (5.21) one obtainsto order O(ε2)

∂Nη∂ξ

=ε2

(X20η

+ Y 20η

)32[

(X1τY0η +X0τY1η −X1ηY0τ −X0ηY1τ )(X20η + Y 2

0η )

−(X0τY0η −X0ηY0τ )(X0ηX1η + Y0ηY1η )]

= O(ε2)

so that equation (5.20) becomes

Nη = Nη|ξ=0 +O(ε2). (5.22)

34

Combining (5.17) and (5.22) with the Taylor series of Nη + aεKη around ξ = 0,we nd

Nη + aεKη = (Nη + aεKη)|ξ=0 + ξ∂

∂ξ(Nη + aεKη)

∣∣∣∣ξ=0

= (Nη + aεKη)|ξ=0 +O(ε2). (5.23)

It follows that to order O(ε2), Nη + aεKη is independent of ξ.Now write u = u0 + εu1 + O(ε2) and v = vf + εv1 + O(ε2), and substituteinto (5.13).

0 =∂2u0

∂ξ2+ ε

∂2u1

∂ξ2+O(ε2) + (Nη + aεKη)

(∂u0

∂ξ+ ε

∂u1

∂ξ+O(ε2)

)+f(u0 + εu1 +O(ε2), vf + εv1 +O(ε2)) (5.24)

We assumed that ∂f∂u and ∂f

∂v are continuous. Since the solutions to (5.1) remain

bounded (see Section 3), ∂f∂u and ∂f

∂v are also bounded on the set of values(u(t), v(t)) | t ∈ R+. Then

f(u(t), v(t)) = f(u0(t), vf (t)) +∂f

∂u(u0(t), vf (t))(u(t)− u0(t))

+∂f

∂v(u0(t), vf (t))(v(t)− vf (t)) + h.o.t.

so

f(u0 + εu1 +O(ε2), vf + εv1 +O(ε2)) = f(u0, vf ) +∂f

∂u(u0, vf )(εu1 +O(ε2))

+∂f

∂v(u0, vf )(εv1 +O(ε2))

= f(u0, vf ) +O(ε). (5.25)

Thus, the O(1) equation of (5.24) becomes

∂2u0

∂ξ2+ ((Nη + aεKη)|ξ=0

∂u0

∂ξ+ f(u0, vf ) = 0. (5.26)

This is precisely equation (4.5). As we have seen, an up-wave solution existswhen

(Nη + aεKη)|ξ=0 = c(vf ).

Substituting into equation (5.16) one nds the following eikonal equation

Nη + aεKη = c(vf ) +O(ε2), (5.27)

orNη = c(vf )− aεKη +O(ε2).

In other words, the normal velocity of the wave front at controller level vfequals the corresponding velocity c(vf ) of a one-dimensional wave, corrected byan amount proportional to the curvature of the front, εKη.This eikonal equation describes the propagation of wave fronts in excitable me-dia, by determining the spatial location of the wave front as a function of time,given an initial condition.

35

5.2 Spiral waves

Following [3] and [4] curvature plays an important role in determining the prop-erties of spiral waves in two-dimensional media.

Let Ft = (x, y)|u(x, y, t) = uth be the front of a spiral wave, where uthrepresents a particular threshold value for the propagator species. We assumethat it rotates around (0, 0) with xed angular velocity ω. We are interestedin a spiral wave that does not extend in space, i.e. Ft remains contained in axed annulus of inner and outer radius r0 and r1 respectively. Because it doesnot extend, the normal velocities at the end points should be tangential to thecircles of radius r0 and r1.Assume that F0 can be parametrized by r:

F0 = (x, y)|x = r cos(θ(r)), y = r sin(θ(r)), r ∈ [r0, r1],

where θ(r) gives the shape of the the spiral wave front.This part at time t is then parametrized as

Ft = (x, y)|x = r cos(θ(r) + ωt), y = r sin(θ(r) + ωt), r ∈ [r0, r1]. (5.28)

The wave front coordinate η is chosen to be the radial variable r =√x2 + y2

on Ft. Our goal is to nd an expression for the shape of the spiral wave frontθ(r) and the angular velocity ω of the spiral wave.

From experimental data we expect that when the spiral F0 rotates counterclockwise in space, then the wave front propagates through the medium clock-wise in time, so that ω < 0. θ(r) increases as a function of the radius r, so thatθ′(r) > 0. Therefore θ′(r)ω < 0 is expected for all r ∈ (r0, r1). This situationcan be found in Figure 5.2.

Figure 5.2: Situation sketch for a counter clockwise rotating spiral wave in spacefor r ∈ [r0, r1]. Rotation of this front is clockwise in time.

The normal velocity and curvature can be calculated directly, using (5.12)and (5.15). One nds

Xt = −rω sin(θ(r) + ωt),

Yt = rω cos(θ(r) + ωt),

Xr = cos(θ(r) + ωt)− rθ′(r) sin(θ(r) + ωt), (5.29)

Yr = sin(θ(r) + ωt) + rθ′(r) cos(θ(r) + ωt), (5.30)

36

so that X2r + Y 2

r = 1 + r2θ′(r)2. Put ψ(r) := rθ′(r). Moreover,

XηYηη − YηXηη = 2θ′(r) + rθ′′(r) + r2θ′(r)3

= ψ′(r) + θ′(r) + r2θ′(r)3

= ψ′(r) + θ′(r)(1 + ψ(r)2) (5.31)

Then we have

Nr =−ωr√1 + ψ2

, Kr =ψ′

(1 + ψ2)32

+ψ

r√

1 + ψ2. (5.32)

Substituting equation (5.32) into equation (5.27), we obtain a rst order non-linear ODE for the unknown function ψ(r):

rψ′ = (1 + ψ2)

(c(r)r

aε

√1 + ψ2 +

ωr2

aε− ψ

), (5.33)

in which c(r) = c(vf (r)) > 0. The tangential condition for the normal veloci-ties at r0 and r1 result into boundary conditions ψ(r0) = 0 = ψ(r1), see (5.29)and (5.30). Then in principal the shape of the spiral wave front can be deter-mined by θ(r) =

∫r−1ψ(r) dr.

Assume that c(r) = c, to rst order. This is true as long as r0 is not too small,so that vf remains xed along the spiral wave front.

The ordinary dierential equation for ψ(r) as given in equation (5.33) canbe rewritten into a system of ordinary dierential equations for r and ψ:

r′ = 1,

ψ′ = (1 + ψ2)(caε

√1 + ψ2 + ωr

aε −ψr

).

(5.34)

Dene

h(r, ψ) := (1 + ψ2)

(c

aε

√1 + ψ2 +

ωr

aε− ψ

r

), (5.35)

for ψ 6= 0.The ψ-nullcline given by h(r, ψ) = 0, is determined by

ω

aεr2 +

c

aε

√1 + ψ2r − ψ = 0.

Hence there are two branches, where in each r is a function of ψ:

r±(ψ) =|ω|ω

[− c

2|ω|√

1 + ψ2 ± 1

2

√c2

ω2(1 + ψ2) + 4

aε

ωψ

]

= −1

2

c

ω

[√1 + ψ2 ∓

√1 + ψ2 +

4aεω

c2ψ

]. (5.36)

We will consider two cases: ω > 0 and ω < 0.If ω > 0 the nullclines of system (5.34) can be found in Figure 5.3.Let the boundary condition at r0 be ψ(r0) = 0 (so that the shape of the

wave front is minimal). Then h(r, ψ) > 0 for r > r0 so that ψ → ∞. Hence, ifω > 0 there does not exist a r1 such that ψ(r1) = 0.

37

Figure 5.3: Nullclines for system (5.34) with a = 1, cε = 3 and ωε = 2.8.

Let ω < 0. The branches r±(ψ) as given in (5.36) exist as long as

1 + ψ2 +4aεω

c2ψ ≥ 0. (5.37)

This is equal to zero when

ψ± = −2aεω

c2±√

4a2ε2ω2

c4− 1.

Hence if(cε

)4 ≥ 4a2(ωε

)2condition (5.37) holds for all ψ.

If(cε

)4< 4a2

(ωε

)2there exist values of ψ such that 1 + ψ2 + 4aεω

c2 ψ < 0. Itfollows that r+(ψ+) = r−(ψ+) and r+(ψ−) = r−(ψ−). The nullclines for thissituation are sketched in Figure 5.4.

Figure 5.4: Nullclines for system (5.34) with a = 1, cε = 1 and ωε = −1.

If r0 < r−(0) = c|ω| there exists r∗1 > c

|ω| such that for all r1 ≥ r∗1 , there

cannot exist a solution with ψ(r0) = 0 = ψ(r1).

38

Let(cε

)4 ≥ 4a2(ωε

)2. For this situation the nullclines of system (5.34) are

sketched in Figure 5.5. The boundary condition at r0 is ψ(r0) = 0. Since we

Figure 5.5: Nullclines for system (5.34) with a = 1, cε = 3 and ωε = −2.8.

assume that c(vf ) = c along the wave front, r0 cannot be too small. Thereforeit is possible to nd values of c and ω such that the boundary condition on r1,ψ(r1) = 0, can be satised. Therefore our expectation that ω < 0 is correct.

Example 2. The trajectory for a = 1, cε = 3, ω

ε = −2.8 and ε = 0.1 can befound in Figure 5.6.

Figure 5.6: Phase portrait and trajectory connecting (r0, 0) and (r1, 0) (red line)for system (5.34) with a = 1, cε = 3, ε = 0.1 and ω

ε = −2.8.

The trajectory in Figure 5.6 ts approximately the parabola

ψ(r) = −1.4(r − 0.3)(r − 1.74).

39

Since ψ(r) = rθ′(r), we can nd an explicit expression for θ(r):

θ(r) =

∫ r

r0

ψ(s)

sds

≈ −0.7r2 + 2.86r − 0.73 ln(r)− 1.67. (5.38)

Since drdt = 1, r(t) = t. Then the parametrization (5.28) becomes

x = t cos(θ(t)− 0.28t), y = t sin(θ(t)− 0.28t). (5.39)

This results into a spiral as given in Figure 5.7.

Figure 5.7: Spiral wave for (5.39) with θ(r) as given in (5.38).

r′±(ψ) is found to be

r′±(ψ) =1

2

c

|ω|ψ√

1 + ψ2∓ 1

2

c2

ω2ψ − 2 aε|ω|√c2

ω2 (1 + ψ2)− 4 aε|ω|ψ, (5.40)

so that

limψ→±∞

r′+(ψ) = 0,

limψ→±∞

r′−(ψ) = ± c

|ω|.

Moreover

limψ→±∞

r+(ψ) = ±aεc,

limψ→±∞

r−(ψ) = ±∞,

which is in agreement with Figure 5.5: there are vertical asymptotes for r+(ψ)and oblique asymptotes for r−(ψ).For existence of solutions that satisfy the boundary conditions ψ(r0) = 0 =

40

ψ(r1) we should have 0 < r0 <c|ω| . If c is xed, we have the following relation-

ship between r1 and ωε

Ω′(r0,

c

ε

)=(r1,

ω

ε

)|r1 > r0, ω < 0, ∃ solutions with ψ(r0) = 0 = ψ(r1)

6= ∅.

Hence for every ω ∈ [−ω∗,− cr0

], with ω∗ the minimal value such that thesolution curve intersects r−(ψ), there is at most one r0 < r1 < r∗1 , with

r∗1 := maxr1(ω)|ω ∈ [−ω∗,− c

r0], such that

(r1,

ωε

)∈ Ω′

(r0,

cε

).

Let r be suciently large. In a rotating spiral the relation between the radiusr, the angular velocity ω and the tangential speed v is given by

v = ωr. (5.41)

Let φ be the angle between the normal velocity c and the velocity v tangentialto the rotation orbit. After one time step ∆t the wave front moved a distance∆s, so that ∆s = c∆t

cosφ . Since |v| ≈∆s∆t , it follows that∣∣∣ωrc

∣∣∣ ≈ 1

cosφ. (5.42)

Consider the tangent to F 0t0 :

τ =

(dxdrdydr

)=

(cos(θ(r) + ωt0)− rθ′(r) sin(θ(r) + ωt0)sin(θ(r) + ωt0) + rθ′(r) cos(θ(r) + ωt0)

), (5.43)

and the parametrization at t = t0

R =

(cos(θ(r) + ωt0)sin(θ(r) + ωt0)

). (5.44)

Then φ is also the angle between τ and R, so that

cosφ =R · τ

||R|| · ||τ ||, (5.45)

withR · τ = 1,

and||R|| = 1.

Therefore

1

cosφ= ||τ ||

=√

1 + ψ(r)2. (5.46)

Combining equation (5.42) with equation (5.46) gives the approximation∣∣∣ωrc

∣∣∣ ≈√1 + ψ(r)2. (5.47)

41

Substituting equation (5.33) into the curvature equation forKr as given in (5.32)we nd

εKr =1√

1 + ψ(r)2

(c√

1 + ψ(r)2 + ωr − εψ(r)

r

)+

εψ(r)

r√

1 + ψ(r)2

= c

(1 +

ωr

c

1√1 + ψ(r)2

). (5.48)

Since ω < 0,√

1 + ψ(r)2 ≈ −ωrc for r large, hence

εKr ≈ c(

1− ωr

c

c

ωr

)= 0. (5.49)

Therefore if the spiral wave front is suciently far from the origin, the cur-vature of the front is negligible. In that case the normal velocity of the spiralwave front equals c(vf ).

Remark 3. In this region the wave fronts propagate like periodic wave trains(with period T = 2π

ω ), which are described in more detail in [12]. It follows thatthere is a relation between the speed c and the period T . This relation is calledthe dispersion relation and is represented by c = σ(T ).

Remark 4. For r1 suciently large, ω is independent of r1 since spirals rotatewith the same angular velocity on all suciently large bounded and unboundeddomains, following [12]. Therefore we have the relationship

ω = εΩ(r0,

c

ε

).

Hence, for suciently large r, c and ω should satisfy both the dispersionrelation c = σ(T ) and the curvature relation ω = εΩ( cε , r0). If both relation aresatised simultaneously, the unique angular velocity ω and corresponding speedc of the spiral waves rotating around a hole of radius r0 can be determined.

5.2.1 Multi-armed spiral waves

In most of the biological media the spiral wave front contains a single arm.However, there are some biological excitable media in which a multi-armed spiralcan develop spontaneously from a single-armed spiral, see [14]. An example ofsuch an excitable medium is in the signalling waves of cAMP in Dictyosteliumdiscoideum.

A multi-armed spiral wave can only be formed in media with a low excitabil-ity, i.e. a low threshold value for which the medium becomes excited. Moreover,the number of arms increases with a decrease of excitability. Vasiev et al. as-sumes that in Dictyostelium the excitability of the cAMP relay system decreases.This change may be caused by a switch in the expression of high to low anitycAMP receptors. Therefore the spiral waves in Dictyostelium can have a dier-ent number of arms. The results of an experiment with Dictyostelium with achange of excitability can be found in Figure 5.8.

Vasiev et al. also show that the arms of a spiral wave front should all rotatewith the same angular velocity counter clockwise. If one of the arms wouldrotate clockwise, it will eventually be repelled by the other arms which rotate

42

Figure 5.8: Experimental study from [14]. Depending on the excitability of themedium there are (A) two-armed, (B) three-armed or (C) ve-armed propagat-ing spiral waves of cAMP in Dictyostelium discoideum.

in the same direction.Moreover the arms of the spiral waves are angularly equally spaced. If this isnot the case, the tip of one arm is close to the tip of another arm so that theywould interact and form one spiral.

It is also possible to determine the shape of a k-armed spiral wave fronts.The spiral wave front is then parametrized by

Xj(r, t) = r cos

(θ(r) + ωt+

2πj

k

)Yj(r, t) = r sin

(θ(r) + ωt+

2πj

k

), (5.50)

with j = 1, 2, . . . , k. However, just as for the one-armed spiral wave fronts,the shape of the spiral wave front can be determined by θ(r) =

∫r−1ψ(r) dr,

with ψ(r) given by the ODE (5.33) and the angular velocity ω is again givenby the relation ω

ε = Ω(cε , r0, r1

). For large r, the dispersion relation is given by

c = σ(T ), with T = 2πωk .

43

6 Discussion

In this master thesis we considered the construction of waves in one and twospatial dimensions in detail. Based on an experimental study of Dictyosteliumdiscoideum, see Chapter 2, we found in Chapter 3 the general form for the wavepropagation in excitable media in the form of a pair of reaction-diusion equa-tions. With singular perturbation theory we proved in Chapter 4 the existenceand uniqueness of travelling waves in one spatial dimension. The speed of thesetravelling waves depends on the level of controller species.In the articles about the travelling waves in one spatial dimension, most of thestatements made there were based on experimental results. Moreover, someimportant assumptions, in general assumptions on the function f and g whichdescribe the nonlinear kinetics of the system, were missing. In this master thesiswe included all assumptions and details needed in order to prove existence anduniqueness of travelling waves in one spatial dimension, which could make thearticles more valuable.

Besides the existence and uniqueness, the stability of waves should also beconsidered. In this master thesis we decided not to consider the stability, sinceit can be seen as a separate topic. Therefore it would be an interesting topic asa follow-up.

In two spatial dimensions, see Chapter 5, the curvature of a wave plays an im-portant role in the normal velocity of a spiral wave front. With the parametriza-tion of a spiral wave as given in Section 5.2, our goal was to nd an expressionfor the shape of the spiral wave front θ(r) and the angular velocity ω < 0 of thespiral wave. For θ(r) we have found an expression in the form of an integral:

θ(r) =∫ ψ(r)

r dr. In most cases this integral should be evaluated numerically,since it is not possible to nd an explicit solution. Based on other articles wegave an example for which we tted ψ(r) with a parabola so that we were ableto nd an explicit solution for θ(r).

We found a restriction on ωε depending on c

ε and a,(cε

)4 ≥ 4a2(ωε

)2, for

which there could exist a solution ψ(r), and therefore θ(r), that satises theboundary conditions ψ(r0) = 0 = ψ(r1).Instead of an explicit expression for ω we found a pair

(r1,

ωε

)as a function of

r0 and cε , so that Ω′

(r0,

cε

). The details of this relationship are not yet clear

and are not given in the articles.According to [3] for r1 suciently large, ω should be independent of r1, so thatthe relationship can be represented as

ω =ε

r21

Ω

(r1c

ε,r0

r1

).

Unfortunately the reason for this relationship is not clear to us. There is aninteresting analysis possible for this relationship in a follow-up.

44

A Stability using the Center Manifold Theorem

Theorem 2 (Theorem 18.1.2 (Existence), see [15] ). There exists a Cr manifoldfor

x = Ax+ f(x, y)

y = By + g(x, y). (A.1)

The dynamics of (A.1) restricted to the center manifold is, for u sucientlysmall, given by the following c-dimensional vector eld

u = Au+ f(u, h(u)), u ∈ Rc. (A.2)

Theorem 3 (Theorem 18.1.3 (Stability), see [15]). i) Suppose the zero so-lution of (A.2) is stable (asymptotically stable) (unstable); then the zerosolution of (A.1) is also stable (asymptotically stable) (unstable).

ii) Suppose the zero solution of (A.2) is stable. Then if (x(t), y(t)) is a so-lution of (A.1) with (x(0), y(0)) suciently small, there is a solution u(t)of (A.2) such that as t→∞

x(t) = u(t) +O(e−γt),

y(t) = h(u(t)) +O(e−γt),

where γ > 0 is a constant.

Consider system (4.34)

u′ = w

w′ = −cw − f(u, v−), (A.3)

Theorem 4. If v0 = v−, the steady state (u,w) = (h−(v−), 0) is locally stable.

Proof. (4.34) has as steady state solutions (u,w) = (h±(v−), 0). Note that inthis case, the third steady state (h0(v−), 0) has coincided with (h−(v−), 0).

Consider the steady state (u,w) = (h−(v−), 0). This steady state can betransformed to the origin via the translation (u, w) = (u − h−(v−), w). In thiscase (4.34) becomes

u′ = w

w′ = −cw − f(u+ h−(v−), v−), (A.4)

so that the steady state becomes (u, w) = (0, 0).Let f u) := f(u+ h−(v−), v−). Note that f ′ = 0 and f ′′ > 0. For simplicity wewill write

f ′ := f ′(0), and f ′′ := f ′′(0).

The Taylor expansion of f(u) around u = 0 is:

f(u) = f(h−(v−), v−) + uf ′ +1

2u2f ′′ +O(u3)

=1

2u2f ′′ +O(u3).

45

System (A.4) can be rewritten as(u′

w′

)= A

(uw

)+ F (u, w), (A.5)

with

A =

(0 10 −c

),

and

F (u, w) =

(0

− 12 u

2f ′′

).

We can now nd a linear transformation T which transforms A into block diag-onal form. The eigenvalues of A are λ− = −c and λ+ = 0, and has eigenvectorsev− = (− 1

c , 1)T and ev+ = (1, 0)T respectively. Therefore we can write

D =

(0 00 −c

), T =

(1 − 1

c0 1

), T−1 =

(1 1

c0 1

),

and A can be written asA = TDT−1.

Let T−1(u, w)T = (x, y)T . Then (A.5) transforms into(x′

y′

)= D

(xy

)+ T−1F (T (x, y)), (A.6)

with

T−1F (T (x, y)) =

(− 1

2c

(x− y

c

)2f ′′

− 12

(x− y

c

)2f ′′

).

Now we can use the theory of center manifolds of[15], in particular Theorem18.1.2 and 18.1.3. Let y = k(x). Dierentiating with respect to time impliesthat y′ = k′(x)x′. Substituting (A.6) gives

k′(x)x′ = y′

−k′(x)1

2c

(x− k(x)

c

)2

f ′′ = −ck(x)− 1

2

(x− k(x)

c

)2

f ′′ (A.7)

. (A.8)

We assume that k(x) has the form k(x) = a1x2 + a2x

3 + O(x4). Substitutingk(x) into (A.8) gives

f ′′

2c(2a1x+ 3a2x

2 + . . .)

(x− a1x

2 + a2x3 + . . .

c

)2

=

c(a1x2 + a2x

3 + . . .) +f ′′

2

(x− a1x

2 + a2x3 + . . .

c

)2

. (A.9)

In order for (A.9) to hold, the coecients of corresponding powers of x on bothsides have to be equal:

O(x2) : 0 = ca1 +f ′′

2⇒ a1 = −f

′′

2c

O(x3) :f ′′

ca1 = ca2 −

f ′′

ca1 ⇒ a2 = − (f ′′)2

c4,

46

so we have

k(x) = −f′′

2cx2 − (f ′′)2

c4x3 +O(x4).

Using Theorem 18.1.2, the vector eld restricted to the center manifold is givenby

x′ = −f′′

2c

(x+

f ′′

2c2x2 +O(x3)

)2

= −f′′

2cx2 +O(x3).

Hence there exists a δ 1 such that for |x| < δ, x = 0 is stable. Hence, byTheorem 2, (x, y) = (0, 0) is locally stable.

In original coordinates (u,w) the ow near (h−(v−), 0) is given by

(u,w) =

(x+

f ′′

2c2x2 + h−(v−) +O(x3),−f

′′

2cx2 +O(x3)

).

Hence (h−(v−), 0) is stable for all |u| < h−(v−) + δ.

47

References

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[3] J.P. Keener, A geometrical theory for spiral waves in excitable media. SIAMJournal on Applied Mathematics, Volume 46, Number 2, 10391056, De-cember 1986.

[4] J.P. Keener and J.J. Tyson, Spiral waves in the Belousov-Zhabotinskii re-action. Physica D 21, 307324, 1986.

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[9] L.S. Schulman and P.E. Seiden, Percolation and galaxies. Science, Volume233, 425-431, 1986.

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[11] J.J. Tyson, K.A. Alexander, V.S. Manoranjan and J.D. Murray, Spiralwaves of cyclic AMP in a model of slime mold aggregation. Development106, 421-426, 1989.

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[13] J.J. Tyson and J.D. Murray, Cyclic AMP waves during aggregation of Dic-tyostelium amoebae. Physica D 34, 193-207, 1989.

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