CHAPTER 3 Core conductor theory and cable properties of neurons Mathematical Research Branch, National Institute of Arthritis, Metabolism, I I A and Digestive Diseuses, National Institutes of Health, Bethesda, Maryland CHAPTER CONTENTS Ohm's law for core current Conservation of current Introduction Relation of membrane current to V, Core conductor concept Effect of assuming extracellular isopotentiality Perspective Passive membrane model Comment Resulting cable equation for simple case Reviews and monographs Physical meaning of cable equation terms Brief Historical Notes Physical meaning of T Early electrophysiology Physical meaning of A Electrotonus Electrotonic distance, length, and decrement Passive membrane electrotonus Effect of placing axon in oil Passive versus active membrane Effect of applied current Cable theory Comment on sign conventions Core conductor concept Effect of synaptic membrane conductance Core conductor theory Effect of active membrane properties Estimation of membrane capacitance Input Resistance and Steady Decrement with Distance Resting membrane resistivity Note on correspondence with experiment Passive cable parameters of invertebrate axons Cable of semi-infinite length Importance of single axon preparations Comments about R,, G,, core current, and input current Estimation of parameters for myelinated axons Doubly infinite length Space and voltage clamp Case of voltage clamps at XI and Xp Dendritic Aspects of' Neurons Relations between axon parameters Axon-dendrite contrast Finite length: effect of boundary condition at X = XI Microelectrodes in motoneurons Sealed end at X = XI: case of B, = 0 Theoretical neuron models and parameters Voltage clamp (V, = 0) at X = X,: case of B, = cc Class of trees equivalent to cylinders Semi-infinite extension at X = X,:case ofB, = 1 Motoneuron membrane resistivity and dendritic domiilance Input conductance for finite length general case Dendritic electrotonic length Branches at X = XI Membrane potential transients and time constants Comment on branching equivalent to a cylinder Spatiotemporal effects with dendritic synapses Comment on membrane injury at X = XI Excitatory postsynaptic potential shape index loci Comment on steady synaptic input at X = XI Comments on extracellular potentials Case of input to one branch of a dendritic neuron model Additional comments and references Passive Membrane Potential Transients and Time Constants Cable Equations Defined Passive decay transients Usual cable equation Time constant ratios and electrotonic length Steady-state cable equations Effect of large L and infinite L Augmented cable equations Transient response to applied current step, for finite length Comment: cable versus wave equation Applied current step with L large or infinite Modified cable equation for tapering core Voltage clamp at X = 0, with infinite L General solution of steady-state cable equation Voltage clamp with finite length Basic transient solutions of cable equation Transient response to current injected at one branch of model Solutions using separation of variables Relations Between Neuron Model Parameters Fundamental solution for instantaneous point charge Input resistance and membrane resistivity List of Symbols Dendritic tree input resistance and membrane resistivity Assumptions and Derivation of Cable Theory Results for trees equivalent to cylinders One dimensional in space Result for neuron equivalent to cylinder Intracellular core resistance Estimation of motoneuron parameters 39 40 HANDBOOK OF PHYSIOLOGY 8 THE NERVOUS SYSTEM I INTRODUCTION The use of intracellular micropipettes began (ca. 1950) to provide a wealth of new electrophysiological Core Conductor Concept data from neuromuscular junctions and from moto- A simple core conductor can be described as a long neuron somas. Correct interpretation of these data depended on a careful consideration of the cable prop- thin tube of membrane that is filled with a core of erties to be expected in these experimental situa- electrically conducting medium (e.g., axoplasm) and tions. With the nerve-muscle preparations, it was is bathed on the outside by another electrically con- found that the cable properties of the muscle fiber ducting medium (e.g., extracellular fluid). This membrane tube is typically a cylinder whose length corresponded (at least in first approximation) to a core conductor of effectively infinite length; that is, is very much greater than its diameter. For nerve axons or dendrites, the resistance to electric current the length was many times the length constant (A), as was also true for axons. With motoneurons, how- flow across the membrane is much greater than the ever, the situation was complicated by the unknown core resistance for short length (i.e., small, compared contribution of the dendrites and by the fact that the with the length constant A) increments along the intracellular recording site was usually restricted to cylinder. Because of these relative resistances, it fol- the soma. lows that electric current inside the core conductor It was necessary to apply cable theory to extensively branched dendritic trees and to consider tends to flow parallel to the cylinder axis for consider- able distance before significant fraction can leak such problems as the following. How significant is ii the contribution of dendritic cable properties to obser- out across the membrane. It is this simple physical vations recorded at the soma? How should the den- concept that provides the basis for a cable theory drites be included in our efforts to estimate motoneu- treatment of steady-state distributions of current and ron membrane properties? How important are den- potential in neuronal core conductors; for transient dritic synapses to the integrative performance of the cable properties, the membrane capacitance must neuron? Also, how well can a dendritic tree be repre- also be taken into consideration. An explicit mathe- sented as an equivalent cylinder? How long are the matical derivation of cable theory from these physical concepts is provided later in this chapter (see the individual dendritic branches relative to their length constant (A) values, and what is the effective length section ASSUMPTIONS AND DERIVATION OF CABLE THE- of a dendritic tree or its most nearly equivalent cylin- ORY). der? The theoretical and experimental efforts of the past Perspective 15 years have provided some of the answers for moto- Both the concepts and the mathematical theory of neurons of cat spinal cord and some general results core conductors have played an important role in and conceptual models that should be useful with neuroscience for over 100 years. They have provided a other neuron types as well. basis for the interpretation of electrophysiological observations in terms of the underlying anatomic Commen t structures. The early mathematical theory was a Considerations of space, time, and the differing remarkable achievement that arose (ca. 1870) from a needs of various readers all share responsibility for need to interpret early experiments made on whole the fact that different portions of this chapter are nerve trunks (see references in section BRIEF HISTORI- written at different levels. The historical notes skip CAL NOTES).Not until the introduction of single axon lightly over the efforts of many people. The mathe- preparations and electronic instrumentation (ca. matical derivation of the cable equation is rather 1930) did detailed quantitative testing of theoretical detailed (probably too pedantic for some readers); it predictions become possible. represents an attempt to meet a need that has been Both theory and experiment underwent comple- expressed to me by numerous colleagues. In contrast, mentary development during the period before and the mathematical solutions €or various particular after World War I1 (1930-1950). Cable theory predic- boundary conditions and initial conditions are pre- tions were elaborated mathematically, computed nu- sented with less explanatory comment, but further merically, and displayed graphically and thus pro- details are available in the recent literature. Also vided the basis for improved experimental designs. many interesting topics and examples have been This led to remarkable success in the characteriza- mentioned only very briefly or not at all. tion of axonal membrane properties and cable proper- ties. It is relevant here to note that the most sophisti- Reviews and Monographs cated studies of active (i.e., nonlinear) membrane properties were made under experimental conditions Taylor (182) has reviewed many aspects of cable (space clamp and voltage clamp) designed to elimi- theory and also has provided references to the older nate cable properties. Although this was highly suc- reviews of the 1920's and 1930's. The bibliographies cessful with excised giant axons, such space clamping and comments on historical aspects of cable theory by was not applicable to cells with dendritic trees. Brazier (121, Harmon & Lewis (64a), Hodgkin (71- CHAPTER 3: CABLE THEORY FOR NEURONS 41 76), Katz (97, 99), Lewis (107), Lorente de NO (113), tions and his theory of the electrotonic state of nerve Scott (169a) Stampfli (1761, and Tasaki (179, 181) tissue (des elektrotonischen Zustandes des Nerven). are useful. A valuable monograph by Cole (24) pro- To him this meant the state of changed electromotive vides unique insights, knowledge, and review of forces (emf‘s) in the tissue during steady applied membrane biophysics and cable theory. current. His theory involved polarizable