Analysis of an excitable dendritic spine with an activity-dependent stem conductance

Journal of Mathematical Biology

Volumn 36, Issue 6, 1998, Pages 569-592

  Wu, H Y ^a   Baer, S M ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

Author keywords

Bursting oscillations; Dendritic spines; Neuronal modeling; Synapse

Indexed keywords

ARTICLE; BIOLOGICAL MODEL; DENDRITE; ELECTRIC CONDUCTIVITY; KINETICS; MATHEMATICAL COMPUTING; PHYSIOLOGY; SYNAPTIC TRANSMISSION; ULTRASTRUCTURE;

DENDRITES; ELECTRIC CONDUCTIVITY; KINETICS; MODELS, NEUROLOGICAL; NUMERICAL ANALYSIS, COMPUTER-ASSISTED; SYNAPTIC TRANSMISSION;

EID: 0032088845     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002850050115     Document Type: Article

References (60)

1. Baer, S. M., Erneux, T.: Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Math. 46, 721-739 (1986)
2. Baer, S. M., Erneux, T.: Singular Hopf bifurcation to relaxation oscillations II. SIAM J. Appl. Math. 52, 1651-1664 (1992)
3. Baer, S. M., Erneux, T., Rinzel, J.: The slow passage through Hopf bifurcation: delay, memory effects, and resonance. SIAM J. Appl. Math. 49, 55-71 (1989)
4. Baer, S. M., Rinzel, J.: Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J. Neurophysiol. 65, 874-890 (1991)
5. Baer, S. M., Rinzel, J., Carrillo, H.: Analysis of an autonomous phase model for neuronal parabolic bursting. J. Math. Biol. 33, 309-333 (1995)
6. Baer, S. M., Tier, C.: An analysis of a dendritic neuron model with an active membrane site. J. Math. Biol. 23, 137-161 (1986)
7. Barinaga, M.: Dendrites shed their dull image. Science 268, 200-201 (1995)
8. Collonier, M.: Synaptic patterns on different cell types in the different laminae of the cat visual cortex. An electron microscope study. Brain Res. 9, 268-287 (1968)
9. Coss, R. G., Perkel, D. H.: The function of dendritic spines: a review of theoretical issues. Behav. Neural Biol. 44, 151-185 (1985)
10. Diamond, J., Gray, E. G., Yasargil, E. N.: The function of a dendritic spine: an hypothesis. In: Andersen, P., Jansen, J. K. S. (eds.) Excitatory Synaptic Mechanisms (pp. 213-222) Oslo: Universitetsforlaget 1970
11. Doedel, E.: AUTO: A program for the automatic bifurcation analysis of autonomous systems. Cong. Num. 30, 265-284 (1981)
12. Ermentrout, G. B., Kopell, N.: Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46, 233-253 (1986)
13. Fifkova, E.: A possible mechanism of morphometric changes in dendritic spines induced by stimulation. Cell. Mol. Neurobiol. 5, 47-63 (1985)
14. Fifkova, E., Van Harreveld, A.: Long lasting morphological changes in dendritic spines of dendate granule cells following stimulation of the entorhinal area. J. Neurocytol. 6, 211-230 (1977)
15. FitzHugh, R.: Mathematical models for excitation and propagation in nerve. In: Schwan, H. P. (ed.) Biological Engineering. New York: McGraw Hill 1969
16. Gear, C.: The numerical integration of ordinary differential equations. Math. Comp. 21, 146-156 (1967)
17. Garthwaite, J.: Glutamate, nitric oxide and cell-cell signalling in the nervous system. TINS 14, 60-67 (1991)
18. Geinisman, Y.: Perforated axospinous synapses with multiple, completely partitioned transmission zones: probable structural intermediates in synaptic plasticity. Hippocampus 3, 417-434 (1993)
19. Grasman, J.: Asymptotic methods for relaxaton oscillations and applications, applied Mathematical Sciences, 63, Berlin, New York: Spinger-Verlag 1987
20. Gray, E. G.: Axo-somatic and axo-dendritic synapses of the cerebral cortex: an electron microscopic study. J. Anat. 93, 420-433 (1959)
21. Greenough, W. T.: Experiential modification of the developing brain. Am. Sci. 63, 37-46 (1975)
22. Hadeler, K. P., An der Heiden, U., Schumacher, K.: Generation of nervous impulse and periodic oscillations. Biol. Cybernet. 23, 211-218 (1976)
23. Harris, K. M., Kater, S. B.: Dendritic spines: cellular specializations imparting both stability and flexibility to synaptic function. Annu. Rev. Neurosci. 17, 341-371 (1994).
24. Harris, K. M., Stevens, J. K.: Dendritic spines of rat cerebellar Purkinje cells; serial electron microscopy with reference to their biophysical characteristics. J. Neurosci., 8, 4455-4469 (1988)
25. Hille, B.: Ionic Channels of Excitable Membranes (Second Edition), Sunderland, MA: Sinauer Associates 1992
26. Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500-544 (1952)
27. Holden, L., Erneux, T.: Slow passage through a Hopf bifurcation: from oscillatory to steady state solutions. SIAM J. Appl. Math. 53, 1045-1058 (1993)
28. Holden, L., Erneux, T.: Understanding bursting oscillations as periodic slow passages through bifurcation and limit points. J. Math. Biol. 31, 351-365 (1993)
29. Hosokawa, T., Bliss T. V. P., Fine, A.: Persistence of individual spines in living brain slices. NeuroReport 3, 477-480 (1992)
30. Hsu, I., Kazarinoff, N. K.: An applicable Hopf bifurcation formula and stability of small periodic solutions of the Field-Noyes Model. J. Math. Anal. Appl. 55, 61-89 (1976)
31. Jack, J. J. B., Noble, D., Tsien, R. W. Electric Current Flow in Excitable Cells. Oxford, UK: Clarendon 1975
32. Kawato, M., Tsukahara, N.: Electrical properties of dendritic spines with bulbous end terminals. Biophys. J. 46, 115-166 (1984)
33. Koch, C., Poggio, T.: A theoretical analysis of electrical properties of spines. Proc. R. Soc. Lond. B Biol. 218, 455-477 (1983)
34. Koch, C., Zador, A., Brown, T. H.: Dendritic spines: convergence of theory and experiment. Science 256, 973-974 (1992)
35. LeMasson, G. Marder, E., Abbott, L. F.: Activity-dependent regulation of conductances in model neurons. Science 259, 1915-1917 (1993)
36. Miller, J. P., Rall, W., Rinzel, J.: Synaptic amplification by active membrane in dendritic spines. Brain Res. 325, 325-330 (1985)
37. Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193-213 (1981)
38. Perkel, D. H., Perkel, D. J.: Dendritic spines: role of active membrane in modulating synaptic efficacy. Brain Res. 325, 331-335 (1985)
39. Peters, A.: Kaiserman-Abramof, I. R.: The small pyramidal neuron of the rat cerebral cortex. The perikaryon, dendrites and spines. Am. J. Anat. 127, 321-356 (1970)
40. Rall, W.: Dendritic spines and synaptic potency. In: Porter, R. (ed.) Studies in Neurophysiology. (pp. 203-209) New York: Cambridge Univ. Press 1978
41. Rall, W., Rinzel, J.: Dendritic spines and synaptic potency explored theoretically. Progc. Int. Union Physiol. Sci. (XXV Int. Congr.) IX, 466 (1971)
42. Rall, W., Rinzel, J.: Dendritic spine function and synaptic attenuation calculations. Prog. Abstr. Soc. Neurosci. 1, 64 (1971)
43. Rall, W., Segev, I.: Functional possibilities for synapses on dendrites and on dendritic spines. In: Edelman, G. M., Gall, W. E., Cowan, W. M. (eds.) Synaptic Function, (pp. 605-636) New York: Wiley 1987
44. Rinzel, J.: Neuronal plasticity (learning). In: Miura, R.M. (ed.) Some Mathematical Questions in Biology-Neurobiology. Lectures on Mathematics in the Life Sciences, (pp. 7-25) Providence, RI: Am. Math. Soc. 1982
45. Rinzel, J., Baer, S. M.: Threshold for repetitive activity for a slow stimulus ramp: a memory effect and its dependence on fluctuations. Biophys. J. 54, 551-555 (1988)
46. Rinzel, J., Ermentrout, G. B.: Analysis of neural excitability and oscillations. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modeling: From Synapses to Networks. (pp. 135-169) Cambridge, London: MIT Press 1989
47. Rinzel J., Lee, Y. S.: Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25, 653-675 (1987)
48. Rinzel, J.: A formal classification of bursting mechanisms in excitable systems. In: Teramoto, E., Yamaguti, M. (eds.) Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences. (Lect. Notes Biomath. vol. 71) New York: Springer-Verlag 1987
49. Segev, I., Rall, W.: Computational study of an excitable dendritic spine. J. Neurophysiol. 60, 499-523 (1988)
50. Shepherd, G. M., Greer, C. A.: The dendritic spine: adaptations of structure and function for different types of synaptic integration. In: Lasek, R., Black, M.M. (eds.) Intrinsic determinants of neuronal form and function. New York: Liss 1988
51. Shepherd, G. M.: The dendritic spine: A multifunctional integrative unit. J. Neurophysiol. 75, 2197-2210 (1996)
52. Shepherd, G. M.: The Synaptic Organization of the Brain. (Third Edition), New York, NY: Oxford Univ. Press 1990
53. Shepherd, G. M., Brayton, R. K., Miller, J. P., Segev, I., Rinzel, J., Rall, W.: Signal enhancement in distal cortical dendrites by means of interaction between active dendritic spines. Proc. Natl. Acad. Sci. USA 82, 2192-2195 (1985)
54. Troy, W. C.: The bifurcation of periodic solutions in the Hodgkin-Huxley equations. Quart. Appl. Math. 36, 73-83 (1978)
55. Turner, P. J.: XMGR: A plotting tool for workstations. Version 3.01 (1994)
56. Van harreveld, A., Fifkova, E.: Swelling of dendritic spines in the fascia dentata after stimulation of the perforant fibers as a mechanism of post-tetanic potentiation. Exp. Neurol. 49, 736-749 (1975)
57. Wang, X.-J., Rinzel, J.: Oscillatory and bursting properties of neurons. In: Arbib, M. (ed.) The Handbook of Brain Theory and Neural Networks. Cambridge, MA: MIT Press 1995
58. Wilson, C. J., Groves, P. M., Kital, S. T., Linder, J. C.: Three-dimensional structure of dendritic spines in the rat neostriatum. J. Neurosci. 3, 383-398 (1983)
59. Wilson, C. J.: Passive cable properties of dendritic spines and spiny neurons. J. Neurosci. 4, 281-297 (1984)
60. Yuste, R., Denk, W.: Dendritic spines as basic functional units of neuronal integration. Nature. 375, 682-684 (1995)