Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers.

The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial interstitial potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm approximately 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-microns radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function.     

Action currents, internodal potentials, and extracellular records of myelinated mammalian nerve fibers derived from node potentials.

The potential distribution within the internodal axon of mammalian nerve fibers is derived by applying known node potential waveforms to the ends of an equivalent circuit model of the internode. The complete spatial/temporal profile of action potentials synthesized from the internodal profiles is used to compute the node current waveforn, and the extracellular action potential around fibers captured within a tubular electrode. For amphibia, the results agreed with empirical values. For mammals, the amplitude of the node currents plotted against conduction velocity was fitted by a straight line. The extracellular potential waveform depended on the location of the nodes within the tube. For tubes of length from 2 to 8 internodes, extracellular wave amplitude (mammals) was about one-third of the product of peak node current and tube resistance (center to ends). The extracellular potentials developed by longitudinal and radial currents in an anisotropic medium (fiber bundle) are compared.     

Interstitial potentials and their change with depth into cardiac tissue.

The electrical source strength for an isolated, active, excitable fiber can be taken to be its transmembrane current as an excellent approximation. The transmembrane current can be determined from intracellular potentials only. But for multicellular preparations, particularly cardiac ventricular muscle, the electrical source strength may be changed significantly by the presence of the interstitial potential field. This report examines the size of the interstitial potential field as a function of depth into a semi-infinite tissue structure of cardiac muscle regarded as syncytial. A uniform propagating plane wave of excitation is assumed and the interstitial potential field is found based on consideration of the medium as a continuum (bidomain model). As a whole, the results are inconsistent with any of the limiting cases normally used to represent the volume conductor, and suggest that in only the thinnest of tissue (less than 200 micron) can the interstitial potentials be ignored.     

Modification of a cylindrical bidomain model for cardiac tissue.

Previous models based on a cylindrical bidomain assumed either that the ratio of intracellular and interstitial conductivities in the principal directions were the same or that there was no radial variation in potential (i.e., a planar front, delta Vm/delta rho = 0). This paper presents a formulation and the expressions for the intracellular, interstitial, extracellular, and transmembrane potentials arising from nonplanar propagation along a cylindrical bundle of cardiac tissue represented as a bidomain with arbitrary anisotropy. For unequal anisotropy, the transmembrane current depends not only on the local change of the transmembrane potential but also on the nature of the transmembrane potential throughout the volume.     

The propagation potential. An axonal response with implications for scalp-recorded EEG

An electrophysiological response of axons, referred to as the "propagation potential," was investigated. The propagation potential is a sustained voltage that lasts as long as an action potential propagates between two widely spaced electrodes. The sign of the potential depends on the direction of action potential propagation. The electrode towards which the action potential is propagating is positive with respect to the electrode from which it is receding. For normal frog sciatic nerves the magnitude of the propagation potential was 17% of the peak of the extracellular action potential; TEA increased it to 32%. For normal earthworm median or lateral giant fibers it was 30%. A ripple pattern on the propagation potential was attributed to variation in resistance along the length of the worm. Cooling increased the duration of the propagation potential and attenuated the higher frequency components of the ripple pattern. Differential records from two widely spaced intracellular microelectrodes in the same axon differed from the propagation potential. The amplitude of the plateau relative to the peak was smaller, it decreased as the action potential propagated from one electrode site to the other, and the potential did not return to zero as rapidly as for extracellular records. When propagation was blocked by heat, the propagation potential slowly decayed. There was no ripple pattern during the decay. In a volume conductor, electrodes contacting the worm did not show the typical propagation potential, but electrodes located a few centimeters away from the worm did. Simple core-conductor models based on classical action potential theory did not reproduce the propagation potential. More complex, modified core-conductor models were needed to accurately simulate it. The results suggest that long, slowly conducting fibers can contribute to the scalp-recorded EEG.     

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.     

A mathematical model for calculating the vector magnetic field of a single muscle fiber.

A mathematical model is described for calculating the volume-conducted magnetic field from active muscle fibers in an anisotropic bundle. With earlier models, the azimuthal magnetic field of a nerve bundle was calculated and the results were compared with the fields measured by toroidal pickup coils. The present model is capable of evaluating all three of the magnetic field components and is thus applicable for analyzing SQUID magnetometer recordings of fields from a muscle bundle. The component of the magnetic field parallel to the fiber axis is more than an order of magnitude smaller than either of the other two components. The amplitude of the magnetic signal is strongly dependent upon the anisotropy of the muscle bundle, the intracellular conductivity, the radius of the muscle fiber, the radius of the muscle bundle, and the location of the fiber in the muscle bundle. The peak-to-peak amplitude of the single-muscle-fiber action field increases linearly with increasing intracellular conductivity, as the square of the radius of the muscle fiber, and exponentially with the distance between the location of the fiber and the center of the bundle.     

Computer simulation of action potential propagation in septated nerve fibers.

The nonlinear, core-conductor model of action potential propagation down axisymmetric nerve fibers is adapted for an implicit, numerical simulation by computer solution of the differential equations. The calculation allows a septum to be inserted in the model fiber; the thin, passive septum is characterized by series resistance Rsz and shunt resistance Rss to the grounded bath. If Rsz is too large or Rss too small, the signal fails to propagate through the septum. Plots of the action potential profiles for various axial positions are obtained and show distortions due to the presence of the septum. A simple linear model, developed from these simulations, relates propagation delay through the septum and the preseptal risetime to Rsz and Rss. This model agrees with the simulations for a wide range of parameters and allows estimation of Rsz and Rss from measured propagation delays at the septum. Plots of the axial current as a function of both time and position demonstrate how the presence of the septum can cause prominent local reversals of the current. This result, not previously described, suggests that extracellular magnetic measurements of cellular action currents could be useful in the biophysical study of septated fibers.     

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.     

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.     

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.     

An electrodiffusive neuron-extracellular-glia model for exploring the genesis of slow potentials in the brain

Within the computational neuroscience community, there has been a focus on simulating the electrical activity of neurons, while other components of brain tissue, such as glia cells and the extracellular space, are often neglected. Standard models of extracellular potentials are based on a combination of multicompartmental models describing neural electrodynamics and volume conductor theory. Such models cannot be used to simulate the slow components of extracellular potentials, which depend on ion concentration dynamics, and the effect that this has on extracellular diffusion potentials and glial buffering currents. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine compartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for action potentials and dendritic calcium spikes in neurons, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. The edNEG model accounts for the concentration-dependent effects on ECS potentials that the standard models neglect. Using the edNEG model, we analyze these effects by splitting the extracellular potential into three components: one due to neural sink/source configurations, one due to glial sink/source configurations, and one due to extracellular diffusive currents. Through a series of simulations, we analyze the roles played by the various components and how they interact in generating the total slow potential. We conclude that the three components are of comparable magnitude and that the stimulus conditions determine which of the components that dominate.     

Intracellular and extracellular activity of cathepsin D in the liver in cirrhosis and its involution

The distribution of cathepsin D in liver with CCl4 induced cirrhosis and its involution in rats was investigated by ultrastructural cytochemistry. Besides intracellular, it was revealed the extracellular activity of cathepsin D. The reaction product was on collagen fibers near the hepatocytes and connective tissue cells as well as on the hepatocytes microvilli and on the outside part of cellular membrane of connective tissue cells (macrophage, fibroblast, Ito cells). Hence the source of extracellular cathepsin D in liver are the parenchymatous as well as nonparenchymal cell elements. The results testify that under the cirrhosis and its involution, the cathepsin D takes part in intracellular proteolysis and is secreted by hepatocytes and connective tissue cells in the intracellular space; it also takes part in extracellular catabolism of connective tissue.     

Electric potential in cylindrical syncytia and muscle fibers

The model developed in an earlier paper using two coupled partial differential equations for calculating the intracellular and extracellular electric potentials in a syncytium is applied here to cylindrical geometry. Eigenfunction expansions are obtained for the potentials resulting from an intracellular point source of current. The required orthogonality relations for the two sets of coupled radial eigenfunctions are derived. The model is applied to the structure composed of the interior and the transverse tubules of a muscle fiber. Asymtotic expansions for ζ and β→0 are obtained, where ζ is the product of the effective intracellular resistivity, the fiber radius and the outer surface membrane admittance per unit area, and β is the ratio of the effective intracellular resistivity to that of the tubular lumen. Earlier results from the distributed circuit model of a muscle fiber are recovered when ζ and β are small, and for a nerve axon when β=0.     

Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model

This work presents a discrete multidomain model that describes ionic diffusion pathways between connected cells and within the interstitium. Unlike classical models of impulse propagation, the intracellular and extracellular spaces are represented as spatially distinct volumes with dynamic/static boundary conditions that electrically couple neighboring spaces. The model is used to investigate the impact of nonuniform geometrical and electrical properties of the interstitial space surrounding a fiber on conduction velocity and action potential waveshape. Comparison of the multidomain and bidomain models shows that although the conduction velocity is relatively insensitive to cases that confine 50% of the membrane surface by narrow extracellular depths (≥2 nm), the action potential morphology varies greatly around the fiber perimeter, resulting in changes in the magnitude of extracellular potential in the tight spaces. Results also show that when the conductivity of the tight spaces is sufficiently reduced, the membrane adjacent to the tight space is eliminated from participating in propagation, and the conduction velocity increases. Owing to its ability to describe the spatial discontinuity of cardiac microstructure, the discrete multidomain can be used to determine appropriate tissue properties for use in classical macroscopic models such as the bidomain during normal and pathophysiological conditions.     

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.     

Sodium, Potassium, and Chloride Fluxes in Intercostal Muscle from Normal Goats and Goats with Hereditary Myotonia

In isolated bundles of external intercostal muscle from normal goats and goats with hereditary myotonia the following were determined: concentrations and unidirectional fluxes of Na⁺, K⁺, and Cl^-, extracellular volume, water content, fiber geometry, and core-conductor constants. No significant difference between the two groups of preparations was found with respect to distribution of fiber size, intracellular concentrations of Na⁺ or Cl^-, fiber water, resting membrane potential, or overshoot of action potential. The intracellular Cl^- concentration in both groups of preparations was 4 to 7 times that expected if Cl^- were distributed passively between intracellular and extracellular water. The membrane permeability to K (P_K) calculated from efflux data was (a) at 38°C, 0.365 x 10^-6 cm sec^-1 for normal and 0.492 x 10^-6 for myotonic muscle, and (b) at 25°C, 0.219 x 10^-6 for normal and 0.199 x 10^-6 for myotonic muscle. From Cl^- washout curves of normal muscle usually only three exponential functions could be extracted, but in every experiment with myotonic muscle there was an additional, intermediate component. From these data PP_cl could be calculated; it was 0.413 x 10^-6 cm sec^-1 for myotonic fibers and was 0.815 x 10^-6 cm sec^-1 for normal fibers. The resting membrane resistance of myotonic fibers was 4 to 6 times greater than that of normal fibers.     

A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.     

A tissue-engineered human dermal construct utilizing fibroblasts and transforming growth factor β1 to promote elastogenesis

Numerous studies have shown that extracellular matrix (ECM)-based scaffolds are suitable for dermal constructs for the differentiation of various cell types in vitro and for constructive tissue remodeling after implantation in vivo. However, a shortcoming of these ECM materials is its limited elastogenesis. Elastic fibers constitute an essential component of mammalian connective tissue and the presence of elastic fibers is crucial for the proper function of the cardiovascular, pulmonary, and intestinal systems. Since it is still largely unknown how cells coordinate the molecular events of elastic-fiber assembly, understanding the ability to regenerate elastic fibers in tissues remains a significant challenge. For this reason, human neonatal dermal fibroblasts (HDFneo) were analyzed for their potential to serve as a cell culture model for elastic fiber assembly. Using optical technologies such as multiphoton laser-scanning microscopy (MPSLM) we demonstrate that HDFneo stimulated with transforming growth factor β1 (TGF-β1) are able to produce a distinct and complex elastic fiber system in vitro. As shown by the desmosine and isodesmosine content, crosslinked elastic fibers were formed within the 3D ECM-based scaffold. This tissue-engineered dermal construct may prove to be an effective template for the development of medicinal approaches in regenerative soft skin tissue reconstruction through TGF-β1 induction.