In this paper, a model proposed by Marhl et al. is considered to investigate the effect of several parameters on the calcium bursting oscillation behavior. Different types of bursting are presented. Fast-slow burster analysis and first return map are used to explain the mechanism of the four types of bursting. The results are instructive for understanding the role of these parameters played in complex dynamics in the Marhl-Haberichter calcium oscillation model

Quanbao, Ji

Yi, Zhou

Yuanhua, Li

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Appl. Math. Inf. Sci.7, No. 6, 2515-2518 (2013) 2515Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.12785/amis/070646Different Types of Periodic Activities in a CalciumOscillation ModelLi Yuanhua, Zhou Yi and Ji Quanbao∗Dept. of Math., Huainan Normal University, Huainan 232038, Anhui, P. R. ChinaReceived: 12 Apr. 2013, Revised: 17 Aug. 2013, Accepted: 19 Aug. 2013Published online: 1 Nov. 2013Abstract: In this paper, a model proposed by Marhl et al. is considered to investigate the effect of several parameters on the calciumbursting oscillation behavior. Different types of bursting are presented. Fast-slow burster analysis and first return map are used toexplain the mechanism of the four types of bursting. The results are instructive for understanding the role of these parameters played incomplex dynamics in the Marhl-Haberichter calcium oscillation model.Keywords: Calcium oscillation, bursting, fast-slow dynamics, return map.1 IntroductionIn excitable as well as in non-excitable cells, manyprocesses, such as cell secretion and egg fertilization, areperformed by the oscillatory changing of free cytosoliccalcium concentration. Calcium oscillations were foundexperimentally in the 1980s [1]. A large number ofexperimental works have confirmed the significant role ofbursting oscillations in cell signaling. Due to theimportance of oscillations, several mathematical modelswere established in order to explain themechanism [2–10].The first model was established by Shen andLarter [5], and its functioning bases on two mainmechanisms, i.e. the calcium-induced calcium release(CICR) and the inositol trisphosphate crosscoupling(ICC). A more detailed research in explaining thecomplex calcium oscillations in non-excitable cells havebeen given by Borghans et al. [6] and Houart et al. [7]followed. In this paper, we focus on effects of differentparameters to study their physiological roles in generatingcomplex Ca2+ oscillations. Another model demonstratingbursting oscillations was proposed by Kummer et al. [8],which incorporates the feedback inhibition on the initialagonist receptor complex by Ca2+ and activatedphospholipase C (PLC), as well as receptortype-dependent self-enhanced behavior of the activatedGa subunit. In the present article, the Marhl-HaberichterCa2+ oscillation model [2, 3, 9, 10] is analyzed by usingthe so-called fast-slow burster analysis and first returnmap [11, 12]. The dynamic mechanism of differentcalcium oscillations in non-excitable cells has beenextensively investigated. For more details and analysisabout these results see Ref. [13].2 Materials and methodsThe mathematical model we used was proposed by Marhlet al. [2]. The model is described by the followingdifferential equations:dCacytdt= Jch− Jpump+ Jleak+ Jout− Jin + JCaPr− JPr (1)dCaerdt=βerρer(Jpump− Jch− Jleak) (2)dCamdt=βmρm(Jin − Jout) (3)whereJch = kchCa2cytCa2cyt+K21(Caer−Cacyt), Prtot = Pr+CaPr,Jpump = kpumpCacyt, JPr = k+CacytPr, JCaPr = k−CaPr,Jleak= kleak(Caer−Cacyt), Jout = (kmCa2cytCa2cyt+K21+ kmit)Cam,∗ Corresponding author e-mail:jqb 2001@163.comc© 2013 NSPNatural Sciences Publishing Cor.2516 L. Yuanhua et al: Different Types of Periodic Activities in a Calcium...Catot = Cacyt +ρerβerCaer +ρmβmCam + CaPr,Jin = kinCa8cytCa8cyt+K82.Three variables in the system are: the free Ca2+concentration in the cytosol (Cacyt), free Ca2+concentration in the endoplasmic reticulum(ER) (Caer),and the free Ca2+ concentration in the mitochondria(Cam).Parameters for which all calculations are made if nototherwise stated:kleak = 0.01s−1, kpump = 20.0s−1,kin = 300µMs−1, km = 125s−1, k+ = 0.09µMs−1,k− = 0.01s−1, K1 = 5.0µM, K2 = 0.8µM, Catot = 90µM,Prtot = 120µM, ρer = 0.01, βer = 0.0025, ρm = 0.01,ρm = 0.0025.Equations (1)-(3) construct a full system with the fastsubsystem (FS.) and the slow subsystem (SS.). In thefast-slow analysis, we determine the type of bursting bybehavior of FS, where the slow variableCam isconsidered as bifurcation parameter.3 ResultsWhen kch = 1500, there is a point-cycle bursting ofsubHopf-suHopf type, as shown in Fig.1 and 2. HB andLPC are subcritical Hopf bifurcation and fold limit cyclebifurcation respectively. Solid and doted lines (right andleft side of HB) are stable and unstable steady state. Dash(thin solid) lines are stable (unstable) periodic solutions.The closed line represents the limit cycle trajectory of thecomplete system.Fig. 1. Time series ofCacyt with kch = 1500 of point-cyclebursting.Fig. 2. Fast-slow dynamical analysis of point-cycle bursting.The main character of this type is the active and silentphases of bursting depend on a stable steady state and astable limit cycle related to a subHopf bifurcation at HB,see Fig.2. When the trajectory passes the subcritical Hopfbifurcation (HB), the stable steady state branch turnsunstable. Therefore, the trajectory unfolds from theunstable steady state to stable periodic attractors. Thisstarts the active phase of bursting. As time progresses thetrajectory passes through the unstable limit cycle. Due tothe attractive stable foci, the trajectory turns to the stablesteady state on the other side of the subcritical Hopfbifurcation. The silent phase of bursting starts again.When kch = 2225 andkleak = 0.03, this system hastwo attractors. The trajectory tends to one of them,depending on the different initial conditions. Thisphenomenon is called birhythmicity. For one initialcondition (Cacyt = 0.5,Caer = 0.5,Cam = 0.85), there isonly regular spiking ofCam shown in Fig.3, so thetrajectory is a simple unfolded limit cycle, as illustratedinFig.4. However, for another initial condition(Cacyt = 0.5,Caer = 0.5,Cam = 1) one period ofoscillations ofCam composes of two spikes differed in theamplitude (as shown in Fig.5, which leads to a doublefolded limit cycle (see Fig.6. Bifurcation analysis issimilar to the above and omitted here).Fig. 3. Time series of birhythmic bursting withkch = 2225 andkleak = 0.03.Fig. 4. Unfolded limit cycle trajectory of the complete system in(Cacyt ,Cam)-plane.c© 2013 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.7, No. 6, 2515-2518 (2013) /www.naturalspublishing.com/Journals.asp 2517Fig. 5. Time series ofCam with kch = 2225 andkleak = 0.03 fordifferent initial conditions.Fig. 6. Folded limit cycle trajectory of the complete system in(Cacyt ,Cam)-plane.When kch = 2000 andkpump = 17.8, the full systemhas four attractors. We call this phenomenonquadric-rhythmicity (see Fig.7). For some strictconditions, the trajectory of the complete system canshow quadruple folded limit cycle (as shown in Fig.8).Fig.9 shows such a return map for an example ofquadric-rhythmic bursting. In this figure, the successivemaxima of Cam are plotted against their predecessors.Points of the bisector line represent cycles of constantamplitudes. Each single point of the return map representsone cycle of the system trajectory.Fig. 7. Time series of quadric-rhythmic bursting withkch = 2000 andkpump = 17.8.Fig. 8. Trajectory of the complete system in(Cacyt ,Cam)-planewith kch = 2000 andkpump = 17.8.Fig. 9. Return map ofCam with kch = 2000 andkpump = 17.8.We start at the point A (marked by an arrow in theanti-clockwise direction), along the grey line, then we passpoint B, C and D. At last we return to point A. This is astrong evidence for quadric-rhythmic bursting.Fig. 10. Time series of chaotic bursting withkch = 1800 andkpump = 17.8.Fig. 11. Return map ofCam with kch = 1800 andkpump = 17.8.When kch = 1800 andkpump = 17.8, the full systemdisplays a chaotic bursting. These chaotic behaviors arec© 2013 NSPNatural Sciences Publishing Cor.2518 L. Yuanhua et al: Different Types of Periodic Activities in a Calcium...characterized by a positive value of the largest Lyapunovexponent and neglected here. Forkch = 1800 andkpump= 17.8, the time course of chaos is shown in Fig.10.Note that the amplitudes of theCacyt spikes keep almostconstant all the time. The corresponding return map to thetime series is presented in Fig.11. In this figure, thesuccessive maxima ofCam are plotted against theirpredecessors. From this figure we can see that thesepoints is very messy, which means the occurrence ofchaos.4 DiscussionIn this paper, the effects of different parameters on theCa2+ oscillations have been studied. Four types ofbursting Ca2+ oscillations are given. Separately, weanalyze point-cycle bursting of subHopf-subHopf type,birhythmicity, quadric-rhythmicity and chaos. Lots ofstudies show that frequency encoding and amplitudeencoding play an extraordinary role in informationprocessing and signal transduction in many biologicalsystems. From this point of view, it is very important toanalyze different types of bursting calcium oscillations,since different types of bursting behavior could reflectdifferent encoding of biologically relevant information.Therefore, more types of bursting behavior should befound and further studies will be necessary to determinemore precisely effects of different parameters on the Ca2+oscillations.AcknowledgementThis work is partially supported by the National NaturalScience Foundation of China under Grant no. 11202083,the Natural Science Foundation of the Anhui HigherEducation Institutions of China under Grant no.KJ2013A240, KJ2013Z309 and the Youth Foundation ofHuainan Normal University (2012LK17).References[1] M. J. Berridge, Neuron,21, 13-26 (1998).[2] M. Marhl, T. Haberichterv and M. Brumen, Bio. Systems,5775-86 (2000).[3] M. Marhl, S. Schuster and M. Brumen, Biophys. Chem.,71,125-132 (1998).[4] J. P. Newman and R. J. Butra, Chaos,20 1-8 (2010).[5] P. Shen and R. Larter, Biophys. Chem.,17 225-232 (1995).[6] J. M. Borghans, G. Dupont and A. Goldbeter, Biophys.Chem.,66, 25-41 (1997).[7] G. Houart, G. Dupont and A. Goldbeter, Bursting, Bull. Math.Biol., 61, 507-530 (1999).[8] U. Kummer, L. F. Olsen, C. J. Dixon, A. K. Green, E.Borhberg-Bauer and G. Baier, Biophys.,79, 1188-1195(2000).[9] T. Haberichter, M. Marhl and R. Heinrich, Biophys. Chem.,90, 17-30 (2001).[10] M. Perc and M. Marhl,18, 759-773 (2003).[11] J. Rinzel,Bursting oscillations in an excitable membranemodel. Lecture Notes in Mathematic, Springer-Verlag, 304-316 (1985).[12] P. Meng and Q. S. Lu, Chin. Phys. Lett.,27, 010502-0010502 (2010).[13] E. G. Roger and P. Mark, Society for Industrial and AppliedMathematics,66, 1917-1948 (2006).[14] E. Harvey, V. Kirk, M. Wechselberger and J. Sneyd, J.Nonlinear Sci.,21, 639-683 (2011).[15] P. C. Rech,Dynamics in the parameter space of a neuronmodel, Chin. Phys. Lett.,29, 60506-060506 (2012).Yuanhua Ligraduated from AnhuiNormal University in1987. He is currentlyan associate professor inHuainan Normal University.His research interestsare in the areas of differentialequations and algebra.Yi Zhou received the MSdegree in Mathematics fromAnhui Normal Universityin 2012. He is currently anassistant in Huainan NormalUniversity. His researchinterests are in the areasof differential equations,and nonlinear dynamics.Quanbao Ji received theMS degree in Mathematicsfrom Shanghai NormalUniversity in 2006 andthe PhD degree in MechanicalEngineering from BeihangUniversity in 2009. Heis currently an associateprofessor in Huainan NormalUniversity. His researchinterests are in the areas of nonlinear dynamics andinformation systems.c© 2013 NSPNatural Sciences Publishing Cor.

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