Abstract
The theory of electrotonus, which has been well developed for small cylinders, is extended: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.
Full text
PDF
Selected References
These references are in PubMed. This may not be the complete list of references from this article.
- Agin D. Electroneutrality and electrodiffusion in the squid axon. Proc Natl Acad Sci U S A. 1967 May;57(5):1232–1238. doi: 10.1073/pnas.57.5.1232. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Barnard A. C., Duck I. M., Lynn M. S. The application of electromagnetic theory to electrocardiology. I. Derivation of the integral equations. Biophys J. 1967 Sep;7(5):443–462. doi: 10.1016/S0006-3495(67)86598-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Clark J., Plonsey R. A mathematical evaluation of the core conductor model. Biophys J. 1966 Jan;6(1):95–112. doi: 10.1016/S0006-3495(66)86642-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
- HAGIWARA S., TASAKI I. A study on the mechanism of impulse transmission across the giant synapse of the squid. J Physiol. 1958 Aug 29;143(1):114–137. doi: 10.1113/jphysiol.1958.sp006048. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Ito M., Oshima T. Electrical behaviour of the motoneurone membrane during intracellularly applied current steps. J Physiol. 1965 Oct;180(3):607–635. doi: 10.1113/jphysiol.1965.sp007720. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Lux H. D., Pollen D. A. Electrical constants of neurons in the motor cortex of the cat. J Neurophysiol. 1966 Mar;29(2):207–220. doi: 10.1152/jn.1966.29.2.207. [DOI] [PubMed] [Google Scholar]
- PLONSEY R. VOLUME CONDUCTOR FIELDS OF ACTION CURRENTS. Biophys J. 1964 Jul;4:317–328. doi: 10.1016/s0006-3495(64)86785-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Plonsey R. An extension of the solid angle potential formulation for an active cell. Biophys J. 1965 Sep;5(5):663–667. doi: 10.1016/S0006-3495(65)86744-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
- RALL W. Branching dendritic trees and motoneuron membrane resistivity. Exp Neurol. 1959 Nov;1:491–527. doi: 10.1016/0014-4886(59)90046-9. [DOI] [PubMed] [Google Scholar]
- RALL W. Electrophysiology of a dendritic neuron model. Biophys J. 1962 Mar;2(2 Pt 2):145–167. doi: 10.1016/s0006-3495(62)86953-7. [DOI] [PMC free article] [PubMed] [Google Scholar]
- RALL W. Membrane potential transients and membrane time constant of motoneurons. Exp Neurol. 1960 Oct;2:503–532. doi: 10.1016/0014-4886(60)90029-7. [DOI] [PubMed] [Google Scholar]
- Rall W. Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input. J Neurophysiol. 1967 Sep;30(5):1138–1168. doi: 10.1152/jn.1967.30.5.1138. [DOI] [PubMed] [Google Scholar]
- Rosenthal F., Woodbury J. W., Patton H. D. Dipole characteristics of pyramidal cell activity in cat postcruciate cortex. J Neurophysiol. 1966 Jul;29(4):612–625. doi: 10.1152/jn.1966.29.4.612. [DOI] [PubMed] [Google Scholar]