Bifurcation structure of a model of bursting pancreatic cells

BioSystems 63 (2001) 3–13

Bifurcation structure of a model of bursting pancreatic cells

Erik Mosekilde a,*, Brian Lading a, Sergiy Yanchuk b, Yuri Maistrenko b

a Department of Physics, Center for Chaos and Turbulence Studies, Building 309, The Technical Uni�ersity of Denmark,2800 Kgs. Lyngby, Denmark

b lnstitute of Mathematics, National Academy of Sciences of Ukraine, Kie� 252601, Ukraine

Abstract

One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic �-cellsreveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one sideof the arms and saddle-node bifurcations on the other. The transition from this structure to the so-calledperiod-adding structure is found to involve a subcritical period-doubling bifurcation and the emergence of type-IIIintermittency. The period-adding transition itself is not smooth but consists of a saddle-node bifurcation in which(n+1)-spike bursting behavior is born, slightly overlapping with a subcritical period-doubling bifurcation in whichn-spike bursting behavior loses its stability. © 2001 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Bursting cells; Period-adding; Bifurcations; Chaos; Basins of attraction

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1. Introduction

By virtue of the far-from-equilibrium condi-tions in which they are maintained through thecontinuous action of ion pumps, many biologicalcells display an excitable electrical activity, or themembrane potential exhibits complicated patternsof slow and fast oscillations associated with varia-tions in the ionic currents across the membrane.This dynamics plays an essential role for thefunction of the cell as well as for its communica-tion with neighboring cells. It is well known, forinstance, that pancreatic �-cells under normal cir-cumstances display a bursting behavior with alter-

nations between an active (spiking) state and asilent state (Dean and Matthews, 1970; Atwaterand Beigelman, 1976; Meissner and Preissler,1980). It is also established (Ozawa and Sand,1986; Miura and Pernarowski, 1995) that the se-cretion of insulin depends on the fraction of timethat the cells spend in the active state, and thatthis fraction increases with the concentration ofglucose in the extracellular environment. Thebursting dynamics controls the influx of Ca2+

ions into the cell, and calcium is considered anessential trigger for the release of insulin. In thisway, the bursting dynamics organizes the responseof the �-cells to varying glucose concentrations.At glucose concentrations below 5 mM, the cellsdo not burst at all. For high glucose concentra-tions (�22 mM), on the other hand, the cellsspike continuously, and the secretion of insulinsaturates (Satin and Cook, 1989).

* Corresponding author. Tel.: +45-45-253-104; Fax: +45-45-931-669.

E-mail address: [email protected] (E. Mosekilde).

0303-2647/01/$ - see front matter © 2001 Elsevier Science Ireland Ltd. All rights reserved.

PII: S 0303 -2647 (01 )00142 -3

E. Mosekilde et al. / BioSystems 63 (2001) 3–134

A number of experimental studies have shownthat neighboring �-cells in an islet of Langerhanstend to synchronize their membrane activity(Sherman et al., 1988), and that cytoplasmic Ca2+

oscillations can propagate across clusters of �-cells in the presence of glucose (Gylfe et al., 1991).The precise mechanism underlying this interactionis not fully known. It is generally considered,however, that the exchange of ions via lowimpedance gap junctions between the cells play asignificant role (Sherman and Rinzel, 1991). Suchsynchronization phenomena are important be-cause not only do they influence the activity of theindividual cell, but they also affect the overallinsulin secretion. Actually, it appears that theisolated �-cell does not burst but shows disorga-nized spiking behavior as a result of the randomopening and closing of potassium channels (Chayand Kang, 1988; Sherman et al., 1988; Smolen etal., 1993). A single �-cell may have of the order ofa few hundred such channels. However, with typi-cal opening probabilities as low as 5–10%, only afew tens will open during a particular spike. Orga-nized bursting behavior arises for clusters of 20 ormore closely coupled cells that share a sufficientlylarge number of ion channels for stochastic effectsto be negligible.

Models of pancreatic �-cells are usually basedon the standard Hodgkin–Huxley formalism withelaborations to account, for instance, for the in-tracellular storage of Ca2+, for aspects of theglucose metabolism, or for the influence of ATPand other hormonal regulators. Over the years,many such models have been proposed with vary-ing degrees of detail (Chay and Keizer, 1983;Chay, 1985a, 1990; Sherman and Rinzel, 1992).At the minimum, a three-dimensional model withtwo fast variables and one slow variable is re-quired to generate a realistic bursting behavior. Inthe earliest models, the slow dynamics was oftenconsidered to be associated with changes in theintracellular Ca2+ concentration. It appears,however, that the correct biophysical interpreta-tion of the slow variable remains unclear. The fastvariables are usually the membrane potential Vand the opening probability n of the potassiumchannels. More elaborate models with a couple ofadditional variables have also been proposed.

Although the different models have beenaround for quite some time, their bifurcationstructure is so complicated as to not yet be under-stood in full. Conventional analyses (Sherman etal., 1988; Sherman, 1994) are usually based on aseparation of the dynamics into a fast and a slowsubsystem, whereafter the slow variable is treatedas a bifurcation parameter for the fast dynamics.How compelling such an analysis may appear,particularly when one considers the large ratio ofthe involved time constants, the analysis fails toaccount for the more interesting dynamics of themodels. Simulations with typical �-cell modelsdisplay chaotic dynamics and period-doubling bi-furcations for biologically interesting parametervalues (Chay, 1985b; Fan and Chay, 1994), andsuch phenomena clearly cannot occur in a two-di-mensional, time-continuous system such as thefast subsystem. The simplified analyses can alsoprovide an incorrect account of the so-called pe-riod-adding transitions in which the systemchanges from a bursting behavior with n spikesper burst into a behavior with n+1 spikes perburst. Finally, the simplified analyses lead to anumber of misperceptions with respect to thehomoclinic bifurcations that control the onset ofbursting.

Wang (1993) has proposed a combination oftwo different mechanisms to explain the emer-gence of chaotic bursting. First, the continuousspiking state undergoes a period-doubling cascadeto a state of chaotic firing, and this state isdestabilized in a boundary crisis. Bursting thenarises through the realization of a homoclinicconnection that serves as a reinjection mechanismfor the chaotic saddle created in the boundarycrisis. In this picture, the bursting oscillations aredescribed as a form of intermittency with thesilent state corresponding to the reinjection phaseand the firing state to the normal laminar phase.Wang supports his analysis by a calculation of theescape rate from the chaotic saddle and he out-lines a symbolic dynamics to characterize the var-ious bursting states. As it appears, however, thequestion of how the homoclinic connection arisesis left unanswered.

Terman (1991, 1992) has performed a moredetailed analysis of the onset of bursting. He has

E. Mosekilde et al. / BioSystems 63 (2001) 3–13 5

obtained a two-dimensional flow-defined map forthe particularly complicated case where the equi-librium point of the full system falls close to asaddle point of the fast subsystem, which has ahomoclinic orbit. By means of this map, Termanproved the existence of a hyperbolic structure (achaotic saddle) similar in many ways to a Smalehorseshoe. This represents an essential step for-ward in understanding the complexity involved inthe emergence of bursting. However, since Ter-man’s set is non-attracting, it cannot be relateddirectly to the observed stable chaotic burstingphenomena.

More recently, Belykh et al. (2000) presented aqualitative analysis of a generic model structurethat can reproduce the bursting and spiking dy-namics of pancreatic �-cells. They consider fourmain scenarios for the onset of bursting. It isemphasized that each of these scenarios involvesthe formation of a homoclinic orbit that travelsalong the route of the bursting oscillations and,hence, cannot be explained in terms of bifurca-tions in the fast subsystem. In one of the scenar-ios, the bursting oscillations arise in a homoclinicbifurcation in which the one-dimensional stablemanifold of a saddle point becomes attracting toits whole two-dimensional unstable manifold.This type of homoclinic bifurcation, and the com-plex behavior that it can produce, does not ap-pear to have been examined previously.

Most recently, Lading et al. (2000) have studiedchaotic synchronization (and the related phenom-ena of riddled basins of attraction, attractor bub-bling, and on–off intermittency) for a model oftwo coupled, identical �-cells, and Yanchuk et al.(2000) have investigated the effects of a smallparameter mismatch between the coupled chaoticoscillators. In the limit of strong interaction, itwas found that such a mismatch gives rise to ashift of the synchronized state away from thesymmetric synchronization manifold, combinedwith occasional bursts out of synchrony. Thiswhole class of phenomena is of significant interestto the biological sciences (Kaneko, 1994) whereone often encounters the situation that a largenumber of cells (or functional units), which eachperforms complicated non-linear oscillations, in-teract to produce a coordinated function on ahigher organizational level.

The purpose of the present paper is to give asomewhat simpler account of the bifurcationstructure of the individual �-cell. Our analysisreveals the existence of a squid-formed area ofchaotic dynamics in parameter plane with period-doubling cascades along one side of the arms andsaddle-node bifurcations along the other. Thetransition from this structure to the so-called pe-riod-adding structure involves a subcritical pe-riod-doubling bifurcation and the emergence oftype-III intermittency. The period-adding transi-tion itself is found to be non-smooth and toconsist of a saddle-node bifurcation in which sta-ble (n+1)-spike behavior is born, overlappingslightly with a subcritical period-doubling bifurca-tion in which stable n-spike behavior ceases toexist. The two types of behavior follow each otherclosely in phase space over a major part of theorbit to suddenly depart and allow one of thesolutions to perform an extra spike.

Bursting behavior similar to the dynamics thatwe have described for pancreatic �-cells is knownto occur in a variety of other cell types as well.Pant and Kim (1976), for instance, have devel-oped a mathematical model to account for experi-mentally observed burst patterns in pacemakerneurons, and Morris and Lecar (1981) have mod-elled the complex firing patterns in barnacle giantmuscle fibers. Braun et al. (1980) have investi-gated bursting patterns in discharging cold fibersof the cat, and Braun et al. (1994) have studiedthe effect of noise on signal transduction in sharksensory cells. Although the biophysical mecha-nism underlying the bursting behavior may varysignificantly from cell type to cell type, we expectmany of the basic bifurcation phenomena to re-main the same.

2. The bursting cell model

As the basis for the present analysis, we shalluse the following model suggested by Sherman etal. (1988):

�dVdt

= −ICa(V)−IK(V,n)−gSS(V−VK) (1)

with


�dndt

=� [n�(V)−n ] (2)

�S

dSdt

=S�(V)−S (3)

ICa(V)=gCam�(V)(V−VCa) (4)

IK(V,n)=gKn(V−VK) (5)

w�(V)=�

1+exp�Vw−V

�w

�n−1

for w=m, n and S (6)

Here, V represents the membrane potential, n maybe interpreted as the opening probability of thepotassium channels, and S accounts for the pres-ence of a slow dynamics in the system. As previ-ously noted, the correct biophysical interpretationof this variable remains uncertain. ICa and IK arethe calcium and potassium currents, gCa=3.6 andgK=10.0 are the associated conductances, andVCa=25 and VK= −75 mV are the respectiveNernst (or reversal) potentials. �/�S defines theratio of the fast (V and n) and the slow (S) timescales. The time constant for the membrane po-tential is determined by the capacitance and thetypical total conductance of the cell membrane.With �=0.02 s and �S=35 s, the ratio kS=�/�S

is quite small, and the cell model is numericallystiff.

The calcium current ICa is assumed to adjustinstantaneously to variations in V. For fixed val-ues of the membrane potential, the gating vari-ables n and S relax exponentially towards thevoltage-dependent steady-state values n�(V) andS�(V). Together with the ratio kS of the fast tothe slow time constant, VS will be used as themain bifurcation parameter. This parameter de-termines the membrane potential at which thesteady-state value for the gating variable S attainsone-half of its maximum value. The otherparameters are gS=4.0, Vm= −20 mV, Vn= −16 mV, �m=12 mV, �n=5.6 mV, �S=10 mV,and �=0.85. These values are all adjusted to fitexperimentally observed relationships. In accor-dance with the formulation used by Sherman etal. (1988), there is no capacitance in Eq. (1), andall the conductances are dimensionless. To elimi-nate a dependence on the cell size, all conduc-

tances have been scaled with the typicalconductance. Hence, we may consider the modelas a model of a cluster of closely coupled �-cellsthat share the combined capacity and conduc-tance of the entire membrane area.

Fig. 1 shows an example of the temporal varia-tions of the variables V, n and S as obtained bysimulating the cell model under conditions whereit exhibits continuous chaotic spiking. Here, kS=0.57×10−3 and VS= −38.34 mV. We notice theextremely rapid opening and closing of some ofthe potassium channels. The opening probabilityn changes from nearly nothing to about 10% atthe peak of each spike. We also notice how theslow variable increases during the bursting phaseto reach a value just below 310, whereafter the cellswitches into the silent phase, and S graduallyrelaxes back. If the slow variable is assumed torepresent the intracellular Ca2+ concentration,this concentration is seen to increase during eachspike until it reaches a threshold, and the burstingphase is switched off. S hereafter decreases asCa2+ is continuously pumped out of the cell.

Let us start our bifurcation analysis with a fewcomments concerning the equilibrium points ofthe �-cell model. The zero points of the vectorfield Eqs. (1)– (3) are given by:

gCam�(V)(V−VCa)+gKn(V−VK)

+gSS(V−VK)=0 (7)

n=n�(V)=�

1+exp�Vn−V

�n

�n−1

(8)

S=S�(V)=�

1+exp�VS−V

�S

�n−1

(9)

so that the equilibrium values of n and S areuniquely determined by V. Substituting Eqs. (8)and (9) into Eq. (7), the equation for the equi-librium potential becomes

f(V)�gCam�(V)(V−VCa)+gKn�(V)(V−VK)

+gSS�(V)(V−VK)=0 (10)

with m�(V) as given by Eq. (6). Assuming VCa�VK and considering the conductances gCa, gK andgS to be positive by definition, we observe thatany equilibrium point of the �-cell model musthave a membrane potential in the interval VK�


V�VCa, and that there must be at least one suchpoint. This assertion follows directly from thecontinuity of f(V) as defined from Eq. (10). ForV�VK, all terms in the expression for f(V) arenegative, and f(V)�0. For V�VCa, all terms inf(V) are positive. Hence, there is at least one rootof Eq. (10) in the interval between VK and VCa,and no roots outside this interval.

Evaluated at such an equilibrium point,the Jacobian matrix for the �-cell model has theform:

J=

��

J11 J12 J13

J21 J22 0J31 0 J33

��

(11)

with

J11= −gCa

�m�(V)�V

(V−VCa)

−gCam�(V)VCa−VK

V−VK

(12)

J12= −gK(V−VK) (13)

J13= −gSkS(V−VK) (14)

J21=��n�(V)

�V(15)

J22= −� (16)

J31=�S�(V)

�V(17)

and

J33= −kS (18)

Applying Hurwitz’ theorem (which gives theconditions for all solutions of the characteristicequation to have negative real parts), we obtainthe following criteria for the equilibrium point tobe asymptotically stable:

J11+J22+J33�0 (19)

and

(J11+J22+J33)(J11 J33+J11 J22+J22 J33)

+J31 J22 J13+J21 J12 J33�0 (20)

Fig. 1. Example of the temporal variations of the membranepotential V(t), the opening probability n(t) for the potassiumchannels, and the slow variable S(t) that controls the switch-ing between the active and the silent phases. kS=0.57×10−3

and VS= −38.34 mV. The model exhibits continuous chaoticspiking. Here and in all of the following figures, S(t) has beenmultiplied by a factor of �S/�.


Fig. 2. Bifurcation diagram for the bursting cell model asobtained by means of one-dimensional continuation methods.The equilibrium point undergoes a Hopf bifurcation forVS� −42 mV. kS=0.1.

3. Bifurcation diagrams for the cell model

Fig. 3 shows a one-dimensional bifurcation dia-gram for the cell model with VS as the controlparameter. Here, kS=0.57×10−3. The figure re-sembles figures that one can find in early papersby Chay (1985b). The diagram was constructedfrom a Poincare section in phase space with n=0.04. With this section, we can ensure that allspikes performed by the model are recorded. ForVS� −37.8 mV, the model exhibits continuousperiodic spiking. As VS is reduced, the spikingstate undergoes a usual period-doubling cascadeto chaos with periodic windows. Each window isterminated by a saddle-node bifurcation to theright and by a period-doubling cascade to the left.

Around VS= −38.3 mV, a dramatic change inthe size of the chaotic attractor takes place. Thismarks the transition to bursting dynamicsthrough the formation of a homoclinic connectionin three-dimensional phase space (Belykh et al.,2000). Below VS� −38.5 mV the bifurcation sce-nario is reversed, and for VS� −38.9 mV abackwards period-doubling cascade leads the sys-tem into a state of periodic bursting with fivespikes per burst. The interval of periodic burstingends near VS = −39.7 mV in a saddle-nodeleading to chaos in the form of type-I intermit-tency (Berge et al., 1984). With further reductionof VS, the chaotic dynamics develops via a new

For a characteristic equation of third-ordera0x

3+a1x2+a2x+a3=0 with a0�0, we have

the Hurwitz’ conditions

a1�0,��

a1 a0

a3 a2

��

�0, and a3�0

(21)

To derive the conditions of Eqs. (19) and (20),we only have to substitute the coefficients of ourcharacteristic equation into Eq. (21), noting thatthe condition a3�0 will always be satisfied withthe assumed parameter values.

For a Hopf bifurcation to occur, the secondcondition in Eq. (21) must be violated. As shownin the bifurcation diagram of Fig. 2, this mayhappen as VS is increased. Here, we have plottedthe equilibrium membrane potential as a functionof VS for kS =0.1. All other parameters assumetheir standard values. For low values of VS (fully-drawn curve in the bifurcation diagram), the equi-librium point is asymptotically stable. AtVS� −42 mV, however, the model undergoes aHopf bifurcation, and the equilibrium point turnsinto an unstable focus. The stable as well as theunstable branch of the bifurcation curve was fol-lowed by means of standard continuation meth-ods. Due to the stiffness of the model, suchmethods are not always easy to apply. In the nextsection, we shall investigate the main structure ofthe subsequent bifurcations.

Fig. 3. One-dimensional bifurcation diagram for kS=0.57×10−3. The model displays chaotic dynamics in the transitionintervals between continuous periodic spiking and bursting,and between the main states of periodic bursting.


Fig. 4. Two-dimensional bifurcation diagram outlining the main bifurcation structure in the (VS, kS) parameter plane. Note thesquid-formed black region with chaotic dynamics.

reverse period-doubling cascade into periodicbursting with four spikes per burst. It is clearfrom this description that chaotic dynamics tendsto arise in the transitions between continuousspiking and bursting, and between the differentbursting states.

To establish a more complete picture of thebifurcation structure, we have applied a largenumber of such one-dimensional scans to identifythe main periodic solutions (up to period 10) andto locate and classify the associated bifurcations.The results of this investigation are displayed inthe two-dimensional bifurcation diagram of Fig.4. To the left in this figure, we observe the Hopfbifurcation curve discussed in Section 2. Belowthis curve, the model has one or more stableequilibrium points. Above the curve, we find aregion of complex behavior delineated by theperiod-doubling curve PD1–2. Along this curve,the first period-doubling of the continuous spikingbehavior takes place. In the heart of the regionsurrounded by PD1–2 we find an interesting squid-formed structure with arms of chaotic behavior(indicated black) stretching down towards theHopf bifurcation curve.

Each of the arms of the squid-formed structureseparates a region of periodic bursting behavior

with n spikes per burst from a region with regular(n+ 1)-spikes per burst behavior. Each arm has aperiod-doubling cascade leading to chaos on oneside and a saddle-node bifurcation on the other. Itis easy to see that the number of spikes per burstmust become very large as kS approaches zero.Fig. 5 is a magnification of part of the two-dimen-sional bifurcation diagram. Here, we observe howthe chaotic region in the arms narrows down asthe bifurcation curves on both sides approach oneanother with decreasing values of VS. This leadsto the so-called period-adding structure (Chay,1985b). Along the curves of this structure, a peri-odic bursting state with n spikes per burst appearsto be directly transformed into a state with n+1spikes per burst.

To illustrates what happens in this transition,Figs. 6 and 7 show one-dimensional bifurcationdiagrams obtained by scanning from the twospikes per burst regime into the three spikes perburst regime for kS=1.0×10−3 and kS=0.84×10−3, respectively. To the left in Fig. 6, we findtwo spikes per burst behavior, and to the right wehave periodic bursting with three spikes per burst.As VS is gradually increased from −40.86 mV,the two spikes per burst behavior remains stableall the way up to VS� −40.827 mV, where it


Fig. 5. Magnification of part of the bifurcation diagram in Fig. 4. Note how the chaotic region in each squid arm narrows downas the bifurcation curves on both sides approach one another and intersect.

undergoes a subcritical period-doubling. In theabsence of another attracting state in the neighbor-hood, the system explodes into a state of Type-IIIintermittency (Berge et al., 1984). The reinjectionmechanism associated with this intermittency be-havior may correspond to the reinjection mecha-nism proposed by Wang (1993).

If we go backwards in the bifurcation diagramof Fig. 6, the unstable period-4 solution generatedin the subcritical period-doubling bifurcation stabi-lizes in a saddle-node bifurcation near VS = −40.851 mV, and with increasing values of VS thestable period-4 solution undergoes a period-dou-bling cascade to chaos. Around VS= −40.841mV, the chaotic attractor disappears in a boundarycrisis as it collides with the inset to the unstableperiod-4 solution. This process is likely to leave achaotic saddle that can influence the dynamics inthe intermittency regime for VS� −40.827 mV.For higher values of VS, the chaotic state (withperiodic windows) continues to exist until thesaddle-node bifurcation at VS� −40.765 mVwhere periodic bursting with three spikes per burstemerges.

Fig. 7 shows a brute-force bifurcation diagramobtained by scanning VS in both directions acrossthe PA2–3 period-adding ‘‘curve’’ for kS=0.84×

10−3. Inspection of this figure clearly reveals thenarrow interval around VS= −41.43 mV wherethe two-spike and three-spike solutions coexist.Evaluation of the eigenvalues shows that thetwo-spike solution disappears in a (subcritical)period-doubling bifurcation and that the stablethree-spike solution arises in a saddle-node bifurca-tion. In the next period-adding transition (PA3–4),

Fig. 6. One-dimensional bifurcation diagram obtained by scan-ning from the two spikes per burst regime into the three spikesper burst regime for kS=1.0×10−3. Note the subcriticalperiod-doubling and the associated transition to type-III inter-mittency for VS� −40.827 mV.


Fig. 7. One-dimensional bifurcation diagram obtained by scan-ning VS in both directions across the PA2–3 period-addingcurve in Fig. 4. kS=0.84×10−3. Note the interval of coexist-ing two-spike and three-spike solutions.

to (n+1)-spike behavior described by Terman(1991, 1992).

Fig. 8 shows a phase space projection of thecoexisting two-spike and three-spikes solutionsthat one can observe for VS = −42.0 mV andkS=0.669×10−3. Note how these solutions fol-low one another very closely for part of the cycleto sharply depart at a point near V= −57 mVand S=264. Hence, with a slightly smaller nu-merical accuracy, it may appear as if the twosolutions smoothly transform into one another.Fig. 9 displays the basins of attraction for the twocoexisting solutions. Here, initial conditions at-tracted to the two-spike solution are markedblack, and initial conditions from which the tra-jectory approaches the three-spike solution areleft blank. The figure was constructed for initialvalues of the fast gate variable of n=0.04. Fi-nally, Fig. 10 shows a magnification of part of thebasin boundary in Fig. 9 around V= −50 mVand S=210.2. Inspection of this magnificationclearly reveals the fractal structure of the basinboundary with the characteristic set of bands ofrapidly decreasing width.

the three-spike solution undergoes a subcriticalperiod-doubling, and a four-spike solutionemerges in a saddle-node bifurcation. Again, thereis a small interval of coexistence between the twosolutions. This is a very different scenario fromthe continuous transition from n-spike behavior

Fig. 8. Phase space projection of the coexisting two-spike and three-spike solutions for VS= −42.0 mV and kS=0.669×10−3.Note the sharp point of departure between the two solutions.


Fig. 9. Basins of attraction for the coexisting two-spike andthree-spike solutions for kS=0.84×10−3. Initial conditionsfrom which the trajectory approaches the two spikes per burstsolution are marked black.

surrounded by the first period-doubling curvefor the periodic spiking behavior. The arms ofthis squid separate regions of different numberof spikes per burst;

2. each arm has a structure with a period-dou-bling cascade on one side and a saddle-nodebifurcation on the other;

3. towards the end of the arm, the first period-doubling bifurcation tends to become subcriti-cal: in a certain parameter region, this givesrise to a chaotic boundary crisis followed by atransition to type-III intermittency; and

4. the so-called period-adding structure ariseswhen the subcritical period-doubling curve in-tersects the saddle-node bifurcation curve onthe other side of the arm. This leads to aregion of coexistence of stable n-spikes and(n+1)-spike behavior.

These results appear to be at odds with theresults usually found in the literature (Terman,1991). It is obvious that different results may beobtained with different models and differentparameter settings. However, the consistency inour bifurcation scenarios seems to imply thatsimilar scenarios may also be found in otherbursting cell models.

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4. Conclusion

We have presented a bifurcation analysis of athree-variable model that can produce the charac-teristic bursting and spiking behavior of pancre-atic �-cells. A more mathematically orienteddescription of the homoclinic bifurcations leadingto bursting has been given elsewhere (Belykh etal., 2000). Our main observations were:1. a squid-formed regime of chaotic dynamics

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Bifurcation structure of a model of bursting pancreatic cells