The cylindrical cell with a time-variant membrane resistance. Measuring passive parameters.

The passive electrical properties of a cable can be measured by injecting a step of current at a point and fitting the resulting potentials at several positions along the cable with analytic solutions of the cable equation. An error analysis is presented for this method (which is based on constant membrane resistance) when the membrane resistance is not constant, but increases linearly with time. The increase of rm produces a "creep" in the membrane potential at long times, as observed in cardiac, skeletal, and smooth muscle. The partial differential equation describing the time-varying cable was solved numberically for a step of current and these "data" were fit by standard constant-resistance methods. Comparing the resulting parameter values with the known true values, we suggest that a correction of the standard methods is not satisfactory for resistance changes of the kind observed; instead, the cable equation must be solved again for the particular form of rm(t). The practical implementation of a method by Adrian and Peachey for measuring the membrane capacitance and an approximate method for estimating the rate-of-change of membrane resistance are discussed in appendices.     

The effect of calcium on the desensitization of membrane receptors at the neuromuscular junction

Desensitization, as represented by the progressive decline in the electromotive effects of depolarizing agents at the neuromuscular junction, was studied by observing the time course of changes in effective transmembrane resistance during the prolonged application of 0.27 mM carbamylcholine to the postjunctional region of frog skeletal muscle fibers. The effective transmembrane resistance was measured by means of two intracellular microelectrodes implanted in the junctional region of single muscle fibers. When carbamylcholine was applied to the muscle there was an immediate decrease in the effective membrane resistance followed by a slower return toward control values which was identified as the phase of desensitization. When the calcium concentration was increased from 0 to 10 mM there was an approximately sevenfold increase in the rate of desensitization. On the other hand, an increase in the concentration of sodium from 28 to 120 mM caused a slowing of the rate of desensitization. Even in muscles depolarized by potassium sulfate, calcium increased the rate of desensitization while high concentrations of potassium tended to prolong the process. Some mechanisms by which calcium might exert these effects are discussed.     

Segmented and "equivalent" representation of the cable equation

The linear cable theory has been applied to a modular structure consisting of n repeating units each composed of two subunits with different values of resistance and capacitance. For n going to infinity, i.e., for infinite cables, we have derived analytically the Laplace transform of the solution by making use of a difference method and we have inverted it by means of a numerical procedure. The results have been compared with those obtained by the direct application of the cable equation to a simplified nonmodular model with "equivalent" electrical parameters. The implication of our work in the analysis of the time and space course of the potential of real fibers has been discussed. In particular, we have shown that the simplified ("equivalent") model is a very good representation of the segmented model for the nodal regions of myelinated fibers in a steady situation and in every condition for muscle fibers. An approximate solution for the steady potential of myelinated fibers has been derived for both nodal and internodal regions. The applications of our work to other cases dealing with repeating structures, such as earthworm giant fibers, have been discussed and our results have been compared with other attempts to solve similar problems.     

Effects of Fiber Type and Size on the Heterogeneity of Oxygen Distribution in Exercising Skeletal Muscle

The process of oxygen delivery from capillary to muscle fiber is essential for a tissue with variable oxygen demand, such as skeletal muscle. Oxygen distribution in exercising skeletal muscle is regulated by convective oxygen transport in the blood vessels, oxygen diffusion and consumption in the tissue. Spatial heterogeneities in oxygen supply, such as microvascular architecture and hemodynamic variables, had been observed experimentally and their marked effects on oxygen exchange had been confirmed using mathematical models. In this study, we investigate the effects of heterogeneities in oxygen demand on tissue oxygenation distribution using a multiscale oxygen transport model. Muscles are composed of different ratios of the various fiber types. Each fiber type has characteristic values of several parameters, including fiber size, oxygen consumption, myoglobin concentration, and oxygen diffusivity. Using experimentally measured parameters for different fiber types and applying them to the rat extensor digitorum longus muscle, we evaluated the effects of heterogeneous fiber size and fiber type properties on the oxygen distribution profile. Our simulation results suggest a marked increase in spatial heterogeneity of oxygen due to fiber size distribution in a mixed muscle. Our simulations also suggest that the combined effects of fiber type properties, except size, do not contribute significantly to the tissue oxygen spatial heterogeneity. However, the incorporation of the difference in oxygen consumption rates of different fiber types alone causes higher oxygen heterogeneity compared to control cases with uniform fiber properties. In contrast, incorporating variation in other fiber type-specific properties, such as myoglobin concentration, causes little change in spatial tissue oxygenation profiles.     

On the electrotonic spread in cardiac muscle of the mouse

As an appropriate model which can simulate the cardiac working muscle with respect to the passive electrical spread, a lattice whose sides have linear cable properties is presented, and the passive potential spread on the model is mathematically analyzed in the fiber direction. Distribution of electrotonic potential in the fiber direction was measured with a pair of intracellular microelectrodes in the cardiac muscle fiber of mouse. By employing “pencil type” microelectrodes potential distribution in the transverse direction within a fiber was also measured. This transverse effect was differentiated from the longitudinal potential distribution. A tonically applied potential at any point of a cell interior spreads continuously in a manner described by a Bessel function. Using appropriate electrical and morphological parameters the experimental results proved to fit the curve obtained from numerical calculation on the model. The apparent length constant obtained for smaller distances (less than 100 μ) from the current source was 70 μ, and it increases as the distance becomes larger. At a point inside the fiber the resistance to the extracellular fluid ranged from 200 to 600 KΩ. The influence of coupling resistance between current and recording electrodes on the measurement of electrotonic potential was examined for small interelectrode distance.     

Some electrical properties of the membrane of the barnacle muscle fibers under internal perfusion

Summary Intracellular perfusion technique has been applied to the muscle fibers of the barnacle species,Balanus nubilus. In these fibers, generation and the form of the calcium spike was governed by the frequency of stimulation and intra- and extracellular calcium concentrations. Voltage-clamp experiments showed that the magnitude of the potassium outward current was controlled by the intracellular calcium concentration whose increase, nearly 10³-fold, raised the resting membrane conductance and the outward potassium current. On the other hand, application of 10mm zinc ions inside the muscle fiber had no effect on either the resting potential or the outward potassium current but suppressed the early inward calcium current. Similarly, the inward calcium current was decreased by low concentration of sodium ions in the extracellular fluid only when its ionic strength was made low by substituting sucrose for the sodium salt. Measurement of outward current with the muscle fiber in calcium-free ASW solution and intracellularly perfused with several cationic solutions established the selectivity sequence TEA相似文献   

An analysis of the relationship between the current and potential generated by a quantum of acetylcholine in muscle fibers without transverse tubules

Summary Miniature end plate potentials (MEPPs) were recorded in glyceroltreated muscle fibers with four microelectrodes which were used to determine the passive electrical characteristics of the same fibers. Voltage responses which were computed from miniature end plate currents (MEPCs) and the passive cable properties of a fiber, agreed very closely with experimentally recorded MEPPs confirming the hypothesis that MEPPs spread passively along a muscle fiber. The model was used to analyze the effect of variations in synaptic current and the properties of a muscle fiber on the postsynaptic response. The decrement of MEPPs was exponential for distances up to 1 to 2 mm from an origin but then deviated from the initial exponential. Variations in the growth time of the input current up to 1 msec had little effect on computed MEPPs whereas an increase in the decay time constant caused a significant increase in MEPP amplitude and effective space constant. An increase in the internal resistivity of a muscle fiber increased MEPP amplitude at the origin but decreased the effective space constant. The amplitude of MEPPs was inversely proportional to the 1.5 power of the diameter of a muscle fiber, and the MEPP space constant increased as the square root of the diameter. The amplitude of MEPPs is not necessarily determined by the input resistance of the muscle fiber. Changes in input resistance caused by changes in membrane resistance would have little effect on te amplitude or decrement of MEPPs.     

Effect of vacuolization on the sodium concentration in an isolated muscle fiber]

Using the Perkin Elmer flame photometer sodium and potassium concentrations have been measured in muscle fibers from the m. ileofibularis of Rana temporaria. After 30 minutes preincubation in the Ringer solution, made hypertonic by the addition of 0.22M glycerol, the muscle fibers were incubated in the normal Ringer solution for 30 min. These fibers showed a vacuolation and an increase in total fiber sodium up to 37.2 mmol/l +/- 5.9 S. E., or 45.8 mmol/kg H2O +/- 7.3 S. E. No significant changes in potassium concentration were observed. Then, the fibers were exposed again to the Ringer solution containing 0.22 M glycerol. This procedure caused the disappearance of vacuoles and decrease in fiber sodium concentration down to 17.7 mmol/l +/- 1.6 S. E., or 21.8 mmol/kg H2O +/- 2.0 S. E. The effect of vacuolation was not blocked by ouabain (1.10(-4) M). It is suggested that the vacuoles have a high NaCl concentration. A model for NaCl and water accumulation in T-tubules is presented.     

Some electrical properties of the membrane of the barnacle muscle fibers under internal perfusion.

Intracellular perfusion technique has been applied to the muscle fibers of the barnacle species, Balanus nubilus. In these fibers, generation and the form of the calcium spike was governed by the frequency of stimulation and intra- and extracellular calcium concentrations. Voltage-clamp experiments showed that the magnitude of the potassium outward current was controlled by the intracellular calcium concentration whose increase, nearly 10(3)-fold, raised the resting membrane conductance and the outward potassium current. On the other hand, application of 10 mM zinc ions inside the muscle fiber had no effect on either the resting potential or the outward potassium current but suppressed the early inward calcium current. Similarly, the inward calcium current was decreased by low concentration of sodium ions in the extracellular fluid only when its ionic strength was made low by substituting sucrose for the sodium salt. Measurement of outward current with the muscle fiber in calcium-free ASW solution and intracellularly perfused with several cationic solutions established the selectivity sequence TEA less than Cs less than Li less than Tris less than Rb less than Na less than K for the potassium channel.     

An analysis of the electrical properties of a skeletal muscle fiber containing a helicoidal T system

The linear electrical properties of skeletal muscle fibers have been analyzed using lumped circuit analogues of helicoidal T system. The geometry of a helicoid is assumed to produce two electrical effects, modeled separately. One model is motivated by the pitch or tilt of the T system, which forces the current flowing in the lumen of the tubules to have a longitudinal projection. The second model is motivated by the longitudinal continuity of a helicoid, which forms a structure similar to a cable within the fiber. The pitch or tilting of the T system plane modified the longitudinal resistance of the fiber, making it slightly frequency dependent; however, the magnitude of the change was less than 0.1%. The longitudinal connections between T system networks had a more complicated effect; the magnitude of the correction was again less than 0.1%. The conclusion from this analysis is that a helicoidal T system, whose pitch is constrained by the sarcomere spacing, will not affect electrical signals recorded intracellularly in intact fibers.     

The effects of ADP and phosphate on the contraction of muscle fibers.

The products of MgATP hydrolysis bind to the nucleotide site of myosin and thus may be expected to inhibit the contraction of muscle fibers. We measured the effects of phosphate and MgADP on the isometric tensions and isotonic contraction velocities of glycerinated rabbit psoas muscle at 10 degrees C. Addition of phosphate decreased isometric force but did not affect the maximum velocity of shortening. To characterize the effects of ADP on fiber contractions, force-velocity curves were measured for fibers bathed in media containing various concentrations of MgATP (1.5-4 mM) and various concentrations of MgADP (1-4 mM). As the [MgADP]/[MgATP] ratio in the fiber increases, the maximum velocity achieved by the fiber decreases while the isometric tension increases. The inhibition of fiber velocities and the potentiation of fiber tension by MgADP is not altered by the presence of 12 mM phosphate. The concentration of both MgADP and MgATP within the fiber was calculated from the diffusion coefficient for nucleotides within the fiber, and the rate of MgADP production within the fiber. Using the calculated values for the nucleotide concentration inside the fiber, observed values of the maximum contraction velocity could be described, within experimental accuracy, by a model in which MgADP competed with MgATP and inhibited fiber velocity with an effective Ki of 0.2-0.3 mM. The average MgADP level generated by the fiber ATPase activity within the fiber was approximately 0.9 mM. In fatigued fibers MgADP and phosphate levels are known to be elevated, and tension and the maximum velocity of contraction are depressed. The results obtained here suggest that levels of MgADP in fatigued fibers play no role in these decreases in function, but the elevation of both phosphate and H+ is sufficient to account for much of the decrease in tension.     

Refractoriness of cardiac muscle as affected by intercalated disks: a model study implications for fibrillation and defibrillation

It is generally agreed that inhomogeneities of the recovery process in cardiac tissue play an important role in the genesis of reentrant arrhythmias. Regarding cardiac muscle as an assembly of discrete cells connected by gap junctions, differences in recovery may result from a nonuniformity of membrane or cable properties. In this study, a computer model of a one-dimensional cardiac muscle fiber including a periodic intercalated disk structure is used to study the influence of disk resistance (Rj) and stimulus strength (J) on refractoriness. Stimulating currents are applied externally in a bipolar arrangement. The basic effect of a current pulse is local de- and hyperpolarizations at the ends of an individual cell. Polarization develops very rapidly and increases with increasing values of Rj or J so that an interaction with membrane current kinetics becomes possible. When a premature stimulus is applied during repolarization of a conditioning action potential, multiple Na currents can occur, either caused by depolarization of the cathodal end of a cell or in the form of anode break excitation at the hyperpolarized end. Those currents affect the response of a fiber such that, at a given value of J, the refractory period is shortened by an increase in Rj. In a ring fiber model with different Rj values in the two halves of ring an extrastimulus timed between the refractory periods of the two branches results in a sustained circus movement. Varying stimulus strength yields an upper limit of vulnerability characterized by a "synchronized extrasystole". The ring model also implies the suppression of circus movement by an external shock. The minimal shock strength required for suppression is close to the upper limit of vulnerability. The simulations suggest that discrete effects of junctional resistance may be involved in fibrillation and defibrillation.     

Chloride conductance of denervated gastrocnemius fibers from normal goats.

Membrane potentials, cable parameters, and component resting ionic conductances of gastrocnemius fibers from normal goats were measured in vitro at six to 32 days following denervation by section of the tibial nerve. Denervated fibers were depolarized an average of 11.6 +/- 1.5 mV (six preparations) from the control mean of 62.1 +/- 1.0 mV (124 fibers) over the period studied. Fibrillation, tetrodotoxin-resistant action potentials, and anode-break excitation were present in the denervated preparations after 13 days. The control cable parameters from 124 fibers (13 preparations) were membrane resistance, 1052 +/- 70 omega-cm2 and membrane capacitance, 6.2 muF/cm2. In denervated fibers membrane resistance increased two to three times in the 13 to 32 day period; membrane capacitance increased about 50% in normal solution at eight to nine, 27-28, and 32 days. Myoplasmic resistivity was assumed to be 112 omega-cm. Measurements were made at 38 degrees C. Component resting conductances were determined from the cable parameters in normal and chloride-free solution. Mean chloride conducantance GC1 and mean potassium conductance GK of control fibers were 776 +/- 49 mumhos/cm2 and 175 +/- 15 mumhos/cm2 (92 fibers), respectively. Following denervation GC1 increased slightly at six to nine days then fell to low values at 16 to 32 days that were close to or indistinguishable from zero. GK increased significantly to 372 +/- 40 mumhos/cm2 and 499 +/- 90 mumhos/cm2 at 16 to 20 and 32 days, respectively. It was concluded from these findings that GC1 and GK of mammalian skeletal muscle are controlled by factors from the nerve and/or muscle action potentials. Goat muscle is different from frog muscle in which GC1 does not change and GK decreases during denervation.     

End-plate potentials in a model muscle fiber. Corrections for the effects of membrane potential on currents and on channel lifetimes.

At the neuromuscular junction, the end-plate potential is generated by a conductance increase in the end-plate membrane. The end-plate depolarization brings the membrane potential toward the reversal potential, which diminishes the driving force for inward current flow. A. R. Martin (1955, J. Physiol. [Lond.]. 130:114-122) devised a simple formula to correct end-plate potential amplitudes for a diminished driving force based on a purely resistive model of the end-plate membrane. The model ignores the membrane capacity, the complexity of the equivalent circuit for a muscle fiber, the variation in channel lifetimes with changes in membrane potential, and the extension of the end plate along a length of the cable. We have developed a model that incorporates all of these features. The calculations show that Martin's correction is, in theory, quite satisfactory for a cable that has the characteristics of a muscle fiber unless the recording is made at a distance from the site of inward current flow. However, there is a discrepancy between models of the frog neuromuscular junction and the available experimental data, which suggests that the end-plate depolarization produced by a given current is greater than expected from their model.     

Potassium ion accumulation in a periaxonal space and its effect on the measurement of membrane potassium ion conductance

Summary Potassium currents of various durations were obtained from squid giant axons voltage-clamped in artificial seawater solutions containing sufficient tetrodotoxin to block the sodium conductance completely. From instantaneous potassium current-voltage relations, the reversal potentials immediately at the end of these currents were determined. On the basis of these reversal potential measurements, the potassium ion concentration gradient across the membrane was shown to decrease as the potassium current duration increased. The kinetics of this change was shown to vary monotonically with the potassium ion efflux across the membrane estimated from the integral over time of the potassium current divided by the Faraday, and to be independent of both the external sodium ion concentration and the presence or absence of membrane series resistance compensation. It was assumed that during outward potassium current flow, potassium ions accumulated in a periaxonal space bounded by the membrane and an external diffusion barrier. A model system was used to describe this accumulation as a continuous function of the membrane currents. On this basis, the mean periaxonal space thickness and the permeability of the external barrier to K⁺ were found to be 357 Å and 3.21×10^–4 cm/sec, respectively. In hyperosmotic seawater, the value of the space thickness increased significantly even though the potassium currents were not changed significantly. Values of the resistance in series with the membrane were calculated from the values of the permeability of the external barrier and these values were shown to be roughly equivalent to series resistance values determined by current clamp measurements. Membrane potassium ion conductances were determined as a function of time and voltage. When these were determined from data corrected for the potassium current reversal potential changes, larger maximal potassium conductances were obtained than were obtained using a constant reversal potential. In addition, the potassium conductance turn-on with time at a variety of membrane potentials was shown to be slower when potassium conductance values were obtained using a variable reversal potential than when using a constant reversal potential.     

Localization of ionic conductances in crayfish muscle fibers

Summary Analysis of the changes in membrane potential and conductance of isolated crayfish muscle fibers caused by rapid solution changes leads to the following conclusions. First, the extensive invagination system of this fiber presents a barrier for diffusion between bath and sarcolemma that accounts for the time lag of electrical responses to changes in bath chloride concentration. Morphological data regarding these invaginations were used in a model which simulated the fiber response on an analog computer. Second, the potassium conductance is effectively localized on the sarcolemma in direct contact with the bath (superficial sarcolemma), whereas the chloride conductance is restricted to the invaginations. This distribution of conductances is the reverse of that found in frog muscle.     

Force deficit during the onset of muscle hypertrophy

The purpose was to study selected structural changes associated with the deficit in maximum specific force (N/cm2) during the early development of skeletal muscle hypertrophy. Ablation of gastrocnemius and plantaris muscles was performed bilaterally in 35-day-old rats (n = 41), and the soleus muscle was studied from days 1 to 30 thereafter. Compared with control muscles from age-matched unoperated rats (n = 48), muscle mass and cross-sectional area increased in parallel from 28 to 52% over the 30-day postoperative period. Specific force of hypertrophied muscle was depressed 38% at days 1 and 3, and by 28% from days 5 to 30 after synergistic muscle ablation compared with age-matched control values. Interstitial space was 38% greater than the control value of 20.4 +/- 1 microliters/100 mg at day 1 only. Protein concentration was depressed 15% for 7 days after the ablation operation, and connective tissue protein concentration was unchanged. The relative magnitude of increased mean fiber cross-sectional area was less than that of muscle mass until day 7 after ablation. Mononuclear cell infiltration in interfascicular spaces occurred from days 3 to 30 without light microscopic evidence of muscle fiber injury. Initial functional deficits are explained in part by an enlarged interstitial space and decreased protein concentration; later deficits are likely accounted for by intracellular changes.     

Reduced concentrations of potassium,magnesium, and sodium-potassium pumps in human skeletal muscle during treatment with diuretics

Animal studies have shown that potassium depletion induced by diuretics or potassium deficient fodder leads to a selective decrease in the concentrations of potassium and in the concentration of sodium-potassium pumps in skeletal muscle. In 25 patients who had received diuretics for 2-14 years the mean concentrations of potassium, magnesium, and sodium-potassium pumps were measured in skeletal muscle biopsy specimens and were significantly lower than in those from a group of age matched controls. The reductions in all three variables were significant in those patients receiving diuretics for arterial hypertension as well as in those being treated for congestive heart failure. In 14 patients the mean muscle potassium concentration was below the control range, but only one of those was hypokalaemic (3·4 mmol/l), and 13 were receiving potassium supplements. In 15 patients the mean muscle magnesium concentration was below normal, and the mean muscle potassium and magnesium concentrations showed a linear correlation. In 12 patients in whom the mean muscle potassium concentration was below 80 μmol/g wet weight there was a linear correlation between the cellular potassium:sodium ratio and the concentration of ³H-ouabain binding sites indicating that potassium deficiency also leads to a down regulation of sodium-potassium pumps in human skeletal muscle.In spite of potassium supplements long term treatment with diuretics may lead to potassium and magnesium deficiencies, which are not detectable using the standard methods of serum analysis. The changes in concentrations of electrolytes and sodium-potassium pumps associated with treatment with diuretics may impair muscle function and potassium homoeostasis and interfere with the distribution of digitalis glycosides.