Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells

  • Richard BertramEmail author
  • Joël Tabak
  • Wondimu Teka
  • Theodore Vo
  • Martin Wechselberger
Part of the Frontiers in Applied Dynamical Systems: Reviews and Tutorials book series (FIADS, volume 1)


Dynamical systems theory provides a number of powerful tools for analyzing biological models, providing much more information than can be obtained from numerical simulation alone. In this chapter, we demonstrate how geometric singular perturbation analysis can be used to understand the dynamics of bursting in endocrine pituitary cells. This analysis technique, often called “fast/slow analysis,” takes advantage of the different time scales of the system of ordinary differential equations and formally separates it into fast and slow subsystems. A standard fast/slow analysis, with a single slow variable, is used to understand bursting in pituitary gonadotrophs. The bursting produced by pituitary lactotrophs, somatotrophs, and corticotrophs is more exotic, and requires a fast/slow analysis with two slow variables. It makes use of concepts such as canards, folded singularities, and mixed-mode oscillations. Although applied here to pituitary cells, the approach can and has been used to study mixed-mode oscillations in other systems, including neurons, intracellular calcium dynamics, and chemical systems. The electrical bursting pattern produced in pituitary cells differs fundamentally from bursting oscillations in neurons, and an understanding of the dynamics requires very different tools from those employed previously in the investigation of neuronal bursting. The chapter thus serves both as a case study for the application of recently-developed tools in geometric singular perturbation theory to an application in biology and a tutorial on how to use the tools.


Hopf Bifurcation Pituitary Cell Slow Manifold Fast Subsystem Critical Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by NSF grants DMS 0917664 to RB, DMS 1220063 to RB and JT, and NIH grant DK 043200 to RB and JT.


  1. Baer SM, Gaekel EM (2008) Slow acceleration and deacceleration through a Hopf bifurcation: Power ramps, target nucleation, and elliptic bursting. Phys Rev 78:036205MathSciNetGoogle Scholar
  2. Baer SM, Erneux T, Rinzel J (1989) The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance. SIAM J Appl Math 49:55–71zbMATHMathSciNetCrossRefGoogle Scholar
  3. Benoit E (1983) Syst‘emes lents-rapids dans r3 et leur canards. Asterique 109–110:159–191MathSciNetGoogle Scholar
  4. Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bull Math Biol 57:413–439zbMATHCrossRefGoogle Scholar
  5. Bertram R, Sherman A, Satin LS (2010) Electrical bursting, calcium oscillations, and synchronization of pancreatic islets. In: Islam MS (ed) The Islets of Langerhans, Springer, pp 261–279Google Scholar
  6. Brons M, Krupa M, Wechselberger M (2006) Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst Commun 49:39–63MathSciNetGoogle Scholar
  7. Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic β-cell. Biophys J 42:181–190CrossRefGoogle Scholar
  8. Clayton TF, Murray AF, Leng G (2010) Modelling the in vivo spike activity of phasically-firing vasopressin cells. J Neuroendocrinology 22:1290–1300CrossRefGoogle Scholar
  9. Coombes S, Bressloff PC (2005) Bursting: The Genesis of Rhythm in the Nervous System. World ScientificGoogle Scholar
  10. Crunelli V, Kelly JS, Leresche N, Pirchio M (1987) The ventral and dorsal lateral geniculate nucleus of the rat: Intracellular recordings in vitro. J Physiol 384:587–601CrossRefGoogle Scholar
  11. Dean PM, Mathews EK (1970) Glucose-induced electrical activity in pancreatic islet cells. J Physiol 210:255–264CrossRefGoogle Scholar
  12. Del Negro CA, Hsiao CF, Chandler SH, Garfinkel A (1998) Evidence for a novel bursting mechanism in rodent trigeminal neurons. Biophys J 75:174–182CrossRefGoogle Scholar
  13. Desroches M, Krauskopf B, Osinga HM (2008a) The geometry of slow manifolds near a folded node. SIAM J Appl Dyn Syst 7:1131–1162zbMATHMathSciNetCrossRefGoogle Scholar
  14. Desroches M, Krauskopf B, Osinga HM (2008b) Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system. Chaos 18:015107MathSciNetCrossRefGoogle Scholar
  15. Desroches M, Guckenheimer J, Krauskopf B, Kuehn C, Osinga HM, Wechselberger M (2012) Mixed-mode oscillations with multiple time scales. SIAM Rev 54:211–288zbMATHMathSciNetCrossRefGoogle Scholar
  16. Doedel EJ (1981) AUTO: A program for the automatic bifurcation analysis of autonomous systems. Congr Numer 30:265–284MathSciNetGoogle Scholar
  17. Doedel EJ, Champneys DJ, Fairgrieve TF, Kuznetov YA, Oldeman KE, Paffenroth RC, Sandstede B, Wang XJ, Zhang C (2007) AUTO-07P: Continuation and bifurcation software for ordinary differential equations Available at
  18. 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 276:22–34MathSciNetCrossRefGoogle Scholar
  19. Erchova I, McGonigle DJ (2008) Rhythms in the brain: An examination of mixed mode oscillation approaches to the analysis of neurophysiological data. Chaos 18:015115MathSciNetCrossRefGoogle Scholar
  20. Fenichel N (1979) Genometric singular perturbation theory. J Differ Equ 31:53–98zbMATHMathSciNetCrossRefGoogle Scholar
  21. FitzHugh R (1961) Impulses and physiological states in theoretic models of nerve membrane. Biophys J 1:445–466CrossRefGoogle Scholar
  22. Fletcher PA, Li YX (2009) An integrated model of electrical spiking, bursting, and calcium oscillations in GnRH neurons. Biophys J 96:4514–4524CrossRefGoogle Scholar
  23. Freeman ME (2006) Neuroendocrine control of the ovarian cycle of the rat. In: Neill JD (ed) Knobil and Neill’s Physiology of Reproduction, 3rd edn, Elsevier, pp 2327–2388Google Scholar
  24. Guckenheimer J (2008) Singular Hopf bifurcation in systems with two slow variables. SIAM J Appl Dyn Syst 7:1355–1377zbMATHMathSciNetCrossRefGoogle Scholar
  25. Guckenheimer J, Haiduc R (2005) Canards at folded nodes. Mosc Math J 5:91–103zbMATHMathSciNetGoogle Scholar
  26. Guckenheimer J, Scheper C (2011) A gemometric model for mixed-mode oscillations in a chemical system. SIAM J Appl Dyn Syst 10:92–128zbMATHMathSciNetCrossRefGoogle Scholar
  27. Harvey E, Kirk V, Osinga H, Sneyd J, Wechselberger M (2010) Understanding anomalous delays in a model of intracelular calcium dynamics. Chaos 20:045104MathSciNetCrossRefGoogle Scholar
  28. Harvey E, Kirk V, Sneyd J, Wechselberger M (2011) Multiple time scales, mixed-mode oscillations and canards in models of intracellular calcium dynamics. J Nonlinear Sci 21:639–683zbMATHCrossRefGoogle Scholar
  29. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conductance and excitation in nerve. J Physiol 117:500–544CrossRefGoogle Scholar
  30. Izhikevich EM (2007) Dynamical Systems in Neuroscience. MIT PressGoogle Scholar
  31. Keener K, Sneyd J (2008) Mathematical Physiology, 2nd edn. SpringerGoogle Scholar
  32. Krupa M, Wechselberger M (2010) Local analysis near a folded saddle-node singularity. J Differ Equ 248:2841–2888zbMATHMathSciNetCrossRefGoogle Scholar
  33. Kukuljan M, Rojas E, Catt KJ, Stojilković SS (1994) Membrane potential regulates inositol 1,4,5-trisphosphate-controlled cytoplasmic Ca2+ oscillations in pituitary gonadotrophs. J Biol Chem 269:4860–4865Google Scholar
  34. Kuryshev YA, Childs GV, Ritchie AK (1996) Corticotropin-releasing hormone stimulates Ca2+ entry through L- and P-type Ca2+ channels in rat corticotropes. Endocrinology 137:2269–2277Google Scholar
  35. LeBeau AP, van Goor F, Stojilković SS, Sherman A (2000) Modeling of membrane excitability in gonadotropin-releasing hormone-secreting hypothalamic neurons regulated by Ca2+-mobilizing and adenylyl cyclase-coupled receptors. J Neurosci 20:9290–9297Google Scholar
  36. 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 30:6214–6224CrossRefGoogle Scholar
  37. Li YX, Rinzel J (1994) Equations for InsP3 receptor-mediated [Ca 2+] oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J theor Biol 166:461–473CrossRefGoogle Scholar
  38. Li YX, Rinzel J, Keizer J, Stojilković SS (1994) Calcium oscillations in pituitary gonadotrophs: Comparison of experiment and theory. Proc Natl Acad Sci USA 91:58–62CrossRefGoogle Scholar
  39. Li YX, Keizer J, Stojilković SS, Rinzel J (1995) Ca2+ excitability of the ER membrane: An explanation for IP3-induced Ca2+ oscillations. Am J Physiol 269:C1079–C1092Google Scholar
  40. Lyons DJ, Horjales-Araujo E, Broberger C (2010) Synchronized network oscillations in rat tuberoinfundibular dopamine neurons: Switch to tonic discharge by thyrotropin-releasing hormone. Neuron 65:217–229CrossRefGoogle Scholar
  41. Milescu LS, Yamanishi T, Ptak K, Mogri MZ, Smith JC (2008) Real-time kinetic modeling of voltage-gated ion channels using dynamic clamp. Biophys J 95:66–87CrossRefGoogle Scholar
  42. Milik A, Szmolyan P (2001) Multiple time scales and canards in a chemical oscillator. In: Jones C, Khibnik A (eds) Multiple-Time-Scale Dynamical Systems, Springer-Verlag, IMA Vol. Math. Appl., vol 122, pp 117–140Google Scholar
  43. Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35: 193–213CrossRefGoogle Scholar
  44. Nowacki J, Mazlan S, Osinga HM, Tsaneva-Atanasova K (2010) The role of large-conductance calcium-activated K+ (BK) channels in shaping bursting oscillations of a somatotroph cell model. Physica D 239:485–493zbMATHMathSciNetCrossRefGoogle Scholar
  45. Nunemaker CS, DeFazio RA, Moenter SM (2001) Estradiol-sensitive afferents modulate long-term episodic firing patterns of GnRH neurons. Endocrinology 143:2284–2292CrossRefGoogle Scholar
  46. Osinga HM, Sherman A, Tsaneva-Atanasova K (2012) Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting. Discret Contin Dyn S 32:2853–2877zbMATHMathSciNetCrossRefGoogle Scholar
  47. Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. In: Teramoto E, Yamaguti M (eds) Lecture Notes in Biomathematics, vol 71, Springer, pp 267–281Google Scholar
  48. Rinzel J, Lee YS (1985) On different mechanisms for membrane potential bursting. In: Othmer HG (ed) Nonlinear Oscilations in Biology, vol 66, Springer-Verlag, pp 19–33Google Scholar
  49. Rinzel J, Keizer J, Li YX (1996) Modeling plasma membrane and endoplasmic reticulum excitability in pituitary cells. Trends Endocrinol Metab 7:388–393CrossRefGoogle Scholar
  50. Rossoni E, Feng J, Tirozzi B, Brown D, Leng G, Moos F (2008) Emergent synchronous bursting of oxytocin neuronal network. PLoS Comp Biol 4(7):1000123MathSciNetCrossRefGoogle Scholar
  51. Rubin J, Wechselberger M (2007) Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model. Biol Cybern 97:5–32zbMATHMathSciNetCrossRefGoogle Scholar
  52. Rubin J, Wechselberger M (2008) The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. Chaos 18:015105MathSciNetCrossRefGoogle Scholar
  53. Shangold GA, Murphy SN, Miller RJ (1988) Gonadotropin-releasing hormone-induced Ca2+ transients in single identified gonadotropes require both intracellular Ca2+ mobilization and Ca2+ influx. Proc Natl Acad Sci USA 85:6566–6570CrossRefGoogle Scholar
  54. Sharp AA, O’Neil MB, Abbott LF, Marder E (1993) Dynamic clamp–computer-generated conductances in real neurons. J Neurophysiol 69:992–995Google Scholar
  55. Sherman A, Keizer J, Rinzel J (1990) Domain model for Ca2+-inactivation of Ca2+ channels at low channel density. Biophys J 58:985–995CrossRefGoogle Scholar
  56. Sherman A, Li YX, Keizer JE (2002) Whole-cell models. In: Fall CP, Marland ES, Wagner JM, Tyson JJ (eds) Computational Cell Biology, 1st edn, Springer, pp 101–139Google Scholar
  57. Sneyd J, Tsaneva-Atanasova K, Bruce JIE, Straub SV, Giovannucci DR, Yule DI (2003) A model of calcium waves in pancreatic and parotid acinar cells. Biophys J 85:1392–1405CrossRefGoogle Scholar
  58. Sneyd J, Tsaneva-Atanasova K, Reznikov V, Sanderson MJ, Yule DI (2006) A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations. Proc Natl Acad Sci USA 103:1675–1680CrossRefGoogle Scholar
  59. Stern JV, Osinga HM, LeBeau A, Sherman A (2008) Resetting behavior in a model of bursting in secretory pituitary cells: Distinguishing plateaus from pseudo-plateaus. Bull Math Biol 70:68–88zbMATHMathSciNetCrossRefGoogle Scholar
  60. Stojilković SS, Tomić M (1996) GnRH-induced calcium and current oscillations in gonadotrophs. Trends Endocrinol Metab 7:379–384CrossRefGoogle Scholar
  61. Stojilković SS, Kukuljan M, Iida T, Rojas E, Catt KJ (1992) Integration of cytoplasmic calcium and membrane potential oscillations maintains calcium signaling in pituitary gonadotrophs. Proc Natl Acad Sci USA 89:4081–4085CrossRefGoogle Scholar
  62. Stojilković SS, Kukuljan M, Tomić M, Rojas E, Catt KJ (1993) Mechanism of agonist-induced [Ca2+]i oscillations in pituitary gonadotrophs. J Biol Chem 268:7713–7720Google Scholar
  63. Stojilković SS, Tabak J, Bertram R (2010) Ion channels and signaling in the pituitary gland. Endocr Rev 31:845–915CrossRefGoogle Scholar
  64. Szmolyan P, Wechselberger M (2001) Canards in \(\mathbb{R}^{3}\). J Diff Eq 177:419–453zbMATHMathSciNetCrossRefGoogle Scholar
  65. Szmolyan P, Wechselberger M (2004) Relaxation oscillations in \(\mathbb{R}^{3}\). J Diff Eq 200:69–144zbMATHMathSciNetCrossRefGoogle Scholar
  66. Tabak J, Toporikova N, Freeman ME, Bertram R (2007) Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents. J Comput Neurosci 22:211–222MathSciNetCrossRefGoogle Scholar
  67. Tabak J, Tomaiuolo M, Gonzalez-Iglesias AE, Milescu LS, Bertram R (2011) Fast-activating voltage- and calcium-dependent potassium (BK) conductance promotes bursting in pituitary cells: A dynamic clamp study. J Neurosci 31:16,855–16,863CrossRefGoogle Scholar
  68. Teka W, Tabak J, Vo T, Wechselberger M, Bertram R (2011a) The dynamics underlying pseudo-plateau bursting in a pituitary cell model. J Math Neurosci 1:12, DOI 10.1186/2190-8567-1-12 MathSciNetCrossRefGoogle Scholar
  69. Teka W, Tsaneva-Atanasova K, Bertram R, Tabak J (2011b) From plateau to pseudo-plateau bursting: Making the transition. Bull Math Biol 73:1292–1311zbMATHMathSciNetCrossRefGoogle Scholar
  70. Teka W, Tabak J, Bertram R (2012) The relationship between two fast-slow analysis techniques for bursting oscillations. Chaos 22, DOI 10.1063/1.4766943
  71. Tomaiuolo M, Bertram R, Leng G, Tabak J (2012) Models of electrical activity: calibration and prediction testing on the same cell. Biophys J 103:2021–2032CrossRefGoogle Scholar
  72. Toporikova N, Tabak J, Freeman ME, Bertram R (2008) A-type K + current can act as a trigger for bursting in the absence of a slow variable. Neural Comput 20:436–451zbMATHMathSciNetCrossRefGoogle Scholar
  73. Tsaneva-Atanasova K, Sherman A, Van Goor F, Stojilković SS (2007) Mechanism of spontaneous and receptor-controlled electrical activity in pituitary somatotrophs: Experiments and theory. J Neurophysiol 98:131–144CrossRefGoogle Scholar
  74. Tse A, Hille B (1992) GnRH-induced Ca2+ oscillations and rhythmic hyperpolarizations of pituitary gonadotropes. Science 255:462–464CrossRefGoogle Scholar
  75. Tse FW, Tse A, Hille B (1994) Cyclic Ca2+ changes in intracellular stores of gonadotropes during gonadotropin-releasing hormone-stimulated Ca2+ oscillations. Proc Natl Acad Sci USA 91:9750–9754CrossRefGoogle Scholar
  76. Tse FW, Tse A, Hille B, Horstmann H, Almers W (1997) Local Ca2+ release from internal stores controls exocytosis in pituitary gonadotrophs. Neuron 18:121–132CrossRefGoogle Scholar
  77. Van Goor F, Li YX, Stojilković SS (2001a) Paradoxical role of large-conductance calcium-activated K+ (BK) channels in controlling action potential-driven Ca2+ entry in anterior pituitary cells. J Neurosci 21:5902–5915Google Scholar
  78. Van Goor F, Zivadinovic D, Martinez-Fuentes AJ, Stojilković SS (2001b) Dependence of pituitary hormone secretion on the pattern of spontaneous voltage-gated calcium influx. Cell-type specific action potential secretion coupling. J Biol Chem 276:33,840–33,846Google Scholar
  79. Vo T, Bertram R, Tabak J, Wechselberger M (2010) Mixed mode oscillations as a mechanism for pseudo-plateau bursting. J Comput Neurosci 28:443–458zbMATHMathSciNetCrossRefGoogle Scholar
  80. Vo T, Bertram R, Wechselberger M (2012) Bifurcations of canard-induced mixed mode oscillations in a pituitary lactotroph model. Discret Contin Dyn S 32:2879–2912zbMATHMathSciNetCrossRefGoogle Scholar
  81. Wechselberger M (2005) Existence and bifurcation of canards in \(\mathbb{R}^{3}\) in the case of a folded node. SIAM J Dyn Syst 4:101–139zbMATHMathSciNetCrossRefGoogle Scholar
  82. Wechselberger M (2012) A propos de canards (apropos canards). Trans Am Math Sci 364: 3289–3309zbMATHMathSciNetCrossRefGoogle Scholar
  83. Wechselberger M, Weckesser W (2009) Bifurcations of mixed-mode oscillations in a stellate cell model. Physica D 238:1598–1614zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Richard Bertram
    • 1
    Email author
  • Joël Tabak
    • 2
  • Wondimu Teka
    • 3
  • Theodore Vo
    • 4
  • Martin Wechselberger
    • 5
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of Mathematics and Biological ScienceFlorida State UniversityTallahasseeUSA
  3. 3.Department of MathematicsIndiana University – Purdue University IndianapolisIndianapolisUSA
  4. 4.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  5. 5.Department of MathematicsUniversity of SydneySydneyAustralia

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