Extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor

The extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor were studied. The spatial distribution of the transmembrane potential was obtained by integrating the system of partial differential equations characterizing the electric processes in the active myelinated nerve fiber. The spatial distribution of the extracellular potentials at various radial distances in the volume conductor were calculated using the line source model. Up to a certain radial distance (500 m) the discontinuity of the action potential propagation is reflected in the extracellular potentials, while further in the volume conductor the potentials are smooth. The effect of the fiber diameter and the internodal distance on the volume conductor potentials as well as the changes in the magnitude of the extracellular potential (in the time domain) between two adjacent nodes at various radial distances were studied. The radial decline of the peak-to-peak amplitude of the extracellular potential depends on the radial coordinater of the field point and increases with the increase ofr.     

The extracellular potential of a myelinated nerve fiber in an unbounded medium and in nerve cuff models

A model is presented for the calculation of single myelinated fiber action potentials in an unbounded homogeneous medium and in nerve cuff electrodes. The model consists of a fiber model, used to calculate the action currents at the nodes of Ranvier, and a cylindrically symmetrical volume conductor model in which the fiber's nodes are represented as point current sources. The extracellular action potentials were shown to remain unchanged if the fiber diameter and the volume conductor geometry are scaled by the same factor (principle of corresponding states), both in an unbounded homogeneous medium and in an inhomogeneous volume conductor. The influence of several cuff electrode parameters, among others, cuff length and cuff diameter, were studied, and the results were compared, where possible, with theoretical and experimental results as reported in the literature.     

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.     

Relation between inter- and intracellular action potentials of frog isolates muscle fiber at various temperatures

Extra- and intracellular action potentials (AP) of isolated muscle fibres plunged into volume conductor were studied at different temperatures. Changes of the first and second AP derivatives and their temperature dependence were described. The changes are explained by temperature effect on the density of input and output ion currents. Changes of the shape of extracellular potential with temperature increase were described. They were concerned with the changes of the first and second AP derivatives and are due to the peculiar distribution of the potential field in the volume conductor around an excitable fibre of the finite length.     

Mathematical modelling of intra- and extracellular potentials generated by active structures: effects of a step change in structure diameter

A mathematical model developed in our laboratory is used to estimate and analyse extracellular potentials generated in a volume conductor by a geometrically inhomogeneous structure with a step increase or a step decrease in its diameter. The transmembrane potentials were calculated using the model of Hodgkin and Huxley (1952) and the method of Joyner et al. (1978). Variations in waveforms of the transmembrane and extracellular potentials were described and discussed. Differences in waveforms of the extracellular potentials and in declines of their components are due to changes in the source which generates these potentials. In case of a propagation block the peak-to-peak amplitude of the extracellular potentials calculated over the area of the block may be higher than that over the area of propagation of action potentials. The possible applications of the results to the analysis of extracellular potentials recorded around actual motoneurons during their orthodromic or antidromic activation are discussed.     

Extracellular potential field of excited isolated frog muscle fibres immersed in a volume conductor

The extracellular potential field of isolated frog muscle fibres immersed in a volume conductor was studied at radial distances up to 3 mm during excitation. The shape of the field distant from both the point of the origin of the excitation and the end of the fibre as well as changes in the field when depolarization wave approached the fibre end were described. Different amplitude decrease rates in individual phases of the extracellular potential and the peak-to-peak amplitude at different temperatures were found. Extracellular potentials at long radial distances were recorded using an averaging technique. The shape of the extracellular potentials at long radial distances over the fibre and beyond its end were very similar to the shape of extraterritorial potentials of a single motor unit.     

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.     

Disentanglement of local field potential sources by independent component analysis

The spontaneous activity of working neurons yields synaptic currents that mix up in the volume conductor. This activity is picked up by intracerebral recording electrodes as local field potentials (LFPs), but their separation into original informative sources is an unresolved problem. Assuming that synaptic currents have stationary placing we implemented independent component model for blind source separation of LFPs in the hippocampal CA1 region. After suppressing contaminating sources from adjacent regions we obtained three main local LFP generators. The specificity of the information contained in isolated generators is much higher than in raw potentials as revealed by stronger phase-spike correlation with local putative interneurons. The spatial distribution of the population synaptic input corresponding to each isolated generator was disclosed by current-source density analysis of spatial weights. The found generators match with axonal terminal fields from subtypes of local interneurons and associational fibers from nearby subfields. The found distributions of synaptic currents were employed in a computational model to reconstruct spontaneous LFPs. The phase-spike correlations of simulated units and LFPs show laminar dependency that reflects the nature and magnitude of the synaptic currents in the targeted pyramidal cells. We propose that each isolated generator captures the synaptic activity driven by a different neuron subpopulation. This offers experimentally justified model of local circuits creating extracellular potential, which involves distinct neuron subtypes.     

Functional capacities of marsupial hearts: size and mitochondrial parameters indicate higher aerobic capabilities than generally seen in placental mammals

This study of marsupial hearts explored the aerobic capacities of this group of mammals; recent information suggests that marsupials possess higher aerobic abilities than previously accepted. Characteristics such as heart mass, mitochondrial features and capillary parameters were examined. A comprehensive study of the heart of red kangaroos was included because of the high maximum oxygen consumption of this species. Goats were also included as a reference placental mammal. Marsupials have a heart that is generally larger than that of placentals. The allometric equation for the relationship between heart mass and body mass for marsupials was M_h=7.5M_b^0.944 (M_h in g and M_b in kg); the equivalent equation for placental mammals was M_h=6.0M_b^0.97. Mitochondrial volume density and inner mitochondrial surface density do not differ between the two mammal groups; although capillary parameters indicated a lower capillary volume in marsupials. Heart size appears to be the major difference between the two groups. The overall pattern seen in marsupials is similar to that of "athletic" placentals and indicates a relatively high aerobic potential.Abbreviations BMR basal metabolic rate - c(K,0) tortuosity factor - Jv(c,f) capillary length density - M_b body mass - M_h heart mass - N_A(c,f) numerical capillary density - r_c mean capillary radius - S(im,m) total surface area of inner mitochondrial membranes in the heart - Sv(im,m) surface density of the inner mitochondrial membranes - Sv(im,mt) surface density of inner mitochondrial membranes per unit volume of mitochondria - TEM transmission electron microscope - O₂max maximum aerobic capacity - V(mt,m) total mitochondrial volume - Vv(f,m) volume fraction of muscle occupied by muscle fibres - Vv(mt,f) mitochondrial volume densityCommunicated by I.D. Hume     

On the Application of Widom's Test Particle Method to Homogeneous and Inhomogeneous Fluids

Abstract

Chemical potentials of a homogeneous and an inhomogeneous Lennard-Jones fluid have been determined by molecular dynamics simulations on the vector computer CYBER 205 by applying essentially the fictitious test particle method of Widom. For the homogeneous fluid we find, contrary to the previous result of Guillot and Guissani, that the simulated chemical potential is independent of the particle number. The crucial point, however, is a sufficiently large cut-off radius in the evaluation of the Boltzmann factor. Comparing with our WCA-type perturbation theory, we get agreement in the chemical potentials within 0.1 kT up to the density n[sgrave]³ = 0.80 and a difference of 0.2 kT at n[sgrave]³ = 0.85. For the inhomogeneous case we consider a fluid in a cylindrical pore and integrate Widom's equation over a certain probe volume as suggested earlier by us. Chemical potentials are then calculated independently in five different probe volumes, which are cylindrical shells. The results agree well from the second to the fourth shell. Inaccuracies in the innermost cylinder can be easily explained by bad statistics. In the shell close to the wall the extremely high local density is responsible for the inaccuracies. Extending the probe volume over all cylindrical shells besides the one closest to the wall is thought to yield rather reliable results for the chemical potential. As a by-product of the simulations we also obtained diffusion coefficients, which are given in an appendix.     

The radial decline of nerve impulses in a restricted cylindrical extracellular space

In order to increase the potentials recorded extracellularly from nerve fibres, peripheral nerves are often placed in restricted space with cylindrical geometry. Equations are derived for computing the potentials expected at the surface of the cylinder, based on the potentials at the external surface of a small nerve fibre located on the long axis of the cylinder. These equations are evaluated numerically, using two formulae for a nerve impulse given in the literature. In both cases there is little attenuation for cylinders with radii less than 0.5 mm, but the potential declines approximately as a power of radius b for 1<b<10 mm. Various factors which might affect these results under different experimental conditions are discussed.     

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.     

Molecular Dynamics Simulation Studies of the Density and Temperature Dependence of Self-diffusion in a Cylindrical Micropore

We report Molecular Dynamics calculations of radial density profiles and self-diffusion coefficients of Lennard-Jones fluids in a cylindrical pore of radius 2σ, for a wide range of temperatures and densities. At n _p σ³ = 0.825 the self-diffusion coefficient parallel to the pore walls D _∥*. follows a monotonic (nearly linear) increase with kT/ε and is very similar to that of the bulk self-diffusion coefficient D _b *. At n _p σ³ = 0.4 and kT/ε ≤ 1.0 the curve of D _∥* vs. kT/ε shows a distinct inflection in the region 0.7 ≤ kT/ε ≤ 0.9 and values of D _∥* are much less than D _b * decreasing to near solid state values at very low temperatures. At the highest temperature studied, kT/ε = 2.98, D _∥* is almost inversely proportional to density and in a fairly close agreement with that of D _b *. At KT/ε = 0.49, D _∥* is much smaller than D _b *. The motion of adsorbate particles normal to the walls is also discussed.     

Sheet conductor model of brain slices for stimulation and recording with planar electronic contacts

Current and voltage in a brain slice are considered, taking into account the boundary conditions at the surface to an electrolyte bath and at the substrate of an electron conductor. A sheet conductor model is introduced with ohmic leak conductance to the bath and capacitive coupling to the substrate. It assigns a current-source density of neuronal activity to extracellular field potentials recorded by planar contacts, and it relates the current of planar capacitive contacts to the field potential that elicits neuronal activity. Two examples are analytically solved: the recording across a layered brain slice and the stimulation by a circular electrode. The study forms the basis for neurophysical experiments with brain slices or retinae on microelectronic chips.     

Theoretical study of transmembrane and extracellular potentials under propagation block due to geometrical inhomogeneity

A mathematical model was used to study transmembrane and extracellular potentials produced by active geometrically inhomogeneous excitable structures under conditions of propagation block. The structures were electrical analogues of intact or damaged unmyelinated nerve fibres, of the soma to axon transition, or of branching axons or dendrites. It was shown that: (1) damage to a cell is equivalent to the presence of a geometrical inhomogeneity, namely of a region of increased diameter; (2) propagation block caused by a geometrical inhomogeneity, results in; (a) a sharp decrease in the calculated transmembrane potential amplitude not only for the blocked region but also before it; (b) a considerable increase in the amplitude of both the negative phase of extracellular potentials at the points of the volume conductor preceding the blocked region and the first positive phase at points in the proximity of the region; (c) a more pronounced increase in the first positive phase amplitude at small radial distances, if the geometrical inhomogeneity is short compared with the length constant (gamma); (3) the membrane damage results in recording of potentials resembling "giant" ones.     

Current source-density analysis in the mushroom bodies of the honeybee (Apis mellifera carnica)

Summary Information processing in the mushroom bodies which are an important part of most invertebrate central nervous systems was analysed by extracellular electrophysiological techniques. The mushroom bodies consist of layers of parallel intrinsic neurons which make synaptic contact with extrinsic input and output neurons. The intrinsic neurons (approximately 170,000/mushroom body) have very small axon diameters (0.1–1 m) which makes it difficult to record their activity intracellularly. In order to analyse the functional properties of this neuropil field potentials were measured extracellularly.Series of averaged evoked potentials (AEPs) were recorded along electrode tracks at consecutive depth intervals in different parts of the mushroom bodies of the bee. These potentials were elicited by olfactory, mechanical and visual stimuli.In order to locate the synaptic areas generating these potentials, current source-densities (CSD) were calculated using the consecutively measured evoked potentials. The conductivities of the extracellular space along the electrode tracks in the pedunculus and calyx and in part of the alpha-lobe of the mushroom bodies were found to be constant.The CSD analysis reveals a complex pattern of source-sink distributions in the mushroom bodies. There is a high degree of correlation between current sinks and sources detected by CSD analysis and the morphological distribution of neurons.The CSD analysis shows that the inputs and outputs of the mushroom bodies involve multimodal synaptic interactions, whereas information processing in the intrinsic Kenyon-cells is limited to sensory inputs from the antenna.Comparison of the electrophysiological with the histological results shows that the intrinsic cells of the mushroom bodies are physiologically not a homogeneous group as is often proposed. Among the intrinsic neurons clearly defined areas of current sources and sinks can be identified and attributed to Kenyon-cells in different layers.Abbreviations AEP averaged evoked potentials - AGT antennoglomerular tract - CSD current source-density - PCT antennoglomerular tract     

Osmotic response of mammalian cells: Effects of permeating cryoprotectants on nonsolvent volume

The nonsolvent volume, b, of a cell permits calculation of cell water volume from measurements of total cell volume, and, consequently, it is used extensively in the determination of membrane permeability coefficients for water and solutes and also in simulations of water and solute fluxes during freezing of cells. The nonsolvent volume is most commonly determined from the ordinate intercept of plots of cell volume as a function of the reciprocal of extracellular nonpermeating solute concentration (so-called Boyle-van't Hoff plots). Once derived, b is often assumed to be constant even under conditions that may differ markedly from those under which it was determined. Our aim was to investigate whether this assumption was valid when cells were exposed to the cryoprotectants glycerol, dimethyl sulphoxide (Me₂SO), or propane-1,2-diol. Rabbit corneal keratocytes, a fibroblastic cell type, were exposed to 10% (v/v) cryoprotectant for 30 min at 22°C in solutions containing a range of nonpermeating solute concentrations. Cell volumes were determined by an electronic particle sizer and mode volume plotted as an inverse function of the concentration of nonpermeating solute. The cells behaved as osmometers under all conditions studied, but we found no evidence to suggest that the nonsolvent volume of cells was altered by Me₂SO or propane-1,2-diol. Glycerol, however, reduced the slope of the Boyle-van't Hoff plot, but this could be ascribed to the failure of the cells to equilibrate fully with the glycerol over the 30 min exposure time; thus, b was unaffected by glycerol. It may be assumed, therefore, that the nonsolvent volume was not influenced by the presence inside cells of any of these nonelectrolyte cryoprotectants. © 1996 Wiley-Liss, Inc.     

On proper linearization, construction and analysis of the Boyle-van't Hoff plots and correct calculation of the osmotically inactive volume

The Boyle–van’t Hoff (BVH) law of physics has been widely used in cryobiology for calculation of the key osmotic parameters of cells and optimization of cryo-protocols. The proper use of linearization of the Boyle–vant’Hoff relationship for the osmotically inactive volume (v_b) has been discussed in a rigorous way in (Katkov, Cryobiology, 2008, 57:142–149). Nevertheless, scientists in the field have been continuing to use inappropriate methods of linearization (and curve fitting) of the BVH data, plotting the BVH line and calculation of v_b. Here, we discuss the sources of incorrect linearization of the BVH relationship using concrete examples of recent publications, analyze the properties of the correct BVH line (which is unique for a given v_b), provide appropriate statistical formulas for calculation of v_b from the experimental data, and propose simplistic instructions (standard operation procedure, SOP) for proper normalization of the data, appropriate linearization and construction of the BVH plots, and correct calculation of v_b. The possible sources of non-linear behavior or poor fit of the data to the proper BVH line such as active water and/or solute transports, which can result in large discrepancy between the hyperosmotic and hypoosmotic parts of the BVH plot, are also discussed.     

Evidence for a folded conformation of the cell wall protein fromAcetabularia (Polyphysa) cliftonii

Summary The cell wall protein fromAcetabularia has a non-random structure in aqueous solution at pH 5.3, as determined on the basis of intrinsic viscosity, sedimentation velocity and small angle X-ray scattering experiments. This non-random structure is stable in a pH range of 4.5–6.8, as observed on the basis of circular dichroism and viscosity measurements, supporting that the cell wall protein has a specific folded structure. All hydrodynamic measurements, including small angle X-ray scattering in solution, in this pH range are consistent with a prolate ellipsoid model for the shape of this protein, with overall dimensions ofc=86.0 Å,b=7.0 Å, anda=7.5 Å, and with a radius of gyration ofR=39.5 Å. The possibility of a coiled shape was investigated using a worm-like chain model, but it was inconsistent with the experimental data. Instead, a filled particle with uniform density which is equivalent in the scattering behavior is proposed. By a comparison of the observed radius of gyration, R_g=39.5 Å, and the radius of gyration of the cross section,R _c =7.5 Å, we were able to describe the cell wall protein in terms of a prolate ellipsoid of revolution. Comparisons of the experimental scattering curve, plotted as logl (h) versus logh, with the corresponding plots of normalized intensities, calculated for particles of particular shape and various axial ratios indicate a very asymmetric shape for the cell wall protein fromAcetabularia.This research was supported by a grant of the Deutsche Forschungsgemeinschaft.     

Interstitial Ascorbate in Turtle Brain Is Modulated by Release and Extracellular Volume Change

Abstract: The isolated turtle cerebellum was used as a model system to study effects of depolarizing conditions on interstitial ascorbic acid concentration. The depolarizing stimulus was Leão's spreading depression, which is characterized by transient negative extracellular potentials, high potassium levels (20–60 μM), and local depression of neuronal activity. Interstitial concentrations of ascorbate (200–400 μM) and other electroactive species were monitored voltammetrically, using graphite fiber microelectrodes. Total tissue ascorbate (1,810 nmol/g tissue wet weight) was similar to mammalian levels and was several orders of magnitude higher than catecholamine and indoleamine content. During spreading depression, a large (up to 200 μM) increase in concentration of interstitial electroactive species was monitored. Use of Nafion-and ascorbate oxidase-coated electrodes and uricase confirmed that ascorbate was the only substance detected. Simultaneous monitoring of ascorbate, extracellular potential, and extracellular volume (using tetramethylammonium and ion-selective microelectrodes) indicated that (a) the ascorbate increase began with the decrease in extracellular volume during spreading depression, and (b) much of the increase was the result of extracellular volume decrease. In sucrose-substituted medium, in which volume changes are eliminated, a 50 μM increase in interstitial ascorbate, caused by release from intracellular stores, was also seen. The ascorbate concentration increase was prolonged in sucrose medium, suggesting that an uptake process involving sodium may further regulate interstitial ascorbate concentration.