Journal of Computational Neuroscience

, Volume 19, Issue 3, pp 325–356 | Cite as

Generation of Very Slow Neuronal Rhythms and Chaos Near the Hopf Bifurcation in Single Neuron Models

  • Shinji DoiEmail author
  • Sadatoshi Kumagai


We have presented a new generation mechanism of slow spiking or repetitive discharges with extraordinarily long inter-spike intervals using the modified Hodgkin-Huxley equations (Doi and Kumagai, 2001). This generation process of slow firing is completely different from that of the well-known potassium A-current in that the steady-state current-voltage relation of the neuronal model is monotonic rather than the N-shaped one of the A-current. In this paper, we extend the previous results and show that the very slow spiking generically appears in both the three-dimensional Hodgkin-Huxley equations and the three dimensional Bonhoeffer-van der Pol (or FitzHugh-Nagumo) equations. The generation of repetitive discharges or the destabilization of the unique equilibrium point (resting potential) is a simple Hopf bifurcation. We also show that the generation of slow spiking does not depend on the stability of the Hopf bifurcation: supercritical or subcritical. The dynamics of slow spiking is investigated in detail and we demonstrate that the phenomenology of slow spiking can be categorized into two types according to the type of the corresponding bifurcation of a fast subsystem: Hopf or saddle-node bifurcation.


slow oscillation chaos Hopf bifurcation homoclinic orbit singular perturbation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams P (1982) Voltage-dependent conductances of vertebrate neurones. TINS-April, pp. 116–119.Google Scholar
  2. Baer SM, Erneux T (1986) Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Math. 46: 721–739.CrossRefGoogle Scholar
  3. Baer SM, Erneux T, Rinzel J (1989) The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance. SIAM J. Appl. Math. 49: 55–71.CrossRefGoogle Scholar
  4. Benoit E, Callot JL, Diener F, Diener M (1981) Chasse au canards. Collect. Math. 31: 37–119.Google Scholar
  5. Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57: 413–439.CrossRefPubMedGoogle Scholar
  6. Braaksma B (1998) Singular Hopf bifurcation in systems with fast and slow variables. J. Nonlinear Sci. 8: 457–490.CrossRefGoogle Scholar
  7. Canavier CC, Clark JW, Byrne JH (1991) Simulation of the bursting activity of neuron R15 in Aplisia: Role of ionic currents, calcium balance, and modulatory transmitters. J. Neurophysiol. 66: 2107–2124.PubMedGoogle Scholar
  8. Carpenter GA (1977) A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Diff. Eqn. 23: 335–367.CrossRefGoogle Scholar
  9. Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J. 42: 181–190.PubMedGoogle Scholar
  10. Connor JA, Walter D and McKown R (1977) Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons. Biophys. J. 18: 81–102.PubMedGoogle Scholar
  11. Crill WE, Schwindt PC (1983) Active currents in mammalian central neurons. TINS-June, pp. 236–240.Google Scholar
  12. Cronin, J (1987) Mathematical Aspects of Hodgkin-Huxley neural theory. Cambridge Univ. Press.Google Scholar
  13. Doedel E, Wang X, Fairgrieve T (1995) AUTO94–Software for continuation and bifurcation problems in ordinary differentialequations. CRPC-95-2, California Inst. of Tech.Google Scholar
  14. Doi S, Inoue J, Sato S, Smith CE (1999) Bifurcation analysis of neuronal excitability and oscillations. In: Poznanski R, ed. Modeling in the Neurosciences: From Ionic Channels to Neural Networks, Chapter 16, Harwood Academic Publishers, pp. 443–473.Google Scholar
  15. Doi S, Inoue J, Kumagai S (2004) Chaotic spiking in the Hodgkin-Huxley nerve model with slow inactivation of the sodium current. J. Integrative Neurosci. 3: 207–225.Google Scholar
  16. Doi S, Kumagai S (2001) Nonlinear dynamics of small-scale biophysical neural networks. In: Poznanski R, ed. Biophysical Neural Networks: Foundations of Integrative Neuroscience, Chapter 10, Mary Ann Liebert, Inc., pp. 261–301.Google Scholar
  17. Doi S, Nabetani S, Kumagai S (2001) Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes. Biol. Cybern. 85: 51–64.Google Scholar
  18. Drover J, Rubin J, Su JH, Ermentrout B (2004) Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies. SIAM J. Appl. Math. 65: 69–92.CrossRefGoogle Scholar
  19. Ermentrout GB (1996) Type I membranes, phase resetting curves, and synchrony. Neural Comp. 8: 979–1001.Google Scholar
  20. Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eqn. 31: 53–98.Google Scholar
  21. FitzHugh R (1959) Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol. 43: 867–896.Google Scholar
  22. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophy. J. 1: 445–466.Google Scholar
  23. Fukai H, Doi S, Nomura T, Sato S (2000a) Hopf bifurcations in multiple parameter space of the Hodgkin-Huxley equations. I. Global Organization of bistable periodic solutions. Biol. Cybern. 82: 215–222.Google Scholar
  24. Fukai H, Nomura T, Doi S, Sato S (2000b) Hopf bifurcations in multiple parameter space of the Hodgkin-Huxley equations. II. Singularity theoretic approach and highly degenerate bifurcations. Biol. Cybern. 82: 223–229.Google Scholar
  25. Gerber B, Jakobsson E (1993) Functional significance of the A-current. Biol. Cybern. 70: 109–114.CrossRefPubMedGoogle Scholar
  26. Guckenheimer J (1996) Towards a global theory of singularly perturbed dynamical systems. Prog. Non. Diff. Eqn. Appl. 19: 213–225.Google Scholar
  27. Guckenheimer J, Harris-Warrick R, Peck J, Willms A (1997) Bifurcation, bursting, and spike frequency adaptation. J. Comp. Neurosci. 4: 257–277.Google Scholar
  28. Guckenheimer J, Hoffman K, Weckesser W (2000) Numerical computation of canards. Int. J. Bif. Chaos 10: 2669–2687.CrossRefGoogle Scholar
  29. Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer.Google Scholar
  30. Guckenheimer J, Labouriau IS (1993) Bifurcation of the Hodgkin and Huxley equations: A new twist. Bull. Math. Biol. 55: 937–952.CrossRefGoogle Scholar
  31. Guckenheimer J, Oliva A (2002) Chaos in the Hodgkin-Huxley model. SIAM J. Appl. Dynam. Sys. 1:105–114.Google Scholar
  32. Guckenheimer J, Willms AR (2000) Asymptotic analysis of subcritical Hopf-homoclinic bifurcation. Physica D 139: 195–216.CrossRefGoogle Scholar
  33. Gutkin BS, Ermentrout GB (1998) Dynamics of membrane excitability determine inter-spike interval variability: A link between spike generation mechanisms and cortical spike train statistics. Neural Comp. 10: 1047–1065.Google Scholar
  34. Hassard B (1978) Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon. J. Theor. Biol. 71: 401–420.CrossRefPubMedGoogle Scholar
  35. Hayashi H, Ishizuka S (1992) Chaotic nature of bursting discharges in the Onchidium pacemaker neuron. J. Theor. Biol. 156: 269–291.Google Scholar
  36. Hille B (1992) Ionic Channels of Excitable Membranes, 2nd ed. Sinauer Associates.Google Scholar
  37. Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. 107: 165–181.Google Scholar
  38. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its applications to conduction and excitation in nerve. J. Physiol. 117: 500–544.PubMedGoogle Scholar
  39. Honerkamp J, Mutschler G Seitz R (1985) Coupling of a slow and a fast oscillator can generate bursting. Bull. Math. Biol. 47: 1–21.CrossRefGoogle Scholar
  40. Izhikevich EM (2000) Neural excitability, spiking, and bursting. Int. J. Bifn. Chaos 10: 1171–1266.Google Scholar
  41. Jones C (1996) Geometric singular perturbation theory. In: Johnson R, ed. Dynamical Systems, Lecture Notes in Mathematics, Vol.1609, Springer.Google Scholar
  42. Jones C, Kopell N (1994) Tracking invariant manifolds with differential forms. J. Diff. Eqn. 108: 64–88.Google Scholar
  43. Keener J, Sneyd J (1998) Mathematical Physiology. Interdisciplinary Applied Mathematics, Vol. 8. Springer-Verlag, New York.Google Scholar
  44. Llinas RR (1988) The intrinsic electrophysiological properties of mammalian neurons: Insights into central nervous system function. Science 242: 1654–1664.PubMedGoogle Scholar
  45. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line stimulating nerve axon. Proc. Inst. Radio Eng. 50: 2061–2070.Google Scholar
  46. Neishtadt AI (1987) Persistence of stability loss for dynamical bifurcations I. Diff. Eqn. 23: 1385–1391.Google Scholar
  47. Neishtadt AI (1988) Persistence of stability loss for dynamical bifurcations II. Diff. Eqn. 24: 171–176.Google Scholar
  48. Noble D (1995) The development of mathematical models of the heart. Chaos. Sol. Frac. 5: 321–333.CrossRefGoogle Scholar
  49. Plant RE (1976) The geometry of the Hodgkin-Huxley model. Comp. Prog. Biomed. 6: 85–91.CrossRefGoogle Scholar
  50. Rinzel J (1978) On repetitive activity in nerve. Fed. Proc. 37: 2793–2802.PubMedGoogle Scholar
  51. Rinzel J (1990) Discussion: Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and update. Bull. Math. Biol. 52: 5–23.CrossRefGoogle Scholar
  52. Rinzel J, Ermentrout GB (1989) Analysis of neuronal excitability and oscillations. In: Koch C, Segev I, eds. Methods in Neuronal Modeling: From Synapses to Networks, MIT Press, London.Google Scholar
  53. Rinzel J, Lee YS (1986) On different mechanisms of membrane potential bursting. In: Othmer HG, ed. Nonlinear Oscillations in Biology and Chemistry, Springer, pp. 19–33.Google Scholar
  54. Rinzel J, Miller RN (1980) Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Math. Biosci. 49: 27–59.CrossRefGoogle Scholar
  55. Rinzel J, Troy WC (1983) A one-variable map analysis of bursting in the Belousov-Zhabotinskii reaction. Contem. Math. 17: 411–427.Google Scholar
  56. Rush ME, Rinzel J (1995) The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models. Bull. Math. Biol. 57: 899–929.CrossRefPubMedGoogle Scholar
  57. Szmolyan P, Wechselberger M (2001) Canards in R3. J. Diff. Eqs. 177: 419–453.CrossRefGoogle Scholar
  58. Traub RD, Wong RKS, Miles R, Michelson H (1991) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J. Neurophysiol. 66: 635–650.PubMedGoogle Scholar
  59. Troy WC (1978) The bifurcation of periodic solutions in the Hodgkin-Huxley equations. Quart. Appl. Math. 36: 73–83.Google Scholar
  60. Wang XJ, Rinzel J (1995) Oscillatory and bursting properties of neurons. In: Arbib MA, ed. The Handbook of Brain Theory and Neural Networks, MIT Press, pp. 686–691.Google Scholar
  61. Wechselberger M (2005) Existence and bifurcation of canards in R3 in the case of a folded node. SIAM J. Appl. Dynam. Sys. 4: 101–139.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Electrical Engineering, Graduate School of EngineeringOsaka UniversitySuitaJapan

Personalised recommendations