Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

An analysis of a dendritic neuron model with an active membrane site

  • 73 Accesses

  • 14 Citations

Abstract

We formulate and analyze a mathematical model that couples an idealized dendrite to an active boundary site to investigate the nonlinear interaction between these passive and active membrane patches. The active site is represented mathematically as a nonlinear boundary condition to a passive cable equation in the form of a space-clamped FitzHugh-Nagumo (FHN) equation. We perform a bifurcation analysis for both steady and periodic perturbation at the active site. We first investigate the uncoupled space-clamped FHN equation alone and find that for periodic perturbation a transition from phase locked (periodic) to phase pulling (quasiperiodic) solutions exist. For the model coupling a passive cable with a FHN active site at the boundary, we show for steady perturbation that the interval for repetitive firing is a subset of the interval for the space-clamped case and shrinks to zero for strong coupling. The firing rate at the active site decreases as the coupling strength increases. For periodic perturbation we show that the transition from phase locked to phase pulling solutions is also dependent on the coupling strength.

This is a preview of subscription content, log in to check access.

references

  1. Baer, S.M., Erneux, T., Tier, C.: Singular Hopf bifurcations arising in neuron models. In preparation, 1985

  2. Bell, J., Cosner, C.: Threshold conditions for a diffusive model of a myelinated axon. J. Math. Biol. 18, 39–52 (1983)

  3. Cartwright, M. L.: Forced oscillations in nearly sinusoidal systems. J. Inst. Elec. Engr. 95, 88–96 (1948)

  4. Dodge, F. A., Cooley, J. W.: Action potential of the motorneuron. IBM J. Res. Devel. 17, 219–229 (1973).

  5. Doedel, E. J.: AUTO: A program for the automatic bifurcation analysis of autonomous systems. Proc. 10th Manitoba Conf. on Num. Math, and Comp., Winnipeg, Cong. Num. 30, 265–284 (1981)

  6. Eckhaus, W.: Relaxation oscillations including a standard chase on french ducks. In: Verhulst, F. (ed.) Lect. Notes in Math. Asymptotic Analysis II 985, 432–449, (1983)

  7. FitzHugh, R.: Computation of impulse initiation and saltatory conduction in a myelinated nerve fiber. Biophys. J. 2, 11–21 (1962).

  8. Goldman, L., Albus, J. S.: Computation of impulse conduction in myelinated fibres; theoretical basis of the velocity-diameter relation. Biophys. J. 8, 596–607 (1968)

  9. Hadeler, K. P., an der Heiden U., Schumacher, K.: Generation of nervous impulse and periodic oscillations. Biol. Cyber. 23, 211–218 (1976)

  10. Hsu, I., Kazarinoff, N. K.: An applicable Hopf bifurcation and stability of small periodic solutions of the Field-Noyes model. J. Math. Anal. Appl. 55, 61–89 (1976).

  11. Kath, W. L.: Resonance in periodically perturbed Hopf bifurcation. Stud. Math. 65, 95–112 (1981)

  12. Kogelman, S., Keller, J. B.: Transient behavior of unstable nonlinear systems with applications to the Benard and Taylor problems. SIAM J. Appl. Math. 20, 619–637 (1971)

  13. Matkowsky, B. J.: A simple nonlinear dynamic stability problem. Bull. Am. Math. Soc. 76, 620–625 (1970).

  14. Nagumo, J. S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE. 50, 2061–2070 (1962)

  15. Pellionisz, A., Llinas, R.: A computer model of cerebellar Purkinje cells. Neurosci. 2, 37–48 (1977)

  16. Rall, W.: Membrane potential transients and membrane time constants of motoneurons. Exp. Neurol. 2, 503–532 (1960)

  17. Rall, W.: Theory of physiological properties of dendrites. Ann. N.Y. Acad. Sci. 96, 1071–1092 (1962)

  18. Rall, W., Shepherd, G. M.: Theoretical reconstruction of field potentials and dendrodentritic synaptic interactions in olfactory bulb. J. Neurophysiol. 31, 884–915 (1968)

  19. Rall, W.: Core conduction theory and cable properties of neurons. In: Kandel, E. R. (ed.) Handbook of physiology, sect. 1: The nervous system, vol. 1: Cellular biology of neurons. Bethesda, Am. Physiol. Soc., Washington, D.C., 1976

  20. Rinzel, J.: Integration and propagation of neuroelectrical signals. In: Levin, S. A. (ed.) Studies in mathematical biology. MAA Studies in Mathematics 15, 1978

  21. Rinzel, J., Keener, J. P.: Hopf bifurcation and repetitive activity in nerve. SIAM J. Appl. Math. 43: 907–922 (1983)

  22. Rosenblat, S., Cohen, D. S.: Periodically perturbed bifurcation-1. Simple bifurcation. Stud. Appl. Math. 63, 1–23 (1980)

  23. Rosenblat, S., Cohen, D. S.: Periodically perturbed bifurcation-II. Hopf bifurcation. Stud. Appl. Math. 64, 143–175 (1981)

  24. Troy, W. C.: Mathematical modeling of excitable media in neurobiology and chemistry. In: Eyring, H., Henderson, D. (eds.) Theoretical chemistry periodicities in chemistry and biology. Academic Press, New York 1978

  25. Tuckwell, H. C.: Introduction to mathematical neurobiology. Pitman, London 1985

Download references

Author information

Additional information

This work was supported in part by NSF Grants MCS 83-00562 and MDS 85-01535

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baer, S.M., Tier, C. An analysis of a dendritic neuron model with an active membrane site. J. Math. Biology 23, 137–161 (1986). https://doi.org/10.1007/BF00276954

Download citation

Key words

  • Threshold
  • Dendrite
  • Cable theory
  • Resonance
  • Bifurcation theory