Abstract
The focus of the earlier chapters of this book has been on the construction of mathematical models based on information about the underlying physiology, and on comparison of model predictions with experimental data. A recurrent theme has been that a relatively small number of broad principles are applicable in the construction of a wide variety of different Ca2+ models in different physiological contexts. This chapter has a different focus, and instead looks at some mathematical methods that have proved useful for the analysis of Ca2+ models. However, there is an analogous theme, which is that a relatively small number of mathematical methods underlie much of current practice in model analysis.
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References
Atri A, Amundson J, Clapham D, Sneyd J (1993) A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte. Biophys J 65(4):1727–39, DOI: 10.1016/S0006-3495(93)81191-3
Bell D, Deng B (2002) Singular perturbation of N-front travelling waves in the FitzHugh–Nagumo equations. Nonlinear analysis: Real world applications 3(4):515–41, DOI: 10.1016/s1468-1218(01)00046-3
Bertram R, Sherman A (2000) Dynamical complexity and temporal plasticity in pancreatic beta-cells. J Biosci 25(2):197–209
Bertram R, Egli M, Toporikova N, Freeman ME (2006) A mathematical model for the mating-induced prolactin rhythm of female rats. Am J Physiol Endocrinol Metab 290(3):E573–82, DOI: 10.1152/ajpendo.00428.2005
Bertram R, Sherman A, Satin LS (2010) Electrical bursting, calcium oscillations, and synchronization of pancreatic islets. Adv Exp Med Biol 654:261–79, DOI: 10.1007/978-90-481-3271-3_12
Boie S, Kirk V, Sneyd J, Wechselberger M (2015) Effects of quasi-steady-state reduction on biophysical models with oscillations. submitted
Carter P, Sandstede B (2014) Fast pulses with oscillatory tails in the FitzHugh–Nagumo system
Champneys A, Kirk V, Knobloch E, Oldeman B, Sneyd J (2007) When Shil’nikov meets Hopf in excitable systems. SIAM J Appl Dyn Syst 6:663–93, DOI: 10.1137/070682654
Doedel EJ (1981) AUTO: A program for the automatic bifurcation analysis of autonomous systems. Congr Numer 30:265–84
Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang X (1998) Auto97: Continuation and bifurcation software for ordinary differential equations. Available for download from http://indy.cs.concordia.ca/auto
Domijan M, Murray R, Sneyd J (2006) Dynamical probing of the mechanisms underlying calcium oscillations. J Nonlin Sci 16(5):483–506, DOI: 10.1007/s00332-005-0744-z
Duan W, Lee K, Herbison AE, Sneyd J (2011) A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis. J Theor Biol 276(1):22–34, DOI: 10.1016/j.jtbi.2011.01.035
Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM
Erneux T, Goldbeter A (2006) Rescue of the quasi-steady-state approximation in a model for oscillations in an enzymatic cascade. SIAM J Appl Math 67(2):305–20, DOI: 10.1137/060654359
Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations. J Diff Eq 31:53–98, DOI: 10.1016/0022-0396(79)90152-9
Flach EH, Schnell S (2006) Use and abuse of the quasi-steady-state approximation. IEE Proceedings-Systems Biology 153(4):187–91, DOI: 10.1049/ip-syb:20050104
Fletcher PA, Li YX (2009) An integrated model of electrical spiking, bursting, and calcium oscillations in GnRH neurons. Biophys J 96(11):4514–24, DOI: 10.1016/j.bpj.2009.03.037
Harvey E, Kirk V, Osinga HM, Sneyd J, Wechselberger M (2010) Understanding anomalous delays in a model of intracellular calcium dynamics. Chaos 20(4):045,104, DOI: 10.1063/1.3523264
Harvey E, Kirk V, Wechselberger M, Sneyd J (2011) Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics. J Nonlin Sci 21(5):639–83, DOI: 10.1007/s00332-011-9096-z
Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond B 221:87–102, DOI: 10.1098/rspb.1984.0024
Izhikevich E (2000) Neural excitability, spiking and bursting. Int J Bif Chaos 10(6):1171–266, DOI: 10.1142/s0218127400000840
Jones C (1984) Stability of the traveling wave solutions of the FitzHugh-Nagumo system. Trans Amer Math Soc 286:431–69
Jones CKRT (1995) Geometric singular perturbation theory. In: Dynamical Systems, Springer Verlag, pp 44–118
Keener J, Sneyd J (2008) Mathematical Physiology, 2nd edn. Springer-Verlag, New York, DOI: 10.1007/978-0-387-79388-7
Krupa M, Sandstede B, Szmolyan P (1997) Fast and slow waves in the FitzHugh-Nagumo equation. J Diff Eq 133(1):49–97, DOI: 10.1006/jdeq.1996.3198
Krupa M, Popovic N, Kopell N (2008) Mixed-mode oscillations in three time-scale systems: a prototypical example. SIAM J Appl Dyn Syst 7(2):361–420, DOI: 10.1137/070688912
Krupa M, Vidal A, Desroches M, Clément F (2012) Mixed-mode oscillations in a multiple time scale phantom bursting system. SIAM J Appl Dyn Syst 11:1458–98, DOI: 10.1137/110860136
LeBeau AP, Robson AB, McKinnon AE, Donald RA, Sneyd J (1997) Generation of action potentials in a mathematical model of corticotrophs. Biophys J 73(3):1263–75, DOI: 10.1016/s0006-3495(97)78159-1
Lee K, Duan W, Sneyd J, Herbison AE (2010) Two slow calcium-activated afterhyperpolarization currents control burst firing dynamics in gonadotropin-releasing hormone neurons. J Neurosci 30(18):6214–24, DOI: 10.1523/JNEUROSCI.6156-09.2010
Li YX, Rinzel J, Keizer J, Stojilković S (1994) Calcium oscillations in pituitary gonadotrophs: comparison of experiment and theory. Proc Natl Acad Sci USA 91:58–62, DOI: 10.1073/pnas.91.1.58
Maginu K (1985) Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion equations. SIAM J Appl Math 45:750–74, DOI: 10.1137/0145044
Nan P, Wang Y, Vivien K, Rubin JE (2015) Understanding and distinguishing three time scale oscillations: case study in a coupled Morris–Lecar system. SIAM J Appl Dyn Syst, in press
Pedersen MG, Bersani AM, Bersani E (2008) Quasi steady-state approximations in complex intracellular signal transduction networks–a word of caution. J Math Chem 43(4):1318–44, DOI: 10.1007/s10910-007-9248-4
Rinzel J (1985) Bursting oscillations in an excitable membrane model. In: Sleeman B, Jarvis R (eds) Ordinary and partial differential equations, Springer-Verlag, New York, pp 304–16, DOI: 10.1007/bfb0074739
Romeo MM, Jones CKRT (2003) The stability of traveling calcium pulses in a pancreatic acinar cell. Physica D 177(1):242–58, DOI: 10.1016/S0167-2789(02)00772-8
Roper P, Callaway J, Armstrong W (2004) Burst initiation and termination in phasic vasopressin cells of the rat supraoptic nucleus: a combined mathematical, electrical, and calcium fluorescence study. J Neurosci 24(20):4818–31, DOI: 10.1523/JNEUROSCI.4203-03.2004
Simpson D, Kirk V, Sneyd J (2005) Complex oscillations and waves of calcium in pancreatic acinar cells. Physica D: Nonlinear Phenomena 200(3–4):303–24, DOI: 10.1016/j.physd.2004.11.006
Sneyd J, Tsaneva-Atanasova K, Yule DI, Thompson JL, Shuttleworth TJ (2004) Control of calcium oscillations by membrane fluxes. Proc Natl Acad Sci USA 101(5):1392–6, DOI: 10.1073/pnas.0303472101
Szmolyan P, Wechselberger M (2001) Canards in R3. J Diff Eq 177(2):419–53, DOI: 10.1006/jdeq.2001.4001
Tsai JC, Zhang W, Kirk V, Sneyd J (2012) Traveling waves in a simplified model of calcium dynamics. SIAM J Appl Dyn Syst 11(4):1149–99, DOI: 10.1137/120867949
Tsaneva-Atanasova K, Osinga HM, Riess T, Sherman A (2010a) Full system bifurcation analysis of endocrine bursting models. J Theor Biol 264(4):1133–46, DOI: 10.1016/j.jtbi.2010.03.030
Tsaneva-Atanasova K, Osinga HM, Tabak J, Pedersen MG (2010b) Modeling mechanisms of cell secretion. Acta Biotheoretica 58(4):315–27, DOI: 10.1007/s10441-010-9115-8
Vo T, Bertram R, Wechselberger M (2013) Multiple geometric viewpoints of mixed mode dynamics associated with pseudo-plateau bursting. SIAM J Appl Dyn Syst 12(2):789–830, DOI: 10.1137/120892842
Wechselberger M (2005) Existence and bifurcation of canards in R3 in the case of a folded node. SIAM J Appl Dyn Syst 4(1):101–39, DOI: 10.1137/030601995
Zhang M, Goforth P, Bertram R, Sherman A, Satin L (2003) The Ca2+ dynamics of isolated mouse beta-cells and islets: implications for mathematical models. Biophys J 84(5):2852–70, DOI: 10.1016/S0006-3495(03)70014-9
Zhang W, Kirk V, Sneyd J, Wechselberger M (2011) Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales. J Math Neurosci 1(1):9, DOI: 10.1186/2190-8567-1-9
Zhang W, Krauskopf B, Kirk V (2012) How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete Cont Dyn S 32(8):2825–51, DOI: 10.3934/dcds.2012.32.2825
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Dupont, G., Falcke, M., Kirk, V., Sneyd, J. (2016). Nonlinear Dynamics of Calcium. In: Models of Calcium Signalling. Interdisciplinary Applied Mathematics, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-29647-0_5
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DOI: https://doi.org/10.1007/978-3-319-29647-0_5
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