Extracellular currents and potentials of the active myelinated nerve fiber.

This paper is concerned with the accurate and rapid calculation of extracellular potentials and currents from an active myelinated nerve fiber in a volume conductor, under conditions of normal and abnormal conduction. The neuroelectric source for the problem is characterized mathematically by using a modified version of the distributed parameter model of L. Goldman and J. S. Albus (1968, Biophys. J., 8:596-607) for the myelinated nerve fiber. Solution of the partial differential equation associated with the model provides a waveform for the spatial distribution of the transmembrane potential V(z). This model-generated waveform is then used as input to a second model that is based on the principles of electromagnetic field theory, and allows one to calculate easily the spatial distribution for the potential everywhere in the surrounding volume conductor for the nerve fiber. In addition, the field theoretic model may be used to calculate the total longitudinal current in the extracellular medium (I0L(z)) and the transmembrane current per unit length (im(z)); both of these quantities are defined in connection with the well-known core conductor model and associated cable equations in electrophysiology. These potential and current quantities may also be calculated as functions of time and as such, are useful in interpreting measured I0L(t) and im(t) data waveforms. An analysis of the accuracy of conventionally used measurement techniques to determine I0L(t) and im(t) is performed, particularly with regard to the effect of electrode separation distance and size of the volume conductor on these measurements. Also, a simulation of paranodal demyelination at a single node of Ranvier is made and its effects on potential and current waveforms as well as on the conduction process are determined. In particular, our field theoretic model is used to predict the temporal waveshape of the field potentials from the active, non-uniformly conducting nerve fiber in a finite volume conductor.     

On Bioelectric Potentials in an Inhomogeneous Volume Conductor

Green's theorem is used to derive two sets of expressions for the quasi-static potential distribution in an inhomogeneous volume conductor. The current density in passive regions is assumed to be linearly related instantaneously to the electric field. Two equations are derived relating potentials to an arbitrary distribution of impressed currents. In one, surfaces of discontinuity in electrical conductivity are replaced by double layers and in the other, by surface charges. A multipole equivalent generator is defined and related both to the potential distribution on the outer surface of the volume conductor and to the current sources. An alternative result involves the electric field at the outer surface rather than the potential. Finally, the impressed currents are related to electrical activity at the membranes of active cells. The normal component of membrane current density is assumed to be equal at both membrane surfaces. One expression is obtained involving the potentials at the inner and outer surfaces of the membrane. A second expression involves the transmembrane potential and the normal component of membrane current.     

Lipid-mediated interactions between intrinsic membrane proteins: a theoretical study based on integral equations

This study of lipid-mediated interactions between proteins is based on a theory recently developed by the authors for describing the structure of the hydrocarbon chains in the neighborhood of a protein inclusion embedded in a lipid membrane [Lagüe et al., Farad. Discuss. 111:165-172, 1998]. The theory involves the hypernetted chain integral equation formalism for liquids. The exact lateral density-density response function of the hydrocarbon core, extracted from molecular dynamics simulations of a pure dipalmitoylphosphatidylcholine bilayer based on an atomic model, is used as input. For the sake of simplicity, protein inclusions are modeled as hard repulsive cylinders. Numerical calculations were performed with three cylinder sizes: a small cylinder of 2.5-A radius, corresponding roughly to an aliphatic chain; a medium cylinder of 5-A radius, corresponding to a alpha-helical polyalanine protein; and a large cylinder of 9-A radius, representing a small protein, such as the gramicidin channel. The calculations show that the average hydrocarbon density is perturbed over a distance of 20-25 A from the edge of the cylinder for every cylinder size. The lipid-mediated protein-protein effective interaction is calculated and is shown to be nonmonotonic. In the case of the small and the medium cylinders, the lipid-mediated effective interaction of two identical cylinders is repulsive at an intermediate range but attractive at short range. At contact, there is a free energy of -2k(B)T for the 2.5-A-radius cylinder and -9k(B)T for the 5-A-radius cylinder, indicating that the association of two alpha-helices of both sizes is favored by the lipid matrix. In contrast, the effective interaction is repulsive at all distances in the case of the large cylinder. Results were obtained with two integral equations theories: hypernetted chain and Percus-Yevick. For the two theories, all results are qualitatively identical.     

Study of ionic currents across a model membrane channel using Brownian dynamics.

Brownian dynamics simulations have been carried out to study ionic currents flowing across a model membrane channel under various conditions. The model channel we use has a cylindrical transmembrane segment that is joined to a catenary vestibule at each side. Two cylindrical reservoirs connected to the channel contain a fixed number of sodium and chloride ions. Under a driving force of 100 mV, the channel is virtually impermeable to sodium ions, owing to the repulsive dielectric force presented to ions by the vestibular wall. When two rings of dipoles, with their negative poles facing the pore lumen, are placed just above and below the constricted channel segment, sodium ions cross the channel. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength does not increase the channel conductance further. When only those ions that acquire a kinetic energy large enough to surmount a barrier are allowed to enter the narrow transmembrane segment, the channel conductance decreases monotonically with the barrier height. This barrier represents those interactions between an ion, water molecules, and the protein wall in the transmembrane segment that are not treated explicitly in the simulation. The conductance obtained from simulations closely matches that obtained from ACh channels when a step potential barrier of 2-3 kTr is placed at the channel neck. The current-voltage relationship obtained with symmetrical solutions is ohmic in the absence of a barrier. The current-voltage curve becomes nonlinear when the 3 kTr barrier is in place. With asymmetrical solutions, the relationship approximates the Goldman equation, with the reversal potential close to that predicted by the Nernst equation. The conductance first increases linearly with concentration and then begins to rise at a slower rate with higher ionic concentration. We discuss the implications of these findings for the transport of ions across the membrane and the structure of ion channels.     

Active muscle fiber as an equivalent cardiac current generator

Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.     

Two-dimensional Fourier representation used in the bioelectric forward problem.

The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.     

A Few Comments on Electrostatic Interactions in Cell Physiology

The role of fixed charges present at the surface of biological membranes is usually described by the Gouy-Chapman-Grahame theory of the electric double-layer where the Grahame equation is applied independently on each side of the membrane and where the capacitive charges (linked to the transmembrane ionic currents) are disregarded. In this article, we generalize the Gouy-Chapman-Grahame theory by taking into account both intrinsic charges (resulting from the dissociation of membrane constituents) and capacitive charges, in the density value of the membrane surface charges. In the first part, we show that capacitive charges couple electrostatic potentials present on both sides of the membrane. The intensity of this coupling depends both on the value of the membrane specific capacitance and the transmembrane electric potential difference. In the second part, we suggest some physiological implications of membrane electric double-layers.     

Modeling the electrical behavior of anatomically complex neurons using a network analysis program: Passive membrane

We describe the application of a popular and widely available electrical circuit simulation program called SPICE to modeling the electrical behavior of neurons with passive membrane properties and arbitrarily complex dendritic trees. Transient responses may be calculated at any location in the cell model following current, voltage or conductance perturbations at any point. A numbering method is described for binary trees which is helpful in transforming complex dendritic structures into a coded list of short cylindrical dendritic segments suitable for input to SPICE. Individual segments are modeled as isopotential compartments comprised of a parallel resistor and capacitor, representing the transmembrane impedance, in series with one or two core resistors. Synaptic current is modeled by a current source controlled by the local membrane potential and an alpha-shaped voltage, thus simulating a conductance change in series with a driving potential. Extensively branched test cell circuits were constructed which satisfied the equivalent cylinder constraints (Rall 1959). These model neurons were perturbed by independent current sources and by synaptic currents. Responses calculated by SPICE are compared with analytical results. With appropriately chosen model parameters, extremely accurate transient calculations may be obtained. Details of the SPICE circuit elements are presented, along with illustrative examples sufficient to allow implementation of passive nerve cell models on a number of common computers. Methods for modeling excitable membrane are presented in the companion paper (Bunow et al. 1985).     

Magnetic field of a single muscle fiber. First measurements and a core conductor model.

We present the first measurements of the magnetic field from a single muscle fiber of the frog gastrocnemius, obtained by using a toroidal pickup coil coupled to a room-temperature, low-noise amplifier. The axial currents associated with the magnetic fields of single fibers were biphasic and had peak-to-peak amplitudes ranging between 50 and 100 nA, depending primarily on the fiber radius. With an intracellular microelectrode, we measured the action potential of the same fiber, which allowed us to determine that the intracellular conductivity of the muscle fiber in the core conductor approximation was 0.20 +/- 0.09 S/m. Similarly, we found that the effective membrane capacitance was 0.030 +/- 0.011 F/m2. These results were not significantly affected by the anisotropic conductivity of the muscle bundle. We demonstrate how our magnetic technique can be used to determine the transmembrane action potential without penetrating the membrane with a microelectrode, thereby offering a reliable, stable, and atraumatic method for studying contracting muscle fibers.     

Transmembrane electrical potential of excitable membranes: a pore analysis influence of surface charges and surface dipoles

A treatment is proposed in order to establish the general expression of the zero-current transmembrane potential of excitable membranes. The membrane model considered here is that of a hydrocarbon layer which is impermeable to ions and which represents the lipid bilayer matrix. In this matrix are incorporated ionic channels. The ion transport process through the channels is described by the absolute-rate theory applied to pores which are seen as chains of potential energy maxima and minima. Only one of the energy barriers corresponds to the gate step, and it is strongly dependent on the transmembrane potential. The kinetic equation is related to the zero current, to electrostatic boundary conditions and to the Gouy-Chapman equation for the aqueous diffuse layer.     

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.     

Influence of the membrane potential on the free energy of an intrinsic protein

A modified Poisson-Boltzmann equation is developed from statistical mechanical considerations to describe the influence of the transmembrane potential on macromolecular systems. Using a Green's function formalism, the electrostatic free energy of a protein associated with the membrane is expressed as the sum of three terms: a contribution from the energy required to charge the system's capacitance, a contribution corresponding to the interaction of the protein charges with the membrane potential, and a contribution corresponding to a voltage-independent reaction field free energy. The membrane potential, which is due to the polarization interface, is calculated in the absence of the protein charges, whereas the reaction field is calculated in the absence of transmembrane potential. Variations in the capacitive energy associated with typical molecular processes are negligible under physiological conditions. The formulation of the theory is closely related to standard algorithms used to solve the Poisson-Boltzmann equation and only small modifications to current source codes are required for its implementation. The theory is illustrated by examining the voltage-dependent membrane insertion of a simple polyalanine alpha-helix and by computing the electrostatic potential across a 60-A-diameter sphere meant to represent a large intrinsic protein.     

Bacteriorhodopsin in liposomes. II. Experimental evidence in support of a theoretical model

In the preceding article equations describing relevant ion flows in illuminated suspensions of bacteriorhodopsin liposomes have been derived. Here these equations are subjected to experimental tests. Changes in permeability characteristics of the liposomal membrane are brought about by addition of specific ionophores and change of medium composition. Using light-driven proton uptake and electrochemical potential differences for protons across the membrane as observation parameters, ridig attempts to falsify the derived equations are unsuccessful.Agreement between equations and experimental results is established on the point of: (i) the antagonistic effect of valinomycin and nigericin on the two components of the proton-motive force, (ii) the time dependence of the changes in transmembrane electrical and chemical potential differences after the onset of illumination.In three independent experimental systems evidence was obtained for the correctness of the postulated dependence of the turnover rate of the photochemical cycle on back pressure by the transmembrane electrochemical potential difference for protons.     

Structural domains of the tight junctional intramembrane fibrils.

Freeze-fracture reveals intramembrane fibrils lying along the intermembrane contacts that characterize tight junctions. Tight junctions from a variety of species are reexamined here by rapid freezing prior to freeze-fracture. The tight junction fibril is uprooted alternatively from either the cytoplasmic or the exoplasmic hemibilayer during freeze-cleavage, exposing two distinct but complementary views of its hybrid structure within the same replica. When the transmembrane fibril is uprooted from the exoplasmic hemibilayer it appears on the P-fracture face as a smooth-surfaced cylinder which is sometimes resolved into periodic globular structures. The lack of indication that the P-face cylinder has been pulled out through the opposite membrane half indicates that this domain of the fibril is, in large part, buried in the hydrophobic interior of the membrane. However, when the transmembrane fibril is uprooted from the cytosolic hemibilayer it appears on the E-fracture face as a row of irregular intramembrane particles. The irregular particles on the E-face aspect of the fibril are interpreted as corresponding to transmembrane protein segments that may very well make projections onto the cytosolic surface of the bilayer. En face views of the outermost junction strand between adjacent epithelial cells show periodic lines on the bilayer on each side of the junction which are interpreted as periodic transmembrane protein segments arising from the core structure of the tight junction fibril. If the backbone of the tight junction strand is an inverted cylindrical micelle, it must typically include proteins, which might anchor it to structures outside the membrane bilayer.     

Bacteriorhodopsin in liposomes. II. Experimental evidence in support of a theoretical model.

In the preceding article equations describing relevant ion flows in illuminated suspensions of bacteriorhodopsin liposomes have been derived. Here these equations are subjected to experimental tests. Changes in permeability characteristics of the liposomal membrane are brought about by addition of specific ionophores and change of medium composition. Using light-driven proton uptake and electrochemical potential differences for protons across the membrane as observation parameters, ridig attempts to falsify the derived equations are unsuccessful. Agreement between equations and experimental results is established on the point of: (i) the antagonistic effect of valinomycin and nigericin on the two components of the proton-motive force, (ii) the time dependence of the changes in transmembrane electrical and chemical potential differences after the onset of illumination. In three independent experimental systems evidence was obtained for the correctness of the postulated dependence of the turnover rate of the photochemical cycle on back pressure by the transmembrane electrochemical potential difference for protons.     

The causal theory of the resting potential of cells

In this pedagogical article the causal theory of the resting potential of cells is presented, which for given extracellular ion concentrations predicts the intracellular ones simultaneously with the resting potential. In addition to the Na, K-pump, fixed charges on the membrane surfaces are taken into account. The equation determining the resting potential in the causal theory suggests a new explanation of the genesis of the resting potential. The usual criterion for an ion pump to be electrogenic is not relevant for the whole of the resting potential, and may therefore be misleading. The physical meaning of the Goldman-Hodgkin-Katz formula for the membrane potential as a diffusion potential is also explained and tested with numbers for the giant axon of the squid. A significant discrepancy between theory and experiment is found which calls for an experimental re-examination of the constitutive equations for passive potassium and sodium currents.     

Electrical behavior of a skeletal muscle fiber in a volume conductor of finite extent

A model study of the spatial distribution of the extracellular potentials and current densities arising from an active single skeletal muscle fiber in a cylindrical volume conductor of finite radial extent is presented. The paper examines the influence of the radius of the volume conductor,b, on the extracellular potentials and currents at different field points. The equivalent sources with respect to the extracellular potential are investigated as well. The axial source density associated with the primary and secondary sources is calculated using the expressions for the intracellular and extracellular potentials. The density of the secondary sources is a decreasing function of the radius of the conducting medium and approaches zero whenb becomes infinitely big.     

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.