Potassium ion accumulation at the external surface of the nodal membrane in frog myelinated fibers.

Potassium accumulation associated with outward membrane potassium current was investigated experimentally in myelinated fibers and analyzed in terms of two models-three-compartment and diffusion in an unstirred layer. In the myelinated fibers, as in squid giant axons, the three-compartment model satisfactorily describes potassium accumulation. Within this framework the average space thickness, theta, in frog was 5,900 +/- 700 A, while the permeability coefficient of the external barrier, PK, was (1.5 +/- 0.1) X 10(-2) cm/s. The model of ionic diffusion in an unstirred aqueous layer adjacent to the axolemma, as an alternative explanation for ion accumulation, was also consistent with the experimental data, provided that D, the diffusion constant, was (1.8 +/- 0.2) X 10(-6) cm/s and l, the unstirred layer thickness, was 1.4 +/- 0.1 micron, i.e., similar to the depth of the nodal gap. An empirical equation relating the extent of potassium accumulation to the amplitude and duration of depolarization is given.     

Fluctuation and linear analysis of Na-current kinetics in squid axon

The power spectrum of current fluctuations and the complex admittance of squid axon were determined in the frequency range 12.5 to 5,000 Hx during membrane voltage clamps to the same potentials in the same axon during internal perfusion with cesium. The complex admittance was determined rapidly and with high resolution by a fast Fourier transform computation of the current response, acquired after a steady state was attained, to a synthesized signal with predetermined spectral characteristics superposed as a continuous, repetitive, small perturbation on step voltage clamps. Linear conduction parameters were estimated directly from admittance data by fitting an admittance model, derived from the linearized Hodgkin-Huxley equations modified by replacing the membrane capacitance with a "constant-phase-angle" capacitance, to the data. The constant phase angle obtained was approximately 80 degrees. At depolarizations the phase of the admittance was 180 degrees, and the real part of the impedance locus was in the left-half complex plane for frequencies below 1 kHz, which indicates a steady-state negative Na conductance. The fits also yielded estimates of the natural frequencies of Na "activation" and "inactivation" processes. By fitting Na-current noise spectra with a double Lorentzian function, a lower and an upper corner frequency were obtained; these were compared with the two natural frequencies determined from admittance analysis at the corresponding potentials. The frequencies from fluctuation analyses ranged from 1.0 to 10.3 times higher than those from linear (admittance) analysis. This discrepancy is consistent with the concept that the fluctuations reflect a nonlinear rate process that cannot be fully characterized by linear perturbation analysis. Comparison of the real part of the admittance and the current noise spectrum shows that the Nyquist relation, which generally applies to equilibrium conductors, does not hold for the Na process in squid axon. The Na-channel conductance, gamma Na, was found to increase monotonically from 0.1 to 4.8 pS for depolarizations up to 50 mV from a holding potential of -60 mV, with no indication of a maximum value.     

Determinants of time-dependent membrane conductance. The nonrole of classical ion-membrane molecule interactions

We have examined the steady-state and time-dependent electrical properties of a model membrane system. The model assumes that the directed velocity and energy of ions moving through the membrane are determined by the applied electric field, ionic diffusion forces, and central elastic collisions between ions and membrane molecules. A simple analysis of the steady-state electrical properties of the model yields results identical with ones obtained previously using a more complex analysis procedure. The time-dependent conductance changes of the model in response to a step change in electric field strength when there is solution symmetry display three qualitative patterns dependent on the nature of the ion-membrane molecule interaction. One of the patterns of conductance change is quite similar to that observed in the sodium conductance system of a number of excitable tissues: an initial conductance rise to a maximum (activation) followed by a decay to a final steady-state value (inactivation). However, the correspondence between the time-dependent model behavior and known experimental behavior of excitable systems is only qualitative. We conclude that the classical ion-membrane molecule interactions we consider are not involved in determining time-dependent conductance processes in the excitable systems for which comparison is possible.     

Kinetics of the association of potential-sensitive dyes with model and energy-transducing membranes: Implications for fast probe response times

Summary The second-order rate constants characterizing the association of potential-sensing dyes of the cyanine, merocyanine, and oxonol classes with glycerylmonooleate suspensions, azolectin vesicles, or submitochondrial particles have been measured and the implications for redistribution type mechanisms proposed to explain the potential-dependent optical signals of these probes considered. The second-order rate constants obtained for the cyanines and oxonols are compatible with microsecond probe response times only on the assumption that a high local dye concentration exists in the aqueous phase immediately adjacent to the membrane surface. Calculations based on a surface charge density induced by a bias potential suggest that the necessary local concentration cannot be attained by a diffusion polarization mechanism. A model based on the rapid recombination of ejected dye with the membrane bilayer seems capable of explaining microsecond probe response times in systems where the potential is rapidly changing polarity; calculations suggest that an ejected dye molecule would not diffuse out of an unstirred layer of 100 microns thickness on a millisecond time scale. Microsecond probe responses are also compatible with a first-order potential-dependent dye ejection from the membrane with no rapid recombination when the potential is not changing polarity. The apparent first-order rate constants describing the interaction of merocanine M-540 with a glycerylmonooleate suspension are independent of dye concentration; the reaction may be diffusion limited. The high local dye concentration need not be met in this case for a mechanism based on the transfer of dye onto the membrane from the aqueous phase to describe the microsecond signals of this dye, but other mechanisms have been proposed to explain such signals. The mechanism leading to potentialdependent signals from optical probes appear to differ substantially between suspensions of energy-transducing biological membranes and those involving excitable membranes such as the squid giant axon or model black lipid membranes.     

Diffusion Models for the Squid Axon Schwann Cell Layer

The Schwann cell, basement membrane, and connective tissue layers that surround the squid giant axon and constitute barriers to diffusion, were modeled in a number of ways to analyze various experimental results. The experiments considered are (a) the time-course of the potassium concentration in the space between the Schwann cell and the axon membrane (from now on referred to as the F-H space) after an initial loading, (b) the time-course of sodium concentration in the F-H space after a sudden change in the sodium concentration in the external fluid; (c) the time-course of the concentration of tetrodotoxin (TTX) or saxitoxin (STX) in the F-H space after a sudden change in external concentration, including (or not) the effects of specific binding of TTX or STX to sites on the axon membrane and nonsaturable binding to sites in the F-H space or in the spaces (clefts) between Schwann cells; (d) the effects of the F-H space, clefts, and diffusion into the clefts from the outside (from now on referred to as convergence into the clefts) on the measured series resistance.

The analysis shows that (1) in no case is it necessary to include the effects of the convergence into the clefts from the outside; (2) in case a, the basement membrane, connective tissue layers, and the unstirred layer may be neglected, i.e., the clefts are rate limiting; (3) in case b the clefts may be neglected, i.e., the unstirred layer is rate limiting; (4) in most cases the clefts may be replaced by an equivalent thin diffusion barrier.

     

Models of membrane resonance in pigeon semicircular canal type II hair cells

Pigeon vestibular semicircular canal type II hair cells often exhibit voltage oscillations following current steps that depolarize the cell membrane from its resting potential. Currents active around the resting membrane potential and most likely responsible for the observed resonant behavior are the Ca⁺⁺-insensitive, inactivating potassium conductance I _A (A-current) and delayed rectifier potassium conductance I _K. Several equivalent circuits are considered as representative of the hair cell membrane behavior, sufficient to explain and quantitatively fit the observed voltage oscillations. In addition to the membrane capacitance and frequency-independent parallel conductance, a third parallel element whose admittance function is of second order is necessary to describe and accurately predict all of the experimentally obtained current and voltage responses. Even though most voltage oscillations could be fitted by an equivalent circuit in which the second order admittance term is overdamped (i.e., represents a type of current with two time constants, one of activation and the other of inactivation), the sharpest quality resonance obtained with small current steps (around 20 pA) from the resting potential could be satisfactorily fit only by an underdamped term.     

Simulation of the electrical activity of an excitable membrane with an electronic analog (author's transl)

1. The sodium and potassium conductances of the HODGKIN-HUXLEY model are simulated by a field effect transistor with a series resistor. This arrangement leads to a simple analog model of the excitable membrane (fig. 1 and 2). 2. Normally, the model is silent (fig. 3), but it becomes automatic (fig. 4) when the decay time (de-activation) of the potassium conductance is at least twice the recovery from inactivation time of the sodium conductance (taud greater than 2 tauri). 3. The effects of changes in sodium (fig. 5 and 6) and potassium (fig. 7, 8 and 9) concentration gradients upon the membrane potential and the ionic currents are easily studied when the model is silent or automatic. 4. When automatic, an increase in the potassium concentration gradient induces a lengthening of the period and ultimately, when the gradient is very high, spontaneous activity is blocked (fig. 9). On the other hand, increases of sodium gradient over 30% of normal value do not modify the period (fig 6). 5. The potassium concentration gradient modifies the excitability solely through membrane polarization (fig. 8), while sodium concentration has no effect on it (fig. 5). 6. Results with the model strengthen the hypothesis that tetraethylammonium (TEA) acts on both the maximum potassium conductance (gK) and the mechanism of sodium conductance inactivation (Tauh) to lengthen the action potential as observed on the Ranvier node (fig. 10). Effects of TEA on potassium conductance activation are also discussed. 7. Because of its simplicity and accuracy, this model lends itself easily to many other simulations.     

Potassium ion accumulation slows the closing rate of potassium channels in squid axons.

Potassium ion accumulation in the periaxonal space between squid axonal membrane and the Schwann cell surrounding the axon slows the rate of potassium channel closing to a degree that is consistent with the effect on channel closing of an equivalent change in the bulk external potassium concentration. The alteration of channel gating is independent of membrane potential, V, for V less than or equal to -60 mV, which suggests that the effect is mediated at a site on the outer surface of the membrane, rather than a site within the channel.     

Multiple-state model of ionic channel in reproducing the time course of electrophysiological events.

BACKGROUND: The predictions of the Hodgkin-Huxley model do not accurately fit all the measurements of voltage-clamp currents, gating charge and single-channel currents. There are many quantitative differences between the predicted and measured characteristics of the sodium and potassium channels. For example, the two-state gate model has exponential onset kinetics, whereas the sodium and potassium conductances show S-shaped activation and the sodium conductance shows an exponential inactivation. In this paper we shall examine a more general channel model that can more faithfully represent the measured properties of ionic channels in the membrane of the excitable cell. METHODS: The model is based on the generalisation of the notion of a channel with a discrete set of states. Each state has state attributes such as the state conductance, state ionic current and state gating charge. These variables can have quite different waveforms in time, in contrast with a two-state gate channel model, in which all have the same waveforms. RESULTS: The kinetics of all variables are equivalent: gating and ionic currents give equivalent information about channel kinetics; both the equilibrium values of the current and the time constants are functions of membrane potential. The results are in almost perfect concordance with the experimental data regarding the characteristics of nerve impulse. CONCLUSIONS: The expected values of the gating charge and the ionic conductance are weighted sums of the state occupancy probabilities, but the weights differ: for the expected value of the gating charge the weights are the state gating charges and for the expected value of the ionic conductance the weights are the state conductances. Since these weights are different, the expected values of the gating charge and the ionic conductance will differ.     

A model for axonal propagation incorporating both radial and axial ionic transport.

We present an axonal model that explicitly includes ionic diffusion in the intracellular, periaxonal, and extracellular spaces and that incorporates a Hodgkin-Huxley membrane, extended with potassium channel inactivation and active ion transport. Although ionic concentration changes may not be significant in the time course of one action potential, they are important when considering the long-term behavior (seconds to minutes) of an axon. We demonstrate this point with simulations of transected axons where ions are moving between the intra- and extracellular spaces through an opening that is sealing with time. The model predicts that sealing must occur within a critical time interval after the initial injury to prevent the entire axon from becoming permanently depolarized. This critical time interval becomes considerably shorter when active ion transport is disabled. Furthermore, the model can be used to study the effects of sodium and potassium channel inactivation; e.g., sodium inactivation must be almost complete (within 0.02%) to obtain simulation results that are realistic.     

The effect of potassium diffusion through the Schwann cell layer on potassium conductance of the squid axon

Summary The accumulation of K⁺ ions in the intermembranous spaces of the Schwann cell layer during K⁺ ion current flow may lead to appreciable changes of the K⁺ equilibrium potential. Thus, for an evaluation of the K⁺ conductance of the axolemma, the transport of K⁺ ions through the Schwann cell layer has to be characterized quantitatively. In the present work this is done for a simplified model of the geometrical arrangement of the slit-like channels traversing the Schwann cell region.The K⁺ transport through the slits is treated for two cases: (a) Assuming that electro-kinetic volume flow does not affect K⁺ transport. In this case, pure diffusion of K⁺ ions accounts for their removal from the intermembranous spaces. Estimates on electro-kinetic volume flow show that this case applies to axons ofLoligo forbesi in voltage clamps of fairly small depolarizations. (b) For the case of appreciable electro-kinetic volume flow, evidence is adduced that its main effect is a widening of the slits through the Schwann cell layer. This physical situation could be treated only for the steady-state of convective diffusion of K⁺ ions in the slits.This case is applied to experiments on large depolarizing voltage clamps forLoligo pealii axons. It is shown that a widening of the slits to up to eight times the resting width is to be expected.In both cases (a) and (b), marked deviations of the K⁺ conductance of the axolemma from the Hodgkin-Huxley conductance result. The series resistance of the Schwann cell layer and the decay of after-effects of trains of action potentials are described by the theory.     

Autorhythmicity and entrainment in excitable membranes

Low calcium increases the excitability of neurones and can induce autorhythmicity in excitable cells. Numerical solutions of the Hodgkin-Huxley membrane equations, and numerical evaluations of the small-signal impedance and admittance are used to illustrate the increase in resonance produced by low [Ca²⁺]₀. The resonant frequency may be located either by the peak of the amplitude of the impedance, or by the frequency at which the phase angle is zero for 1:1 entrained action potentials. Autorhythmicity is produced by any mechanism which increases the resonant peak of the amplitude of the membrane impedance.     

Excitability changes in the crustacean motor axon following activity

It has been shown experimentally that the crustacean motor axon is supernormally excitable following a train of action potentials (Zucker 1974). Such a phenomenon can lead to recruitment of terminals which are unexcited at low rates of stimulation. Although currents underlying the crustacean motor axon have been characterized (Connor et al. 1977), it is not known whether this membrane model accounts for a supernormal period, what might cause superexcitablity in this model, or how excitability might change during repetitive stimulation. In present study, it is demonstrated that the crustacean motor axon model does predict a supernormal period, that the supernormal period results from slow recovery from inactivation of the transient potassium, or A, current, and that supernormal excitability is enhanced by repetitive stimulation.     

Initiation and entrainment of multicellular automaticity via diffusion limited extracellular domains

Electrically excitable cells often spontaneously and synchronously depolarize in vitro and in vivo preparations. It remains unclear how cells entrain and autorhythmically activate above the intrinsic mean activation frequency of isolated cells with or without pacemaking mechanisms. Recent studies suggest that cyclic ion accumulation and depletion in diffusion-limited extracellular volumes modulate electrophysiology by ephaptic mechanisms (nongap junction or synaptic coupling). This report explores how potassium accumulation and depletion in a restricted extracellular domain induces spontaneous action potentials in two different computational models of excitable cells without gap junctional coupling: Hodgkin-Huxley and Luo-Rudy. Importantly, neither model will spontaneously activate on its own without external stimuli. Simulations demonstrate that cells sharing a diffusion-limited extracellular compartment can become autorhythmic and entrained despite intercellular electrical heterogeneity. Autorhythmic frequency is modulated by the cleft volume and potassium fluxes through the cleft. Additionally, inexcitable cells can suppress or induce autorhythmic activity in an excitable cell via a shared cleft. Diffusion-limited shared clefts can also entrain repolarization. Critically, this model predicts a mechanism by which diffusion-limited shared clefts can initiate, entrain, and modulate multicellular automaticity in the absence of gap junctions.     

Effects of inhibition of proteoglycan synthesis on the differentiation of cultured rat Schwann cells

Schwann cells synthesize two heparan sulfate proteoglycans, one that is a component of the Schwann cell basement membrane and a smaller one that is an integral component of the Schwann cell plasma membrane. To determine the functions of these molecules, Schwann cell-nerve cell cultures were grown in medium containing a specific inhibitor of proteoglycan biosynthesis, 4-methylumbelliferyl-beta-D-xyloside. Treatment with 1 mM beta-D-xyloside caused a 90% reduction in the accumulation of 35SO4-labeled proteoglycans in the cell layer of the cultures. Gel filtration analysis revealed that both the basement membrane and plasma membrane proteoglycans were affected. Inhibition of proteoglycan biosynthesis was accompanied by an inhibition of laminin deposition into extracellular matrix as determined by immunostaining of cultures and by immunoblotting of cell-associated proteins. This occurred even though there was no decrease in the amount of laminin detected in the medium of beta-D-xyloside-treated cultures. Deposition of collagen type IV was similarly affected. In addition, there was no myelin produced in beta-D-xyloside treated cultures. However, when beta-xyloside-treated cultures were supplied with exogenous basement membrane, Schwann cells produced numerous myelin segments. These results indicate that Schwann cell proteoglycans play an essential role in basement membrane assembly, and that the integral plasma membrane proteoglycan is not required for the basement membrane to exert its effects on Schwann cell differentiation.     

Analyzing an electrogenic cotransporter

A model for the sodium-dependent accumulation of glutamate by synaptosomes has been presented which fits the data of Wheeler and his coworkers and supports their hypothesis of an electrogenic cotransporter. Since their hypothesis was based on experimental data on the operation of the cotransporter on the outer membrane, the model was expanded to predict events when the cotransporter was operating on both sides of the membrane. The model predicts that the accumulation of glutamate is sensitive to the synaptosomal sodium and emphasizes the importance of the sodium/potassium pump to maintain this value. A model which uses only an electrogenic form of the cotransporter on the external membrane and a neutral form on the inside of the membrane predicts too much or too little accumulation of glutamate at different membrane potentials. A model which uses an electrogenic cotransporter on the external membrane and a concentration-dependent sodium glutamate leak would require a significant increase in the permeability of sodium glutamate when the membrane depolarizes. Only the operation of all four mentioned mechanisms will fit experimental data at two different external sodium concentrations and over the range of membrane potentials measured experimentally.     

Membrane Ion Conductances of Mammalian Skeletal Muscle in the Post-Decompression State after High-Pressure Treatment

Exposure of excitable tissues to hyperbaric environments has been shown to alter membrane ion conductances, but only little is known about the state of the membranes of intact cells in the post-decompression phase following a prolonged high-pressure treatment. Furthermore, almost nothing is known about high-pressure effects on skeletal muscle membranes. Therefore, we investigated changes to the input resistances, membrane potentials and voltage-gated membrane currents for sodium (INa), potassium (IK) and calcium (ICa) ions under voltage-clamp conditions in enzymatically isolated intact mammalian single fibers following a 3-hr high-pressure treatment up to 25 MPa at +4 degrees C. After a 3-hr 20 MPa treatment, the input resistance was increased but declined again for treatments with higher pressures. The resting membrane potentials were depolarized in the post-decompression phase following a 20-MPa high-pressure treatment; this could be explained by an increase in the Na+- over K+-permeability ratio and in intracellular [Na+]i. Following a 10-MPa high-pressure treatment, INa, IK and ICa amplitudes were similar compared to controls but were significantly reduced by 25 to 35% after a 3-hr 20-MPa high-pressure treatment. Interestingly, the voltage-dependent inactivation of INa and ICa seemed to be more stable at high pressures compared to the activation parameters, as no significant changes were found up to a 20-MPa treatment. For higher pressure applications (e.g., 25 MPa), there seemed to be a marked loss of membrane integrity and INa, IK and ICa almost disappeared.     

Electrical Properties of Mitochondrial Membranes

The electrical capacity of the membrane of rat liver mitochondria is 0.5 to 0.6 µ./cm². This membrane capacity is obtained from the analysis of the frequency dependence of the admittance of a suspension of swollen mitochondria. In potassium chloride media the mitochondrial membrane capacity does not depend on the ion concentration. The internal conductance of the mitochondria was approximately one-half that of the external medium; the same applies if the mitochondria are equilibrated in a medium with a 10-fold difference in potassium chloride concentration. Hence the swollen mitochondria investigated here appear to be able to adjust their internal ion concentration in proportion with that of the external phase. The similarity of the membrane capacity of isolated mitochondria with the range of values known for other membranes suggests a common molecular structure. The analysis of experimental data suggests an anisotropic electrical behavior of the interior of mitochondria. This anisotropy is readily explained by the existence of internal membranes.     

Structural stability,alternate descriptions and information

The inactivation properties of a model of the nerve membrane are examined. The inactivation kinetics are closely first order and may be characterized by Hodgkin-Huxley (H-H) parameters h_∞ and τ_h which depend on potential in agreement with experiments. Some differences from the H-H equations are identified. The forms predicted for τ_h variation with hyper-polarization and change of external [K⁺] agree with available data. While the inactivation time delay predicted by the model is too small to be detected experimentally, there are grounds for expecting that it may be larger in other tissues, as observed in Myxicola giant axons. The variation of the delay with test potential is predicted to be exponential. Although the model is coupled in the sense defined by Hoyt, it gives rise to an inactivation shift of negligible magnitude. However, introducing a simple variability in one physical parameter leads to the observed form of both the peak transient current voltage relation and the inactivation shift. Inactivation shift thus does not unambiguously indicate coupling; that it results from parametric heterogeneity may be a better hypothesis, and is readily testable. The inactivation shift dependence on current ratio, from experimental data, can be used to correct for the effects of parametric heterogeneity and obtain the value of a previously predicted fundamental parameter of excitable membranes. It is suggested that the effects of parametric heterogeneity must be considered in interpreting experiments and designing models for excitable systems.     

Rapid evaluation of a protein-based voltage probe using a field-induced membrane potential change

The development of a high performance protein probe for the measurement of membrane potential will allow elucidation of spatiotemporal regulation of electrical signals within a network of excitable cells. Engineering such a probe requires a functional screen of many candidates. Although the glass-microelectrode technique generally provides an accurate measure of a given test probe, throughputs are limited. In this study, we focused on an approach that uses the membrane potential changes induced by an external electric field in a geometrically simple mammalian cell. For quantitative evaluation of membrane voltage probes that rely on the structural transition of the S1–S4 voltage sensor domain and hence have non-linear voltage dependencies, it was crucial to introduce exogenous inwardly rectifying potassium conductance to reduce cell-to-cell variability in resting membrane potentials. Importantly, the addition of the exogenous conductance drastically altered the profile of the field-induced potential. Following a site-directed random mutagenesis and the rapid screen, we identified a mutant of a voltage probe Mermaid, exhibiting positively shifted voltage sensitivity. Due to its simplicity, the current approach will be applicable under a microfluidic configuration to carry out an efficient screen. Additionally, we demonstrate another interesting aspect of the field-induced optical signals, ability to visualize electrical couplings between cells.