Biological Cybernetics

, Volume 50, Issue 1, pp 15–33

Cable theory in neurons with active, linearized membranes

  • Christof Koch


This investigation aims at exploring some of the functional consequences of single neurons containing active, voltage dependent channels for information processing. Assuming that the voltage change in the dendritic tree of these neurons does not exceed a few millivolts, it is possible to linearize the non-linear channel conductance. The membrane can then be described in terms of resistances, capacitances and inductances, as for instance in the small-signal analysis of the squid giant axon. Depending on the channel kinetics and the associated ionic battery the linearization yields two basic types of membrane: a membrane modeled by a collection of resistances and capacitances and membranes containing in addition to these components inductances. Under certain specified conditions the latter type of membrane gives rise to a membrane impedance that displays a prominent maximum at some nonzero resonant frequency fmax. We call this type of membrane quasi-active, setting it apart from the usual passive membrane. We study the linearized behaviour of active channels giving rise to quasi-active membranes in extended neuronal structures and consider several instances where such membranes may subserve neuronal function: 1. The resonant frequency of a quasi-active membrane increases with increasing density of active channels. This might be one of the biophysical mechanisms generating the large range over which hair cells in the vertebrate cochlea display frequency tuning. 2. The voltage recorded from a cable with a quasi-active membrane can be proportional to the temporal derivative of the injected current. 3. We modeled a highly branched dendritic tree (δ-ganglion cell of the cat retina) using a quasi-active membrane. The voltage attenuation from a given synaptic site to the soma decreases with increasing frequency up to the resonant frequency, in sharp contrast to the behaviour of passive membranes. This might be the underlying biophysical mechanism of receptive fields whose dimensions are large for rapid signals but contract to a smaller area for slow signals as suggested by Detwiler et al. (1978).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams, P.R., Constanti, A., Brown, D.A., Clark, R.B.: Intracellular Ca2+ activates a fast voltage-sensitive K+ current in vertebrate sympathetic neurones. Nature 296, 746–749 (1982)Google Scholar
  2. Ashmore, J.: Listening with one cell. Nature 304, 489–490 (1983)Google Scholar
  3. Barrett, E.F., Barrett, J.N., Crill, W.E.: Voltage-sensitive outward currents in cat motoneurones. J. Physiol. 304, 251–276 (1980)Google Scholar
  4. Boycott, B.B., Wässle, H.: The morphological types of ganglion cells of the domestic cat's retina. J. Physiol. 240, 397–419 (1974)Google Scholar
  5. Brown, D.A., Adams, P.R.: Muscarinic suppression of a novel voltage-sensitive K+-current in a vertebrate neurone. Nature 283, 673–675 (1980)Google Scholar
  6. Brown, T.H., Perkel, D.H., Norris, J.C., Peacock, J.H.: Electrotonic structure and specific membrane properties of mouse dorsal root ganglion neurons. J. Neurophysiol. 45, 1–15 (1981)Google Scholar
  7. Brühl, G., Jansen, W., Vogt, H.-J.: Nachrichtenübertragungstechnik. Stuttgart: Kohlhammer Verlag 1979Google Scholar
  8. Buxton, B.F., Buxton, H.: Monocular depth perception from optical flow by space time signal processing. Proc. R. Soc. London B218, 27–47 (1983)Google Scholar
  9. Chandler, W.K., Fitzhugh, R., Cole, K.S.: Theoretical stability properties of a space-clamped axon. Biophys. J. 2, 105–127 (1962)Google Scholar
  10. Clapham, D.E., DeFelice, L.J.: The theoretical small signal impedance of the frog node, Rana pipiens. Pflügers Arch. 366, 273–276 (1976)Google Scholar
  11. Clapham, D.E., DeFelice, L.J.: Small signal impedance of heart cell membranes. J. Membrane Biol. 67, 63–71 (1982)Google Scholar
  12. Cole, K.S.: Rectification and inductance in the squid giant axon. J. Physiol. 25, 2951 (1941)Google Scholar
  13. Cole, K.S., Baker, R.F.: Longitudinal impedance of the squid giant axon. J. Gen. Physiol. 24, 771–788 (1941)Google Scholar
  14. Connors, B.W., Gutnick, M.J., Prince, D.A.: Electrophysiological properties of neocortical neurons in vitro. J. Neurophysiol. 48, 1302–1420 (1982)Google Scholar
  15. Cooley, J.W., Dodge, F.A., Jr.: Digital computer solutions for excitation and propagation of the nerve impulse. Biophys. J. 6, 583–599 (1966)Google Scholar
  16. Crawford, A.C., Fettiplace, R.: The frequency selectivity of auditory nerve fibres and hair cells in the cochlea of the turtle. J. Physiol. 306, 79–125 (1980)Google Scholar
  17. Crawford, A.C., Fettiplace, R.: An electrical tuning mechanism in turtle cochlear hair cells. J. Physiol. 312, 377–412 (1981)Google Scholar
  18. Crill, W.E., Schwindt, P.C.: Active currents in mammalian central neurons. Trends Neurosci. 6, 236–240 (1983)Google Scholar
  19. DeHaan, R.L., DeFelice, L.J.: Oscillatory properties and excitability of the heart cell membrane. In: Theoretical chemistry, periodicities in chemistry and biology. pp. 181–233. Eyring, H., Henderson, D. (eds.) New York: Academic Press 1978Google Scholar
  20. Derrington, A.M., Lennie, P.: The influence of temporal frequency and adaptation level on receptive field organization of retinal ganglion cells in cat. J. Physiol. 33, 343–366 (1982)Google Scholar
  21. Detwiler, P.B., Hodgkin, A.L., McNaughton, P.A.: A surprising property of electrical spread in the network of rods in the turtle's retina. Nature 274, 562–565 (1978)Google Scholar
  22. Detwiler, P.B., Hodgkin, A.L., McNaughton, P.A.: Temporal and spatial characteristics of the voltage response of rods in the retina of the snapping turtle. J. Physiol. 300, 213–250 (1980)Google Scholar
  23. Eisenberg, R.S., Johnson, E.A.: Three-dimensional electrical field problems in physiology. Prog. Biophys. Mol. Biol. 20, 1–65 (1970)Google Scholar
  24. Enroth-Cugell, C., Robson, J.G.: The contrast sensitivity of retinal ganglion cells of the cat. J. Physiol. 187, 517–552 (1966)Google Scholar
  25. Enroth-Cugell, C., Robson, J.G., Schweitzer-Tong, D.E., Watson, A.B.: Spatio-temporal interactions in cat retinal ganglion cells showing linear spatial summation. J. Physiol. 341, 279–307 (1983)Google Scholar
  26. Gustafsson, B., Galvan, M., Grafe, P., Wigström, H.: A transient outward current in a mammalian central neurone blocked by 4-aminopyridine. Nature 299, 252–254 (1982)Google Scholar
  27. Halliwell, J.V., Adams, P.R.: Voltage-clamp analysis of muscarinic excitation in hippocampal neurons. Brain Res. 250, 71–92 (1982)Google Scholar
  28. Hengstenberg, R.: Spike responses of “non-spiking” visual interneurone. Nature 270, 338–340 (1977)Google Scholar
  29. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)Google Scholar
  30. Holden, A.V., Yoda, M.: The effect of ionic channel density on neuronal function. J. Theor. Neurophysiol. 1, 60–81 (1981)Google Scholar
  31. Hopkins, C.D.: Stimulus filtering and electroreception: Tuberous electroreceptors in three species of gymnotoid fish. J. Comp. Physiol. 111, 171–207 (1976)Google Scholar
  32. Johnston, D., Lam, D.M.-K.: Regenerative and passive membrane properties of isolated horizontal cells from a teleost retina. Nature 292, 451–453 (1981)Google Scholar
  33. Koch, C.: Nonlinear information processing in dendritic trees of arbitrary geometries. Ph. D. Thesis, University of Tübingen (1982)Google Scholar
  34. Koch, C., Poggio, T.: A theoretical analysis of electrical properties of spines. Proc. Roy. Soc. London B218, 455–477 (1983a)Google Scholar
  35. Koch, C., Poggio, T.: A simple algorithm for solving the cable equation in dendritic trees of arbitrary geometry. Submitted (1983b)Google Scholar
  36. Koch, C., Poggio, T.: Velocity: its meaning and application to one-dimensional cables (in preparation) (1984)Google Scholar
  37. Koch, C., Poggio, T., Torre, V.: Retinal ganglion cells: A functional interpretation of dendritic morphology. Phil. Trans. R. Soc. London B298, 227–264 (1982)Google Scholar
  38. Koch, C., Poggio, T., Torre, V.: Nonlinear interaction in a dendritic tree: localization, timing and role in information processing. Proc. Natl. Acad. Sci. USA 80, 2799–2802 (1983)Google Scholar
  39. Korn, G.A., Korn, T.M.: Mathematical handbook for scientists and engineers. New York: McGraw-Hill 1961Google Scholar
  40. Lewis, R.S., Hudspeth, A.J.: Voltage- and ion-dependent conductances in solitary vertebrate hair cells. Nature 304, 538–541 (1983)Google Scholar
  41. Llinas, R., Jahnsen, H.: Electrophysiology of mammalian thalamic neurones in vitro. Nature 297, 406–408 (1982)Google Scholar
  42. Llinas, R., Sugimori, M.: Electrophysiological properties of in vitro Purkinje cell dendrites in mammalian cerebellar slices. J. Physiol. 305, 197–213 (1980)Google Scholar
  43. Llinas, R., Yarom, Y.: Properties and distribution of ionic conductances generating electroresponsiveness of mammalian inferior olivary neurones in vitro. J. Physiol. 315, 569–584 (1981)Google Scholar
  44. Marchiafava, P.L.: The responses of retinal ganglion cells to stationary and moving visual stimuli. Vision Res. 19, 1203–1211 (1979)Google Scholar
  45. Mauro, A., Conti, F., Dodge, F., Schor, R.: Subthreshold behavior and phenomenological impedance of the squid giant axon. J. Gen. Physiol. 55, 497–523 (1970)Google Scholar
  46. Meyer, J.H., Zakon, H.H.: Androgens alter the tuning of electroreceptors. Science 217, 635–637 (1982)Google Scholar
  47. Mirolli, M.: Fast inward and outward current channels in a nonspiking neurone. Nature 292, 251–253 (1981)Google Scholar
  48. Moore, L.E., Tsai, T.D.: Ion conductances of the surface and transverse tubular membranes of skeletal muscle. J. Membrane Biol. 73, 217–226 (1983)Google Scholar
  49. Poggio, T., Torre, V.: A theory of synaptic interactions. In: Theoretical approaches to neurobiology, pp. 28–38. Reichardt, W.E., Poggio, T. (eds.). Cambridge, NJ: MIT Press 1981Google Scholar
  50. Rall, W.: Core conductor theory and cable properties of neurons. In: Handbook of physiology, pp. 39–97. Kandel, E., Geiger, S. (eds.). Washington, DC: American Physiological Society 1977Google Scholar
  51. Reichardt, W.: Autocorrelation: a principle for the evaluation of sensory information by the central nervous system. In: Sensory communication, pp. 303–318. Rosenblith, W.A. (ed.). Cambridge, NJ: MIT Press 1961Google Scholar
  52. Richter, J., Ullman, S.: A model for the temporal organization of X- and Y-type receptive fields in the primate retina. Biol. Cybern. 43, 127–145 (1982)Google Scholar
  53. Rinzel, J.: Integration and propagation of neuroelectric signals. In: Studies in mathematical biology, pp. 1–66. Levin, S.A. (ed.). Math. Assoc. America (1978)Google Scholar
  54. Roberts, A., Bush, B.M.H.: Neurones without impulses: their significance for vertebrate and invertebrate nervous systems. Cambridge, NJ: Cambridge University Press 1981Google Scholar
  55. Sabah, N.H., Leibovic, K.N.: Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the squid giant axon. Biophys. J. 9, 1206–1222 (1969)Google Scholar
  56. Sabah, N.H., Leibovic, K.N.: The effect of membrane parameters on the properties of the nerve impulse. Biophys. J. 12, 1132–1144 (1972)Google Scholar
  57. Sabah, N.H., Spangler, R.A.: Repetitive response of the Hodgkin-Huxley model for the squid giant axon. J. Theor. Biol. 29, 155–171 (1970)Google Scholar
  58. Schmitt, F.O., Dev, P., Smith, B.H.: Electrotonic processing of information by brain cells. Science 193, 114–120 (1976)Google Scholar
  59. Schwartzkroin, P.A., Slawsky, M.: Probable calcium spikes in hippocampal neurons. Brain Res. 135, 157–161 (1977)Google Scholar
  60. Scott, A.C.: Effect of the series inductance of a nerve axon upon its conduction velocity. Math. Biosci. 11, 277–290 (1971)Google Scholar
  61. Sirovich, L., Knight, B.W.: On subthreshold solutions of the Hodgkin-Huxley equations. Proc. Natl. Acad. Sci. USA 74, 5199–5202 (1977)Google Scholar
  62. Smith, K.J., Schauf, C.L.: Size-dependent variation of nodal properties in myelinated nerve. Nature 293, 297–299 (1981)Google Scholar
  63. Swindale, N.V.: Anatomical logic of retinal nerve cells. Nature 303, 570–571 (1983)Google Scholar
  64. Torre, V., Owen, W.G.: High-pass filtering of small signals by the rod network in the retina of the toad, Bufo Marinus. Biophys. J. 41, 305–324 (1983)Google Scholar
  65. Torre, V., Owen, W.G., Sandini, G.: The dynamics of electrically interacting cells. IEEE Trans. Syst. Man, Cybern. (in press)Google Scholar
  66. Torre, V., Poggio, T.: A synaptic mechanism possibly underlying directional selectivity to motion. Proc. R. Soc. London B202, 409–416 (1978)Google Scholar
  67. Wong, R.K.S., Prince, D.A., Basbaum, A.I.: Intradendritic recordings from hippocampal neurons. Proc. Natl. Acad. Sci. USA 76, 986–990 (1979)Google Scholar
  68. Wyatt, H.J., Daw, N.W.: Directionally sensitive ganglion cells in the rabbit: specificity for stimulus direction, size and speed. J. Neurophysiol. 38, 613–626 (1975)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Christof Koch
    • 1
  1. 1.Department of Psychology and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations