Biological Cybernetics

, Volume 50, Issue 1, pp 15–33

Cable theory in neurons with active, linearized membranes

  • Christof Koch

DOI: 10.1007/BF00317936

Cite this article as:
Koch, C. Biol. Cybern. (1984) 50: 15. doi:10.1007/BF00317936


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).

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

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