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Core Conductor Theory and Cable Properties of Neurons

Handbook of Physiology, The Nervous System, Cellular Biology of Neurons

  1. Wilfrid Rall

Published Online: 1 JAN 2011

DOI: 10.1002/cphy.cp010103

Comprehensive Physiology

Comprehensive Physiology

How to Cite

Rall, W. 2011. Core Conductor Theory and Cable Properties of Neurons. Comprehensive Physiology. 39–97.

Author Information

  1. Mathematical Research Branch, National Institute of Arthritis, Metabolism, and Digestive Diseases, National Institutes of Health, Bethesda, Maryland

Publication History

  1. Published Online: 1 JAN 2011

Abstract

The sections in this article are:

  • 1
    Introduction
    • 1.1
      Core Conductor Concept
    • 1.2
      Perspective
    • 1.3
      Comment
    • 1.4
      Reviews and Monographs
  • 2
    Brief Historical Notes
    • 2.1
      Early Electrophysiology
    • 2.2
      Electrotonus
    • 2.3
      Passive Membrane Electrotonus
    • 2.4
      Passive Versus Active Membrane
    • 2.5
      Cable Theory
    • 2.6
      Core Conductor Concept
    • 2.7
      Core Conductor Theory
    • 2.8
      Estimation of Membrane Capacitance
    • 2.9
      Resting Membrane Resistivitiy
    • 2.10
      Passive Cable Parameters of Invertebrate Axons
    • 2.11
      Importance of Single Axon Preparations
    • 2.12
      Estimation of Parameters for Myelinated Axons
    • 2.13
      Space and Voltage Clamp
  • 3
    Dendritic Aspects of Neurons
    • 3.1
      Axon-Dendrite Contrast
    • 3.2
      Microelectrodes in Motoneurons
    • 3.3
      Theoretical Neuron Models and Parameters
    • 3.4
      Class of Trees Equivalent to Cylinders
    • 3.5
      Motoneuron Membrane Resistivity and Dendritic Dominance
    • 3.6
      Dendritic Electrotonic Length
    • 3.7
      Membrane Potential Transients and Time Constants
    • 3.8
      Spatiotemporal Effects with Dendritic Synapses
    • 3.9
      Excitatory Postsynaptic Potential Shape Index Loci
    • 3.10
      Comments on Extracellular Potentials
    • 3.11
      Additional Comments and References
  • 4
    Cable Equations Defined
    • 4.1
      Usual Cable Equation
    • 4.2
      Steady-state Cable Equations
    • 4.3
      Augmented Cable Equations
    • 4.4
      Comment: Cable Versus Wave Equation
    • 4.5
      Modified Cable Equation for Tapering Core
    • 4.6
      General Solution of Steady-state Cable Equation
    • 4.7
      Basic Transient Solutions of Cable Equation
    • 4.8
      Solutions Using Separation of Variables
    • 4.9
      Fundamental Solution for Instantaneous Point Charge
  • 5
    List of Symbols
  • 6
    Assumptions and Derivation of Cable Theory
    • 6.1
      One Dimensional in Space
    • 6.2
      Intracellular Core Resistance
    • 6.3
      Ohm's Law for Core Current
    • 6.4
      Conservation of Current
    • 6.5
      Relation of Membrane Current to Vi
    • 6.6
      Effect of Assuming Extracellular Isopotentiality
    • 6.7
      Passive Membrane Model
    • 6.8
      Resulting Cable Equation for Simple Case
    • 6.9
      Physical Meaning of Cable Equation Terms
    • 6.10
      Physical Meaning of τ
    • 6.11
      Physical Meaning of λ
    • 6.12
      Electrotonic Distance, Length, and Decrement
    • 6.13
      Effect of Placing Axon in Oil
    • 6.14
      Effect of Applied Current
    • 6.15
      Comment on Sign Conventions
    • 6.16
      Effect of Synaptic Membrane Conductance
    • 6.17
      Effect of Active Membrane Properties
  • 7
    Input Resistance and Steady Decrement with Distance
    • 7.1
      Note on Correspondence with Experiment
    • 7.2
      Cable of Semi-infinite Length
    • 7.3
      Comments about R, G, Core Current, and Input Current
    • 7.4
      Doubly Infinite Length
    • 7.5
      Case of Voltage Clamps at X1 and X2
    • 7.6
      Relations Between Axon Parameters
    • 7.7
      Finite Length: Effect of Boundary Condition at X= X1
    • 7.8
      Sealed End at X= X1: Case of B1 = 0
    • 7.9
      Voltage Clamp(V1 = 0) at X = X1: Case of B1 = ∞
    • 7.10
      Semi-infinite Extension at X = X1: Case of B1 = 1
    • 7.11
      Input Conductance for Finite Length General Case
    • 7.12
      Branches at X = X1
    • 7.13
      Comment on Branching Equivalent to a Cylinder
    • 7.14
      Comment on Membrane Injury at X = X1
    • 7.15
      Comment on Steady Synaptic Input at X= X1
    • 7.16
      Case of Input to One Branch of Dendritic Neuron Model
  • 8
    Passive Membrane Potential Transients and Time Constants
    • 8.1
      Passive Decay Transients
    • 8.2
      Time Constant Ratios and Electrotonic Length
    • 8.3
      Effect of Large L and Infinite L
    • 8.4
      Transient Response to Applied Current Step, for Finite Length
    • 8.5
      Applied Current Step with L Large or Infinite
    • 8.6
      Voltage Clamp at X = 0, with Infinite L
    • 8.7
      Voltage Clamp with Finite Length
    • 8.8
      Transient Response to Current Injected at One Branch of Model
  • 9
    Relations Between Neuron Model Parameters
    • 9.1
      Input Resistance and Membrane Resistivity
    • 9.2
      Dendritic Tree Input Resistance and Membrane Resistivity
    • 9.3
      Results for Trees Equivalent to Cylinders
    • 9.4
      Result for Neuron Equivalent to Cylinder
    • 9.5
      Estimation of Motoneuron Parameters