Analytical and numerical construction of equivalent cables

The mathematical complexity experienced when applying cable theory to arbitrarily branched dendrites has lead to the development of a simple representation of any branched dendrite called the equivalent cable. The equivalent cable is an unbranched model of a dendrite and a one-to-one mapping of potentials and currents on the branched model to those on the unbranched model, and vice versa. The piecewise uniform cable, with a symmetrised tri-diagonal system matrix, is shown to represent the canonical form for an equivalent cable. Through a novel application of the Laplace transform it is demonstrated that an arbitrary branched model of a dendrite can be transformed to the canonical form of an equivalent cable. The characteristic properties of the equivalent cable are extracted from the matrix for the transformed branched model. The one-to-one mapping follows automatically from the construction of the equivalent cable. The equivalent cable is used to provide a new procedure for characterising the location of synaptic contacts on spinal interneurons.     

Analytical theory for extracellular electrical stimulation of nerve with focal electrodes. II. Passive myelinated axon.

The cable model of a passive, myelinated fiber is derived using the theory of electromagnetic propagation in periodic structures. The cable may be excited by an intracellular source or by an arbitrary, time-varying, applied extracellular field. When the cable is stimulated by a distant source, its properties are qualitatively similar to an unmyelinated fiber. Under these conditions relative threshold is proportional to the cube of the source distance and inversely proportional to the square of the fiber diameter. Electrical parameters of the model are chosen where possible, from mammalian peripheral nerve and anatomic parameters from cat auditory nerve. Several anatomic representations of the paranodal region are analyzed for their effects on the length and time constants of the fibers. Sensitivity of the model to parameter changes is studied. The linear model reliably predicts the effects of fiber size and electrode-fiber separation on threshold of cat dorsal column fibers to extracellular electrical stimulation.     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

General solution of cable theory with both ends sealed

General solution of the cable theory with both ends sealed when injecting an arbitrary current at an arbitrary point of the cable is presented, which is a time-dependent transient solution. The solution is an infinite series, each term of which is the product of a cosine term including a position variable only and an exponential term including a time variable only. The general solution contains almost all solutions reported hitherto as particular cases and the mutual relations among the various solutions of quite different forms are clarified by this general solution. Moreover the shorter the cable becomes, the more rapidly this solution converges, therefore it is useful for an analysis of the short cable in the case where the relative deviation error may grow large. The truncation error can also be estimated as the solution is an infinite series of simple functions.     

The effects of discrete gap junction coupling on propagation in myocardium

A modified cable theory for a bi-domain model of myocardium that incorporates the effect of gap junctions as discrete objects coupling cardiac cells is derived. The theory is shown to be in agreement with a number of experiments that cannot be explained using standard continuous cable theory, and resolves some apparent contradictions on failure of propagation in two-dimensional anisotropic tissue. In addition, some as yet untested predictions of the theory are mentioned.     

Electrical Interactions via the Extracellular Potential Near Cell Bodies

Ephaptic interactions between a neuron and axons or dendrites passing by its cell body can be, in principle, more significant than ephaptic interactions among axons in a fiber tract. Extracellular action potentials outside axons are small in amplitude and spatially spread out, while they are larger in amplitude and much more spatially confined near cell bodies. We estimated the extracellular potentials associated with an action potential in a cortical pyramidal cell using standard one-dimensional cable theory and volume conductor theory. Their spatial and temporal pattern reveal much about the location and timing of currents in the cell, especially in combination with a known morphology, and simple experiments could resolve questions about spike initiation. From the extracellular potential we compute the ephaptically induced polarization in a nearby passive cable. The magnitude of this induced voltage can be several mV, does not spread electrotonically, and depends only weakly on the passive properties of the cable. We discuss their possible functional relevance.     

Voltage clamp calculations for myelinated and demyelinated axons

Exact cable theory is used to calculate voltage distributions along fully myelinated axons and those with various patterns of demyelination. The model employed uses an R-C circuit for the soma, an equivalent cable for the dendrites, a myelinated axon with n internodes and a cable representing telodendria. For the case of a voltage clamp at the soma, a system of 2n + 1 equations must be solved to obtain the potential distribution and this is done for arbitrary n. An explicit calculation is performed for one internode whereas computer-generated solutions are obtained for several internodes. The relative importance of the position of a single demyelinated internode is determined. An approximate expression is given for the critical internodal length necessary for action potential generation.     

Complementary variational principles for the steady-state finite cable model of nerve membranes

Maximum and minimum principles for the steady-state finite cable model of nerve membranes are derived from the canonical theory of complementary variational principles. An accurate variational solution is obtained in an illustrative calculation.     

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree extending Rall's equivalent cylinder model is presented. It is obtained from a combination of cable theory with the continuum approach. Replacing the fine structure of the branching dendrites by an equivalent, conductive medium characterized by averaged electrical parameters, the one-dimensional cable equations with spatially varying parameters are derived. While these equations can be solved in general only numerically, we were able to formulate a general branching condition (comprising Rall's 3/2 power relationship as a special case) under which analytical solutions can be deduced from those of the equivalent cylinder model. This model allows dendritic trees with a greater variety of branching patterns than before to be analytically treated.     

An efficient algorithm for cable theory,applied to blowfly photoreceptor cells and LMC's

A both simple and efficient algorithm is presented that yields the voltages and currents in an arbitrary cable structure. The algorithm consists of the following steps: 1. The cable structure is divided into homogeneous cable segments; 2. Each cable segment is considered as a two-port, and replaced by an equivalent circuit consisting of discrete elements; 3. The resulting equivalent scheme of the whole cable structure is solved with an algorithm for ladder networks (or, if the structure is not tree-like, with a network analysis program), which yields the input and output voltages and currents of each cable segment; and if required 4. The voltage and current distribution in each segment is determined from the input and output voltages and currents. The algorithm is applied to blowfly photoreceptor cells that are electrically coupled, and to blowfly Large Monopolar Cells. For LMC's it is shown that the loads at the input and output sides of the axon determine whether unidirectional or bidirectional signal transmission occurs.     

Pointwise bounds for the solution of a nonlinear problem in cell membrane theory

Pointwise upper and lower bounds for the solution of a class of nonlinear problems arising in the steady-state finite cable model of cell membranes are presented. Simple analytical bounding curves are obtained for an illustrative example in the theory of nerve membranes.     

Poisson process stimulation of an excitable membrane cable model.

The convergence of multiple inputs within a single-neuronal substrate is a common design feature of both peripheral and central nervous systems. Typically, the result of such convergence impinges upon an intracellularly contiguous axon, where it is encoded into a train of action potentials. The simplest representation of the result of convergence of multiple inputs is a Poisson process; a general representation of axonal excitability is the Hodgkin-Huxley/cable theory formalism. The present work addressed multiple input convergence upon an axon by applying Poisson process stimulation to the Hodgkin-Huxley axonal cable. The results showed that both absolute and relative refractory periods yielded in the axonal output a random but non-Poisson process. While smaller amplitude stimuli elicited a type of short-interval conditioning, larger amplitude stimuli elicited impulse trains approaching Poisson criteria except for the effects of refractoriness. These results were obtained for stimulus trains consisting of pulses of constant amplitude and constant or variable durations. By contrast, with or without stimulus pulse shape variability, the post-impulse conditional probability for impulse initiation in the steady-state was a Poisson-like process. For stimulus variability consisting of randomly smaller amplitudes or randomly longer durations, mean impulse frequency was attenuated or potentiated, respectively. Limitations and implications of these computations are discussed.     

Membrane potential perturbations induced in tissue cells by pulsed electric fields

Pulsed electric fields directly influence the electrophysiology of tissue cells by transiently perturbing their transmembrane potential. To determine the magnitude and time course of this interaction, electrotonic cable theory was used to calculate the membrane potential perturbations induced in tissue cells by a spatially uniform, pulsed electric field. Analytic solutions were obtained that predict shifts in membrane potential along the length of cells as a function of time in response to an electrical pulse. For elongated tissue cells, or groups of tissue cells that are coupled electrotonically by gap junctions, significant hyperpolarizations and depolarizations can result from millisecond applications of electric fields with strengths on the order of 10–100 mV/cm. The results illustrate the importance of considering cellular cable parameters in assessing the effects of transient electric fields on biological systems, as well as in predicting the efficacy of pulsed electric fields in medical treatments. © 1995 Wiley-Liss, Inc.     

Further study of the two-dimensional cable theory: An electric model for a flat thin association of cells with a directional intercellular communication

Summary The two-dimensional cable theory originally presented in relation to the electrotonus along a flat tissue of the rat atrial appendage is improved by taking account of double space constants instead of a single one and an explicit boundary condition at the tip of the current injecting microelectrode. A differential equation is formulated for the membrane potential change which is produced along the tissue by the intracellular injection of a current. The solution is formally expressed in terms of the Green's function. Specific solutions corresponding to the injection of a unit current step or a linearly rising current are discussed in detail.     

Equations for the membrane potential

The problem of an electric signal propagating over a thin layer modeling the cell membrane is considered in the electrostatic approximation. At both sides of the membrane there are bulk conductors with different properties. Various types of boundary conditions are examined. Solutions are obtained for a cylindrical fiber and planar layer and compared with the relationships used in the classical Hodgkin-Huxley theory. It is shown that agreement with the cable theory is obtained only if the system is fully symmetrical and the near-membrane layers are thin.     

Outgrowth of pseudopodial cables induced by all-trans retinoic acid in micromere-derived cells isolated from sea urchin embryos

Cultured cells derived from micromeres of sea urchin embryos underwent pseudopodial cable growth without spicule rod formation in the presence of all-trans retinoic acid (tRA) or insulin. Pseudopodial cable growth caused by tRA or insulin was inhibited by genistein, a protein tyrosine kinase inhibitor. Phosphorylation of protein tyrosine residue was augmented in the cells treated with tRA or insulin and was inhibited by genistein. Probably, protein tyrosine kinase takes an indispensable part in signal transduction systems for tRA and insulin in these cells. In tRA-treated cells, augmentation of the phosphorylation of protein tyrosine residue was accompanied by an increase in the activity of protein tyrosine kinase and was inhibited by actinomycin D, inhibiting cable growth. Activation of this enzyme in tRA-treated cells probably depends on RNA synthesis. In insulin-treated cells, augmentation of tyrosine residue phosphorylation occurred without any appreciable change in this enzyme's activity and was hardly affected by actinomycin D. Phosphorylation of protein tyrosine residue seems to be activated by the binding of insulin to an insulin receptor. Pseudopodial cable growth in these cells treated with tRA or insulin was inhibited by wortmannin. Phosphatidylinositol 3 kinase probably participates in tRA and insulin signal transduction systems.     

Measurement of GABA-evoked conductance changes of lobster muscle fibres by a three-microelectrode voltage clamp technique

The effective membrane conductance and capacity of lobster muscle fibres was measured by a three-intracellular-microelectrode voltage clamp technique. Conductance values agreed well with those determined under current clamp, by means of the 'short' cable equations. Reversible increases in conductance evoked by gamma-aminobutyric acid (GABA) were reflected by differences (delta V) in electrotonic potential amplitude recorded at the centre, and midway between the centre and fibre end respectively. GABA dose--conductance curves derived from cable theory or from delta V measurements were virtually identical. The effective capacity (ceff), determined from the area beneath the 'on' delta V capacity transient, yielded values of the membrane time constant consistently lower than those obtained by the graphical method of E. Stefani & A.B. Steinbach (J. Physiol., London. 203, 383-401 (1969)); one possible explanation for this discrepancy is discussed. In the presence of GABA, the effective capacity was reduced in a dose-related manner. The results were interpreted in terms of an equivalent circuit in which surface membrane was arranged in parallel with cleft-tubular membrane of finite conductance, charged through an access resistance. GABA was though to be decreasing ceff by selectively increasing the conductance of the cleft-tubular membranes.