Journal of Computational Neuroscience

, Volume 40, Issue 3, pp 297–315 | Cite as

A unified model for two modes of bursting in GnRH neurons



Gonadotropin-releasing hormone (GnRH) neurons exhibit at least two intrinsic modes of action potential burst firing, referred to as parabolic and irregular bursting. Parabolic bursting is characterized by a slow wave in membrane potential that can underlie periodic clusters of action potentials with increased interspike interval at the beginning and at the end of each cluster. Irregular bursting is characterized by clusters of action potentials that are separated by varying durations of interburst intervals and a relatively stable baseline potential. Based on recent studies of isolated ionic currents, a stochastic Hodgkin-Huxley (HH)-like model for the GnRH neuron is developed to reproduce each mode of burst firing with an appropriate set of conductances. Model outcomes for bursting are in agreement with the experimental recordings in terms of interburst interval, interspike interval, active phase duration, and other quantitative properties specific to each mode of bursting. The model also shows similar outcomes in membrane potential to those seen experimentally when tetrodotoxin (TTX) is used to block action potentials during bursting, and when estradiol transitions cells exhibiting slow oscillations to irregular bursting mode in vitro. Based on the parameter values used to reproduce each mode of bursting, the model suggests that GnRH neurons can switch between the two through changes in the maximum conductance of certain ionic currents, notably the slow inward Ca2+ current I s, and the Ca2+ -activated K+ current I KCa. Bifurcation analysis of the model shows that both modes of bursting are similar from a dynamical systems perspective despite differences in burst characteristics.


Mathematical model Parabolic bursting Irregular bursting Slow oscillations Slow-fast subsystem analysis Estradiol feedback 



This work was supported by the Natural Sciences and Engineering Council of Canada (NSERC) discovery grant to AK and NIH grants R01HD34860 and R01HD41469 to SMM.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

10827_2016_598_MOESM1_ESM.pdf (633 kb)
ESM 1 (PDF 632 kb)


  1. Achard, P., & De Schutter, E. (2006). Complex parameter landscape for a complex neuron model. PLoS Computational Biology, 2, 0794–0804. doi: 10.1371/journal.pcbi.0020094.CrossRefGoogle Scholar
  2. Amini, B., Clark, J. W., & Canavier, C. C. (1999). Calcium dynamics underlying pacemaker-like and burst firing oscillations in midbrain dopaminergic neurons: a computational study. Journal of Neurophysiology, 82, 2249–2261.PubMedGoogle Scholar
  3. Bekkers, J. M. (2000). Properties of voltage-gated potassium currents in nucleated patches from large layer 5 cortical pyramidal neurons of the rat. Journal of Physiology, 525(Pt 3), 593–609. doi: 10.1111/j.1469-7793.2000.t01-1-00593.x.CrossRefPubMedPubMedCentralGoogle Scholar
  4. Belchetz, P. E., Plant, T. M., Nakai, Y., et al. (1978). Hypophysial responses to continuous and intermittent delivery of hypopthalamic gonadotropin-releasing hormone. Science, 202, 631–633. doi: 10.1126/science.100883.CrossRefPubMedGoogle Scholar
  5. Bertram, R., Butte, M. J., Kiemel, T., & Sherman, A. (1995). Topological and phenomenological classification of bursting oscillations. Bulletin of Mathematical Biology, 57, 413–439. doi: 10.1007/BF02460633.CrossRefPubMedGoogle Scholar
  6. Canavier, C. C., Clark, J. W., & Byrne, J. H. (1991). Simulation of the bursting activity of neuron R15 in Aplysia: role of ionic currents, calcium balance, and modulatory transmitters. Journal of Neurophysiology, 66, 2107–2124.PubMedGoogle Scholar
  7. Carmeliet, E. (1987). Voltage-dependent block by tetrodotoxin of the sodium channel in rabbit cardiac Purkinje fibers. Biophysical Journal, 51, 109–114. doi: 10.1016/S0006-3495(87)83315-5.CrossRefPubMedPubMedCentralGoogle Scholar
  8. Chen X., Sneyd J. (2014). A Computational Model of the Dendron of the GnRH Neuron. Bulletin of Mathematical Biology 904–926. doi:  10.1007/s11538-014-0052-6.
  9. Chen, X., Iremonger, K., Herbison, A., et al. (2013). Regulation of electrical bursting in a spatiotemporal model of a GnRH neuron. Bulletin of Mathematical Biology, 75, 1941–1960. doi: 10.1007/s11538-013-9877-7.CrossRefPubMedGoogle Scholar
  10. Christian, C. A., & Moenter, S. M. (2010). The neurobiology of preovulatory and estradiol-induced gonadotropin- releasing hormone surges. Endocrine Reviews, 31, 544–577. doi: 10.1210/er.2009-0023.CrossRefPubMedPubMedCentralGoogle Scholar
  11. Chu, Z., & Moenter, S. M. (2006). Physiologic regulation of a tetrodotoxin-sensitive sodium influx that mediates a slow afterdepolarization potential in gonadotropin-releasing hormone neurons: possible implications for the central regulation of fertility. Journal of Neuroscience, 26, 11961–11973. doi: 10.1523/JNEUROSCI.3171-06.2006.CrossRefPubMedGoogle Scholar
  12. Chu, Z., Andrade, J., Shupnik, M. A., & Moenter, S. M. (2009). Differential regulation of gonadotropin-releasing hormone neuron activity and membrane properties by acutely applied estradiol: dependence on dose and estrogen receptor subtype. Journal of Neuroscience, 29, 5616–5627. doi: 10.1523/JNEUROSCI.0352-09.2009.CrossRefPubMedPubMedCentralGoogle Scholar
  13. Chu, Z., Takagi, H., & Moenter, S. M. (2010). Hyperpolarization-activated currents in gonadotropin-releasing hormone (GnRH) neurons contribute to intrinsic excitability and are regulated by gonadal steroid feedback. Journal of Neuroscience, 30, 13373–13383. doi: 10.1523/JNEUROSCI.1687-10.2010.CrossRefPubMedPubMedCentralGoogle Scholar
  14. Chu, Z., Tomaiuolo, M., Bertram, R., & Moenter, S. M. (2012). Two types of burst firing in gonadotrophin-releasing hormone neurones. Journal of Neuroendocrinology, 24, 1065–1077. doi: 10.1111/j.1365-2826.2012.02313.x.CrossRefPubMedPubMedCentralGoogle Scholar
  15. Csercsik, D., Farkas, I., Hrabovszky, E., & Liposits, Z. (2012). A simple integrative electrophysiological model of bursting GnRH neurons. Journal of Computational Neuroscience, 32, 119–136. doi: 10.1007/s10827-011-0343-y.CrossRefPubMedGoogle Scholar
  16. DeFazio, R. A., & Moenter, S. M. (2002). Estradiol feedback alters potassium currents and firing properties of gonadotropin-releasing hormone neurons. Molecular Endocrinology, 16, 2255–2265. doi: 10.1210/me.2002-0155.CrossRefPubMedGoogle Scholar
  17. Destexhe, A., Mainen, Z. F., & Sejnowski, T. J. (1994). Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. Journal of Computational Neuroscience, 1, 195–230. doi: 10.1007/BF00961734.CrossRefPubMedGoogle Scholar
  18. Dickson, C. T., Magistretti, J., Shalinsky, M. H., et al. (2000). Properties and role of I(h) in the pacing of subthreshold oscillations in entorhinal cortex layer II neurons. Journal of Neurophysiology, 83, 2562–2579.PubMedGoogle Scholar
  19. Doedel, E. J., & Oldeman, B. E. (2007). AUTO-07P : continuation and bifurcation software for ordinary differential equations. Montreal, Canada: Concordia University. Accessed 1 Sep 2015.Google Scholar
  20. Duan, W., Lee, K., Herbison, A. E., & Sneyd, J. (2011). A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis. Journal of Theoretical Biology, 276, 22–34. doi: 10.1016/j.jtbi.2011.01.035.CrossRefPubMedGoogle Scholar
  21. Dutton, A., & Dyball, R. E. (1979). Phasic firing enhances vasopressin release from the rat neurohypophysis. Journal of Physiology, 290, 433–440.CrossRefPubMedPubMedCentralGoogle Scholar
  22. Ermentrout, B. (2002). Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. Philadelphia: SIAM.CrossRefGoogle Scholar
  23. Fletcher, P. A., & Li, Y. X. (2009). An integrated model of electrical spiking, bursting, and calcium oscillations in GnRH neurons. Biophysical Journal, 96, 4514–4524. doi: 10.1016/j.bpj.2009.03.037.CrossRefPubMedPubMedCentralGoogle Scholar
  24. Fox, R. F., & Lu, Y. N. (1994). Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. Physical Review E, 49, 3421–3431. doi: 10.1103/PhysRevE.49.3421.CrossRefGoogle Scholar
  25. Glanowska, K. M., & Moenter, S. M. (2015). Differential regulation of GnRH secretion in the preoptic area (POA) and the median eminence (ME) in male mice. Endocrinology, 156, 231–241. doi: 10.1210/en.2014-1458.CrossRefPubMedPubMedCentralGoogle Scholar
  26. Hemond, P. J., O’Boyle, M. P., Roberts, C. B., et al. (2012). Simulated GABA synaptic input and L-type calcium channels form functional microdomains in hypothalamic gonadotropin-releasing hormone neurons. Journal of Neuroscience, 32, 8756–8766. doi: 10.1523/JNEUROSCI.4188-11.2012.CrossRefPubMedPubMedCentralGoogle Scholar
  27. Herbison, A. E. (1998). Multimodal influence of estrogen upon gonadotropin-releasing hormone neurons. Endocrine Reviews, 19, 302–330.CrossRefPubMedGoogle Scholar
  28. Hibino, H., Inanobe, A., Furutani, K., et al. (2010). Inwardly rectifying potassium channels : their structure, function, and physiological roles. Physiology Reviews, 90, 291–366. doi: 10.1152/physrev.00021.2009.CrossRefGoogle Scholar
  29. Hiraizumi, Y., Nishimura, I., Ishii, H., et al. (2008). Rat GnRH neurons exhibit large conductance voltage- and Ca2 + −Activated K+ (BK) currents and express BK channel mRNAs. Journal of Physiological Sciences, 58, 21–29. doi: 10.2170/physiolsci.RP013207.CrossRefPubMedGoogle Scholar
  30. Huguenard, J. R., & McCormick, D. A. (1992). Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons. Journal of Neurophysiology, 68, 1373–1383.PubMedGoogle Scholar
  31. Hunter, J. D. (2007). Matplotlib: a 2D graphics environment. Computing in Science and Engineering, 9, 99–104. doi: 10.1109/MCSE.2007.55.CrossRefGoogle Scholar
  32. Izhikevich, E. M. (2000). Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 10, 1171–1266. doi: 10.1142/S0218127400000840.CrossRefGoogle Scholar
  33. Kato, M., Ui-Tei, K., Watanabe, M., & Sakuma, Y. (2003). Characterization of voltage-gated calcium currents in gonadotropin-releasing hormone neurons tagged with green fluorescent protein in rats. Endocrinology, 144, 5118–5125. doi: 10.1210/en.2003-0213.CrossRefPubMedGoogle Scholar
  34. Kuo, C. C., & Bean, B. P. (1994). Na + channels must deactivate to recover from inactivation. Neuron, 12, 819–829. doi: 10.1016/0896-6273(94)90335-2.CrossRefPubMedGoogle Scholar
  35. LeBeau, A. P., Van Goor, F., Stojilkovic, S. S., & Sherman, A. (2000). Modeling of membrane excitability in gonadotropin-releasing hormone-secreting hypothalamic neurons regulated by Ca2 + −mobilizing and adenylyl cyclase-coupled receptors. Journal of Neuroscience, 20, 9290–9297.PubMedGoogle Scholar
  36. Lee, K., Duan, W., Sneyd, J., & Herbison, A. E. (2010). Two slow calcium-activated afterhyperpolarization currents control burst firing dynamics in gonadotropin-releasing hormone neurons. Journal of Neuroscience, 30, 6214–6224. doi: 10.1523/JNEUROSCI.6156-09.2010.CrossRefPubMedGoogle Scholar
  37. Lee, K., Liu, X., & Herbison, A. E. (2012). Burst firing in gonadotrophin-releasing hormone neurones does not require ionotrophic GABA or glutamate receptor activation. Journal of Neuroendocrinology, 24, 1476–1483. doi: 10.1111/j.1365-2826.2012.02360.x.CrossRefPubMedGoogle Scholar
  38. LeMasson G., Maex R. (2001). Introduction to equation solving and parameter Fitting. In E. De Schutter (Ed.), Computational neuroscience: realistic modeling for experimentalists (pp. 1–25). CRC Press.Google Scholar
  39. Liu, X., & Herbison, A. E. (2008). Small-conductance calcium-activated potassium channels control excitability and firing dynamics in Gonadotropin-Releasing Hormone (GnRH) neurons. Endocrinology, 149, 3598–3604. doi: 10.1210/en.2007-1631.CrossRefPubMedGoogle Scholar
  40. Longtin, A. (1997). Autonomous stochastic resonance in bursting neurons. Physical Review E, 55, 868–876. doi: 10.1103/PhysRevE.55.868.CrossRefGoogle Scholar
  41. Longtin, A., Laing, C., & Chacron, M. J. (2003). Correlations and memory in neurodynamical systems. In G. Rangarajan & M. Ding (Eds.), Processes with long-range correlations (pp. 286–308). Berlin: Springer.CrossRefGoogle Scholar
  42. Lu, B., Su, Y., Das, S., et al. (2007). The neuronal channel NALCN contributes resting sodium permeability and is required for normal respiratory rhythm. Cell, 129, 371–383. doi: 10.1016/j.cell.2007.02.041.CrossRefPubMedGoogle Scholar
  43. Magistretti, J., & Alonso, A. (1999). Biophysical properties and slow voltage-dependent inactivation of a sustained sodium current in entorhinal cortex layer-II principal neurons: a whole-cell and single-channel study. The Journal of General Physiology, 114, 491–509. doi: 10.1085/jgp.114.4.491.CrossRefPubMedPubMedCentralGoogle Scholar
  44. Miyasho, T., Takagi, H., Suzuki, H., et al. (2001). Low-threshold potassium channels and a low-threshold calcium channel regulate Ca2+ spike firing in the dendrites of cerebellar Purkinje neurons: a modeling study. Brain Research, 891, 106–115. doi: 10.1016/S0006-8993(00)03206-6.CrossRefPubMedGoogle Scholar
  45. Moenter, S. M. (2010). Identified GnRH neuron electrophysiology: a decade of study. Brain Research, 1364, 10–24. doi: 10.1016/j.brainres.2010.09.066.CrossRefPubMedPubMedCentralGoogle Scholar
  46. Moenter, S. M., Chu, Z., & Christian, C. A. (2009). Neurobiological mechanisms underlying oestradiol negative and positive feedback regulation of gonadotrophin-releasing hormone neurones. Journal of Neuroendocrinology, 21, 327–333. doi: 10.1111/j.1365-2826.2009.01826.x.CrossRefPubMedPubMedCentralGoogle Scholar
  47. Nunemaker, C. S., Defazio, R. A., & Moenter, S. M. (2002). Episodic firing patterns of GnRH neurons. Endocrinology, 143, 2284–2292.CrossRefPubMedGoogle Scholar
  48. Nunemaker, C. S., Straume, M., Defazio, R. A., & Moenter, S. M. (2003). Gonadotropin-releasing hormone neurons generate interacting rhythms in multiple time domains. Endocrinology, 144, 823–831. doi: 10.1210/en.2002-220585.CrossRefPubMedGoogle Scholar
  49. Pielecka, J., & Moenter, S. M. (2006). Effect of steroid milieu on gonadotropin-releasing hormone-1 neuron firing pattern and luteinizing hormone levels in male mice. Biology of Reproduction, 74, 931–937. doi: 10.1095/biolreprod.105.049619.CrossRefPubMedGoogle Scholar
  50. Pielecka-Fortuna, J., DeFazio, R. A., & Moenter, S. M. (2011). Voltage-gated potassium currents are targets of diurnal changes in estradiol feedback regulation and kisspeptin action on gonadotropin-releasing hormone neurons in mice. Biology of Reproduction, 85, 987–995. doi: 10.1095/biolreprod.111.093492.CrossRefPubMedPubMedCentralGoogle Scholar
  51. Plant, R. E. (1981). Bifurcation and resonance in a model for bursting nerve cells. Journal of Mathematical Biology, 11, 15–32. doi: 10.1007/BF00275821.CrossRefPubMedGoogle Scholar
  52. Rinzel, J., & Lee, Y. S. (1987). Dissection of a model for neuronal parabolic bursting. Journal of Mathematical Biology, 25, 653–675. doi: 10.1007/BF00275501.CrossRefPubMedGoogle Scholar
  53. Roberts, C. B., O’Boyle, M. P., & Suter, K. J. (2009). Dendrites determine the contribution of after depolarization potentials (ADPs) to generation of repetitive action potentials in hypothalamic gonadotropin releasing-hormone (GnRH) neurons. Journal of Computational Neuroscience, 26, 39–53. doi: 10.1007/s10827-008-0095-5.CrossRefPubMedGoogle Scholar
  54. Sun, J., Chu, Z., & Moenter, S. M. (2010). Diurnal in vivo and rapid in vitro effects of estradiol on voltage-gated calcium channels in gonadotropin-releasing hormone neurons. Journal of Neuroscience, 30, 3912–3923. doi: 10.1523/JNEUROSCI.6256-09.2010.CrossRefPubMedPubMedCentralGoogle Scholar
  55. Van Geit, W., Achard, P., & De Schutter, E. (2007). Neurofitter: a parameter tuning package for a wide range of electrophysiological neuron models. Frontiers in Neuroinformatics, 1, 1. doi: 10.3389/neuro.11.001.2007.PubMedPubMedCentralGoogle Scholar
  56. Van Goor, F., LeBeau, A. P., Krsmanovic, L. Z., et al. (2000). Amplitude-dependent spike-broadening and enhanced Ca(2+) signaling in GnRH-secreting neurons. Biophysical Journal, 79, 1310–1323. doi: 10.1016/S0006-3495(00)76384-3.CrossRefPubMedPubMedCentralGoogle Scholar
  57. Wang, Y., Garro, M., & Kuehl-Kovarik, M. C. (2010). Estradiol attenuates multiple tetrodotoxin-sensitive sodium currents in isolated gonadotropin-releasing hormone neurons. Brain Research, 1345, 137–145. doi: 10.1016/j.brainres.2010.05.031.CrossRefPubMedPubMedCentralGoogle Scholar
  58. Wildt, L., Häusler, A., Marshall, G., et al. (1981). Frequency and amplitude of gonadotropin-releasing hormone stimulation and gonadotropin secretion in the rhesus monkey. Endocrinology, 109, 376–385. doi: 10.1210/endo-109-2-376.CrossRefPubMedGoogle Scholar
  59. Willms, A. R. (2002). NEUROFIT: software for fitting Hodgkin-Huxley models to voltage-clamp data. Journal of Neuroscience Methods, 121, 139–150. doi: 10.1016/S0165-0270(02)00227-3.CrossRefPubMedGoogle Scholar
  60. Willms, A. R., Baro, D. J., Harris-Warrick, R. M., & Guckenheimer, J. (1999). An improved parameter estimation method for Hodgkin-Huxley models. Journal of Computational Neuroscience, 6, 145–168. doi: 10.1023/A:1008880518515.CrossRefPubMedGoogle Scholar
  61. Wintermantel, T. M., Campbell, R. E., Porteous, R., et al. (2006). Definition of estrogen receptor pathway critical for estrogen positive feedback to gonadotropin-releasing hormone neurons and fertility. Neuron, 52, 271–280. doi: 10.1016/j.neuron.2006.07.023.CrossRefPubMedGoogle Scholar
  62. Zhang, C., Bosch, M. A., Rick, E. A., et al. (2009). 17Beta-estradiol regulation of T-type calcium channels in gonadotropin-releasing hormone neurons. Journal of Neuroscience, 29, 10552–10562. doi: 10.1523/JNEUROSCI.2962-09.2009.CrossRefPubMedPubMedCentralGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Spencer Moran
    • 1
  • Suzanne M Moenter
    • 2
    • 3
    • 4
  • Anmar Khadra
    • 1
  1. 1.Department of PhysiologyMcGill UniversityMontrealCanada
  2. 2.Departments of Molecular and Integrative PhysiologyUniversity of MichiganAnn ArborUSA
  3. 3.Obstetrics and GynecologyUniversity of MichiganAnn ArborUSA
  4. 4.Internal MedicineUniversity of MichiganAnn ArborUSA

Personalised recommendations