localized and asynchronous patterns via canards in coupled calcium...

Physica D 215 (2006) 46–61www.elsevier.com/locate/physd

Localized and asynchronous patterns via canards in coupled calciumoscillators

Horacio G. Rotsteina,∗, Rachel Kuskeb

a Department of Mathematics and Center for Biodynamics, Boston University, Boston, MA, USAb Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Received 8 February 2005; received in revised form 28 November 2005; accepted 11 January 2006Available online 7 March 2006

Communicated by C.K.R.T. Jones

Abstract

In this paper we consider the mechanism for localized behavior in coupled calcium oscillators described by the canonical two-pool model.Localization occurs when the individual cells oscillate with amplitudes of different orders of magnitude. Our analysis and computations show that acombination of diffusive coupling, heterogeneity, and the underlying canard structure of the oscillators all contribute to the localized behavior. Twokey quantities characterize the different states of the system by representing the effects of both the autonomous and the non-autonomous termswhich are due to the coupling. By highlighting the influence of the canard phenomenon, these quantities identify stabilizing and destabilizingeffects of the coupling on the localized behavior. We compare our analysis with computations, describing multi-mode states and asynchronizedlarge oscillations in addition to the localized states.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Calcium oscillators; Coupled oscillators; Localized oscillations; Mixed-mode oscillations; Canard phenomenon

1. Introduction

Localized oscillatory patterns are characterized by one groupof cells displaying large amplitude oscillations (LAOs) and theremaining cells exhibiting small amplitude oscillations (SAOs),an order of magnitude smaller in amplitude than the LAOs.Localization has been observed in chemical, biological andoptical systems [1–5] and can occur by various mechanismsaccording to the nature of the uncoupled oscillators and thetype of coupling. For oscillations governed by underlying limitcycles rather than conservative oscillations, the mechanism oflocalization is typically controlled by effective parameters incritical regimes, for example, near bifurcations or transitionsin qualitative behavior [5,7,8]. It is useful to relate theseparameters to the bifurcation or control parameters of the singleoscillators or to the coupling terms which can be local [5] orglobal [7,8].

∗ Corresponding author. Tel.: +1 617 353 1493.E-mail address: [email protected] (H.G. Rotstein).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.01.007

Our focus here is on different mechanisms that producelocalization in a simple biochemical model of diffusively cou-pled calcium oscillations, also relevant for other systems ofrelaxation oscillators. Two important ingredients here are therelatively small coupling and heterogeneity in the populationof oscillators. A third key factor is the canard phenomenon[9–13]. In two-dimensional (2D) relaxation oscillators undergo-ing a Hopf bifurcation, this phenomenon is sometimes referredto as the canard explosion, appearing as a sudden transitionbetween the SAO and LAO regimes as a parameter passesthrough its canard critical value. In the context of the coupledsystem we compute both an effective canard critical value andan effective bifurcation parameter for each cell. The effectivebifurcation parameters change over time, and for the individualcells it can be compared to the effective critical value to illus-trate whether that cell engages in SAO or LAO activity, or acombination of the two.

The specific calcium oscillators considered in this paper aredescribed by the canonical two-pool model [14]. Although thismodel is too simple to reproduce the calcium dynamics inall its complexity, it has served as a basic prototypical model

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H.G. Rotstein, R. Kuske / Physica D 215 (2006) 46–61 47

which captures the main features of calcium oscillations, welldocumented for the single cell model [15] (see Appendix A).The oscillatory dynamics depend on a control parameter IP3(inositol (1,4,5)-triphosphate) which regulates calcium releaseand controls the oscillations amplitude, thus playing the roleof the bifurcation parameter for the canard phenomenon inthe 2D single cell model. The canard structure is just oneof a number of this model’s features found also in moregeneral types of limit cycle-type oscillators. Then the relativelysimple two-pool model is an ideal candidate for developingnew methods for studying the interaction of coupling and thecanard phenomenon. Through a new asymptotic technique weobtain analytical expressions and reduced systems which can beused to understand the mechanistic aspects of localized patternsand to make predictions about them. The approach does notdepend on the particular model; we expect that it is valuable forthe analysis of other diffusively coupled relaxation oscillators,since it is based primarily on the interaction of the bifurcation(control) parameter (related to IP3 in the original application)and on the coupling parameters. In order to capture the broadrange of effects which may occur in these systems, we haveintroduced a general diffusive coupling; the physiologicallyrelevant one for calcium oscillators corresponds to a specificcase within this general coupling.

The analysis of [7,8] explained the mechanism forspatially localized oscillations for the globally coupledBelousov–Zhabotinsky (BZ) reaction [1–3] and FitzHugh–Nagumo (FHN) models, where the canard phenomenon alsoplayed a critical role. In both models the individual oscillatorsare identical, and heterogeneity in the cluster (set of oscillatorsin the same amplitude regime) size is created through globalinhibitory coupling. In contrast, localization in the two-poolmodel is due to heterogeneity in the individual oscillators and acompeting influence from diffusive coupling. This is surprisingsince, intuitively, diffusion tends to homogenize the system. Infact, we find that relatively small diffusive coupling has a non-trivial influence on localized states, supporting and stabilizingthem in unexpected regimes.

We summarize the historical context for the canardphenomenon, mentioning those results closely related tothis study. The canard phenomenon in two dimensions wasdiscovered by Benoit et al. [11] for the van der Pol (VDP)oscillator. It has been studied by various different techniques:non-standard analysis [11], asymptotic analysis [10,16,17],and geometric singular perturbation theory and blow-uptechniques [9,12,13]. In a single 2D oscillator undergoinga canard explosion there is an exponentially small rangeof the parameter governing the transition between SAO toLAO regimes. We follow Krupa and Szmolyan [12,13] in ourapproach and in the calculation of the canard critical value,denoted by λc, which marks the sudden transition between theSAO and LAO regimes.

The canard phenomenon in three dimensions (or higherdimensions) is more generic than in two dimensions. While 2Dsystems can display SAOs or LAOs but not both for a fixed setof parameters, 3D systems can display mixed-mode oscillations(MMOs) that alternate over time between SAO and LAO

[18–22]. The diffusively coupled two-pool system studied here,a 4D system, is able to exhibit a more complex pattern:localized states of mixed-mode type (localized mixed-modeoscillations) where one oscillator is always in a SAO regimewhile the other alternates over time between LAO and SAOregimes. As we show in this manuscript, this pattern, whichcannot be obtained for 2D and 3D systems, can neverthelessbe explained by decomposing the 4D diffusively coupled two-pool model into two 2D forced (effectively 3D) subsystems,similarly to the analysis of localized patterns obtained in theglobally coupled BZ reaction [7,8].

Some insight into the different types of localized phenomenacan be found by comparing related chemical and biologicalsystems, in which sudden explosions of limit cycles andMMOs are often found [7,8,19,20,23–34]. MMOs have beenstudied both analytically and numerically [18,35–38,21],and in some cases the maximum number of subthresholdoscillations alternating with a large amplitude oscillationcan be estimated [18]. However, global predictions are notalways possible; the appearance and prevalence of subthresholdoscillations in a mixed mode state depends on initial conditionsor the reset value after a large amplitude oscillation. Severalapproaches have been developed in some of the referencesmentioned above to deal with this type of problem by studyingthem as reduced two-dimensional systems evolving in time.

1.1. The model and strategy

Our starting point is the system of diffusively coupledcalcium oscillators described by the two-pool model in adimensionless form [15].{

V ′

k = µk − Vk − γ /εF(Vk, Wk) + Dv(V j − Vk),

W ′

k =1ε

F(Vk, Wk) + Dw(W j − Wk),(1)

for j, k = 1, 2, j 6= k, where

F(V, W ) = β

(V n

V n + 1

)−

(W m

W m + 1

) (V p

V p + α p

)− δW.

(2)

A more detailed description of the dimensional model as wellas the non-dimensionalization is given in Appendix A. In (1)Vk and Wk are the concentrations of calcium in the cytoplasmand the calcium sensitive store or pool, respectively. Theparameter ε is very small, indicating that the rate at whichcalcium flows from the IP3 to the Ca2+ sensitive pools is verylarge. The parameter µk is the bifurcation or control parameterrelated to IP3, defining parameter regimes where oscillationsare stable or unstable. The parameters Dv and Dw give diffusivecoupling between individual cells. This choice of coupling,while not physiologically realistic, provides a simple first stepin understanding the interaction of coupling, heterogeneity,and the canard phenomenon on localization. In particular thecoupling term in the second equation in (2) is not realisticsince the calcium pools in two different cells are not diffusivelyconnected. However, this general form allows the developmentof principles for the influences of coupling, which then can

48 H.G. Rotstein, R. Kuske / Physica D 215 (2006) 46–61

be applied to more complex models. Throughout the paper weconsider the effect of varying the coupling coefficients Dv , Dw,and µk while keeping the other parameters fixed, as given inAppendix A.

A change of variables transforms the single cell two-poolmodel into a FHN-type system [15,39], so even though thecoupling we use here is not global as in [7,8], it is notunexpected that we can find localized behavior in the coupledsystem as in [7,8]. An example of this behavior for (1) isshown in Fig. 1 in terms of the convenient parameter λk , whichis just −µk plus a constant (see (3)). In the top graphs weshow the oscillations for uncoupled non-identical oscillators,Dv = Dw = 0 and λ1 6= λ2 (µ1 6= µ2). A key quantityfor predicting SAO or LAO in each single relaxation oscillatoris the canard critical value λc, which delineates the parameterranges for which the oscillator is in a SAO regime (λk < λc)and a LAO regime (λk > λc). For the uncoupled system shownin the top of Fig. 1, both λ1 > λc and λ2 > λc, consistent withthe observation of LAO for each single oscillator. The value ofλc can be estimated as a function of the parameters of the modelusing a geometric [12] or asymptotic [10] approach, as we showin Section 2. In the bottom graphs, we show the behavior forthe same values of λk , but with a small coupling coefficientDw = .01 and Dv = 0. The first oscillator has LAO whilethe second has SAO, corresponding to localized oscillations.Clearly these localized oscillations have been induced by thediffusive coupling. In our analysis we extend the idea of acanard critical value to the coupled system.

When there is coupling in (1) each oscillator is composed oftwo contributions, the autonomous and non-autonomous parts,labeled with “k” and “ j” respectively. As in [7,8], each memberin the system (1) can be thought of as an autonomous oscillatorforced by the non-autonomous part. The patterns studied hereare the result of contributions from both parts, so that a simpleyet important step is the transformation which allows these twocontributions to be transparently identified. Then we can definetwo quantities which capture their effects on localized, MMO,and asynchronous LAO states.

• The canard critical value, λc(Dv, Dw). We derive ananalytical expression which gives an approximation to thethreshold for LAOs or SAOs for the autonomous part ofa single oscillator, modified from the uncoupled case bythose terms appearing in the coupled system as additionalautonomous contributions. This quantity (see Section 2)λc(Dv, Dw), obtained by an extension of [7], plays a rolesimilar to that of the canard critical value λc for a singleoscillator and reduces to it in the uncoupled case Dv =

Dw = 0. Here we use the theory developed in [13].• The effective bifurcation parameters, λeff

k (Dv, Dw, V j , W j )

for k = 1, 2 ( j 6= k). This quantity can be viewed as ageneralization of the bifurcation parameter, combining λk(−µk+ a constant) with non-autonomous contributions fromthe coupling which vary with time through (V j , W j ) (seeSection 2). Thus it is a dynamical quantity, as forced throughthe coupling with the other oscillator. It can be compared tothe threshold value λc(Dv, Dw), common to both oscillators,to understand and predict (de-)stabilization of localization

Fig. 1. Localized patterns in a diffusively two pool model created via diffusion.(a) Dw = 0. Both (v1, w1) and (v2, w2) are in a LAO regime. (b) Dw = 0.01.(v1, w1) are in a SAO and (v2, w2) are in a LAO regime.

and other phenomena. The effect of forcing on relaxationtype oscillators has been found to result in richer patternsboth experimentally and theoretically [38,40]. A forcingviewpoint has also been used in [7,8] to study localizedpatterns in the BZ reaction. A related important aspect can bethe timing of the variation of λc(Dv, Dw) compared with theposition of Vk and Wk in the phase plane, which we outlinefor specific cases.

In [6], a comparison of synaptic input strength to a time-dependent threshold at a key moment was used to determinewhether a neuron is recruited into an active population ornot. In the system with study in this paper, the comparison isbetween a dynamic variable and a time-independent thresholdλc(Dv, Dw). This threshold still carries information about thecoupling through its dependence on the diffusion coefficients.

The paper is organized as follows. In Section 2 we give areduction of the model which gives a convenient viewpoint for


the canard phenomenon. We also provide a canard analysis andderive the expressions for λc(Dv, Dw) and λeff

k , generalizing[7,13]. In Section 3 we use these two quantities to explainand predict localized phenomena in the coupled case, andcompare to numerical simulations. In Section 4 we discussother asynchronous oscillations in the coupled case, and showhow they are related to the canard analysis.

2. The canard phenomenon: background and developmentof mechanistic tools

First we transform system (1) to a form in which each uncou-pled oscillator is of VDP type: each uncoupled oscillator hasactivator and inhibitor variables whose nullclines are a cubic-like curve and a vertical line, respectively. The transforma-tion we use here is different from the one used in [15] (seeAppendix A) where the transformation resulted in a system ofFHN type rather than VDP type; for the FHN type the inhibitornullcline is linear, non-vertical and non-constant. The form weuse here is more suitable for identifying the autonomous andnon-autonomous contributions. For each value of k the coordi-nate change for the full transformation is given by

vk = −Vk

γ− vmin, wk = γ Wk + Vk − wmin,

λk = −(µk + γ vmin). (3)

The transformation of variables, applied to the uncoupled os-cillators, is a “rotation” of the nullclines followed by a transla-tion which places the local minimum of the resulting cubic-likenullcline, (vmin, wmin), at the origin.

Then we define

f (v, w) = F(−γ (v + vmin), v + vmin + (w + wmin)/γ ), (4)

and by rescaling time, t → εt , and substituting (3) and (4) into(1), we getv′

k = f (vk, wk) − ε/γ [−λk + γ (1 + Dv)vk] + εDvv j ,

w′

k = ε[−λk + γ (1 + Dv − Dw)vk − Dwwk+ γ (Dw − Dv)v j + Dww j ],

(5)

for k = 1, 2.For the parameters used in this paper, we find that vmin ≈

−.3570 and wmin = 1.5503. Note that for Dw = 0 (even forvalues of Dv > 0), system (5) is a VDP-type oscillator. Thisproperty is lost if Dw > 0. However, the form of (5) allows avaluable perspective on the pair of equations for each k; belowwe show that the corresponding nullclines can be viewed asthose from the uncoupled case perturbed by diffusive couplingand forcing by the other oscillator.

For the cases Dv > 0 and/or Dw > 0 system (5) can be seenas two forced autonomous oscillators; that is, two autonomousoscillators (terms with the subscript k), each forced by theother one (terms with subscript j). In order to understand thedynamics of system (5) it is useful to decompose each oneof these oscillators into its autonomous and non-autonomousparts. The autonomous part for (5) is{

v′

k = f (vk, wk) − ε/γ [−λk + γ (1 + Dv)vk],

w′

k = ε[−λk + γ (1 + Dv − Dw)vk − Dwwk],(6)

Fig. 2. Schematic representation of phase planes for system (7) for differentvalues of λ. The nullclines of (7) are given by w = f (v) and w = g(v, λ)

which are the solutions of F(v, w) = G(v, w; λ) = 0.

and clearly, the remaining terms give the non-autonomous part.Note that system (6) is an uncoupled one but it incorporatesdiffusive effects. In this way, system (6) can be seen asan intermediate description between the fully uncoupled andthe fully coupled ones, that partially captures the effects ofdiffusion.

2.1. Canard phenomenon for the uncoupled case

First we review the canard phenomenon for two-dimensionaloscillators, referring the reader to [9,10,12,13] for a morethorough introduction. Consider the following system:{

v′= F(v, w),

w′= ε G(v, w; λ),

(7)

where 0 < ε � 1. The function F is such that its zerolevel curve is a cubic-like function, which can be written asw = f (v), taking its local minimum as (0, 0) (without lossof generality) and such that its local maximum is to the rightof v = 0. The function G is a non-increasing function of w

such that the zero level curve G(v, w; λ) = 0 is an increasingfunction of v for every λ in a given neighborhood of λ = 0, andis also a decreasing function of λ for all v in a neighborhood ofv = 0. In the VDP oscillator case, G does not depend on w; it isa vertical line. In the more general case that includes FHN-typeoscillators, we assume that G can be written w = g(v; λ).

We further assume that the nullclines w = f (v) and w =

g(v; λ) (F = 0 and G = 0) intersect at (v0, w0) = (0, 0) whenλ = 0, and that for λ > 0, (v0, w0) lies on the central branch off . Precise mathematical statements of the former assumptionsare given in Appendix B. We schematically illustrate this inFig. 2. The autonomous part (6) of system (5) satisfies the aboveconditions. In the special case of Dw = 0 in (6), the nullclineG = 0 is the vertical line v = λγ −1(1 + Dv)

−1.The nullclines for the uncoupled model studied in this

manuscript are presented in Fig. 3 for two different valuesof λ. Keeping all other parameters fixed, the dynamics ofsystem (7) depends on the value of λ, which determines the


Fig. 3. Phase plane for a single two-pool oscillator. (a) λ < λc: the system is in a SAO regime. (b) λ > λc: the system is in a LAO regime.

relative position of the w-nullcline with respect to the v-nullcline. For some value λ = λH = O(ε) > 0, system(7) undergoes a Hopf bifurcation in a neighborhood of (0, 0).Then it has a stable fixed point for values of λ < λH and alimit cycle for values of λ > λH (see [7,13] for more details).For the parameters of the model studied in this manuscript,λ > λH and the Hopf bifurcation is supercritical, so that thelimit cycle created is stable. As λ increases, the amplitudeof the limit cycle created at the Hopf bifurcation increasesslowly for small enough values of λ; for these values, part ofthe trajectory is very close to the unstable middle (unstable)branch of the v nullcline for a while, then crosses the unstablebranch and moves toward the left branch of the v nullcline,as illustrated in Fig. 3(a). At the critical value λc > λH thetrajectory moves toward the right branch of the v-nullclineinstead of moving toward the left branch. Then the limit cycleexpands sharply, becoming a relaxation oscillator [41–44], asseen in the transition from Fig. 3(a) to (b), together with thecorresponding time series for v and w presented in Fig. 4.The amplitude of the limit cycle either increases slowly orremains constant as λ is increased. The dramatic transitionfrom a “small” amplitude limit cycle to a “large” amplitudelimit cycle is known as the canard phenomenon [9–13,45],which occurs for an exponentially small (in ε) interval of valuesof λ in a neighborhood of λc. Following [13] and [7], wecalculate in Appendix B an asymptotic expansion λc in powersof ε1/2, obtaining an analytical approximation λc ∼ .014 forthe parameters used in this study. For the single uncoupledoscillators (Dv = Dw = 0) this analytical approximation iswithin the O(ε3/2) correction in (8) of the numerically obtainedvalue, λc ≈ 0.015330 ∼ 1.4ε + O(ε3/2) (taking ε = .01). InFigs. 3 and 4 we show results for λ < λc and λ > λc, and in thefollowing sections we use the numerically obtained value of λcfor comparisons with the coupled case.

In this case the canard phenomenon has been induced bychanges in the value of λ, which shifts both the v- and w-nullclines. As seen in Eqs. (5), the shift in the v-nullcline is an

order of magnitude smaller due to a factor of ε. The differenteffect on the w- and v-nullclines means that their relativeposition can change significantly as we vary λ. By symmetry,for larger values of λ the w-nullcline is placed near the localmaximum of w = f (v) and a similar effect is seen. In thefollowing we concentrate on the canard phenomenon near theminimum v = 0.

2.2. Calculation of the canard critical value λc(Dv, Dw)

For a single oscillator of the form (7) satisfying the canardconditions (see Appendix B and [12,13]), Krupa and Szmolyangave an approximate expression for λc, the critical value of λ

dividing between the SAO and LAO regimes [13].For Dv ≥ 0 or Dw ≥ 0, the singularity and non-degeneracy

assumptions listed in Appendix B are satisfied by system (6),the autonomous part of system (5). So one expects the existenceof a canard critical value λc(Dv, Dw) such that for values ofλ < λc(Dv, Dw) (λ > λc(Dv, Dw)), system (6) is in a SAO(LAO) regime.

In Appendix B we derive λc(Dv, Dw) following [7,12,13]for a canonical system. System (6) is brought to the canonicalform (17) by substituting f (v, w) from (4) and −λ+γ (1+Dv−

Dw)v−Dww into F(v, w) and G(v, w; λ) in (17), respectively.Then one gets the following expression for the canard criticalvalue:

λc(Dv, Dw) = −εγ (1 + Dv − Dw)[γ (1 + Dv − Dw) fvw fvv

+ γ (1 + Dv − Dw)| fw| fvvv − Dw f 2vv

− 2(1 + Dv) f 2vv]/(2 f 3

vv) +O(ε3/2).

Note that λc for an uncoupled oscillator becomes λc(0, 0) inthis notation. Note also that λc(Dv, Dw) is independent of λ;although the two oscillators may be heterogeneous (λ1 and λ2are not necessarily equal), they still have a common value forλc(Dv, Dw).

In the coupled case of (5), the autonomous part resultingfrom the diffusion in (6) may change the canard critical value in


Fig. 4. v and w traces for a single two-pool oscillator. (a) λ < λc: the systemis in a SAO regime. (b) λ > λc: the system is in a LAO regime.

each oscillator to a value λc(Dv, Dw). Then this quantity can beused to test for the influence of the coupling on localization. Forexample, suppose that λ1 < λc(Dv, Dw) and λc(Dv, Dw) <

λ2 < λc(0, 0) in the absence of any other contribution such asnon-autonomous forcing effects. Then the first oscillator is in aSAO regime while the second one is still in a LAO regime, eventhough λ2 < λc(0, 0); that is, the system exhibits a localizedstate due to diffusion.

In Fig. 5 we show the graphs of the analytical predictionof λc(Dv, Dw) as a function of Dv and Dw. In general,λc(Dv, Dw) increases with Dv and decreases with increasedDw. This value is useful in predicting localized behavior, whenthe non-autonomous part of the equation consists of SAOswhich can be neglected to leading order.

2.3. The effective bifurcation parameters: λeffk

As noted above, the coupling in the oscillators influencesthe dynamics through two aspects, namely, the autonomous and

Fig. 5. λc(Dv, Dw) as a function of Dv for different values of Dw .λc(Dv, Dw) is given by Eq. (8).

non-autonomous parts of the oscillator. The non-autonomouspart can be viewed as the forcing exerted to each oscillatorby the other one which moves the nullclines in a dynamicfashion, thus effectively transforming the bifurcation parameterto an oscillating function depending not only on thediffusion coefficients but also on the dynamic variables ofthe forcing oscillator. We denote this dynamical quantity asλeff

k (Dv, Dw, V j , W j ) and compare it against the canard criticalvalue λc(Dv, Dw) at some specific times in order to understandthe nature of localized solutions. Specifically, in Sections 3 and4 we show a number of cases in which fluctuations in λeff

k aboveor below λc(Dv, Dw) lead to LAOs or SAOs, respectively. Thenthis quantity lends a viewpoint complementary to the canardcritical value, λc(Dv, Dw).

We define

λeffk (Dv, Dw, v j , w j ) ≡ λk + γ (Dv − Dw)v j − Dww j . (8)

Substituting the value λeffk in Eq. (5) we get

v′

k = f (vk, wk) − ε[−λeffk (Dv, 0, v j , w j )

+ γ (1 + Dv)vk]/γ,

w′

k = ε[−λeffk (Dv, Dw, v j , w j ) + γ (1 + Dv − Dw)vk

− Dwwk].

(9)

Note that (9) has the same form as (6) with λeffk replacing

λk . From (8) it is clear that λeffk is time-dependent through v j

and w j for j 6= k. Since the model includes the more generalform of diffusive coupling with both Dv and Dw, the expressionfor λeff

k captures several general effects. Fluctuations in v j canincrease or decrease λeff

k , depending on the difference Dv − Dw.In contrast Dw > 0 also decreases λeff

k since the significantfluctuations in w j are positive. In the following sections weillustrate the different influences of these coupling coefficients.For example, Dv alone provides positive feedback, contributingto localized and asynchronous MMOs, as well as transitionsto both LAOs and SAOs. Contributions from Dw can promotelocalization and anti-phased oscillations. Combinations of these


contributions provide additional routes to localized oscillationsand asynchronous LAOs.

The definition of λeffk is based on perturbations to the value

of λk in the wk equation, since it is clear from (5) that a shiftin λk translates directly into a shift in the w-nullcline withthe same order of magnitude. As discussed in Section 2.1,small variations in the w-nullcline can result in O(1) transitionsbetween SAOs and LAOs. Perturbations to λ in the vk equationdo not in general trigger the canard explosion, since thesechanges have a coefficient of O(ε) and are in practice higherorder corrections to the nullcline.

3. Analysis for localized solutions

In this section we analyze several mechanisms by whichlocalized solutions are created as a consequence of diffusivecoupling. We relate the quantities λc(Dv, Dw) and λeff

k tostabilization and destabilization of certain dynamics by thecoupling coefficients Dw and Dv . For example, when Dv >

0 or Dw > 0 there can be a shift in the canard criticalvalue λc(Dv, Dw) which effectively places λ1 or λ2 indifferent oscillatory regimes (LAO or SAO) as compared to theuncoupled case. In addition, there can be a dynamic variationin λeff

k relative to λc(Dv, Dw) and/or relative to the location ofoscillator k in the phase plane, which plays a role similar toselecting parameter regimes for SAOs or LAOs.

3.1. Destabilization of SAO by Dw

In Fig. 6 we consider values λ1 = .015 and λ2 = .01,Dv = 0 and Dw = .05. Since λk < λc(0, 0) for k = 1, 2, theoscillators exhibit only SAO when uncoupled. The key featuresof this case are:

(i) λ1 > λc(Dv, Dw),(ii) λ2 < λc(Dw, Dv),

(iii) the fluctuations in λeff2 do not drive v2 and w2 to LAOs.

Conditions (ii) and (iii) maintain the SAOs in the secondoscillator, so that v2 and w2 can be neglected in predicting thebehavior of v1 and w1. Then condition (i) causes LAOs in thefirst oscillator, thus yielding the localized state.

The localization considered in Fig. 6 above holds for a rangeof values of λ1 > λc, holding λ2 fixed. However, for largervalues of λ2, keeping all other parameters fixed, the localizedoscillation is lost to an asynchronous LAO pattern due to theviolation of condition (iii) (see Fig. 13 in Section 4.1). If bothDw and Dv are non-zero, one can observe localized oscillationsover a larger range of λ2 and λ1, as we show next.

3.2. Stabilization of SAOs by Dv and Dw

We explore two types of localized behavior which resultfrom a stabilization of SAOs by the coupling. The differencebetween the two types is the variation in non-autonomouseffects captured in λeff

k .

Controlled variation in λeffk

In Fig. 7 we show localized behavior for λ1 = .02, λ2 = .016,and Dw = Dv = .05. Since λk > λc(0, 0) for k = 1, 2, in

Fig. 6. Localized solutions for a diffusively coupled two-pool model. (v1, w1)

is in a LAO regime and (v2, w2) is in a SAO regime.

Fig. 7. Localized solutions for a diffusively coupled two-pool model. (v1, w1)

is in a LAO regime and (v2, w2) is in a SAO regime.

the absence of coupling both oscillators are in the LAO regime.The key factors for this localized phenomenon are:

(i) λ1 > λc(Dv, Dw),(ii) λ2 < λc(Dv, Dw),

(iii) the fluctuations in λeff2 are not sufficient to drive v2 and w2

to LAOs.

In the absence of non-autonomous effects, conditions (i)and (ii) would be sufficient to guarantee a localized state,with oscillator 1 in a LAO regime and oscillator 2 in aSAO regime. However, as mentioned in Section 3.1 andshown in Section 4.1, the fluctuations in λeff

2 may exceedλc(Dv, Dw), providing a possibility for a shift to LAOs foroscillator 2. To confirm condition (iii) the range of λeff

2 can beapproximated by substituting the range of v1 and w1 for LAOs


into (8). Then the approximate range of λeff2 is (λc(Dv, Dw) −

Dw max(w1), λc(Dv, Dw) − Dw min(w1)), neglecting v2 andw2 for SAOs. Since w1 > 0 for all but a very small timeinterval, effectively λeff

2 < λc(Dv, Dw), confirming condition(iii). Note also that Dv = Dw in (8) so that the variation in λeff

2is decreased compared with the case Dv = 0 and Dw > 0, sothat Dv has a stabilizing effect on the localized state.

This phenomenon is also observed for smaller values ofλ2, in particular, for values of Dv and Dw for which λeff

2 <

λc(Dv, Dw) as in Fig. 6. For λ2 ≥ λc(Dv, Dw) there isa transition to asynchronous LAOs for both oscillators, asexpected.

Increased variation in λeffk

Fig. 1 shows a second example of stabilized localizedoscillations, with λ1 = .023, λ2 = .016, Dv = 0 and verysmall Dw = .01. The mechanism for localized oscillations isdifferent from that above, owing primarily to four factors:

(i) λ2 slightly larger than λc(Dv, Dw),(ii) λ1 significantly larger than λc(Dv, Dw),

(iii) significant variation in λeff2 , driven by v1 and w1,

(iv) the coupling is very small, so that λc(Dv, Dw) ≈ λc(0, 0).

Conditions (i) and (ii) imply that large enough oscillationsin v1 and w1 cause (iii), since λeff

2 falls below λc for significanttime intervals, driving the second oscillator to SAOs. Thenthe fluctuations in λeff

1 are negligible, so that (ii) implies thatthe first oscillator remain in a LAO regime. Condition (iv)guarantees that the forcing of the second oscillator by the firstis not too large, even though v1 and w1 exhibit LAOs. Forincreased coupling λc(Dv, Dw) is shifted and the fluctuationsin λeff

k increase for both k = 1, 2, destabilizing the localizedoscillations to anti-phased LAOs, discussed in Section 4.

3.3. Destabilization of localized solutions to SAOs by Dv

In Fig. 8 we demonstrate how a localized state for theuncoupled system, corresponding to λ1 > λc(0, 0) > λ2, canbe destabilized for Dv > 0. We take λ1 = .02, λ2 = .012and Dv = .1, Dw = 0. For very small coupling, we obtainlocalized solutions; this is not surprising since λ1 is in the LAOregime and λ2 is in the SAO regime. When we increase thecoupling to Dv = .1 with Dw = 0, we find that the localizedoscillations lose stability to SAOs for both oscillators. (Thereis typically a long transient of mixed mode oscillations (seeSection 4.2) before both oscillators reach a SAO regime.) Herethe key factors are:

(i) λc(Dv, Dw) increases with Dv ,(ii) λeff

1 < λc(Dv, Dw) for part of the trajectory,(iii) v1 increases through zero when λeff

1 < λc(Dv, Dw),(iv) λ2 is well below λc(Dv, Dw).

Condition (i) would be enough to cause SAOs in the firstoscillator for values of λ1 closer to λc(0, 0). For larger valuesof λ1 as in Fig. 8 a complete description can be obtainedonly by considering the phase plane and λeff

1 , as describedby conditions (ii) and (iii). As shown in Fig. 8(b), λeff

1 is

Fig. 8. SAO solutions for a diffusively coupled two-pool model. (a) Both(v1, w1) and (v2, w2) are in a SAO regime. (b) Evolution of λeff

k for k = 1, 2,compared with λc(Dv, Dw) and with the dynamics of v1 and v2.

subthreshold (λeff1 < λc(Dv, Dw)) for the part of the trajectory

on which the oscillations either return to the left branch of thev-nullcline (SAO), or follow the LAO cycle (see Fig. 3(a)). Thesubthreshold value of λeff

1 determines that the system takes theSAO route. Due to condition (iv) oscillator 2 is also in a SAOregime so that the localized state is lost.

In Section 4, we show how the system in the regime λ1 >

λc(0, 0) > λ2 can be sensitive to parameter changes, exhibitingmixed-mode asynchronous patterns for slight variations in λk ,Dv and Dw.

3.4. Mixed-mode localized solutions

In these patterns one of the oscillators remains in the SAOregime, while the second exhibits mixed-mode oscillations(MMOs), alternating between periods of SAO and LAO. Thedynamics are sensitive to small parameter changes, but theycan be observed for different values of the coupling and thebifurcation parameters. Synchronized MMOs have been found


Fig. 9. Regular mixed-mode localized solutions for the diffusively coupledtwo-pool model. (v2, w2) is in a SAO regime and (v1, w1) is in a mixed-modeoscillation regime with alternating single LAO and single SAO. (b) Evolutionof λeff

k for k = 1, 2, compared with λc(Dv, Dw) and with the dynamics of v1and v2.

in strongly diffusively coupled neural models [20]. To ourknowledge, localized MMOs of the type shown here have notyet been reported.Regular localized MMOs

In Figs. 9 and 10 we show two types of regular localizedMMOs, where there is a fixed number of SAOs and LAOs ineach period. In both cases λ1 = .0168 and λ2 = .0135, butdifferent solutions are observed by varying Dv and Dw. Recallthat these coupling coefficients compete in the destabilizing andstabilizing effects on SAOs.

In Fig. 9 we show regular localized MMOs for Dv = .1,Dw = .05. The main features are:

(i) λ1 6= λ2, resulting in phase differences when bothoscillators have SAOs,

(ii) λ1 > λc(Dv, Dw), but λ1 is close to λc(Dv, Dw),(iii) λ2 < λc(Dv, Dw), with λ2 a significant distance from

λc(Dv, Dw),

(iv) v1 increases through zero when λeff1 < λc(Dv, Dw)

following LAOs.

We begin with the system in the state where bothoscillators exhibit SAOs. Conditions (i) and (ii) imply thatλeff

1 fluctuates about λc(Dv, Dw), with a net effect of λeff1 >

λc(Dv, Dw). Then the SAOs of oscillator 1 gradually growin amplitude, with an eventual transition to LAOs dueto condition (ii). To complete the MMO cycle, conditions(ii) and (iv) combine to drive the first oscillator backto SAOs: the forcing of (v1, w1) via the coupling with(v2, w2) is large enough to cause fluctuations in λeff

1 , sothat it drops sufficiently below λc(Dv, Dw) as v1 increasesthrough zero following the LAOs (see Fig. 9(b)). As inSection 3.3, this forces v1 back to the left branch of thev-nullcline and back into the SAO regime. Then the cyclerepeats. Condition (iii) allows (v2, w2) to remain in a SAOregime, as long as the coupling is not large. Similar localizedMMOs can be observed for Dv 6= 0 and Dw = 0, e.g. forλ1 = .0165, λ2 = .011 and Dv = .046.

Another mechanism for regular MMOs is shown in Fig. 10for Dv = .05 and Dw = .1, with λ1 and λ2 as above. Thecanard critical value is reduced as compared to the previouscase (specifically, λc(.05, .1) < λc(.1, .05)), the key conditionsare similar, but the variations in λeff

1 and λeff2 have different

effects.

(i) λ1 6= λ2, resulting in phase differences when bothoscillators have SAOs,

(ii) λ1 > λc(Dv, Dw), but λ1 is close to λc(Dv, Dw),(iii) λ2 < λc(Dv, Dw), with λ2 closer to λc(Dv, Dw) as

compared with the previous case,(iv) following LAOs, v2 increases through zero when λeff

2 <

λc(Dv, Dw).

Again we begin with the initial state where both oscillatorsexhibit SAOs. Conditions (i) and (ii) again yield gradualgrowth in these SAOs for (v1, w1). The difference as comparedwith the regular MMOs is condition (iii) which allowsincreasing oscillations in (v1, w1) to drive λeff

2 sufficientlyabove λc(Dv, Dw). Then (v2, w2) exhibits one LAO per cycle,which forces λeff

1 well below λc(Dv, Dw) so that (v1, w1)

remains in a SAO regime.Condition (iv) shows that the feedback through fluctuations

in λeff2 is also sufficient to force the second oscillator back to

SAOs; as v2 passes through zero it returns to the left branch ofthe nullcline since λeff

2 < λc(Dv, Dw), and the cycle repeats.

Irregular mixed-mode localizationIn Figs. 11 and 12 we show irregular localized MMOs. In

Fig. 11 the values are the same as in Fig. 9 (for which regularMMOs are observed), except for Dw = 0.072. In Fig. 12, λ1is slightly smaller than in Fig. 9, with the other parametersthe same. Transitions between LAOs and SAOs in the MMOpattern can be observed for λeff

1 < λc(Dv, Dw) as v1 increasesthrough 0 (not shown), similar to conditions (iv) in the regularlocalized MMOs of Figs. 9(b) and 10(b). The difference in thenumber of SAOs and LAOs can be explained through secondarybifurcations in canards in systems with more than two degrees


Fig. 10. Regular mixed-mode localized solutions for the diffusively coupledtwo-pool model. (v1, w1) is in a SAO regime and (v2, w2) is in a mixed-modeoscillation regime with one LAO alternating with three SAOs.

of freedom, with the number of SAOs and LAOs dependenton initial conditions and reset values following LAOs [18].However, a detailed analysis of this phenomenon is outside thescope of this work.

4. Asynchronous LAO patterns

In this section we demonstrate a number of patterns in whichboth oscillators exhibit LAOs. For most of these phenomena, acareful phase plane analysis is required to completely describethe behavior. Nevertheless, we outline some cases in whichthe expressions for λc and λeff

k can lend some insight into thedynamics (see Table 2).

4.1. Destabilization of localization and SAOs by Dw

In Fig. 13 we show the case for λ1 = .015, λ2 = .012, Dw =

.05, Dv = 0, that is, λ1 > λc(Dw, Dv) > λ2. Note that theparameter values are the same as in Fig. 6, with the exceptionof an increase in λ2, yielding a shift of the localized solutions

Fig. 11. Irregular mixed-mode localized solutions for a diffusively coupledtwo-pool model. (v2, w2) is in a SAO regime and (v1, w1) is in a mixed-modeoscillation regime with a single LAO alternating with 1–4 SAOs.

Fig. 12. Irregular mixed-mode localized solutions for a diffusively coupledtwo-pool model. (v2, w2) is in a SAO regime and (v1, w1) is in a mixed-modeamplitude regime with a single LAO alternating with 2–3 SAOs on average.

shown in Fig. 6 to a LAO antiphased pattern. The shift is due toa violation of condition (iii) in Section 3.1; for increased λ2 andv1 < 0, λeff

2 exceeds λc(Dv, Dw) as v2 crosses through zero,forcing the system to asynchronous LAOs. Therefore a phaseplane analysis is required to completely predict this transition.

One can also obtain anti-phased LAOs for λ1 = λ2 <

λc(Dw, Dv), as shown in Fig. 14 for λ1 = λ2 = .01, Dv = 0,and Dw = .1. Here the main factors are:

(i) break of symmetry in the initial condition (here we haveused v1(0) = .03, w1(0) = .01, v2(0) = −.3, andw2(0) = −.1),

(ii) λeffk > λc(Dv, Dw) for sufficiently large values of |γ vk +

wk | in (8),(iii) λeff

k > λc(Dv, Dw) when vk increases through 0.


Table 1Significant factors in the routes to and from localized states

Uncoupled Coupling Effect λk near λc(0, 0) λeffk Figure #

SAO/SAO Dw ↑ S 1 – 6LAO/LAO Dw ≈ Dv , both ↑ S 2 – 7LAO/LAO Dw � 1 S 2 – 1LAO/SAO Dv ↑ D – 1 8LAO/SAO Dv ↑ or Dw ↑ S 1 1, 2 9–12

Column 1 gives the state of system without coupling for oscillator 1 and 2. Column 2 indicates the element of coupling which plays a role in the transition, ↑ forincreasing. Column 3 gives the coupling effect on the localized state, S or D for stabilizing/destabilizing. Column 4 gives the index k for any λk close to the criticalcanard value λc(0, 0) in the uncoupled case. Column 5 gives the index k for variation in any λeff

k which plays a significant role in the transition. Column 6 gives thenumber of the figure in which the behavior is shown.

Table 2Significant factors in the routes to and from asynchronous LAO

Uncoupled Coupling Effect λk near λc(0, 0) λeffk Figure #

LAO/SAO Dw ↑ S (LAO) 1 2 13SAO/SAO Dw ↑ S (LAO) – 1, 2 14LAO/SAO Dv S (MMO(c)) 1 1 15LAO/SAO Dv ↑ S (MMO(i)) 1 1, 2 16LAO/LAO Dv ↑ or Dw ↑ S (LAO) 1, 2 1, 2 17

Elements represented in each column are the same as in Table 1, with the exception of Column 3, which lists the stabilizing (S) or destabilizing (D) effect onasynchronous LAOs or MMOs. MMO(c) and MMO(i) refer to coherent and incoherent MMO, respectively.

The asymmetric initial conditions of condition (i) lead toanti-phased oscillations, which are necessary for condition (ii).Even though both λk’s are well within the SAO regime for theuncoupled system, the coupling is sufficient to drive the systemto LAOs, via condition (iii) (see Fig. 14(b)).

In contrast, if we consider identical, in-phase oscillationswith λ1 = λ2 < λc(Dv, Dw) as in Fig. 14, or if the initialconditions are similar for the two oscillators, then from (8) wefind that λeff

k ∼ λk and conditions (ii) and iii) are not satisfied.Then in-phase LAOs cannot be sustained, and the system movesto SAOs.

4.2. Mixed-mode patterns and irregular LAO

The behavior of λeffk suggests conditions under which MMO

and irregular LAO are observed, complementing a detailedstudy of the phase plane necessary to completely predict thesepatterns.

Mixed-mode patternsIn this section we show MMOs of two types: coherent

MMOs, where the oscillators show phase locked behavior whenin the LAO part of the cycle, and incoherent MMOs, where theLAO part of the cycle is not phase locked. In Fig. 15 we showcoherent MMOs for λ1 = .017, λ2 = .01, Dv = .1, Dw = 0,where a single LAO follows a fixed period of SAO. The threekey factors in this case are:

(i) λ1 > λc(Dv, Dw),(ii) λ1 is close to λc(Dv, Dw), so that fluctuations in λeff

1 maycause transitions in behavior,

(iii) λ2 is well below λc(Dv, Dw), so that λeff2 < λc(Dv, Dw)

when oscillator 1 exhibits SAOs.

As also seen in Section 3.4, condition (i) has a net effectof λeff

1 > λc(Dv, Dw), so that if oscillator 1 is initially in

the SAO regime, the amplitude of its oscillations grows untilit eventually makes the transition to LAOs due to condition(ii). Condition (iii) implies that the second oscillator maintainsSAOs until the first oscillator is in a LAO regime. TheLAO in the second oscillator are slaved to the first oscillator,immediately following LAO in oscillator 1. The LAO in thesecond oscillator creates a significant variation in λeff

1 , so thatoscillator 1 returns to SAO, as does the second oscillator due tocondition (iii), and the cycle repeats.

Fig. 16 shows incoherent MMOs, for λ1 = .02, λ2 =

.012, Dv = .1, Dw = 0. Incoherent MMOs have interactionsthat are qualitatively similar to that of coherent MMOs, bothdepending on conditions (i) and (ii) above. Condition (iii) aboveis modified for incoherent MMOs.(iii)∗ λ2 is well below λc(Dv, Dw), so that λeff

2 is belowλc(Dv, Dw) on average when oscillator 1 has SAOs.

Here the λk are close to λc(Dv, Dw), for both k = 1, 2, sothat the influence of λeff

k plays a crucial role in the differencebetween the coherent and incoherent MMOs shown in Figs. 15and 16, respectively. In Fig. 16 the fluctuations in λeff

1 dropbelow λc(Dv, Dw) less frequently, so that the first oscillatorhas a stronger preference for the LAO state than in Fig. 15.Condition (iii)∗ implies that fluctuations in λeff

2 make LAOsmore likely in the second oscillator in Fig. 16 than in Fig. 15.Then the LAOs of the two are no longer phase locked, and thesecond oscillator enters the LAO regime before the first.

A fourth crucial element in maintaining both coherence andincoherence MMOs:(iv) (De)stabilizing effects on LAOs and SAOs related to themagnitude of Dv and Dw.

As seen in Section 3, increasing Dv and decreasing Dw

favors SAOs, as in Fig. 15. Moderate values of Dv together withconditions (iii) and (iv) lead to regular MMOs where the LAOs


Fig. 13. Antiphased LAO regime for the diffusively coupled two-pool modelwith λ1 > λc(Dv, Dw) and λ2 < λc(Dv, Dw).

in oscillator 2 are slaved to oscillator 1. For larger Dv with theother parameters fixed as in Fig. 15, the MMOs destabilize toSAOs for both oscillators. Decreasing Dv and increasing Dw

destabilizes the MMOs to LAOs for both oscillators.

Irregular LAOsWe show irregular LAOs in Fig. 17 for λ1 = .017 = λ2,

Dv = .2, Dw = .1. This phenomenon can be observedby starting with initial conditions such as the MMO statein Fig. 15, and increasing the values of λ2, Dv , and Dw

accordingly. Several aspects of the parameter values contributeto this phenomenon:

(i) λk is near λc(Dv, Dw) for both k = 1, 2,(ii) moderate values of coupling Dv and Dw,

(iii) Dv > Dw.

Condition (i) allows fluctuations in λeffk that cause occasional

excursions to the SAO regime, which are crucial for theirregularity; for larger values of λ1 = λ2 these oscillationssynchronize to regular LAOs. Factor (ii) refers to the range of

Fig. 14. Antiphased LAO regime for the diffusively coupled two-pool modelwith λ1 = λ2 < λc(Dv, Dw).

the coupling coefficients Dv and Dw which are large enoughto destabilize localization, as seen in previous sections, butnot large enough to encourage synchronization, as in [20].Condition (iii) balances the competitive effects of Dv and Dw:increasing Dw alone contributes to asynchronized LAOs, asshown in Figs. 13 and 14, while increasing Dv destabilizesLAOs, as in Fig. 8 and the MMOs in Figs. 15 and 16. Thus,for these intermediate values of the coupling, while keeping λknear λc(Dv, Dw), fluctuations in λeff

k play a significant role inthe irregularity.

5. Discussion

In this study we demonstrate and analyze the mechanismsof various types of localized patterns in a diffusively coupledmodel of calcium (relaxation) oscillators. In addition to theclassical localized patterns with each oscillator displayingeither LAOs or SAOs but not both, we found MMOlocalized patterns in which one oscillator displays SAOs


Fig. 15. Coherent mixed-mode oscillations for the diffusively coupled two-pool model, with λ2 sufficiently below λc(Dv, Dw).

and the other alternates between LAOs and SAOs eitherin a regular or irregular manner. We demonstrate that thepatterns presented here can be explained in a self-consistentmanner by extending concepts developed for the analysisof the canard phenomenon in two dimensions [10,12,7,8].Two key quantities, the perturbed canard critical value andthe effective bifurcation parameter, capture the autonomousand non-autonomous effects introduced through the coupling.These quantities are valuable diagnostics for explaining andpredicting the pattern dynamics, particularly for the localizedpatterns, since these states appear by a variety of mechanisms.

The coupling, heterogeneity, and the canard phenomenoncombine to support localization and MMOs in a number ofregimes. In some cases, by increasing Dw, SAOs can bedestabilized in only one oscillator in a system where theuncoupled oscillators both display SAOs. In other cases, byincreasing both Dv and Dw, SAOs can be stabilized in onlyone oscillator in a system where the uncoupled oscillators bothdisplay LAOs. Both mechanisms have the common feature that

Fig. 16. Incoherent mixed-mode oscillations for the diffusively coupled two-pool model.

the localized patterns appear when the two oscillators are indifferent states, displaying either LAOs or SAOs but not both.A different type of mechanism is responsible for the generationof mixed-mode oscillatory patterns. There, when uncoupled,the oscillators display either SAOs or LAOs but not both.By increasing both Dv and Dw, the LAO regime is partiallydestabilized and the corresponding oscillator enters a MMOregime.

The key contributing factors for localization are differentfrom other prototypical examples, such as the localized patternsobserved both experimentally [1,2] and theoretically [3,7,8]in the globally coupled Belousov–Zhabotinsky reaction. Thesepatterns, with each oscillator stabilized in a different amplituderegime, are obtained for values of the global feedbackparameter that are neither too large nor too small. In that settinglocalization is a consequence of the inhibitory global couplingrather than diffusive (local) coupling. Localized patterns havebeen obtained in non-relaxation oscillatory systems as well[4,5].


Fig. 17. Irregular oscillations for the diffusively coupled two-pool model withmoderate values of the coupling.

The MMO patterns presented here are an additional con-sequence of the coupling. Single two-dimensional relaxationoscillators undergoing a supercritical Hopf bifurcation displayeither LAOs or SAOs but not both. The possibility of obtainingoscillatory patterns in which both LAOs and SAOs are presentrequires a higher dimensional system with at least one fastand two slow equations. Note that the present coupled systemhas two fast and two slow variables. Three-dimensional sys-tems of this type produce mixed-mode oscillations (MMOs)in which SAOs alternate with LAOs either in a regular or ir-regular way [25,29–33,46] (see also references therein). Quasi-synchronized MMOs have also been obtained in a system ofstrongly diffusively coupled two-dimensional relaxation oscil-lators with two fast and two slow variables [20]. There both os-cillators alternate between LAOs and SAOs and were almostsynchronized as a consequence of the strong coupling. TheMMOs observed in the present study are different from the onesfound in [20]; when one operates away from the strong couplinglimit, a richer variety of patterns appear, including localizedand asynchronous MMOs. The experimental and theoretical

irregular (in amplitude and frequency) patterns observed in ex-periments and simulations of the BZ reaction [1–3] are a higher-dimensional manifestation of irregular MMO patterns that havebeen observed in globally coupled relaxation oscillators. A de-tailed analysis of these patterns in higher-dimensional systemsis outside the scope of this study. For some insight into the tech-niques used to analyze MMOs in three-dimensional systems werefer to [18,20,29,35].

Among the non-localized but still interesting patterns wefound are LAO antiphase patterns for parameter values forwhich both uncoupled oscillators are in a SAO regime.Furthermore, moderate increases in the coupling can alsopartially destabilize SAOs, leading to an antiphased MMOregime. LAO antiphase patterns have been previously found forthe coupled van der Pol oscillator [36] and the globally coupledBelousov–Zhabotinsky reaction [3].

Acknowledgments

The authors want to thank Nancy Kopell, Martin Wechsel-berger and Martin Krupa for reading earlier versions of thismanuscript. This work was partially supported by the Bur-roughs Wellcome Fund (HGR), NSF-DMS 021505 (HGR),NSF-DMS 0072311 (RK) and a NSERC discovery grant (RK).

Appendix A. The two-pool model

We give a brief summary of the two-pool model and acommon transformation used to analyze the oscillations. Wefollow [15]. The two-pool model derives its name from theoriginal assumption of two distinct internal Ca2+ stores in thecell, one sensitive to IP3 (inositol (1,4,5)-triphosphate) and theother sensitive to Ca2+, with concentrations denoted c and cs ,respectively. The equations describe the interaction of c and cs ,which have the dimensional form as summarized in Keener andSneyd [15],

dc

dτ= r − kc − F(c, cs)

dcs

dτ= F(c, cs). (10)

Here τ is time, r is the influx of Ca2+ due to IP3, and Ca2+

is pumped out at a rate kc. Since r is constant for constantIP3, by varying r one can study variations in the behavior ofthe system due to IP3. The function F describes the flux fromthe IP3-sensitive pool to the Ca2+-sensitive pool, and it is acombination of uptake, release, and leakage with rate −k f cs .In particular

F(c, cs) = Juptake − Jrelease − k f cs

=U1cn

K n1 + cn −

U2cms

K m1 + cm

s

cp

K p3 + cp

− k f cs . (11)

The positive feedback of Ca2+ release is evidenced by thedependence of Jrelease on c. The system is non-dimensionalizedusing

Vk = c/K1, Wk = cs/K2, t = τk, ε = kK2/U2,

µ = r/(kK1), α = K3/K1, β = U1/U2,

γ = U1/U2, δ = k f K2/U2, (12)


to obtain (1) with Dw = Dv = 0{V ′

k = µk − Vk − γ /εF(Vk, Wk),

W ′

k =1ε

F(Vk, Wk),(13)

for each k. In [15] the following reduction for a single oscillatorwas proposed:

vk = Wk, wk = Vk + γ Wk . (14)

Substituting (14) into (13) one gets{v′

k =1ε

F(vk, wk),

w′

k = [µk + γ vk − wk](15)

with

F(v, w) = F(w − γ v, v) (16)

for each value of k.System (15) describes an oscillator of FHN type, with the

v- and w-nullclines significantly different from v- and w-nullclines of system (5). However, the form of (16) is not themost convenient for the canard analysis we use in this paper.We propose a better and slightly different approach which is totransform system (13) into a van der Pol (VDP)-type oscillator.(In a VDP-type oscillator the w-nullcline is a vertical line whilein a FHN-type oscillator it makes a positive angle with the v-axis). This is achieved by using transformation (3), which canalso be viewed as a linear transformation of (14).

Throughout the paper we use the parameter values as in [15]:α = .9, β = .13, γ = 2, δ = .004, m = 2, n = 2, p = 4 andε = .01.

Appendix B. The canard critical value for the canonicalform

We follow [7] and extend our result to the case studied inthis manuscript. The calculations are similar to those in [7]. Weconsider a system of the form{

v′= F(v, w) + εΨ(v, w, λ, ε),

w′= εG(v, w, λ)

(17)

where 0 < ε � 1 and Ψ(v, w, λ, ε) = O(v, w, λ, ε). Thefunctions F and G are as in Section 2.1. We make the followingassumptions for ε = 0.

• Singularity: F(0, 0) = 0 and ∂ F/∂v(0, 0) = 0, G(0, 0, 0) =

0.• Non-degeneracy assumptions defining the minimum:

∂2 F/∂v2(0, 0) > 0, ∂ F/∂w(0, 0) < 0.• Non-degeneracy assumptions defining a canard point:

∂G/∂v(0, 0, 0) > 0 ∂G/∂w ≤ 0, ∂G/∂λ < 0.

With these assumptions system (17) satisfies the conditionsstated in [12,13]. The canonical form for system (17) is givenby

v′= −w + v2

− wh1(v, w) + v2h2(v, w)

+ εh6(v, w, λ, ε),

w′= ε[v − λ + vh3(v, w, λ) − λh4(v, w, λ)

+ wh5(v, w, λ)],

(18)

where

h1(v, w) =2G1/2

v Fvw

|Fw|1/2 Fvv

v +O(w), (19)

h2(v, w) =2G1/2

v |Fw|1/2 Fvvv

3F2vv

v +O(w), (20)

h3(v, w, λ) =|Fw|

1/2Gvv

G1/2v Fvv

v +O(w, λ), (21)

h4(v, w, λ) = −2G1/2

v |Fw|1/2Gvλ

Fvv|Gλ|v +O(w, λ), (22)

h5(v, w, λ) =Gw

G1/2v |Fw|1/2

v +O(v, w, λ), (23)

h6(v, w, λ) =2Ψv

Fvv

v +O(v2, w, λ, ε) (24)

and all the function are evaluated at (v, w, λ, ε) = 0.Following [7] and [13] the following expression for λc can

be calculated:

λc = La1 − 3a2 + 2a3 − 2a5 − 4a6

8ε +O(ε3/2), (25)

where

L =2G3/2

v |Fw|1/2

4 Fvv|Gλ|, a1 =

∂h1

∂v= −

2G1/2v Fvw

|Fw|1/2 Fvv

,

a2 =∂h2

∂v=

2 G1/2v |Fw|

1/2 Fvvv

3F2vv

,

a3 =∂h4

∂v=

|Fw|1/2Gvv

G1/2v Fvv

, a5 = h5 =Gw

G1/2v |Fw|1/2

,

a6 =∂h6

∂v= 2Ψv F2

vv, (26)

where all the functions are evaluated at 0.Substituting into (25) we get

λc(√

ε) = −G3/2

v |Fw|1/2

4Fvv|Gλ|[−a1 + 3a2 − 2a3 + 2a5 + 4a6] ε

+O(ε3/2)

= −Gv

2F3vv|Gλ|

[Gv Fvw Fvv + Gv|Fw|Fvvv

− |Fw|Gvv Fvv + Gw F2vv + 2Ψv F2

vv]ε

+O(ε3/2), (27)

where all the functions are calculated at 0.

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