LONGITUDINAL IMPEDANCE OF THE SQUID GIANT AXON

Longitudinal alternating current impedance measurements have been made on the squid giant axon over the frequency range from 30 cycles per second to 200 kc. per second. Large sea water electrodes were used and the inter-electrode length was immersed in oil. The impedance at high frequency was approximately as predicted theoretically on the basis of the poorly conducting dielectric characteristics of the membrane previously determined. For the large majority of the axons, the impedance reached a maximum at a low frequency and the reactance then vanished at a frequency between 150 and 300 cycles per second. Below this frequency, the reactance was inductive, reaching a maximum and then approaching zero as the frequency was decreased. The inductive reactance is a property of the axon and requires that it contain an inductive structure. The variation of the impedance with interpolar distance indicates that the inductance is in the membrane. The impedance characteristics of the membrane as calculated from the measured longitudinal impedance of the axon may be expressed by an equivalent membrane circuit containing inductance, capacity, and resistance. For a square centimeter of membrane the capacity of 1 µf with dielectric loss is shunted by the series combination of a resistance of 400 ohms and an inductance of one-fifth henry.     

Squid Axon Membrane Response to White Noise Stimulation

The current from a white noise generator was applied as a stimulus to a space-clamped squid axon in double sucrose gap. The membrane current and the voltage response of the membrane were then amplified, recorded on magnetic tape, and the stimulus was cross-correlated with the response. With subthreshold stimuli, a cross-correlation function resembling that obtained from a resonant parallel circuit is obtained. As the intensity of the input noise is increased, the cross-correlation function resembles that obtained from a less damped oscillatory circuit. When the noise intensity is further increased so that an appreciable frequency of action potentials is observed, an additional component appears in the experimental cross-correlogram. The subthreshold cross-correlogram is analyzed theoretically in terms of the linearized Hodgkin-Huxley equations. The subthreshold axon approximates a parallel resonant circuit. The circuit parameters are temperature dependent, with resonant frequency varying from approximately 100 Hz at 10°C to approximately 250 Hz at 20°C. The Q₁₀ of the resonant frequency is equal to 1.9. These values are in agreement with values found previously for subthreshold oscillations following a single action potential.     

On the role of subthreshold dynamics in neuronal signaling

The role of subthreshold dynamics in neuronal signaling is examined using periodic pulse train stimulation of the Fitzhugh-Nagumo (FN) model of nerve membrane excitability and results from the squid giant axon as an experimental data base. For a broad range of stimulus conditions the first pulse in a pulse train elicited an action potential, whereas all subsequent pulses elicited subthreshold responses, both in the axon and in the FN model. These results are not well described by the Hodgkin and Huxley 1952 model. Various different patterns of subthreshold responses, including chaotic dynamics, can be observed in both systems-the FN model and the axon-depending upon stimulus conditions. For some conditions action potentials are occasionally interspersed among the subthreshold events with randomly occurring interspike intervals. The randomness is directly attributable to the underlying subthreshold chaos-deterministic chaos-rather than to a stochastic noise source. We conclude that this mechanism may contribute to multimodal interspike interval histograms which have been observed from individual neurons throughout the nervous system.     

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.     

Digital Computer Solutions for Excitation and Propagation of the Nerve Impulse

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.     

Nonlinear cable properties of the giant axon of the cockroach Periplaneta americana

The steady state nonlinear properties of the giant axon membrane of the cockroach Periplaneta americana were studied by means of intracellular electrodes. The resistivity of this membrane markedly decreases in response to small subthreshold depolarizations. The specific slope resistance is reduced by twofold at 5 mV depolarization and by a factor of 14 at 20 mV depolarization. As a result, the spatial decay, V(X), of depolarizing potentials is enhanced when compared with the passive (exponential) decay. This enhancement is maximal at a distance of 1-1.5 mm from a point of subthreshold (0-20 mV) depolarizing perturbation. At that distance, the difference between the actual potential and the potential expected in the passive axon is approximately 30%. The effects of membrane rectification on V(X) were analyzed quantitatively with a novel derivation based on Cole's theorem, which enables one to calculate V(X) directly from the input current-voltage (I0-V) relation of a long axon. It is shown that when the experimental I0-V curve is replotted as (I0Rin)-1 against V (where Rin is the input resistance at the resting potential), the integral between any two potentials (V1 greater than V2) on this curve is the distance, in units of the resting space constant, over which V1 attenuates to V2. Excellent agreement was found between the experimental V(X) and the predicted value based solely on the input I0-V relation. The results demonstrate that the rectifying properties of the giant axon membrane must be taken into account when the electrotonic spread of even small subthreshold potentials is studied, and that, in the steady state, this behavior can be extracted from measurements at a single point. The effect of rectification on synaptic efficacy is also discussed.     

Temperature Characteristics of Excitation in Space-Clamped Squid Axons

Temperature characteristics of excitability in the squid giant axon were measured for the space-clamped axon with the double sucrose gap technique. Threshold strength-duration curves were obtained for square wave current pulses from 10 µsec to 10 msec and at temperatures from 5°C to 35°C. The threshold change of potential, at which an action potential separated from a subthreshold response, averaged 17 mv at 20°C with a Q¹⁰ of 1.15. The average threshold current density at rheobase was 12 µa/cm² at 20°C with a Q¹⁰ of 2.35 compared to 2.3 obtained previously. At short times the threshold charge was 1.5·10^-8 coul/cm². This was relatively independent of temperature and occasionally showed a minimum in the temperature range. At intermediate times and all temperatures the threshold currents were less than for both the single time constant model and the two factor excitation process as developed by Hill. FitzHugh has made computer investigations of the effect of temperature on the excitation of the squid axon membrane as represented by the Hodgkin-Huxley equations. These are in general in good agreement with our experimental results.     

Extension of the neuronal membrane model to account for suppression of the action potential by a constant magnetic field

A biophysical explanation of the reduced excitability in neurons exposed to a constant magnetic field is based on an extended neuronal membrane model. In the presence of a constant magnetic field, reduced excitability is manifested as an increase in the excitation threshold and a decrease in the frequency of action potentials. The proposed explanation for the reduced excitability rests on the well-known Hall effect. The separation of charges resulting from the Lorentz force exerted on moving intracellular ions leads to the formation of a Hall electric field in a direction perpendicular to that of action-potential transmission. Consequently, the ion current for discharging the membrane capacitance is reduced in the presence of a magnetic field, thereby limiting initiation of the action potential. The validity of the proposed biophysical explanation is justified analytically and verified by simulations based on the Hodgkin and Huxley model for the electrical excitability of a neuron. Based on derivation of the current segregation ratio α characterizing the reduction in the stimulating current from first principles, the equivalent circuit model of the neuronal membrane is extended to account for the reduced excitability of neurons exposed to a constant magnetic field.     

The standard Hodgkin-Huxley model and squid axons in reduced external Ca++ fail to accommodate to slowly rising currents.

Accommodation may be defined as an increase in the threshold of an excitable membrane when the membrane is subjected to a sustained subthreshold depolarizing stimulus. Some excitable membranes show accommodation in response to currents which rise linearly at a very slow rate. In this report we point out a theoretical and an experimental counterexample, i.e., a nerve model and an axon which do not accommodate. The nerve model is the standard Hodgkin-Huxley axon, which Hodgkin and Huxley expected not to be excited by a very slowly rising current. This expectation is often quoted as fact, in spite of contrary calculations which we confirm. We have found that squid axons in seawater with reduced divalent cation concentration also do not accommodate to slowly rising currents.     

Processes of excitation in the dendrites and in the soma of single isolated sensory nerve cells of the lobster and crayfish

The stretch receptor organs of Alexandrowicz in lobster and crayfish possess sensory neurons which have their cell bodies in the periphery. The cell bodies send dendrites into a fine nearby muscle strand and at the opposite pole they give rise to an axon running to the central nervous system. Mechanisms of excitation between dendrites, cell soma, and axon have been studied in completely isolated receptor structures with the cell components under visual observation. Two sensory neuron types were investigated, those which adapt rapidly to stretch, the fast cells, and those which adapt slowly, the slow cells. 1. Potentials recorded from the cell body of the neurons with intracellular leads gave resting potentials of 70 to 80 mv. and action potentials which in fresh preparations exceeded the resting potentials by about 10 to 20 mv. In some experiments chymotrypsin or trypsin was used to make cell impalement easier. They did not appreciably alter resting or action potentials. 2. It has been shown that normally excitation starts in the distal portion of dendrites which are depolarized by stretch deformation. The changed potential within the dendritic terminals can persist for the duration of stretch and is called the generator potential. Secondarily, by electrotonic spread, the generator potential reduces the resting potential of the nearby cell soma. This excitation spread between dendrites and soma is seen best during subthreshold excitation by relatively small stretches of normal cells. It is also seen during the whole range of receptor stretch in neurons in which nerve conduction has been blocked by an anesthetic. The electrotonic changes in the cells are graded, reflecting the magnitude and rate of rise of stretch, and presumably the changing levels of the generator potential. Thus in the present neurons the resting potential and the excitability level of the cell soma can be set and controlled over a wide range by local events within the dendrites. 3. Whenever stretch reduces the resting membrane potential, measured in the relaxed state in the cell body, by 8 to 12 mv. in slow cells and by 17 to 22 mv. in fast cells, conducted impulses are initiated. It is thought that in slow cells conducted impulses are initiated in the dendrites while in fast cells they arise in the cell body or near to it. In fresh preparations the speed of stretch does not appreciably influence the membrane threshold for discharges, while during developing fatigue the firing level is higher when extension is gradual. 4. Some of the specific neuron characteristics are: Fast receptor cells have a relatively high threshold to stretch. During prolonged stretch the depolarization of the cell soma is not well maintained, presumably due to a decline in the generator potential, resulting in cessation of discharges in less than a minute. This appears to be the basis of the relatively rapid adaptation. A residual subthreshold depolarization can persist for many minutes of stretch. Slow cells which resemble the sensory fibers of vertebrate spindles are excited by weak stretch. Their discharge rate remains remarkably constant for long periods. It is concluded that, once threshold excitation is reached, the generator potential within slow cell dendrites is well maintained for the duration of stretch. Possible reasons for differences in discharge properties between fast and slow cells are discussed. 5. If stretch of receptor cells is gradually continued above threshold, the discharge frequency first increases over a considerable range without an appreciable change in the firing level for discharges. Beyond that range the membrane threshold for conducted responses of the cell soma rises, the impulses become smaller, and partial conduction in the soma-axon boundary region occurs. At a critical depolarization level which may be maintained for many minutes, all conduction ceases. These overstretch phenomena are reversible and resemble cathodal block. 6. The following general scheme of excitation is proposed: stretch deformation of dendritic terminals → generator potential → electrotonic spread toward the cell soma (prepotential) → dendrite-soma impulse → axon impulse. 7. Following release of stretch a transient hyperpolarization of slow receptor cells was seen. This off effect is influenced by the speed of relaxation. 8. Membrane potential changes recorded in the cell bodies serve as very sensitive detectors of activity within the receptor muscle bundles, indicating the extent and time course of contractile events.     

Effect of a volatile anesthetic upon presynaptic excitability in mammalian hippocampus

Changes in presynaptic terminal axon excitability produced by enflurane in the rat hippocampal slice preparation were investigated by stimulation of Schaeffer collateral terminal axons and by recording single unit antidromic action potentials. Stimulating pulses were preceded by conditioning hyperpolarizing or depolarizing current pulses. A plot of net threshold for action potential generation against the conditioning pulse yields an "accommodation curve;" changes in this curve can be used to assess the mechanism by which changes in excitability are produced. Enflurane, at a concentration equivalent to approximately equal to 1.3 times the minimum alveolar concentration, reduced excitability of terminal axons and increased accommodation in a manner consistent with a possible change in the inactivation of gNa.     

The velocity of conduction in nerve fiber and its electric characteristics

The theory developed in this paper shows that the propagation of spike potential along a nerve fiber and the conduction of an electric wave along an inert inorganic conductor follow a common quantitative relationship. This result gives further support to the belief that propagation of excitation is an electrical process. The basic idea of the theory is derived from the consideration that velocity has, by its mathematical definition, a local meaning; conduction in a nerve is completely determined by the local characteristics of the latter, as well as those of the wave. The final formula derived does not make use of any other field of science beyond the fundamental principles of electricity. It gives the conduction velocity in terms of the electric characteristics of the fiber and of the duration of the spike potential. The formula is in agreement with the known dependence of the conduction velocity on various parameters characterizing the axon. The computed velocity agrees with the measured ones on the squid giant axon, crab nerve axon, frog muscle fiber and Nitella cell. The membrane inductance appears as a velocity controling agent which prevents also a possible distortion of the spike potential during conduction. The structural meaning of the electric characteristics of the axon membrane is discussed from the viewpoint of the diffusion theory. A formula for the velocity of spread of the electrotonus is also derived.     

Oscillatory behavior of the squid axon membrane potential

Squid axons impaled with a microelectrode have been treated with concentrations of xylene and benzene such that there is no change in threshold or resting potential at 20°C., while the spike height declines about 10 mv. A decrease in ambient temperature results in large, reversible, increases in threshold. While neither low temperature nor the added blocking agent induces repetitive firing from a single stimulus, the two treatments when combined do yield repetitive responses which commence at a sharply defined temperature. The alteration in the membrane responsible for the effects observed can be described by saying that there has been a large increase in the inductance of the equivalent electric circuit, and the temperature coefficient of the apparent membrane inductance has a Q₁₀ = 5.     

A model for the polarization of neurons by extrinsically applied electric fields.

A model is presented for the subthreshold polarization of a neuron by an applied electric field. It gives insight into how morphological features of a neuron affect its polarizability. The neuronal model consists of one or more extensively branched dendritic trees, a lumped somatic impedance, and a myelinated axon with nodes of Ranvier. The dendritic trees branch according to the 3/2-power rule of Rall, so that each tree has an equivalent cylinder representation. Equations for the membrane potential at the soma and at the nodes of Ranvier, given an arbitrary specified external potential, are derived. The solutions determine the contributions made by the dendritic tree and the axon to the net polarization at the soma. In the case of a spatially constant electric field, both the magnitude and sign of the polarization depend on simple combinations of parameters describing the neuron. One important combination is given by the ratio of internal resistances for longitudinal current spread along the dendritic tree trunk and along the axon. A second is given by the ratio between the DC space constant for the dendritic tree trunk and the distance between nodes of Ranvier in the axon. A third is given by the product of the electric field and the space constant for the trunk of the dendritic tree. When a neuron with a straight axon is subjected to a constant field, the membrane potential decays exponentially with distance from the soma. Thus, the soma seems to be a likely site for action potential initiation when the field is strong enough to elicit suprathreshold polarization. In a simple example, the way in which orientation of the various parts of the neuron affects its polarization is examined. When an axon with a bend is subjected to a spatially constant field, polarization is focused at the bend, and this is another likely site for action potential initiation.     

Surface charges on membranes

Summary The equations of membrane potential developed by Kobatake and coworkers have been applied to the literature data on the resting membrane potential of the crayfish andMyxicola axons to derive values for the surface charge density present on the axon membranes. Some shortcomings of the method are briefly discussed. The value for the surface charge density derived for the squid axon membrane agreed with a similar value derived from measurements of shifts in Na and/or potassium conductance-voltage relations following changes in the concentration of calcium in the solutions bathing the axons.     

Determinants of excitability in cardiac myocytes: mechanistic investigation of memory effect

The excitability of a cardiac cell depends upon many factors, including the rate and duration of pacing. Furthermore, cell excitability and its variability underlie many electrophysiological phenomena in the heart. In this study, we used a detailed mathematical model of the ventricular myocyte to investigate the determinants of excitability and gain insight into the mechanism by which excitability depends on the rate and duration of pacing (the memory effect). Results: i) The primary determinant of excitability depends upon the duration (T) of the stimulus. ii) For a short T, excitability is determined by the difference between the threshold membrane potential and the resting membrane potential. iii) For a long T, excitability is determined by the resting membrane resistance, R(m). iv) In the case of long T, pacing induced changes in [Na(+)](i) and [Ca(2+)](i) over time affect R(m) and excitability by shifting the current-voltage (IV) curve in the vertical direction and are responsible for the memory effect. CONCLUSIONS: The results have important implications during an arrhythmia, where a cardiac cell may be subjected to rapid repetitive excitation for an extended period of time. Effective anti-arrhythmic strategies may be developed to exploit the R(m) dependence of excitability for a long T.     

Cable analysis of a motor-nerve terminal branch in a volume conductor

An equivalent electrical circuit is given for a branch of an amphibian motor-nerve terminal in a volume conductor. The circuit allows for longitudinal current flow inside the axon as well as between the axon and its Schwann cell sheath, and also for the radial leakage of current through the Schwann cell sheath. Analytical and numerical solutions are found for the spatial and time dependence of the membrane potential resulting from the injection of depolarizing current pulses by external electrodes at one or two separate locations on the terminal. These solutions show that the depolarization at an injection site can cause a hyperpolarization at sites a short distance away. This effect becomes more pronounced in a short terminal with sealed-end boundary conditions. The hyperpolarization provides a possible explanation for recent experimental results, which show that the average quantal release due to a test depolarizing current pulse delivered by an electrode at one site on a nerve terminal is reduced by the application of an identical conditioning pulse at a neighbouring site.     

Kinetics of ion transport in lipid membranes induced by lysine-valinomycin and derivatives.

Lysine-valinomycine and two N epsilon-acyl derivatives are compared with respect to their potency to transport Rb+ ions across thin lipid membranes. Lysine-valinomycin acts as a neutral ion carrier only above a pH of about 7 of the aqueous solutions, while at lower pH the molecules seem to be positively charged due to a protonation of the epsilon-NH2 group of the lysine residue. A kinetic analysis based on voltage jump relaxation experiments and on the nonlinearity of the current-voltage characteristics showed that the conductance increment delta per carrier molecule for uncharged lysine-valinomycin is similar to that of natural valinomycin. The attachment of a rather bulky side group such as the dansyl or para-nitrobenzyloxycarbonyl group reduced delta by approximately one order of magnitude. Some of the relaxation data of the valinomycin analogues were influenced by an unspecific relaxation of the pure lipid membrane. This structural relaxation represents a limitation to the possibility of analyzing specific transport systems in thin lipid membranes by the voltage jump or charge pulse techniques. It is shown that the time dependence of this structural relaxation--which was first published by Sargent (1975)--is at variance with a three capacitor equivalent circuit of the membrane, which was suggested by Coster and Smith (1974) on the basis of a.c. measurements. A modified equivalent circuit has been found to represent a satisfactory analogue for the current relaxation in the presence of valinomycin. It turned out, however, that such an equivalent circuit provides little insight into the molecular mechanism of transport.     

Are axoplasmic microtubules necessary for membrane excitation?

Summary The excitability of the squid giant axon was studied as a function of transmembrane hydrostatic pressure differences, the latter being altered by the technique of intracellular perfusion. When a KF solution was used as the internal medium, a pressure difference of about 15 cm water had very little effect on either the membrane potential or excitability. However, within a few minutes after introducing either a KCl-containing, a KBr-containing, or a colchicine-containing solution as the internal medium, with the same pressure difference across the membrane, the axon excitability was suppressed. In these cases, removal of the pressure difference restored the excitability, indicating that the structure of membrane was not irreversibly damaged. Electron-microscopic observations of these axons revealed that the perfusion with a KF solution or colchicine-containing solution preserves the submembranous cytoskeletal layer, whereas perfusion with a KCl or KBr solution dissolves it. These results suggest that the submembranous cytoskeletons including microtubules provide an important mechanical support to the excitable membrane but are not essential elements in channel activities.