complex oscillations and waves of calcium in pancreatic acinar cells

Physica D 200 (2005) 303–324

Complex oscillations and waves of calcium inpancreatic acinar cells

David Simpson, Vivien Kirk∗, James Sneyd

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand

Received 28 January 2004; received in revised form 9 November 2004; accepted 15 November 2004

Communicated by C.K.R.T. Jones

Abstract

We perform a bifurcation analysis of a model of Ca2+ wave propagation in the basal region of pancreatic acinar cells.The model we consider was first presented in Sneyd et al. [J. Sneyd, K. Tsaneva-Atanasova, J.I.E. Bruce, S.V. Straub, D.R.Giovannucci, D.I. Yule, A model of calcium waves in pancreatic and parotid acinar cells, Biophys. J. 85 (2003) 1392–1405],where a partial bifurcation analysis was given of the model in the absence of diffusion. We obtain more complete informationabout bifurcations of the diffusionless model via numerical studies, then analyse the spatially extended model by numericalinvestigation of the travelling wave equations and direct numerical solution of the model equations. We find solitary waves in themodel equations arising from homoclinic bifurcations in the travelling wave equations. The solitary waves exist and appear tobe stable for a significant interval of the primary bifurcation parameter (i.e., the concentration of inositol trisphosphate) but aree mplicatedm eroclinicb©

K c bifur-

c

1

lm

cellin-

onsy areim-eticaleld

0d

ventually replaced by irregular spatio-temporal behaviour. The homoclinic bifurcations are related to a number of coathematical structures in the travelling wave equations, including an anomalous homoclinic-Hopf bifurcation, hetifurcations between an equilibrium and a periodic orbit, and homoclinic bifurcations of periodic orbits.2004 Published by Elsevier B.V.

eywords:Calcium waves; Pancreatic acinar cells; Travelling waves; Homoclinic bifurcation; Homoclinic-Hopf bifurcation; Homoclini

ation of a periodic orbit

. Introduction

In many cell types, the concentration of free cytoso-ic calcium is organised in a complex spatio-temporal

anner. When stimulated by extracellular agonists

∗ Corresponding author. Tel.: +64 9 3737999; fax: +64 9 3737457.E-mail address:[email protected] (V. Kirk).

(such as hormones or neurotransmitters) manytypes exhibit calcium oscillations and waves, bothtracellular and intercellular. These calcium oscillatiand waves are of particular interest because themathematically complex as well as physiologicallyportant, and thus, over the last decade, the theorstudy of calcium dynamics has grown into a major fiof mathematical physiology[12,17,26,29].

167-2780/$ – see front matter © 2004 Published by Elsevier B.V.oi:10.1016/j.physd.2004.11.006

304 D. Simpson et al. / Physica D 200 (2005) 303–324

One cell type that has been studied extensively isthe pancreatic acinar cell[2,6,16,19,32,36]. Pancreaticacinar cells exhibit calcium oscillations of differenttypes, depending on the agonist used, and these os-cillations can be organised into periodic intercellularwaves that travel around an entire acinus (a ring ofcells arranged around a central duct). In addition tothe complex intercellular behaviour, the waves withineach acinar cell have a complex structure, based uponthe underlying morphological and functional polarityof the acinar cells. Because of these complexities, theexact mechanisms that control oscillations and wavesin pancreatic acinar cells remain unclear.

Detailed descriptions of the physiology of pancre-atic acinar cells can be found in[16,21,22,33–35].Here we give only a very brief description. Addition ofagonist, often acetylcholine (ACh) or cholecystokinin(CCK), stimulates the production of inositol trispho-sphate (IP3), which diffuses through the cytoplasm,binds to IP3 receptors located on the endoplasmic retic-ulum and opens them, leading to the release of largeamounts of calcium into the cytoplasm. This releasedcalcium also affects the open probability of the IP3receptors, leading to sequential positive and negativefeedback. Once the calcium is pumped back into theendoplasmic reticulum, or pumped out of the cell, thecycle can repeat. Thus, the oscillations and waves arecaused by periodic release and reuptake of calciumfrom the internal store, the endoplasmic reticulum. Inaddition to the complex dynamics of calcium release,e ho-l set ly ad salr psa cellt ero-g ares re-g d byc

cre-a col-lt y oft val-u anda wn

physiological complexities of the calcium dynamics.This resulted in a spatially distributed model that con-sisted of seven ordinary differential equations and onereaction–diffusion equation, all with spatially varyingparameters that were determined by fitting to exper-imental data. The full model equations are given inSection 2with the parameters and some supporting in-formation contained inAppendix A.

Although numerical simulations can give some un-derstanding of the model’s behaviour, a more completeunderstanding of the waves and oscillations relies ona detailed study of the bifurcations occurring in themodel. Furthermore, before we can understand the be-haviour of the full spatially distributed system, we needfirst to understand how each region of the cell wouldbehave in isolation. Sneyd et al.[31] performed a par-tial bifurcation analysis of their model in the absenceof diffusion for parameters in both the apical and basalregions. They identified the bifurcations by which thesteady state loses stability in each case, and locatedseveral more local bifurcations in the system modellingthe basal region, but their analysis was not exhaustive,as we will discuss further inSection 2. In this paper,we extend their bifurcation analysis of the model inthe absence of diffusion with parameters appropriateto the basal region, and also obtain results for the casewhere diffusion is included. Further investigation of themodel with the apical parameters, and of the full spa-tially distributed model incorporating both basal andapical regions, is left for future work.

m-i l. Ino lerm a-t tiona hisg re oft InS ves In as ua-t rchf l-l latedt onm rca-t ur-c dic

ach pancreatic acinar cell is functionally and morpogically polarized. The apical region of the cell, cloo the central duct of the acinus, contains not onifferent density of IP3 receptors than does the baegion, but also different densities of calcium pumnd influx channels. Thus, the response of a single

o agonist stimulation shows dramatic spatial heteneity; calcium oscillations in the apical regionignificantly different from oscillations in the basalion, even though the regions are closely connectealcium diffusion.

The most detailed mathematical models of pantic acinar cells are due to Sneyd, Yule, and their

eagues[1,9,18,30,31]. In their most recent model[31],hey incorporated the known spatial heterogeneithe acinar cells by specifying different parameteres for the apical and basal regions of the cell,lso incorporated a significant fraction of the kno

A major aim of our study is to understand the dynacs of the spatially distributed model of Sneyd et arder to do this, it is helpful to look first at two simpodels. InSection 2, we summarise the model equ

ions derived in Sneyd et al. then perform a bifurcanalysis of the model in the absence of diffusion. Tives us important clues about the gross structu

he bifurcation set in the model including diffusion.ection 3, we focus on the existence of travelling waolutions to the partial differential equation model.tandard way, we derive an ordinary differential eqion model (the ‘travelling wave equations’) and seaor homoclinic bifurcations of equilibria in the traveing wave equations, as these correspond to isoravelling waves in the full partial differential equatiodel. In doing so, we uncover a zoo of other bifu

ions, including an anomalous homoclinic-Hopf bifation, numerous homoclinic bifurcations of perio

D. Simpson et al. / Physica D 200 (2005) 303–324 305

orbits, and heteroclinic bifurcations between an equi-librium solution and a periodic orbit. While many ofthese bifurcations may have no physiological signifi-cance, some are of considerable mathematical interest,and we consequently describe the associated dynamicsin more detail than would be appropriate in a solelyphysiological context. Although analysis of the trav-elling wave equations tells us about the existence ofisolated travelling waves in the partial differential equa-tions, it does not determine the stability of the travellingwaves in the full system. InSection 4, we report on nu-merical simulations of the partial differential equationmodel by which we determine whether the branchesof travelling wave solutions located inSection 3areobserved (i.e., are asymptotically stable) in the partialdifferential equations.Section 5summarises our resultsand discusses the implications of our analysis for theunderlying physiological system.

The numerical results presented in this paper werecomputed using the bifurcation and continuation pack-ages AUTO[4] and XPPAUT[5].

2. The model in the absence of diffusion

Sneyd et al.[31] derive the following model equa-tions:

∂c

∂t= Dc

∂2c

∂x2+ (kfPIPR + v1PRyR + Jer)(ce − c)

)

dI2

dt= φ5A − (k−1 + l−2)I2, (7)

dw

dt= k−

c (w∞ − w)

w∞ , (8)

wherec is the concentration of free Ca2+ ions,ce theconcentration of calcium ions in the endoplasmic retic-ulum, and the quantitiesR,O,A, I1, I2, andw representthe fraction of receptors in various states. See[31] fordetails of the derivation of the model and the physio-logical meaning of the variables. The quantitiesJpm,Jmito, Jserca, Jin, PRyR, PIPR, w∞ and theφi are func-tions of the variables; these functions are all specifiedin Appendix A, as are the values of the constants usedin the numerical simulations described below. The pa-rameterp in these equations denotes the concentrationof inositol trisphosphate (IP3) and will be one of themain bifurcation parameters we consider.

Before we examine in detail the behaviour of thefull partial differential equation model, Eqs.(1)–(8), itis helpful to examine the model in the absence of dif-fusion, i.e., withDc = 0, because, as we will see, thisgives information about some of the important large-scale features of the bifurcation structure of the fullmodel. In this case, Eq.(1) reduces to an ordinary dif-ferential equation:

dc

dt= (kfPIPR + v1PRyR + Jer)(ce − c)

T -t t al.p ht-d di-a sto s ofC ast el hto ofp is ofs solu-t ona as ab

ni ly

− Jserca− Jmito + δ(Jin − Jpm), (1)

1

γ

dce

dt= −(kfPIPR+v1PRyR + Jer)(ce − c) + Jserca,

(2

dR

dt= φ−2O − φ2pR + (k−1 + l−2)I1 − φ1R, (3)

dO

dt= φ2pR − (φ−2 + φ4 + φ3)O + φ−4A + k−3S,

(4)

dA

dt= φ4O − (φ−4 + φ5)A + (k−1 + l−2)I2, (5)

dI1

dt= φ1R − (k−1 + l−2)I1, (6)

− Jserca− Jmito + δ(Jin − Jpm). (9)

his equation together with Eqs.(2)–(8) forms a sysem of eight ordinary differential equations. Sneyd eerformed a partial bifurcation analysis of this eigimensional system and produced the bifurcationgram reproduced inFig. 1. They found there is june steady state solution for positive concentrationa2+ and IP3 (i.e.,c ≥ 0,p ≥ 0). This steady state h

wo Hopf bifurcations at positivep, and is stable to theft of the left-most Hopf bifurcation and to the rigf the right-most Hopf bifurcation. For each valuebetween the Hopf bifurcations the steady state

addle type, and there exists at least one periodicion. In the following, we refer to a periodic orbit

branch created in one of the Hopf bifurcationsasic periodic orbit.

A notable feature ofFig. 1is that no stable solutios identified for values ofp between approximate


Fig. 1. Partial bifurcation diagram for Eqs.(2)–(9) with the param-eter values given inAppendix A(after[31]). The curve labelled ‘ss’denotes steady-state solutions; all other curves represent periodic or-bits by plotting the maximum value ofc over the orbit. Solid (resp.dashed) curves denote stable (resp. unstable) solutions. HB denotesa Hopf bifurcation and PD denotes a period-doubling bifurcation.

3.37 and 8.80. We have performed a more detailedbifurcation analysis which reveals more structure inthe bifurcation diagram, as shown inFig. 2. WhereSneyd et al.[31] found two pairs of period-doublingbifurcations, we now find that each pair is in fact onlythe beginning of a longer (perhaps infinite) sequenceof period-doubling bifurcations. Only the first threeperiod-doubling bifurcations in the two sequences andthe resultant periodic orbits are shown inFig. 2.

There are also further complications. As seen inFig. 2, there are several disjoint bubbles of periodic or-bits in the bifurcation diagram, each bubble comprisedof many branches of periodic orbits. Only six of theseare shown inFig. 2, but there are more bubbles lyingto the left of bubble 8. (The reason for labelling thebubbles as we do will become apparent later.)

The detailed structure of bubble 3 is as follows. Ateach end of bubble 3 there is a saddle-node bifurcationof periodic orbits, with the lower branch of periodicorbits created in the bifurcation being of saddle type.The upper branch of periodic solutions is initially sta-ble near each saddle-node bifurcation, but there are se-quences (perhaps infinite) of period-doubling bifurca-

Fig. 2. A more complete bifurcation diagram for Eqs.(2)–(9) (cf.Fig. 1). The lowest curve denotes steady-state solutions; all othercurves represent periodic orbits by plotting the maximum value ofcover the orbit. A dashed line denotes an unstable solution. Six bub-bles are shown, each containing two sequences of period-doublingbifurcations.

tions at both ends, in a similar fashion to that describedabove for the basic periodic orbit. All bubbles we haveobserved have a similar structure.

Fig. 3displays time series for several different val-ues ofp. In Panel A,p = 8.7, which is to the left ofthe first two reverse period-doubling bifurcations ofthe basic periodic orbit. As expected, the time series isperiodic with a repeating section containing four differ-ent peaks, corresponding to a period-quadrupled orbit.Whenp = 7.8, there are at least two distinct attractorscoexisting in the phase space. There is an apparentlystrange attractor which arises from period-doubling bi-furcations of the basic periodic orbit and also a stableperiodic orbit belonging to bubble 3. Panel B shows atime series forp = 7.8. The initial condition used re-sults in a solution that is attracted to the stable periodicorbit on bubble 3 but some solutions do end up at theother attractor. Asp is decreased further, the periodicorbit from bubble 3 also undergoes multiple period-doubling bifurcations and produces a second compli-cated attractor. Byp = 7.0 the formerly distinct at-tractors seem to have merged (see Panel C). The timeseries in Panel C contains many peaks; the peaks with


Fig. 3. Time series corresponding to Eqs.(2)–(9) for various values of the parameterp, after transients have died away: (A)p = 8.7, with initialcondition (c, ce, R,O,A, I1, I2, w) = (0.5,72,0.02,0.5,0.3,0.004,0.2,0.99); (B)p = 7.8 with initial condition (c, ce, R,O,A, I1, I2, w) =(0.25,68,0.06,0.1,0.5,0.002,0.38,0.90); (C)p = 7.0, initial condition as in (A); (D)p = 5.0, initial condition as in (A).

the largest values ofcarise from traversals near the pe-riodic orbits associated with bubble 3 with other peaksresulting from the trajectory passing near the basic pe-riodic orbit and its doubled orbits. This phenomenon,i.e., one solution passing near both of the formerly dis-tinct attractors, is observed over a very long time span,which suggests that byp = 7.0, the two attractors havemerged. Numerical simulations suggest that mergingoccurs when one or both of the attractors collides withthe lower (saddle-type) branch of fundamental periodicorbits in bubble 3. Various scenarios for merging of at-tractors of this type have been seen elsewhere and are

discussed, for instance, in[24] but we do not pursuethis further here. Panel D displays a time series forp = 5.0. This time series also has many peaks and ap-pears to correspond to a trajectory that visits three ormore bubbles that have, presumably, merged.

In Fig. 4, the period of the fundamental orbit in eachbubble is plotted as a function ofp. It can be seen thatthe period of each orbit does not vary greatly withpand is approximately a multiple of 7.5. For instance,the fundamental orbit in bubble 3 has period close to22.5 = 3 × 7.5. Although not shown inFig. 4, periodicorbits on the branch that bifurcates from the Hopf bi-


Fig. 4. Partial bifurcation diagram for Eqs.(2)–(9)showing variationof the period of the fundamental periodic orbit in the six bubbles.Stability is not indicated.

furcations have period close to 7.5 for the values ofpused to plotFig. 4. Thus the name ‘bubblen’ indicatesthat the fundamental periodic orbit in the bubble hasperiod approximatelyn times that of the basic periodicorbit.

In summary, we have now located numerically oneor more stable solutions for each value ofp in the inter-val [1,10], for the model in the absence of diffusion.Depending onp, the stable solution may be periodic,being either the basic periodic orbit or the fundamentalorbit in an bubble or arising in a period-doubling bifur-cation of one of these orbits. Alternatively, the stablesolution may be an attractor created at the end of aperiod-doubling cascade or created by the merging ofseveral other attractors.

3. Travelling wave equations

We now return to the full partial differential equationmodel, Eqs.(1)–(8), to investigate the effect of includ-ing diffusion in the model. We are particularly inter-ested in whether travelling wave solutions exist, sincesuch solutions are thought to be important physiologi-cally, and so rewrite the equations using the travelling

wave coordinate

ξ = x + st, (10)

wheres is the wave speed. Substituting this into(1)yields a second-order ordinary differential equation

Dccξξ − scξ + F = 0, (11)

where

F = (kfPIPR + v1PRyR + Jer)(ce − c)

− Jserca− Jmito + δ(Jin − Jpm), (12)

and where

cξ ≡ ∂c

∂ξ, cξξ ≡ ∂2c

∂ξ2. (13)

Eq. (11) can be rewritten as two ordinary differentialequations by introducing a new variabled:

cξ = d, (14)

Dcdξ = sd − F. (15)

The remaining seven model equations are modifiedonly slightly by changing to travelling wave coordi-nates; the right-hand side of each equation is scaled bys−1 and differentiation is now with respect toξ. The re-sult is a system of nine ordinary differential equations,called the travelling wave equations. These equationshave two bifurcation parameters, i.e.,p, the concentra-t ilyi eree iso-l ua-t rav-e thet fullp bil-iW icso -io e ofd as-i lesd

vee itive

ion of IP3, ands, the wave speed. We are primarnterested in finding parameter values for which thxist homoclinic orbits because these correspond to

ated travelling waves in the partial differential eqions. (We note however, that while existence of tlling waves can be determined from analysis of

ravelling wave equations, we must return to theartial differential equation model to determine sta

ty of any travelling waves we identify. SeeSection 4.).e are also interested in investigating the dynam

f the travelling wave equations ass gets large; Magnu [20] showed that takings → ∞ results in a systemf equations identical to the model in the absenciffusion, so by examining the dynamics for incre

ng values ofs, we can understand how the bubbescribed in the last section are formed.

As in the case of no diffusion, the travelling waquations have just one equilibrium solution at pos


concentrations of calcium ions and IP3. This equilib-rium has a Hopf bifurcation forp ands values on thecurve labelled HB inFig. 5. The bifurcation curvehas vertical asymptotes forp ≈ 2.07 andp ≈ 30.7,and for each value ofp between the asymptotes thereis a unique value ofs giving a Hopf bifurcation. Theequilibrium solution is always of saddle type. Forvalues ofp ands below the Hopf bifurcation curve,the equilibrium has eight eigenvalues with negativereal part and one with positive real part, while abovethe Hopf bifurcation curve, the equilibrium has sixeigenvalues with negative real part. The importantpoint to note here is that the extra dimension in thetravelling wave equations introduces an extra positiveeigenvalue, destroying stability of the equilibrium.In fact, as will be seen in this section, we find nostable solution of any kind in the travelling waveequations.

There are several branches of homoclinic orbits inthe travelling wave equations, as illustrated inFig. 5.We used AUTO to locate these orbits by the standardtechnique of following nearby periodic orbits of fixed,high period (in most cases, using a period of 10,000).

HB-C)

urca-toteselves

Branch A of homoclinic orbits (denoted HC-A inFig. 5) is of particular interest because, as will be seenlater, part of this branch corresponds to stable travel-ling waves in the full model with diffusion, with thetravelling waves existing over a significant interval inp. On the scale ofFig. 5, the lower end of branch Aappears to terminate on the locus of Hopf bifurcationsat p ≈ 2.65. However, careful numerical simulationsreveal that branch A turns around just to the left ofthe locus of Hopf bifurcations and doubles back on it-self, following a path indistinguishable on the scale ofFig. 5from the original path of branch A. Atp ≈ 1.887the branch turns back yet again; thereafter, the branchzigzags between the locus of the Hopf bifurcation andthe position of the turning point atp ≈ 1.887. The na-ture of the homoclinic orbit changes as we zigzag alongbranch A.Fig. 6, Panel A, shows a homoclinic orbit onbranch A the first time the branch approaches the Hopfbifurcation. Panels B and C in the same figure showthe homoclinic orbit as branch A approaches the Hopfbifurcation for the second and tenth times; we see thateach time branch A doubles back the homoclinic orbitgains a new large loop, with the loops accumulatingon the periodic orbit shown inFig. 6. This periodic or-bit is a basic periodic orbit; following the orbit withfixed s = 6.73, we find that it originates in a Hopf bi-furcation on the right arm of the Hopf locus. Thus itseems that the homoclinic connection corresponding tobranch A eventually becomes a heteroclinic connectionbetween the equilibrium and the basic periodic orbit.T dy-n f ho-m ata thateH ex-p in-t ld oft an-i ses lsob tablem nalu c-c theo noti mer a

his phenomenon is somewhat reminiscent of theamics near a homoclinic T-point where a branch ooclinic bifurcations of an equilibrium is terminatedco-dimension 2 heteroclinic bifurcation betweenquilibrium and a second equilibrium (see, e.g.,[11]).owever, in our case the heteroclinic bifurcation isected to be co-dimension 1, since it involves the

ersection of the one-dimensional unstable manifohe equilibrium and the eight-dimensional stable mfold of the periodic orbit in a nine-dimensional phapace. (At the heteroclinic bifurcation, there will ae an intersection between the eight-dimensional sanifold of the equilibrium and the two-dimensionstable manifold of the periodic orbit; this will our in a co-dimension zero way, and will not affectverall co-dimension of the bifurcation.) We have

nvestigated this heteroclinic bifurcation further. Soecent work on heteroclinic bifurcations involving

Fig. 5. Partial bifurcation set for the travelling wave equations.denotes the locus of Hopf bifurcations. HC-A (resp. HC-B and HCdenotes branch A (resp. branches B and C) of homoclinic biftions. Ass increases, the Hopf locus tends to two vertical asympwhereas the homoclinic branches B and C turn back on themsat abouts = 43 ands = 82, respectively.


Fig. 6. Phase portraits for the homoclinic orbit on branch A near the lower end of branch A, atp = 2.2: (A) the first time branch A approachesthe Hopf locus; (B) the second time branch A approaches the Hopf locus; (C) the tenth time branch A approaches the Hopf locus. In each panel,a dashed loop indicates the location of the basic periodic orbit, although in Panel C this periodic orbit is obscured by five large loops on thehomoclinic orbit which lie close together and appear as a single thick loop near the bottom of the panel.

periodic orbit and an equilibrium has been reported byRademacher[23], but in that case, the heteroclinic bi-furcation is co-dimension 2; the results of Rademacherseem not to apply to the phenomenon at the lower endof branch A, but are related to the termination of ho-moclinic branches B and C as discussed below.

We note that in our numerics, branch A doubledback on itself in this way a total of 10 times beforewe stopped the simulation, and it seemed that this pro-cess could continue indefinitely. However, we foundthe path of branch A by following a nearby periodicorbit of fixed high period (actually period 10,000); asmore loops are added to the homoclinic orbit, we ex-

pect the period 10,000 orbit might eventually divergefrom the true homoclinic orbit.

The period 10,000 approximation to branch A can becontinued from its lower end at (p, s) ≈ (2.65,6.73) to(p, s) ≈ (3.0,13.4), with no further back-tracking oc-curring. At (p, s) ≈ (2.18,13.39), this approximationto the homoclinic bifurcation curve crosses the locusof Hopf bifurcations. Since there is only one equilib-rium solution to the travelling wave equations there isa homoclinic-Hopf bifurcation at the crossing point.Such a bifurcation has been studied in three dimen-sions by Hirschberg and Knobloch[13], and by Dengand Sakamoto[3] in three or higher dimensions in the


Fig. 7. Partial bifurcation set and phase portraits for an unfoldingof a homoclinic-Hopf bifurcation, with unfolding parametersµ andσ (after [13]). The Hopf bifurcation is assumed to occur atµ = 0and the homoclinic bifurcation of the equilibrium atσ = 0, µ ≤ 0.The phase portraits show the equilibrium solution and part of theunstable manifold of either the equilibrium solutions (forµ < 0) orthe periodic orbit created in the Hopf bifurcation (forµ > 0).

case that the homoclinic orbit is contained in the cen-tre manifold of the equilibrium at the Hopf bifurca-tion. They found that homoclinic bifurcations of theequilibrium exist on one side only of the homoclinic-Hopf point, with homoclinic orbits of the periodic orbitcreated in the Hopf bifurcation occurring on the otherside of the homoclinic-Hopf point (seeFig. 7). This is

not what we see in numerics; inspection of phase por-traits near the period 10,000 approximation to branchA suggests that there are homoclinic bifurcations ofthe equilibrium on both sides of the Hopf bifurcationcurve. For instance,Fig. 8shows phase portraits com-puted at two points on branch A, one to the left ofthe Hopf bifurcation curve and one to the right; thehomoclinic orbit looks similar in the two cases. Anunexpected feature of the phase portrait inFig. 8(A)is that the homoclinic orbit approaches the equilib-rium without spiralling. At this point, the eigenval-ues of the flow linearised about the equilibrium are−2.26, −2.16, −1.19, −0.069, −0.0075, −0.0039,−0.0017± 0.068i, 0.81; the eigenvalues with nega-tive real part closest to zero are complex, so the lackof spiralling indicates that the homoclinic orbit doesnot approach the equilibrium on the slow stable man-ifold. This observation provides a possible explana-tion for the persistence of the homoclinic bifurcationbeyond the Hopf bifurcation: the Hopf bifurcation isnot ‘seen’ by the homoclinic orbit because the orbitis not tangent to the centre manifold of the equilib-rium near the homoclinic-Hopf point. A more completeexplanation of the apparently anomalous homoclinic-Hopf bifurcation observed here would require a studyof homoclinic-Hopf bifurcations in more than threedimensions (without the restriction that the homo-clinic orbit lie in the centre manifold of the equilib-rium as in[3]), which to our knowledge has not beendone.

Fig. 8. Phase portraits at various points on branch A: (A) Atp = 2.10, to the 0 are0.82,−0.002± 0.068i; (B) atp = 2.60, to the right of the Hopf bifurcation.In (B) the periodic orbit created in the Hopf bifurcation is also shown.

left of Hopf bifurcation. The eigenvalues with real part closest toThe eigenvalues with real part closest to 0 are 0.008± 0.065i,−0.005.


We note that the flow linearised about the equi-librium solution only has complex eigenvalues whenp > 2.44 on the lower part of branch A and whenp > 2.02 on the upper part of the branch. Homo-clinic orbits on this region of the lower branch ex-hibit spiralling, and the eigenvalues with real part clos-est to zero satisfySil’nikov’s condition [27] (i.e., ifthe critical eigenvalues areλ, µ ± iω whereµλ < 0then|µ/λ| < 1), which implies the existence of horse-shoes and other complicated dynamics[10]. As willbe seen, the travelling waves in the spatially extendedmodel corresponding to this part of branch A areunstable, so we do not pursue the possible compli-cated dynamics associated with these homoclinic orbitsfurther.

Sil’nikov’s condition is also satisfied by the criticaleigenvalues of the linearised flow on the upper partof branch A forp > 2.02 but no spiralling is seen forthe corresponding homoclinic orbits until very close tothe computed end of the branch at (p, s) ≈ (3.0,13.4).The AUTO computations terminate abnormally at thispoint (or nearby, with the exact position depending onthe level of accuracy used in the computations and theperiod selected for the high period approximation to thehomoclinic orbit), but in all cases the termination of thebranch seems to be associated with the appearance ofspiralling in the homoclinic orbit. We have not beenable to determine the precise cause of the terminationof branch A at its upper end, but speculate that it islinked to the homoclinic orbit finally ‘seeing’ the parto plexe roacht

itso rsae l-c es)v rnsa ar-i nchB ondt epa sest onb luesw tive

Fig. 9. Bifurcation diagram of homoclinic branches B and C plottedin a coordinate system that best distinguishes different parts of thebranches. The upper (resp. lower) sections of branch B and C arelabelled B1 and C1 (resp. B2 and C2).

real eigenvalue and one pair of complex eigenvalueswith positive real part, and that the homoclinic orbitsall spiral away from the equilibrium (unlike the casefor the upper part of branch A).

Both parts of branch B appear to terminate ats ≈ 6.Numerical simulation suggests that the mechanism issimilar in both cases, i.e., the homoclinic orbit collideswith the basic periodic orbit (seen inFig. 10). Whenthis occurs there is a heteroclinic loop composed of aconnection from the equilibrium solution to the peri-odic orbit and a second connection from the periodicorbit back to the equilibrium. This phenomenon hassome similarities to the way in which branch A termi-nates at its lower end, but a significant difference is thatbranch B does not zigzag as the heteroclinic bifurca-tion is approached. We note that the dimensions of themanifolds involved in the heteroclinic connection at theends of branches B1 and B2 are different from those atthe end of branch A. Specifically, on branches B1 andB2, the equilibrium has a three-dimensional unstablemanifold and a six-dimensional stable manifold (onepair of unstable eigenvalues are complex, all others arereal), while the periodic orbit has an eight-dimensionalstable manifold and a two-dimensional unstable mani-fold (with all Floquet multipliers being real). The het-eroclinic bifurcation is therefore co-dimension 2, cor-

f the unstable manifold associated with the comigenvalues and consequently being unable to app

he equilibrium as before.We now turn our attention to the homoclinic orb

n branch B. On the scale ofFig. 5, branch B appeas a single curve extending froms ≈ 6 tos ≈ 43. How-ver, by plottingcmax (i.e., the maximum value of caium concentration that the homoclinic orbit achieversuss, as inFig. 9, it can be seen that branch B turound ats ≈ 43 and doubles back on itself. Comp

son of phase portraits on the different parts of braconfirms that the two parts of the branch corresp

o distinct phenomena (seeFig. 10, which shows phasortraits for various points on branch B: two ats = 6nd two ats = 20). We note that branch B never cros

he curve of Hopf bifurcations, that at all pointsranch B the critical eigenvalues (i.e., the eigenvaith real part closest to zero) consist of one nega


Fig. 10. Phase portraits of homoclinic orbits on branch B. The equilibrium solution undergoing the homoclinic bifurcation occurs near the topleft of each phase portrait. Whens = 6 the homoclinic orbits seem to approach a nearby periodic orbit (shown as the dashed loop). Labels B1

and B2 correspond to the labels used inFig. 9. (A) s = 20, on branch B1; (B) s = 20, on branch B2; (C) s = 6, on branch B1; (D) s = 6, onbranch B2.

responding to the higher of the co-dimensions of thetwo heteroclinic connections that occur in the bifur-cation. This type of heteroclinic bifurcation has beenstudied by Rademacher[23], who reports that a mono-tone approach to the heteroclinic bifurcation in thetwo-parameter bifurcation set results from the lead-ing Floquet multipliers of the periodic orbit beingreal.

Although both parts of branch B appear to terminateat a heteroclinic bifurcation involving the equilibriumand the basic periodic orbit, there is a difference be-tween the two cases which is hinted at inFig. 10. Thehomoclinic orbit shown in Panel D passes close to thebasic periodic orbittwicebefore returning to the equi-librium; it does not pass near the equilibrium twice, sois not a double-pulse homoclinic orbit of the equilib-

rium but looks as though it may be near a double-pulsehomoclinic orbit of the basic periodic orbit. Phase por-traits obtained at points even closer to the end of thebranch than in Panel D show more of this character.Thus, the heteroclinic orbit at the end of branch B1may lie near a single-pulse homoclinic orbit of the pe-riodic orbit while the heteroclinic orbit near the end ofbranch B2 may be nearer a double-pulse homoclinicorbit.

The homoclinic bifurcations occurring on branchC are similar to those on branch B. In particular, thebranch extends froms ≈ 6 to s ≈ 82. At s ≈ 82 thebranch turns around and doubles back. The ends of thebranch again terminate ats ≈ 6 when the homoclinicorbit collides with the basic periodic orbit, coincidingwith the formation of a heteroclinic connection from


Fig. 11. Bifurcation diagrams for the travelling wave equations whens = 8. (A) The dashed line represents a curve of equilibria and the solid linesrepresent curves of maximum values ofc over a periodic orbit. Stability is not indicated. HB: Hopf bifurcation. HC-A: homoclinic bifurcation,branch A. PD: period-doubling bifurcation. (B) Period vs.p for various branches of periodic solutions. The dynamics associated with each ofthe branches (a)–(f) is explained in the text. Branches (a) and (b) also appear in Panel A, but (c)–(f) do not. The asterisks mark the locations onbranch (b) corresponding to the periodic orbits plotted inFig. 13.

the equilibrium to the periodic orbit. There may alsobe other branches of homoclinic bifurcations of thistype that we have not found.

We note that while the existence of a homoclinicorbit in the travelling wave equations implies the exis-tence of a travelling wave solution in the partial differ-ential equation model, we cannot determine stability ofthe travelling wave solution from the travelling waveequations, but must either solve the partial differentialequations numerically or perform further theoreticalcalculations (for instance, as done for a similar modelby Romeo and Jones[25]). Stability of the travellingwave solutions in our model, corresponding to branchesA, B and C, is discussed inSection 4.

Fig. 5does not contain complete information aboutthe bifurcations corresponding to the travelling waveequations; there are many other bifurcations (such asperiod-doubling bifurcations and homoclinic bifurca-tions of periodic orbits) that are not shown. We nowillustrate some of the more important features of thebifurcation set, missing fromFig. 5, by constructing bi-furcation diagrams for selected fixed values ofs. We donot claim to provide a complete description of the dy-namics associated with the travelling wave equations;there are many bifurcations that we have found but donot include and certainly many that we have not found.We focus on those which arise via a primary or higher-

order bifurcation from the equilibrium solution and onthe origin of the bubbles found in the model in the ab-sence of diffusion.

The minimum value ofs for which there are Hopfbifurcations iss ≈ 4.87 (seeFig. 5). For s just largerthan 4.87, e.g.s = 5, the bifurcation diagram is simple;the branch of periodic orbits that emanates from one ofthe two Hopf bifurcations continues until terminatingat the other Hopf bifurcation. For such a low value ofs, there are no further bifurcations of the basic periodicorbit or the equilibrium. However whens = 8, the pic-ture is very different. The branch of periodic orbits thatoriginates from the right-hand Hopf bifurcation nowterminates at a homoclinic bifurcation, on homoclinicbranch A inFig. 5(seeFig. 11). On the other hand, pe-riodic orbits which bifurcate from the left-hand Hopfbifurcation end up at a homoclinic bifurcation of thebasic periodic orbit. In fact, many of the branches ofperiodic orbits shown inFig. 11 terminate at homo-clinic bifurcations of the basic periodic orbit, as wenow describe.

A homoclinic orbit to a periodic solution exists whenthe stable and unstable manifolds of the periodic orbitintersect. Such an intersection is generally transverseand hence structurally stable. Thus, in a one-parameterbifurcation diagram such asFig. 11, a homoclinic orbitto a periodic orbit exists (if at all) over an interval of


Fig. 12. Possible relative positions of the stable and unstable mani-folds of a periodic orbit, P, when there is a trajectory homoclinic toP. Each picture shows a schematic phase portrait illustrating the dy-namics induced on a cross-section transverse to P. Different relativepositions of the manifolds may be obtained by varying a bifurcationparameter (denotedλ here). (A) Transverse intersections of the man-ifolds create a structurally stable homoclinic orbit, which will existover an interval in the bifurcation parameter (say,λ ∈ (λ1, λ2)). (B)The ‘first’ tangency between the stable and unstable manifolds ofP occurs atλ = λ1. (C) The ‘last’ tangency between the stable andunstable manifolds of P occurs atλ = λ2.

the bifurcation parameter. The ends of the interval cor-respond to first and last tangencies of the manifolds, asin Fig. 12.

The dynamics near single homoclinic tangencies hasbeen studied in[8,7] where it is shown that a countablenumber of saddle-node bifurcations of periodic orbitsoccur near each tangency. When the tangencies are em-bedded in a one-parameter family, the tangencies typi-cally come in pairs, with the periodic orbits created inthe saddle-node bifurcations near one tangency beingdestroyed in the saddle-node bifurcations near the othertangency either by forming loops of periodic orbits[15]or by forming a continuous wiggly curve[13,14]. Thelatter case seems to occur in our system. For instance,the branch of periodic orbits that emerges from the left-most Hopf bifurcation whens = 8 (Fig. 11, Panel B,branch (b)) wiggles back and forth, with the left turningpoints on the wiggly branch accumulating atp ≈ 7.32

and the right turning points accumulating atp ≈ 14.4.As we move up curve (b) in Panel B, the periodic or-bit gains more and more large loops close to the basicperiodic orbit, supporting our claim that we are near ahomoclinic bifurcation of a periodic orbit; we conjec-ture that the points of accumulation of the saddle-nodebifurcations correspond to first and last homoclinic tan-gency of the basic periodic orbit. The phase portraitsfor the periodic orbits at two places on branch (b) areshown inFig. 13. It can be seen that the periodic orbitgains three large loops from one phase portrait to thenext, and the same thing is seen all the way up, i.e.,the periodic orbit gains three loops for each completewiggle of branch (b).

There are 10 places inFig. 11, Panel B, where theperiod of a periodic orbit gets very large. Five of thesecorrespond to homoclinic bifurcations of equilibria,i.e., on branches A (see curve (a)), branches B1 andB2 (which are indistinguishable on the scale of the fig-ure, and occur at the left end of curve (c) and the rightend of curve (d)), and branches C1 and C2 (which arealso indistinguishable on this scale, and occur at theleft end of curve (e) and the right end of curve (f)).The other five places where the period of the periodicorbit gets large correspond to homoclinic bifurcationsof the basic periodic orbit. These bifurcations occur invarious ways. For instance, on the left end of the curvelabelled (d) inFig. 11, Panel B, the periodic orbit gainsone large loop for each complete wiggle of the branchwhere as on curve (b) described above, three large loopsa . Ther ulseh e tot ngh

ofp eri-o so s inF area or-b onds (d)s ofph el tl -

re gained as each complete wiggle is traversedight end of curve (c) corresponds to a double-pomoclinic orbit, i.e., the periodic orbit passes clos

he equilibrium twice before closing, and the limitiomoclinic orbit will presumably do the same.

A scaling for the loci of saddle-node bifurcationseriodic orbits near a homoclinic tangency of a pdic orbit is given in[7]. Comparison of the positionf the saddle-node bifurcations on the wiggly curveig. 11gives further evidence that these branchesssociated with homoclinic bifurcations of periodicits. For instance, we consider the loci of every secaddle-node bifurcation on the wiggly part of curvehown inFig. 11, Panel B. Ifpn denotes the valueat thenth of these saddle-node bifurcations (n = 1aving the leastp value), then[7] predicts that in th

imit of large n, plottingpn versusn yields a straighine with slope ln(1/λs), whereλs is the largest sta


Fig. 13. Phase portraits for periodic orbits at two places on branch (b) inFig. 11(p = 10, s = 8). The positions corresponding to these phaseportraits were marked with asterisks inFig. 11.

ble Floquet multiplier of the basic periodic orbit. Forvalues ofp relevant here,λs is approximately constant,equal to≈ 0.415, so the straight line expected in thepnversusnplot has slope≈ −0.879.Fig. 14plotspn ver-susn for the saddle-node bifurcations on curve (d). A90% confidence interval for the least-squares fit of theslope is [−1.13,−0.47], which is in good correspon-dence with the theoretical prediction, as illustrated inFig. 14.

Fig. 15shows bifurcation diagrams fors = 10. Inthis case, the branch of periodic orbits arising fromthe right-hand Hopf bifurcation again terminates at ho-moclinic branch A (curve (a)), but, unlike the case fors = 8, there now exist two period-doubling bifurca-tions near each other on this branch of periodic orbits.The periodic orbit arising from the left-hand Hopf bi-furcation terminates at one of these period-doubling bi-furcations (curve (b′)), while the other period-doublingbifurcation throws off a branch of periodic orbits whicheventually terminates at a homoclinic bifurcation onbranch B1 (curve (c′). The global bifurcations associ-ated with the terminations of curves (d) and (e) are asin the cases = 8, described above, with each curve be-ing terminated at one end in a homoclinic bifurcationof an equilibrium and at the other end in a homoclinicbifurcation of a periodic orbit. Branch (g) terminatesin a similar manner. Branch (f) seen inFig. 11 canbe continued tos = 10, but for clarity is not shown inFig. 15. There are further branches of periodic orbitsnot shown in this figure. In particular, in going from

s = 8 to s = 10, the wiggly upper half of branch (b)in Fig. 11has combined (in a complicated way) withthe wiggly right half of branch (c) to produce a newcurve of periodic orbits that is not shown inFig. 15.We do not show this new branch of periodic orbits on

Fig. 14. Comparison of the loci of saddle-node bifurcations onbranch (d) inFig. 11, Panel B, with the theoretical prediction from[7], as described in the text. The loci of the saddle-node bifurcationsare plotted as circles, and the least-squares fit through these pointshas slope−0.80 (not shown). The dotted lines have slopes−1.13and−0.47, corresponding to the endpoints of the 90% confidenceinterval of the least-squares fit for the slope, with the intercepts cho-sen to minimise the least-squares error. The solid line has slope equalto the theoretical prediction (i.e.,−0.879) with the intercept againchosen to minimise the least-squares error.


Fig. 15. Bifurcation diagrams fors = 10. (A) The dashed line represents a curve of equilibria and the solid lines represent curves of maximumvalues ofc over a periodic orbit. Stability is not indicated. HB: Hopf bifurcation. HC-A: homoclinic bifurcation, branch A. HC-B1: homoclinicbifurcation, branch B1. PD: period-doubling bifurcation. (B) Magnification of indicated section of Panel A. (C) Period vs.p for various branchesof periodic solutions. The dynamics associated with each of the branches (a)–(e), (g) is explained in the text. Branches (a), (b′) and (c′) alsoappear in Panel A, but (d), (e) and (g) do not.

any further bifurcation diagrams, but note that ass in-creases beyonds = 10 many closed loops of periodicorbits (i.e.,isolas) are pinched off from this branch in astraightforward way, with the loops persisting to larges. These isolas will be discussed further below.

Fig. 16 shows bifurcation diagrams in the cases = 12. It can be seen that the branch of periodic or-bits created in one Hopf bifurcation is destroyed inthe other Hopf bifurcation, with similar pairwise cre-ation and annihilation of any periodic orbits created inperiod-doubling bifurcations arising from the basic pe-riodic orbit. Comparison ofFigs. 15 and 16suggeststhat curves (a), (b′) and (c′) in Fig. 15have been recon-figured into two curves by the times = 12, with the lefthalf of branch (b′) in Fig. 15joining onto the right halfof branch (a), thus forming a branch of basic periodicorbits (labelled (b-a) inFig. 16, Panel B), while theleft half of branch (a) inFig. 15joins onto branch (c′),forming an isolated branch of periodic orbits (labelled

(a-c) inFig. 16). Thus, the periodic orbits arising fromthe homoclinic bifurcations on branches A, B and C arenow disconnected from the Hopf bifurcations, which isreminiscent of the situation in the model without dif-fusion (seeFig. 1). The global bifurcations associatedwith the terminations of curves (e) and (g) are almostthe same as fors = 10, with the difference being that,as a consequence of the reconfiguration of the curvesof periodic orbits described above, the homoclinic bi-furcations of periodic orbits that occur at some endsof these branches now involve periodic orbits on theisolated branch (a-c) rather than periodic orbits aris-ing from a Hopf bifurcation. Branch (h) terminates ina similar way.

Fig. 17shows the bifurcation diagram fors = 14.There is no homoclinic orbit from branch A in thisbifurcation diagram, since branch A has an upper end-point ats ≈ 13.4. The other curves of periodic orbitshave also been reconfigured, so that the periodic orbits


Fig. 16. Bifurcation diagrams fors = 12. (A) The dashed line represents a curve of equilibria and the solid lines represent curves of maximumvalues ofc over a periodic orbit. Stability is not indicated. HB: Hopf bifurcation. HC-A: homoclinic bifurcation, branch A. HC-B1: homoclinicbifurcation, branch B1. (B) Period vs.p for various branches of periodic solutions. The dynamics associated with each of the branches is explainedin the text. Branches (b-a), (a-c) also appear in Panel A, but branches (e), (g), (h) do not.

created on homoclinic branches B1 and B2 now forma single, somewhat convoluted branch (branch (d-h)in Fig. 17). We denote by the termhomoclinic isolaabranch of this type, i.e., a single curve of periodic orbitsthat terminates at each end in a homoclinic bifurcationof an equilibrium solution. The periodic orbits createdon homoclinic branches C1 and C2 also form a homo-clinic isola (branch (e) inFig. 17).

Ass is increased further, the main changes in the bi-furcation structure involve the homoclinic isolas seenin Fig. 17. Specifically, ass is increased, isolas of peri-odic orbits are pinched off from the homoclinic isolas,as illustrated inFig. 18. Panel A shows homoclinicisola 2 fromFig. 17ats = 14; this isola consists of pe-riodic orbits arising from homoclinic branches B1 andB2. The same homoclinic isola computed whens = 16

Fig. 17. Bifurcation diagrams fors = 14. (A) The dashed line represents a curve of equilibria and the solid lines represent maximum values ofc over a periodic orbit. Stability is not indicated. HB: Hopf bifurcation. HC-Bn: homoclinic bifurcation, branch Bn. (B) Period vs.p for variousbranches of periodic solutions. The dynamics associated with each of the branches is explained in the text. All branches except (e) also appearin Panel A.


Fig. 18. The curve (d-e) fromFig. 17and associated isolas of periodic orbits for various values ofs: (A) s = 14; (B) s = 16; (C)s = 18; (D)s = 24.

(seen in Panel B) is topologically equivalent, but wesee an isola of periodic orbits about to pinch off. Whens = 18 (Panel C) the isola of periodic orbits is fullyformed. Increasings further, we observe the birth of asecond isola (seen in Panel D, fors = 24). Homoclinicbranches B1 and B2 come together ats ≈ 43; ass isincreased beyond 24 to about 43, the homoclinic isolagradually diminishes, and disappears ats ≈ 43 with nofurther formation of isolas observed. The two isolas ofperiodic orbits formed by this process persist for allobserved values ofs > 24.

Fig. 19 shows a partial bifurcation diagram fors = 50. Six isolas are shown, although more have beenfound and there may be even more that have not beenlocated.Fig. 19bears a marked resemblance toFig. 2.This is not a coincidence, since, as mentioned earlier,the dynamics associated with the travelling waves mustreduce to the dynamics of the model without diffusionin the limit s → ∞. Thus we find a direct correspon-dence between isolas of periodic orbits observed in thetravelling wave equations with sufficiently largesandthe fundamental orbits in the bubbles identified in Eqs.(2)–(9), i.e., the model in the absence of diffusion. The

isolas in the travelling wave equations are formed ina number of different ways. The formation of isolas5 and 6 has already been described; these pinch offfrom the branch labelled (d-h) inFig. 17as illustratedin Fig. 18. Isolas 3, 4 and 8 are formed in a similarway, with isolas 3 and 4 pinching off from branch (a-c)at s ≈ 11.7 while isola 8 pinches off from branch (e)in Fig. 17. Isola 7 breaks off in a similar manner, butfrom a branch connecting orbits that are homoclinicto a periodic orbit. In fact, two further isolas (isolas9 and 10, not shown) also break off in a similar wayfrom a branch connecting homoclinic bifurcations of aperiodic orbit.

The description just given of the formation of iso-las 3–10 begs the question of whether there is a finiteor infinite number of such isolas, particularly in thelimit s → ∞. We have located isolas 9 and 10 in themodel of the absence of diffusion. However, we con-jecture that isola 10 is the last one in that limit; we havefound an isola 11 existing froms ≈ 14 to s ≈ 29 butfor s larger than about 29 the isola is gone, destroyedwhen it shrinks to a point. In addition to this principalsequence of isolas, there is a subsidiary sequence of


Fig. 19. Bifurcation diagram fors = 50. The dashed line representsa curve of equilibria and the solid lines represent curves of maximumvalues ofc over a periodic orbit. Stability is not indicated. Six isolasof periodic orbits are shown. There are period-doubling bifurcationsnear both ends of all isolas; these are not shown, nor are the period-doubled branches that bifurcate from them.

isolas that pinch off from the branch of periodic orbitsformed from parts of branches (b) and (c) inFig. 11, asdescribed in the discussion ofFig. 11above. These iso-las persist to large values ofsbut occur within the sameinterval of parameterpas the principal sequence of iso-las. It is possible that a pattern of period-doubling cas-cades followed by attractor merging similar to that as-sociated with the principal sequence (and described inSection 2) will occur also for the subsidiary sequence.

In summary, we find an extremely complicated bi-furcation structure associated with the travelling waveequations(2)–(9), with many branches of periodic or-bits occurring and numerous homoclinic bifurcationsof the equilibrium and of periodic orbits being ob-served. We have not described (or studied) all the in-tricate detail of these bifurcations, choosing to focuson the solutions that may have physiological signifi-cance, and in particular on the homoclinic bifurcationsof equilibria which correspond to isolated travellingwaves in the model with diffusion. In the next section,we return to the full partial differential equation model,Eqs.(1)–(8), and report on numerical integrations per-formed with the aim of determining the stability within

the full system of the travelling waves found by study-ing the travelling wave equations(2)–(9).

4. Simulations of the spatially extended model

The existence of isolated travelling wave solutionsto the spatially extended system given by Eqs.(1)–(8) was established inSection 3. In this section, weinvestigate the stability of these solutions in the spa-tially extended system by direct numerical solution ofEqs.(1)–(8). The equations were solved using a sim-ple explicit scheme with no-flux boundary conditionson a domain of length 1000�m. In each simulationpwas held fixed, and a variety of initial conditions wereused as described below. Different grid sizes were used(ranging from 2000 to 4000 points on the domain) butthe results were not significantly altered by the choiceof grid.

At p = 2.0, our initial condition was a localisedpulse inc. Specifically, att = 0 we usedc = 0.5 forx in the leftmost 20�m of our interval andc = c∗ else-where, wherec∗ is the equilibrium value ofc at thisp, obtained by setting all derivatives in Eqs.(1)–(8) tozero and solving forc. All other variables in Eqs.(1)–(8) were initialised to their equilibrium value acrossthe entire interval ofx. With this initial condition, astable travelling wave is obtained; the wave travels tothe right in the domain with wave speed 13.2�m s−1.The shape of this wave across the domain at two dif-f er dst Ai al-u astp n.T r-c qui-l rilye o bes ug-g iodso

d Ca s ofE s atp earlyt le. In

erent times is shown inFig. 20. Comparison with thesults ofSection 3shows that this wave correspono the homoclinic orbit on the upper part of branchn Fig. 5. Similar numerical simulations for other ves ofp suggest that branch A is stable for at le∈ (1.2,2.2), i.e., to the left of the Hopf bifurcatio

he stability of branch A to the right of the Hopf bifuation has not been determined. In this region, the eibrium solution is unstable; we would not necessaxpect the corresponding isolated travelling wave ttable here, although preliminary numerical work sests that solutions stay near branch A for long perf time.

The homoclinic bifurcations on branches B anlso correspond to isolated travelling wave solutionqs.(1)–(8), but numerical solution of the equationvalues corresponding to these branches show cl

hat the corresponding travelling waves are unstab


Fig. 20. A stable isolated travelling wave solution to Eqs.(1)–(8)atp = 2.0 obtained by direct numerical solution of the equations asdescribed in the text: (i)t = 20; (ii) t = 60. A space–time plot of thissolution is shown inFig. 21, Panel A.

Fig. 21. Space–time plots of solutions to Eqs.(1)–(8) obtained bydirect numerical solution: (A)p = 2, initial condition correspondingto branch A ofFig. 5; (B) p = 4.6, initial condition correspondingto branch B ofFig. 5; (C)p = 7.3, initial condition corresponding tobranch C ofFig. 5. Panel (A) shows an apparently unstable isolatedtravelling wave. In (B) and (C), the travelling wave solutions areunstable.

these cases, the variablesc, ce, etc. were initialised att = 0 to their values on the homoclinic orbit of interest.For instance, atp = 7.3 a wave initially travels to theright with wave speed consistent with that of the homo-clinic orbit at the samep value on branch B inFig. 5,but this wave disintegrates after some time, with sub-sequent evolution of the system leading to a disorderedpattern of left- and right-travelling pulses. Similar re-sults are obtained for the travelling wave correspondingto branch C inFig. 5(seeFig. 21(C)).

5. Discussion

This paper has investigated the dynamics associatedwith a mathematical model, originally developed bySneyd et al.[31], for Ca2+ wave propagation in thebasal region of pancreatic acinar cells. In[31], a par-tial analysis was presented of the model in the absenceof diffusion, with an attracting solution identified forsome but not all values of the bifurcation parameter.Our original motivation in studying this problem wasto identify an attracting solution for the diffusionlessmodel for all parameter values; such solutions are ex-pected to exist in a sensible model since calcium con-centrations are finite in the underlying cellular system.Sneyd et al.[31] showed that the single equilibriumsolution of the model is attracting for sufficiently smallor sufficiently large values of the bifurcation parame-ter and determined that the equilibrium solution losess nso theb so-l ningt oma re-a bits,o de ofp ev-e eter,m

essm fu-s fullm tlyl ngles edi-a ated

tability in a Hopf bifurcation, producing small regiof stable periodic motion at intermediate values ofifurcation parameter. We found further attracting

utions for intermediate parameter values, determihat the attractor could be a periodic orbit arising frperiod-doubling bifurcation of the Hopf cycle or cted in one of sequence of bubbles of periodic orr could be a strange attractor created by a cascaeriod-doubling bifurcations or by the merging of sral such attractors. For some values of the paramore than one attractor was observed.The broad bifurcation structure of the diffusionl

odel was found to persist for the model with difion. In particular, direct numerical solution of theodel showed that for sufficiently small or sufficien

arge values of the bifurcation parameter, the siteady-state solution is attracting, while for intermte values of the parameter there is more complic


dynamics, including the existence of isolated travellingwaves and irregular spatio-temporal behaviour. We fo-cussed on finding isolated travelling wave solutions, asthese are thought to be of some physiological interest.In a standard way we derived travelling wave equations,then numerically located homoclinic orbits of the singleequilibrium in these equations; such orbits correspondto isolated travelling waves in the full model, with sta-bility of the travelling waves being determined by di-rect solution of the full model. We found three branchesof homoclinic orbits in the travelling wave equations,one of which (i.e., branch A) exists over a significantinterval of parameter values with part of the branch cor-responding to stable travelling waves in the full model.We were unable to determine the mechanism by whichthis branch of travelling waves terminates, but observethat the branch appears to persist well beyond the lo-cus of a Hopf bifurcation of the equilibrium, where theequilibrium becomes unstable in the full model.

In terms of the underlying physiology, we find thesame broad dynamical behaviour in the model of Sneydet al.[31] as in previous models of calcium wave prop-agation in pancreatic acinar cells[26], namely that asingle branch of stable isolated travelling waves existsfor a wide range ofp values, i.e., of the concentrationof inositol trisphosphate. There are also relatively largeintervals ofpon which more complicated but basicallyoscillatory behaviour is observed. Thus the model sug-gests we should not be surprised to see highly complexoscillatory behaviour in these cell types. However, wee oura ond-i ; ana eent o bed

byt tingm thew heso v-e bi-f nsb nda pfb icals dt the

two-dimensional centre manifold of the equilibrium atthe Hopf bifurcation, which is not the case in our situa-tion. In contrast to the situation seen in[13,3], numer-ical evidence suggests that homoclinic bifurcations ofthe equilibrium may occur on both sides of the Hopfbifurcation in our system. A theoretical study of thehomoclinic-Hopf bifurcation in the case that the ho-moclinic orbit does not lie in the centre manifold of theequilibrium at the Hopf bifurcation would thus be ofinterest.

Acknowledgements

We thank Alan Champneys and Alastair Rucklidgefor helpful conversations about this work. We alsothank Jens Rademacher for making available to us re-sults from his Ph.D. thesis in advance of its comple-tion, and for useful information about the dynamicsexpected near a heteroclinic bifurcation involving anequilibrium and a periodic orbit. David Simpson re-ceived partial financial support from the University ofAuckland, Faculty of Science.

Appendix A

Here we give a complete list of functions that arepresent in the model equations.

J

J

J

J

P

P

w

mphasise that this is a tentative conclusion sincenalysis is restricted to parameter values corresp

ng to the basal region of pancreatic acinar cellsnalysis of the model incorporating coupling betw

he basal and apical regions of the cells remains tone.

We were originally motivated to study this modelhe underlying physiology, but some rather interesathematical phenomena were uncovered alongay. Specifically, in the process of locating brancf homoclinic orbits of the equilibrium in the tralling wave equations, we also found homoclinic

urcations of periodic orbits, heteroclinic bifurcatioetween the equilibrium and a periodic orbit, ahomoclinic-Hopf bifurcation. The homoclinic-Ho

ifurcation is of particular interest; previous theorettudies of this bifurcation[13,3] have been restricteo the case that the homoclinic orbit is contained in

pm(c) = Vpmc2

K2pm + c2

, (A.1)

mito(c) = Vmitoc3

1 + c2, (A.2)

serca(c, ce) = Vsercac

ce(Kserca+ c), (A.3)

in = 1

5+ 1

20p, (A.4)

RyR(c,w) = w(1 + (c/Kb)3)

1 + (Ka/c)4 + (c/Kb)3, (A.5)

IPR(O,A) =(

1

10O + 9

10A

)4

, (A.6)

∞(c) = 1 + (Ka/c)4 + (c/Kb)3

1 + 1/Kc + (Ka/c)4 + (c/Kb)3, (A.7)


and

φ1(c) = (k1L1 + l2)c

L1 + c(1 + L1/L3), (A.8)

φ2(c) = k2L3 + l4c

L3 + c(1 + L3/L1), (A.9)

φ−2(c) = k−2 + l−4c

1 + c/L5, (A.10)

φ3(c) = k3L5

L5 + c, (A.11)

φ4(c) = (k4L5 + l6)c

L5 + c, (A.12)

φ−4(c) = L1(k−4 + l−6)

L1 + c, (A.13)

φ5(c) = (k1L1 + l2)c

L1 + c, (A.14)

also

S = 1 − R − O − A − I1 − I2. (A.15)

Table A.1Parameter values used in Eqs.(1)–(9), from Sneyd et al.[31]; thesecorrespond to the parameters for the basal region of a pancreaticacinar cell, and were determined by fitting to experimental data

Receptor densities

There are further parameters which are related to thosegiven inTable A.1by the following equations:

Ka = 4

√k−

a

k+a

K1 = L1L2,

Kb = 3

√k−

b

k+b

K2 = L3L4,

Kc = k−c

k+c

K4 = L5L6,

Ki = k−i

kiLi = l−i

li.

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kf 0.32 s−1 v1 0.04 s−1

Transport parametersJer 0.002 s−1 γ 5.405Vserca 120 (�M)2 s−1 Kserca 0.18�MVpm 28�M s−1 Kpm 0.425�Mδ 0.1 Dc 20 (�m)2 s−1

Vmito 0

IPR parametersk1 0.64 s−1(�M)−1 k−1 0.04 s−1

k2 37.4 s−1(�M)−1 k−2 1.4 s−1

k3 0.11 s−1(�M)−1 k−3 29.8 s−1

k4 4.0 s−1(�M)−1 k−4 0.54 s−1

L1 0.12�M l2 1.7 s−1

L3 0.025�M l4 1.7 (�M)−1 s−1

L5 54.7�M l6 4707 s−1

RyR parametersk+

a 1500 (�M)4 s−1 k−a 28.8 s−1

k+b 1500 (�M)3s−1 k−

b 385.9 s−1

k+c 1.75 s−1 k−

c 0.1 s−1

IPR—IP3 receptor. RyR—ryanodine receptor.


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complex oscillations and waves of calcium in pancreatic acinar cells

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