Subthreshold Behavior and Phenomenological Impedance of the Squid Giant Axon

The oscillatory behavior of the cephalopod giant axons in response to an applied current has been established by previous investigators. In the study reported here the relationship between the familiar "RC" electrotonic response and the oscillatory behavior is examined experimentally and shown to be dependent on the membrane potential. Computations based on the three-current system which was inferred from electrical measurements by Hodgkin and Huxley yield subthreshold responses in good agreement with experimental data. The point which is developed explicitly is that since the three currents, in general, have nonzero resting values and two currents, the "Na" system and the "K" system, are controlled by voltage-dependent time-variant conductances, the subthreshold behavior of the squid axon in the small-signal range can be looked upon as arising from phenomenological inductance or capacitance. The total phenomenological impedance as a function of membrane potential is derived by linearizing the empirically fitted equations which describe the time-variant conductances. At the resting potential the impedance consists of three structures in parallel, namely, two series RL elements and one series RC element. The true membrane capacitance acts in parallel with the phenomenological elements, to give a total impedance which is, in effect, a parallel R, L, C system with a "natural frequency" of oscillation. At relatively hyperpolarized levels the impedance "degenerates" to an RC system.     

Cation Interdiffusion in Squid Giant Axons

Radiotracer techniques were used to study the influxes and effluxes of various univalent cations in internally perfused squid giant axons. Membrane currents and conductances were determined by the voltage-clamp technique under analogous internal and external conditions. Both sodium-containing and sodium-free internal and external media were studied. Membrane impedance was measured with an ac impedance bridge to determine the general magnitude and time course of the impedance loss which accompanied the excitation process in both varieties of external media. Maximum transmembrane currents were found to be of comparable magnitude to the charge transfer associated with the peak interdiffusion flux measured under the same conditions. The product of the membrane resistance and the interdiffusion flux remained constant over a wide range of resistance and flux values, both at rest and during activity, both in sodium-containing and sodium-free media. The implications of these findings for excitation theory are discussed.     

Discontinuous feedback amplifier: principles and application to the measurement of transmembrane potentials and currents of frog atrial fibers

The study and achievement of a discontinuous feedback amplifier to measure membrane potentials and currents in frog atrial fibres using the double sucrose gap technique was achieved. It was shown that, with the present device, the effects of the resistance in series with the membrane resistance and the membrane capacity on the measure of cardiac membrane potentials and fast currents are markedly reduced.     

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.     

Minimizing the influence of the series resistance in potential clamped Ranvier nodes

Series resistance artifacts in ionic current measurements on single myelinated nerve fibres are commonly minimized by reducing sodium currents. Doing this some deviations from the predictions of the Hodgkin-Huxley-Frankenhaeuser formalism become evident. In the present investigation two methods to reduce sodium currents were used with and without compensated feedback to examine the influence of the nodal series resistance. Changing the availability of sodium permeability by appropriate prepulses peak sodium current-voltage relations obeyed the Hodgkin-Huxley-Frankenhaeuser formalism only provided the amount of compensated feedback was set to give minimum of the current-voltage relation near E = 0. For reduced sodium concentration in the bathing fluid the so-called independence principle predicts a shift of the minimum of the current-voltage relation on the potential axis in negative direction as compared to ordinary Ringer solution, if the effective series resistance is sufficiently small. This was confirmed by experiments only if the above mentioned amount of compensated feedback was used. The results suggest that the "E = 0"-criterion indicates optimum compensation of the influence of the series resistance.     

Subtle Paranodal Injury Slows Impulse Conduction in a Mathematical Model of Myelinated Axons

This study explores in detail the functional consequences of subtle retraction and detachment of myelin around the nodes of Ranvier following mild-to-moderate crush or stretch mediated injury. An equivalent electrical circuit model for a series of equally spaced nodes of Ranvier was created incorporating extracellular and axonal resistances, paranodal resistances, nodal capacitances, time varying sodium and potassium currents, and realistic resting and threshold membrane potentials in a myelinated axon segment of 21 successive nodes. Differential equations describing membrane potentials at each nodal region were solved numerically. Subtle injury was simulated by increasing the width of exposed nodal membrane in nodes 8 through 20 of the model. Such injury diminishes action potential amplitude and slows conduction velocity from 19.1 m/sec in the normal region to 7.8 m/sec in the crushed region. Detachment of paranodal myelin, exposing juxtaparanodal potassium channels, decreases conduction velocity further to 6.6 m/sec, an effect that is partially reversible with potassium ion channel blockade. Conduction velocity decreases as node width increases or as paranodal resistance falls. The calculated changes in conduction velocity with subtle paranodal injury agree with experimental observations. Nodes of Ranvier are highly effective but somewhat fragile devices for increasing nerve conduction velocity and decreasing reaction time in vertebrate animals. Their fundamental design limitation is that even small mechanical retractions of myelin from very narrow nodes or slight loosening of paranodal myelin, which are difficult to notice at the light microscopic level of observation, can cause large changes in myelinated nerve conduction velocity.     

The effect of metacaine on the slow variation of peak sodium currents in potential clamped Ranvier nodes following changes in holding potential

The effect of changes in the holding potential on peak sodium currents in isolated myelinated nerve fibres (peak INa) was investigated with the conventional sodium inactivation being kept at h infinity = 1. In Ringer solution no stationary values of peak INa could be obtained over the potential range tested. Near the normal resting potential, ER, peak INa changed with time clearly even after 10 min. Therefore, the individual values of peak INa as normalized by peak INa at ER and corrected for the unevitable run-down of peak INa could not serve as measure for stationary values of any membrane parameter. Under metacaine (1 mmol/l) peak INa changed comparably faster and proved to be less potential dependent as compared to peak INa of the untreated fibre. The effects observed are not necessarily governed by a specific process located inside the nodal membrane.     

Transfection of HeLa-cells with pEGFP plasmid by impedance power-assisted electroporation

Bioimpedance spectrometry was applied to study cell viability and pEGFP plasmid-transfection efficiency in electroporation (EP) of 20,000 HeLa cells with 0.3 microg DNA in 90 microl low conductivity 0.32 M sucrose medium of pH 7.5. Monopolar rectangular pulses, of field strength 75 V/mm, and pulse length 0.1 ms were applied in 1-16 repetitions with a 10-sec pause interval between pulses. Surviving cells were stained by crystal violet and counted using a confocal microscope. Transfected cells were fixed with 10% formaldehyde and counted as green spots in a fluorescence microscope. In the present investigation we used the method of bioimpedance spectrometry to analyze the effect of EP on survival and transfection ratio of cells in suspension. DC and low-frequency AC currents preferably pass through the medium due to the high impedance of the cell membrane. At frequencies above 10 kHz the impedance of the cell membrane starts to decrease and the impedance value of the cell suspension approach a lower limit value Rinfinity at infinite frequency. Recording of electrical impedance spectra of cells in culture was performed over a frequency range of 10 Hz to 125 kHz, allowing separation of the contribution from extracellular space and that of the cell membranes. A parallel resistance capacitance model of the cell suspension was used to evaluate the response of applying EP pulses. The values of the collective membrane resistance RM decay exponentially (r2=0.995) with the number of applied pulses. The ratio of the extrapolated value of the intact membrane resistance before pulsing, RM,0, and the value RM,N after each pulse makes an index of the effect of electroporation on the cells. The ratio RM,N/RM,0 as well as the relative change of the dissipation factor, tandelta, on the "Loss Change Index" (LCI) fits well a dose-response model (r2=0.98) with the number of applied pulses. The changes in the model parameters membrane resistance DeltaRM=[1-RM,N/RM,o] and loss factor [1-tandelta0/tandeltaN] correlate well with the transfection ratio and fraction of dead cells. Those parameters were used for power-assisted electroporation in monitoring, controlling, and optimizing the EP procedure.     

Voltage Clamp of Cardiac Muscle: A Theoretical Analysis of Early Currents in the Single Sucrose Gap

A theoretical model is presented for the early currents in the voltage clamp of cardiac muscle using the single sucrose gap technique. The preparation is represented by a single one-dimensional active cable with modified Hodgkin-Huxley membrane and the interent imperfections in the technique are also included, e.g., leakage through the sucrose gap and resistance in series with the membrane in the test compartment. The stability of the control system was found to depend on the position of the control point with respect to the sucrose gap border. Computed currents for a stable system closely resembled those in the literature and those from a near-ideal system (e.g., squid axon.) The potential immediately across the membrane, however (not including potential drops across the series resistance external to the membrane), was found to be essentially uncontrolled and the “current-voltage” relationship was shown to be almost independent of membrane properties.     

Increased charge displacement in the membrane of myelinated nerve at reduced extracellular pH.

Asymmetry currents were measured in nodes of myelinated nerve fibers from Rana esculenta at extracellular pH values of 5.2, 7.0, and 8.1 by averaging the currents during and after 1-ms depolarizing and hyperpolarizing voltage pulses. The charge displacement in the nodal membrane was obtained by numerical integration of the asymmetry currents. Lowering the pH from 7.0 to 5.2 significantly slows down the kinetics of the fast charge displacement during depolarization but hardly affects the kinetics after repolarization. The pH reduction increases the maximum charge displacement during depolarization by 46%. No differences between asymmetry currents were found between pH 7.0 and 8.1. It is concluded that protonation by extracellular H+ ions may increase the net charge or the transition range of mobile subunits in the nerve membrane.     

The Electrodynamics of Current-Current Interaction in a Double Helix

General expressions for the tangential and radial forces for two current elements moving contra-wise to each other in a helical manner are derived. The two repulsive forces are calculated as a function of velocity, with explicit values given for parameters, appropriate to DNA. The calculated electrodynamic respulsive forces oppose the stabilizing standard radial and tangential forces in the helix and, if large enough, can result in strand separation. The currents required are of the same order of magnitude as observed intra-cellular currents and as passed recently through short segments of duplex DNA. A speculative mechanism involving a resistance-capacitance membrane network in series with a resistance DNA network is formulated. Strand separation in DNA may well originate in such an electrodynamic system.     

Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents

Many neuron types exhibit preferred frequency responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of biophysical parameters on these properties is not well understood. We propose a general framework to analyze the role of different ionic currents and their interactions in shaping the properties of impedance amplitude and phase in linearized biophysical models and demonstrate this approach in a two-dimensional linear model with two effective conductances g _L and g ₁. We compute the key attributes of impedance and phase (resonance frequency and amplitude, zero-phase frequency, selectivity, etc.) in the g _L ???g ₁ parameter space. Using these attribute diagrams we identify two basic mechanisms for the generation of resonance: an increase in the resonance amplitude as g ₁ increases while the overall impedance is decreased, and an increase in the maximal impedance, without any change in the input resistance, as the ionic current time constant increases. We use the attribute diagrams to analyze resonance and phase of the linearization of two biophysical models that include resonant (I _h or slow potassium) and amplifying currents (persistent sodium). In the absence of amplifying currents, the two models behave similarly as the conductances of the resonant currents is increased whereas, with the amplifying current present, the two models have qualitatively opposite responses. This work provides a general method for decoding the effect of biophysical parameters on linear membrane resonance and phase by tracking trajectories, parametrized by the relevant biophysical parameter, in pre-constructed attribute diagrams.     

Impedance analysis of a tight epithelium using a distributed resistance model.

This paper develops techniques for equivalent circuit analysis of tight epithelia by alternating-current impedance measurements, and tests these techniques on rabbit urinary bladder. Our approach consists of measuring transepithelial impedance, also measuring the DC voltage-divider ratio with a microelectrode, and extracting values of circuit parameters by computer fit of the data to an equivalent circuit model. We show that the commonly used equivalent circuit models of epithelia give significant misfits to the impedance data, because these models (so-called "lumped models") improperly represent the distributed resistors associated with long and narrow spaces such as lateral intercellular spaces (LIS). We develop a new "distributed model" of an epithelium to take account of these structures and thereby obtain much better fits to the data. The extracted parameters include the resistance and capacitance of the apical and basolateral cell membranes, the series resistance, and the ratio of the cross-sectional area to the length of the LIS. The capacitance values yield estimates of real area of the apical and basolateral membranes. Thus, impedance analysis can yield morphological information (configuration of the LIS, and real membrane areas) about a living tissue, independently of electron microscopy. The effects of transport-modifying agents such as amiloride and nystatin can be related to their effects on particular circuit elements by extracting parameter values from impedance runs before and during application of the agent. Calculated parameter values have been validated by independent electrophysiological and morphological measurements.     

Properties of internally perfused, voltage-clamped, isolated nerve cell bodies

The membrane properties of isolated neurons from Helix aspersa were examined by using a new suction pipette method. The method combines internal perfusion with voltage clamp of nerve cell bodies separated from their axons. Pretreatment with enzymes such as trypsin that alter membrane function is not required. A platinized platinum wire which ruptures the soma membrane allows low resistance access directly to the cell's interior improving the time resolution under voltage clamp by two orders of magnitude. The shunt resistance of the suction pipette was 10-50 times the neuronal membrane resistance, and the series resistance of the system, which was largely due to the tip diameter, was about 10(5) omega. However, the peak clamp currents were only about 20 nA for a 60-mV voltage step so that measurements of membrane voltage were accurate to within at least 3%. Spatial control of voltage was achieved only after somal separation, and nerve cell bodies isolated in this way do not generate all-or-none action potentials. Measurements of membrane potential, membrane resistance, and membrane time constant are equivalent to those obtained using intracellular micropipettes, the customary method. With the axon attached, comparable all-or-none action potentials were also measured by either method. Complete exchange of Cs+ for K+ was accomplished by internal perfusion and allowed K+ currents to be blocked. Na+ currents could then be blocked by TTX or suppressed by Tris-substituted snail Ringer solution. Ca2+ currents could be blocked using Ni2+ and other divalent cations as well as organic Ca2+ blockers. The most favorable intracellular anion was aspartate-, and the sequence of favorability was inverted from that found in squid axon.     

The Selective Inhibition of Delayed Potassium Currents in Nerve by Tetraethylammonium Ion

The effect of tetraethylammonium ion (TEA) on the voltage clamp currents of nodes of Ranvier of frog myelinated nerve fibers is studied. The delayed K currents can be totally abolished by TEA without affecting the transient Na currents or the leakage current in any way. Both inward and outward currents disappear. In low TEA concentrations small K currents remain with normal time constants. The dose-response relationship suggests the formation of a complex between TEA and a receptor with a dissociation constant of 0.4 mM. Other symmetrical quaternary ammonium ions have very little effect. There is no competition between TEA and agents that affect the Na currents such as Xylocaine, tetrodotoxin, or Ca ions. The pharmacological data demonstrate that the Na, K, and leakage permeabilities are chemically independent, probably because their mechanisms occupy different sites on the nodal membrane. The data are gathered and analyzed by digital computer.     

ELECTRIC IMPEDANCE OF SUSPENSIONS OF ARBACIA EGGS

Apparatus has been designed and constructed for the measurement of the electric impedance of suspensions of Arbacia eggs in sea water to alternating currents of frequencies from one thousand to fifteen million cycles per second. This apparatus is simple, rugged, compact, accurate, and rapid. The data lead to the conclusions that the specific resistance of the interior of the egg is about 90 ohm cm. or 3.6 times that of sea water, and that the impedance of the surface of the egg is probably similar to that of a "polarization capacity". The characteristics of this surface impedance can best be determined by measurements of the capacity and resistance of suspensions of eggs. No specific change has been found in the interior resistance or the surface impedance which can be related either to membrane formation or to cell division.     

Maturation involves suppression of voltage-gated currents in the frog oocyte

Voltage- and time-dependent currents having slow kinetics have been studied in plasma membranes of immature oocytes of the european frog, Rana esculenta. IK, corresponding to an outward flow of K+, is activated at potentials more positive than about -40 mV, and subserves outward rectification; Iir, corresponding to an outward flow of Cl-, is activated at potentials more negative than about -80 mV and subserves inward rectification. Such currents can act as negative feedback mechanisms in the control of membrane potential in the immature oocyte and limit to a somewhat restricted range its possible deviations from resting values. Besides IK, membrane depolarizations to potentials more positive than about +30 mV are capable of activating INa, corresponding to outflow of Na+. By contrast, the frog mature egg-cell has a single voltage- and time-dependent current, IM, activated at potentials more positive than +30 mV, with properties similar to INa. The disappearance of IK and Iir along with remarkable reduction in leakage lowers impedance in the egg membrane. It seems reasonable to suggest that the observed changes in membrane permeability reflect changes which have taken place along the maturation process and are of importance for successful fertilization.     

Linear electrical properties of passive and active currents in spherical heart cell clusters.

Impedance studies were performed on small spherical clusters of embryonic chick heart cells grown in tissue culture. Each syncytial cluster was impaled with two microelectrodes; one injected low amplitude stochastic current and the other recorded the resulting perturbation of intracellular potential. The current and potential records were digitized, decomposed into their sinusoidal components, and the frequency domain impedance of the cluster was determined. The impedance data were compared with a theory for current flow in a spherical syncytium and values were derived for parameters describing the membranes and intercellular clefts of the tissue. The clusters were spontaneously active but usually became temporarily quiescent when impaled with two electrodes. The potential stabilized at a value close to -30 mV. At this depolarized potential, active slow currents, presumably present in the cardiac action potential, contributed noticeably to the linear impedance, producing a resonant peak in the magnitude of the impedance at a frequency of 1-3 Hz. The linearized impedance functions for these currents were characterized in the presence and absence of tetrodotoxin (TTX) and D-600. TTX had no noticeable effect on the impedance but D-600 essentially abolished the active currents. Although the ionic basis of these currents is not known, frequency domain analysis appears to be a viable technique for studying slow currents in heart muscle.     

Decreased rate of sodium conductance inactivation in the node of Ranvier induced by a polypeptide toxin from sea anemone.

The effects of two toxins extracted from the tentacles of Anemonia sulcata on ionic currents have been tested on the nodal membrane of myelinated nerve fibres from Rana esculenta. While external application of Toxin I at 100 muM leaves both specific ionic currents unmodified, Toxin II at 10 muM reacts with a receptor site associated with the sodium conductance inactivation gating. Since internal application by diffusion of Toxin II at a concentration of 700 muM leaves the ionic currents unchanged, the receptor site is most likely located on the external side of the nodal membrane. An equilibrium dissociation constant for the effects of Toxin II was estimated as 20 muM. The on-reaction is fast (rate constant for the on-reaction roughly equal to 3.103 M-1) suggesting a readily accesible receptor site for the toxin. The kinetics characteristics of the sodium currents recorded in the presence of Toxin II suggest that there are at least two steps in the reaction leading to Na+ -channels with the inactivation gate completely immobilized. The relatively fast reversibility of the intermediate stage of the reaction and the rather slow but, in the end, complete reversal of the toxin effects suggest that the toxin acts by modifying the energy profile for the transition "inactivation gate in the open configuration to inactivation gate in the closed configuration." Toxin II at higher concentrations (greater than 100 muM) also inhibits the potassium currents but these effects were not studied in any detail.     

Characteristics of sodium tail currents in Myxicola axons. Comparison with membrane asymmetry currents.

Sodium currents after repolarization to more negative potentials after initial activation were digitally recorded in voltage-clamped Myxicola axons compensated for series resistance. The results are inconsistent with a Hodgkin-Huxley-type kinetic scheme. At potentials more negative than -50 mV, the Na+ tails show two distinct time constants, while at more positive potentials only a single exponential process can be resolved. The time-course of the tail currents was totally unaffected when tetrodotoxin (TTX) was added to reduce gNa to low values, demonstrating the absence of any artifact dependent on membrane current. Tail currents were altered by [Ca++] in a manner consistent with a simple alteration in surface potential. Asymmetry current "off" responses are well described by a single exponential. The time constant for this response averaged 2.3 times larger than that for the rapid component of the Na+ repolarization current and was not sensitive to pulse amplitude or duration, although it did vary with holding potential. Other asymmetry current observations confirm previous reports on Myxicola.