Analysis of Certain Errors in Squid Axon Voltage Clamp Measurements

Localized membrane current and potential measurements were made on the squid giant axon in voltage clamp experiments. Spatial control of potential was impaired by the use of axial current supplying electrodes with surface resistance greater than 20 ohms for a centimeter length of axon. No region of membrane which was indeed subjected to a potential step showed more than one inward current peak. Other patterns were results of space clamp failure. Membrane current and potential patterns during space clamp failure were approximately reproduced in computations on a model containing two membrane patches obeying the equations of Hodgkin and Huxley. Non-uniformities in the axon or electrodes are not necessary for non-uniform electrical behavior. An extension of the core conductor model which includes the axial wire and external solution has been analyzed. The space constant of electrotonic spread is less than 0.5 mm with a usable electrode. Errors of about 5 per cent are introduced by ignoring the external solution. Resistance between the membrane and the control electrodes reduces the control and a few ohm cm² could lead to serious errors in interpretation.     

Influence of intercellular clefts on potential and current distribution in a multifiber preparation.

A theoretical model is presented for current and voltage clamp of multifiber bundles in a double sucrose gap. Attention is focused on methodological errors introduced by the intercellular cleft resistance. The bundle is approximated by a continuous geometry. Voltage distribution, as a function of radial distance and time, is defined by a parabolic partial differential equation which is specified for different membrane characteristics. Assuming a linear membrane, analytical solutions are given for current step and voltage step conditions. The theoretical relations (based on Bessel functions) may be used to calculate membrane conductance and capacity from experimental clamp data. The case of a nonlinear membrane with standard Hodgkin-Huxley kinetics for excitatory Na current is treated assuming maximum Na conductances (gNa) of 120, 10, and 1 mmho/cm2. Numerical simulations are presented for potential and current distribution in a bundle of 60 microns diameter during depolarizing voltage steps. Adequate voltage control is restricted to the peripheral fibers of the bundle whereas the membrane potential of the inner fibers deviates from the command level during early inward current, tending to the Na equilibrium potential. In the peak current-voltage diagram the loss of voltage control is reflected by an increased steepness of the negative region and a decreased slope conductance of the positive region. With gNa = 120 mmho/cm2, the positive slope conductance is approximately 25% of the slope expected from ideal space clamping. With the lower values of gNa, the slope conductance ratio is in the order of 50%. Implications of the results for an experimental voltage clamp analysis of early inward current on multifiber preparations are discussed.     

Electrical Bistability of the Motoneuronal Dendritic Membrane Determined by Non-inactivating Inward Currents: a Model Study

We investigated features of the spatial pattern of electrical bistable states of dendrites using a computer model of an abducens motoneuron with the dendritic branching reconstructed in detail. The dendritic membrane has an N-shaped current-voltage relation (I-V curve) determined mainly by the presence of L-type calcium channels. Such channels, according to indirect experimental data, are present in the dendrites of these cells together with glutamatergic NMDA-type channels also capable of determining electrical bistability of the membrane and the corresponding specific patterns of electrical activity generated by such neurons. For our model, we obtained steady-state local I-V curves and transferred spatial distribution maps of the membrane potential difference (surface density of transmembrane currents), as well as increments of the axial dendritic current, to three-dimensional images of the reconstructed branching dendrites. The latter increments determine the contribution of a dendritic site in general axial current delivering the charge to the trigger zone of a neuron. The simulation results showed that incorporation of non-inactivating calcium channels into dendritic membrane leads to the origination of a pattern of spatial distribution of bistable electrical states in the dendrites, which were not described earlier. Such features are most important under conditions of a stable state of high depolarization of the relevant parts of the dendrites. In this case, the respective feature was the existence of a zone of maximum density of the inward transmembrane current, which covers areas of first-order branching of all dendrites. Since the greatest relative contribution to the total current belongs to the inward calcium current, the above zone of first branchings can be considered a “hot spot” zone characterized by increased entry of Ca²⁺. This may have important functional consequences for local intracellular calcium signaling.     

Simulated propagation of cardiac action potentials.

We have used numerical methods for solving cable equations, combined with previously published mathematical models for the membrane properties of ventricular and Purkinje cells, to simulate the propagation of cardiac action potentials along a unidimensional strand. Two types of inhomogeneities have been simulated and the results compared with experimentally observed disturbances in cardiac action potential propagation. Changes in the membrane model for regions of the strand were introduced to simulate regions of decreased excitability. Regional changes in the intercellular coupling were also studied. The results illustrate and help to explain the disturbances in propagation which have been reported to occur at regions of decreased excitability, regions with changing action potential duration, or regions with changing intercellular coupling. The propagational disturbances seen at all of these regions are discussed in terms of the changing electrical load imposed upon the propagating impulse.     

Squid giant axons. A model for the neuron soma?

Insertion of electrically floating wires along the axis of a squid giant axon produces an apparent increase in diameter in the region where the wire surface has been treated to give it a low resistance. The shape of action potentials propagating into this region depend upon the surface resistance (and the length) of the wire. As this segment's internal resistance is lowered by reducing the wire's surface resistance, the following characteristic sequence of changes in the action potential is seen at the transition region: (a) the duration increases; (b) two peaks develop, the first one generated in the normal axon region and the second one generated later in the axial wire region, and; (c) blockage occurs (for a very low resistance wire). Action potentials recorded at the membrane region near the tip of the axial wire in (b) resemble those recorded at the initial segment of neurons upon antidromic invasions. Squid axon action potentials propagated from a normal region into that containing the low resistance wire also resemble antidromic invasions recorded in neuron somas. Hyperpolarizing current pulses applied through the wire act as if the wire surface resistance was momentarily reduced. For example, the two components of the action potential recorded at the axial wire membrane region noted in (b) can be sequentially blocked by the application of increasing hyperpolarizing current through the wire. Similar effects are seen when hyperpolarizing currents are injected into motoneuron somas. It is concluded that the geometrical properties of the junction of a neuron axon with its soma may be in themselves sufficient to determine the shape of the action potentials usually recorded by microelectrodes.     

Voltage clamp simulations for multifiber bundles in a double sucrose gap: cable complications

A theoretical model is presented for voltage clamp of a bundle of cylindrical excitable cells in a double sucrose gap. The preparation in the test node is represented by a single one-dimensional cable (length/diameter ratio approximately) with standard Hodgkin-Huxley kinetics for transmembrane Na current. Imperfections of voltage control due to internal (longitudinal) resistivity and external (radial) resistance in series to the membrane are analysed. The electrical behavior of a fiber is described by the cable equation with appropriate boundary conditions and subsidiary equations reflecting the membrane characteristics. Membrane voltage and current distribution in response to a step command was obtained by numerical integration. The results are described in two papers. The present paper deals with the effect of internal resistivity with the external resistance being neglected. The closed loop response of a fiber displays a strong tendency to oscillate. To stabilize the system a phase lead was inserted and the gain of the control amplifier was reduced. Conditions for stability were examined by Nyquist analysis. When the Na system was activated by a command pulse below ENa, a voltage gradient developed between a depolarization (relative to the command signal) at the end where voltage was monitored and a hyperpolarization at the site of current injection. In spite of a poor voltage control the total measured current appeared to have a smooth transient. With large voltage gradients a small, second inward current was seen. At a low (high) Na conductance maximum peak inward current was larger (smaller) that the current expected from ideal space clamping.     

Electrophysiological response of rat atrial myocytes to acidosis

The effect of acidosis on the electrical activity of isolated rat atrial myocytes was investigated using the patch-clamp technique. Reducing the pH of the bathing solution from 7.4 to 6.5 shortened the action potential. Acidosis had no significant effect on transient outward or inward rectifier currents but increased steady-state outward current. This increase was still present, although reduced, when intracellular Ca(2+) was buffered by 1,2-bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid (BAPTA); BAPTA also inhibited acidosis-induced shortening of the action potential. Ni(2+) (5 mM) had no significant effect on the acidosis-induced shortening of the action potential. Acidosis also increased inward current at -80 mV and depolarized the resting membrane potential. Acidosis activated an inwardly rectifying Cl(-) current that was blocked by 4,4'-diisothiocyanostilbene-2,2'-disulfonic acid (DIDS), which also inhibited the acidosis-induced depolarization of the resting membrane potential. It is concluded that an acidosis-induced increase in steady-state outward K(+) current underlies the shortening of the action potential and that an acidosis-induced increase in inwardly rectifying Cl(-) current underlies the depolarization of the resting membrane potential during acidosis.     

Extracellular recording of localized electrical activity in denervated frog slow muscle fibres

Pyriformis muscles of Rana temporaria were denervated by cutting the sciatic nerve in the pelvis. Slow muscle fibres were depolarized with intracellular current pulses, and the electrical activity was recorded simultaneously with intracellular and extracellular recording electrodes. When the extracellular electrode was moved along the fibre surface, outward and inward currents of variable amplitude were recorded. Inward currents coincided with the fast rising phase of the intracellularly recorded action potential; up to four inward current peaks could be detected in single fibres investigated over 3.4--8 mm of their length. The distance between inward current peaks was generally 1--2 mm, but greater distances were also observed. Composite action potentials could be shown to be due to inward currents arising in separate areas of the slow fibre membrane. It is concluded that after denervation Na-channels are incorporated into spatially limited areas of the membrane of slow muscle fibres.     

Opioids activate both an inward rectifier and a novel voltage-gated potassium conductance in the hippocampal formation

Opioid receptors were found to activate two different types of membrane potassium conductance in acutely dissociated neurons from the CA1/subiculum regions of the adult rat hippocampal formation. Opioid-responsive neurons were distinguished based on their morphology and electrophysiological responses. In one population of neurons having a multipolar, nonpyramidal cell shape, mu-selective opioid agonists increased an inward rectifying potassium current. Opioid activation of the inward rectifying conductance resulted in small outward potassium currents at resting membrane potentials and increased inward currents at hyperpolarized potentials. In a second population of nonpyramidal neurons, mu opioid agonists increased a novel voltage-gated potassium current. This current was blocked by internal CsCl2, unaffected by external BaCl2 or CdCl2, irreversibly activated by intracellular GTP-gamma-S, and inactivated by sustained depolarization. In contrast to the inward rectifying conductance, the voltage-gated conductance was not activated at resting membrane potentials or hyperpolarized potentials. The opioid-activated, voltage-gated conductance represents a new class of G protein-regulated potassium current in the brain.     

Correlation of electrical current influx with nuclear position and division inFunaria caulonema tip cells

Summary Ionic currents around caulonema tip cells of the filamentous protonema of the mossFunaria hygrometrica were examined using a nonintrusive vibrating microelectrode to map electrical current before and during mitosis. Tip cells in interphase generate inward electrical currents that are maximal at the nuclear region. These currents remain concentrated over the nucleus as it migrates forward maintaining a constant distance from the growing tip. Just prior to mitosis this inward current increases twofold. During mitosis and cytokinesis current at the nuclear zone increases to four times the resting level and fluctuates, falling to zero after cell plate fusion with parental walls. The locus of outward current could not be dectected. These results suggest that plasma membrane ion currents may regulate both nuclear positioning and subsequent temporal and spatial control of cell division.     

Computational analysis of confined jet flow and mass transport in a blind tube.

A computational analysis of confined nonimpinging jet flow in a blind tube is performed as an initial investigation of the underlying fluid and mass transport mechanics of tracheal gas insufflation. A two-dimensional axisymmetric model of a laminar steady jet flow into a concentric blind-end tube is put forth and the governing continuity, momentum, and convection-diffusion equations are solved with a finite element code. The effects of the jet diameter based Reynolds number (Re(j)), the ratio of the jet-to-outer tube diameters (epsilon), and the Schmidt number (Sc) are evaluated with the determined velocity and contaminant concentration fields. The normalized penetration depth of the jet is found to increase linearly with increasing Re(j) for epsilon = O(0.1). For a given epsilon, a ring vortex that develops is observed to be displaced downstream and radially outward from the jet tip for increasing Re(j). The axial shear stress profile along the inside wall of the outer tube possesses regions of fixed shear stress in addition to a local minimum and maximum in the vicinity of the jet tip. Corresponding regions of axial shear stress gradients exist between the fixed shear stress regions and the local extrema. Contaminant concentration gradients develop across the ring vortex indicating the inward diffusion of contaminant into the jet flow. For fixed epsilon and Sc and Re(j) approximately 900, normalized contaminant flow rate is observed to be approximately twice that of simple diffusion. This model predicts modest net axial contaminant transport enhancement due to convection-diffusion interaction in the region of the ring vortex.     

Electrical bistability in a neuron model with monostable dendritic and axosomatic membranes

In a two-compartment mathematical model, we studied the reason for and conditions of manifestation of electrical bistability in a neuron composed of monostable parts. One compartment of the model simulated the dendrites; their membrane was monostable at high depolarization and characterized by an N-shaped steady current-voltage (I–V) characteristic endowed by inward synaptic current through voltage-dependent channels sensitive to N-methyl-D-aspartate (NMDA). Another compartment simulated the axosomatic region with a positively sloped linearizedI–V characteristic of the membrane monostable at the resting membrane potential. For the whole cell, bistability was obvious at a subcritical intensity of NMDA activation; the reason was the current directed from the more depolarized dendritic region into the somatic region, and the necessary condition was that the above somatopetal core current must exceed the net inward transmembrane current (the latter was the sum of the inward synaptic and outward passive extrasynaptic currents) of the dendritic compartment. This relation essentially depended on the size of the dendrites. Neirofiziologiya/Neurophysiology, Vol. 32, No. 2, pp. 98–101, March–April, 2000.     

Effects of tetraethylammonium and 4-aminopyridine on the plateau potential of circular myometrium from the pregnant rat

In the pregnant rat, spontaneous electrical activity of circular muscle (CM) changes from single, plateau-type action potentials at early and mid-term to repetitive spike trains at term. To examine mechanisms underlying the plateau, we studied the effects of potassium channel blockers tetraethylammonium (TEA) and 4-aminopyridine (4-AP) on membrane potentials in CM from rats on gestation Days 14, 15, 16, 21 (term). Apparent membrane conductance was measured at rest and during the plateau in Day 14 muscles with and without TEA. 4-AP depolarized the resting membrane on all gestation days. Therefore, a direct action of 4-AP on plateau configuration could not be separated from an indirect effect of depolarization. TEA did not affect the resting potential but increased action potential size and depolarization rate on all gestation days. On Day 16, TEA reduced plateau amplitude, unmasking small, repetitive depolarizations. D-600 decreased plateau amplitude and duration and attenuated these effects of TEA. Plateau conductance increased initially then decreased before membrane repolarization. Membrane conductance and outward rectification during the plateau were reduced by TEA. The plateau potential may result from an outwardly rectifying TEA-sensitive current combined with a slow inward current, the plateau magnitude being determined by the relative intensity of each current.     

Hypoxic induction of Ca2+-dependent action potentials in small pulmonary arteries of the cat

Small pulmonary arteries (less than 300 micron) from cats were mounted in myographs to record mechanical and electrical responses to hypoxia. When these preparations were exposed to a PO2 of 30-50 Torr after equilibration at 300 Torr they consistently developed active force, which increased or decreased in amplitude as [Ca2+] was raised or lowered, respectively, and was blocked on addition of verapamil. Intracellular electrical recording with glass microelectrodes demonstrated membrane depolarization and action potential generation when PO2 was lowered. Steady-state voltage vs. applied current curves obtained before and during hypoxia showed a significant reduction in input resistance. The relationship between membrane potential and extracellular K+ was not different during hypoxia compared with control, suggesting that there were not marked changes in K+ permeability under this condition. In the presence of verapamil to block Ca2+ inward current the hypoxia-induced action potentials were abolished concomitant with partial membrane repolarization. The results of these studies suggest that in certain isolated pulmonary arteries hypoxia induces contraction by a mechanism involving an increased Ca2+ conductance. These data suggest that the sensor involved in hypoxic pulmonary vasoconstriction may lie within the vessel wall and somehow mediates changes in smooth muscle ionic conductances.     

Depolarization of electroplax membrane in calcium-free ringer's solution

Summary Electrical stimulation, either cathodal or anodal, of the monocellular electroplax preparation in Ca-free Ringer's solution results in a sustained depolarization which is determined by the amount of current passed through the cell. The membrane potential recovers only when Ca is added again. These changes take place at the innervated side of the electroplax only. This depolarization of the membrane is pH-dependent; it depolarizes more at pH 6.0 than at pH 9.0. The membrane does not depolarize and the action potential is not blocked within an hour in Ca-free solution unless the cell is stimulated. The sustained depolarization is not prevented or reversed by curare, tetracaine, physostigmine, tetrodotoxin, and tetraethylammonium.After stimulation, the outward K current remains unchanged regardless of whether Ca is present. In contrast, the inward current is dependent on Ca in the outside solution on the innervated membrane; in the absence of Ca following stimulation, the inward K current is decreased.The depolarization by carbamylcholine is reduced in Ca-free and increased in Mgfree Ringer's solution. In contrast to the depolarization induced by electrical stimulation, these carbamylcholine depolarizations may be reversed by washing with Ca-free or Ca- and Mg-free Ringer's solution.     

Oöplasmic segregation and secretion in the Pelvetia egg is accompanied by a membrane-generated electrical current

Using an ultrasensitive extracellular vibrating electrode, I have studied the membrane-generated electrical currents around the egg of the brown alga, Pelvetia, between fertilization and germination. During this period, the egg chooses an elongation axis and moves wall-precursor vesicles to the prospective growth region where they are secreted. This results in visible oöplasmic segregation which appears under the light microscope as a 1- to 2-μm-thick clear band at the cortex of the growth region. A steady electrical current enters a small region of the membrane and leaves the remainder of the egg's surface as early as 30 min after fertilization. This early spatial current pattern is unstable and shifts position, often with more than one inward current region. However, current enters mainly on the side where germination will occur and is usually largest at the prospective cortical clearing region. The average measured early current density is 0.06 μA/cm² at 50 μm from the egg's surface, implying a surface current density of between 0.2 and 1 μA/cm² due to the extrapolation uncertainty. At germination the current increases about twofold, resulting in a total transcellular current on the order of 100 pA. Unilateral growth-orienting light reversal stimulates inward current on the new dark side, and subsequent morphological polarity reversal is preceded by electrical polarity reversal. The steady current tends to increase when the external Ca²⁺ concentration is increased or the external Na⁺ concentration is decreased, suggesting that the current is carried in part by Ca²⁺. This current will generate a transcellular electrical field which may be the force driving the observed oöplasmic segregation.     

A distributed-parameter model of the myelinated nerve fiber

This paper presents a new model for the characterization of electrical activity in the nodal, paranodal and internodal regions of isolated amphibian and mammalian myelinated nerve fibers. It differs from previous models in the following ways: (1) in its ability to incorporate detailed anatomical and electrophysiological data; (2) in its approach to the myelinated nerve fiber as a multi-axial cable; and (3) in the numerical algorithm used to obtain distributed model equation solutions for potential and current. The morphometric properties are taken from detailed electron microscopic anatomical studies (Berthold & Rydmark, 1983a, Experientia 39, 964-976). The internodal axolemma is characterized as an excitable membrane and model-generated nodal and internodal membrane action potentials are presented. A system of describing equations for the equivalent network model is derived, based on the application of Kirchoff's Current Law, which take the form of multiple cross-coupled parabolic partial differential equations. An implicit numerical integration method is developed and the numerical solution implemented on a parallel processor. Non-uniform spatial step sizes are used, enabling detailed representation of the nodal region while minimizing the number of total segments necessary to represent the overall fiber. Conduction velocities of 20.2 m sec-1 at 20 degrees C for a 15 microns diameter amphibian fiber and 57.6 m sec-1 at 37 degrees C for a 17.5 microns diameter mammalian fiber are achieved, which agrees qualitatively with published experimental data at similar temperatures (Huxley & St?mpfli, 1949, J. Physiol., Lond. 108, 315-339; Rasminsky, 1973, Arch, Neurol. 28, 287-292). The simulation results demonstrate the ability of this model to produce detailed representations of the transaxonal, transmyelin and transfiber potentials and currents, as well as the longitudinal extra-axonal, periaxonal and intra-axonal currents. Also indicated is the potential contribution of the paranodal axolemma to nodal activity as well as the presence of significant longitudinal currents in the periaxonal space adjacent to the node of Ranvier.     

Propagation through electrically coupled cells. Effects of a resistive barrier

Action potential propagation through cardiac tissue occurs in a spatially inhomogeneous three-dimensional electrical syncytium composed of discrete cells with regional variations in membrane properties and intercellular resistance. In comparison with axons, cardiac tissue presents some differences in the application of core conductor cable theory. We have used analytical and numerical techniques to contrast the propagation of action potentials along nerve axons and along cardiac strands, including an explicit inclusion of cellular anatomical factors (the surface-to-volume ratio), the strand radius, and the regional distribution of longitudinal resistance. A localized decrease in the number of gap junctions will produce a functional resistive barrier, which can lead to unidirectional block of propagation if the tissue on two sides of the barrier in either excitability or passive electrical load. However, in some circumstances, a resistive barrier separating regions of different electrical load can actually facilitate propagation into the region of larger electrical load.     

Membrane conductance of an electroporated cell analyzed by submicrosecond imaging of transmembrane potential.

Transmembrane potential was induced in a sea urchin egg by applying a microsecond electric pulse across the cell. The potential was imaged at a submicrosecond time resolution by staining the cell membrane with the voltage-sensitive fluorescent dye RH292. Under moderate electric fields, the spatial distribution of the induced potential as well as its time dependence were in accord with the theoretical prediction in which the cell membrane was regarded as an insulator. At higher field intensities, however, the potential apparently did not fully develop and tended to saturate above a certain level. The saturation is ascribed to the introduction of a large electrical conductance, in the form of aqueous openings, in the membrane by the action of the induced potential (electroporation). Comparison of the experimental and theoretical potential profiles indicates that the two regions of the membrane that opposed the electrodes acquired a high membrane conductance of the order of 1 S/cm2 within 2 microseconds from the onset of the external field. The conductance was similar in the two regions, although permeability in the two regions of the membrane long after the pulse treatment appeared quite different.     

Computation of Impulse Initiation and Saltatory Conduction in a Myelinated Nerve Fiber

A mathematical model of the electrical properties of a myelinated nerve fiber is given, consisting of the Hodgkin-Huxley ordinary differential equations to represent the membrane at the nodes of Ranvier, and a partial differential cable equation to represent the internodes. Digital computer solutions of these equations show an impulse arising at a stimulating electrode and being propagated away, approaching a constant velocity. Action potential curves plotted against distance show discontinuities in slope, proportional to the nodal action currents, at the nodes. Action potential curves plotted against time, at the nodes and in the internodes, show a marked difference in steepness of the rising phase, but little difference in peak height. These results and computed action current curves agree fairly accurately with published experimental data from frog and toad fibers.