Bifurcation of periodic solutions of Hodgkin-Huxley model for the squid giant axon

The Hodgkin-Huxley model of the space-clamped squid giant axon is shown to admit unstable periodic solutions for current stimuli less than the stimulus at which the rest state becomes linearly unstable. The periodic solutions are demonstrated both by bifurcation theory and by numerical integration. The presence of subcritical unstable oscillations explains the discontinuous behaviour of the amplitude of the repetitive response as a function of current stimulus     

Toward a multi-membrane model for potassium conduction in squid giant axon.

Possible physical mechanisms are considered which come close to a quantitative explanation for features of the potassium admittance magnitude. At 1–30 Hz there is an elevation of [Y] and positive phase above that obtained from the Hodgkin-Huxley model. Moreover there appears to be a slight negative phase for lower frequencies. An additional important feature for model fitting is the movement of the middle zero-phase crossing to the left with depolarization. Two general classes of subsystems are discussed. (1) Extracellular: potassium accumulation, barriers to diffusion near or adjacent to the excitable membrane, diffusion with volume flow, bulklimited diffusion through the Schwann cell layer and adsorption or absorption by the Schwann cells; (2) processes intrinsic to the excitable membrane: cyclic steady state, co-operative, inactivating and second order. A generalized potassium inactivation is treated in detail which provides fairly quantitative fits to transmembrane transfer data with a voltage-dependent inactivation time constant ranging between 40 and 100 ms. However, potassium accumulation coupled with hypothesized sorptive effects of the greater membrane, particularly the Schwann cell layer, also provide reasonable fits. Based on lack of experimental evidence for an inactivation, the choice is made for a multicompartment model. When an HH membrane element is combined with accumulation-depletion in an extracellular space and with a bulk limited or surface limited diffusion through the Schwann cells good agreement is obtained with measured admittance.     

Single sodium channels from the squid giant axon

Since the work of A. L. Hodgkin and A. F. Huxley (1952. J. Physiol. [Lond.].117:500-544) the squid giant axon has been considered the classical preparation for the study of voltage-dependent sodium and potassium channels. In this preparation much data have been gathered on macroscopic and gating currents but no single sodium channel data have been available. This paper reports patch clamp recording of single sodium channel events from the cut-open squid axon. It is shown that the single channel conductance in the absence of external divalent ions is approximately 14 pS, similar to sodium channels recorded from other preparations, and that their kinetic properties are consistent with previous results on gating and macroscopic currents obtained from the perfused squid axon preparation.     

Calcium currents in squid giant axon.

Voltage-clamp experiments were carried out on intracellularly perfused squid giant axons in a Na-free solution of 100 mM CaCl2+sucrose. The internal solution was 25 mM CsF+sucrose or 100 mM RbF+50mM tetraethylammonium chloride+sucrose. Depolarizing voltage clamp steps produced small inward currents; at large depolarizations the inward current reversed into an outward current. Tetrodotoxin completely blocked the inward current and part of the outward current. No inward current was seen with 100 mM MgCl2+sucrose as internal solution. It is concluded that the inward current is carried by Ca ions moving through the sodium channel. The reversal potential of the tetrodotoxin-sensitive current was +54mV with 25 mM CsF+sucrose inside and +10 mV with 100 mM RbF+50 mM tetraethylammonium chloride+sucrose inside. From the reversal potentials measured with varying external and internal solutions the relative permeabilities of the sodium channel for Ca, Cs and Na were calculated by means of the constant field equations. The results of the voltage-clamp experiments are compared with measurements of the Ca entry in intact axons.     

Kinetics of ion movement in the squid giant axon

The loss of Na²², K⁴², and Cl³⁶ from single giant axons of the squid, Loligo pealii, following exposure to an artificial sea water containing these radioisotopes, occurs in two stages, an initial rapid one followed by an exponential decline. The time constants of the latter stage for the 3 ion species are, respectively, 290, 200, and 175 minutes. The outflux of sodium is depressed while that of potassium is accelerated in the absence of oxygen; the emergence of potassium is slowed by cocaine, while that of sodium is unaffected. One cm. ends of the axons take up about twice as much radiosodium as the central segment; this difference in activity is largely preserved during exposure to inactive solution. Such marked differences are not observed with radiopotassium. From the experimental data estimates are given of the influxes and outfluxes of the individual ions. The kinetics of outflux suggests a cortical layer of measureable thickness which contains the ions in different proportions from those in the medium and which governs the rate of emergence of these ions from the axon as though it contained very few but large (relative to ion dimensions) pores.     

A model for conductance changes in the squid giant axon. II. Channel kinetics

The kinetics of the sodium and potassium channels in voltage clamped squid giant axon following a relaxation of the membrane subunits are examined and compared with the Hodgkin-Huxley equations. Mechanisms are suggested for the turn-off of the sodium conductance and a set of kinetic states are proposed for the potassium channel which are consistent with the experimental observations. Determination of the rate constants for relaxation of the surface subunits which triggers the subsequent changes within the independent channels provide information on the equilibrium constant and free energy for this process. The free energy is observed to approach zero as the depolarizing voltage of the clamp approaches $\text{E}{}_{\mathrm{<mtext>Na</mtext>}}$, the voltage for zero sodium current in voltage clamped axons. Analysis of the final rate constants in the kinetic sequence for potassium indicates a symmetry of the channel when it is in its steady-state configuration during clamp in the absence of external gradients.     

Reversible electrical breakdown of squid giant axon membrane

Charge pulse relaxation experiments were performed on squid giant axon. In the low voltage range, the initial voltage across squid axon membrane was a linear function of the injected charge. For voltages of the order of 1 V this relationship between injected charge and voltage across the membrane changes abruptly. Because of a high conductance state caused by these large electric fields the voltage across the membrane cannot be made large enough to exceed a critical value, Vc, defined as the breakdown voltage, Vc has for squid axon membrane a value of 1.1 V at 12 degrees C. During breakdown the specific membrane conductance exceeds 1 S. cm-2. Electrical breakdown produced by charge pulses of few microseconds duration have no influence on the excitability of the squid axon membrane. The resealing process of the membrane is so fast that a depolarizing breakdown is followed by the falling phase of a normal action potential. Thus, membrane voltages close to Vc open the sodium channels in few microseconds, but do not produce a decrease of the time constant of potassium activation large enough to cause the opening of a significant percentage of channels in a time of about 10 mus. It is probable that the reversible electrical breakdown is mainly caused by mechanical instability produced by electrostriction of the membrane (electrochemical model), but the decrease in the Born energy for ion injection into the membrane, accompanying the decrease in membrane thickness, may play also an important role. Because of the high conductance of the membrane during breakdown it seems very likely that this results in pore formation.     

Nuclear magnetic resonance study of squid giant axon

Using a spin-echo technique, the spin-lattice and spin-spin relaxation times (T1 and T2) of water protons in a single nerve fiber (giant axon of squid) were determined. Similar measurements were also carried out on axoplasm extruded from these nerve fibers. It was found that the relaxation times of water protons of both the intact fiber and the extruded axoplasm are approximately equal (and much less than those of a free solution), suggesting that the relaxation times of cellular water are shortened mainly by water-protein interactions rather than by water-membrane interactions.     

Neurofilament protein is phosphorylated in the squid giant axon

We have observed the phosphorylation of neurofilament protein from squid axoplasm. Phosphorylation is demonstrated by 32P labeling of protein during incubation of axoplasm with [gamma-32P]ATP. When the labeled proteins are separated by SDS-polyacrylamide gel electrophoresis (SDS-PAGE), two bands, at 2.0 x 10(5) daltons and greater than 4 x 10(5) daltons, contain the bulk of the 32P. The 2.0 x 10(5)-dalton phosphorylated polypeptide comigrates on SDS-PAGE with one of the subunits of squid neurofilament protein. Both major phosphorylated polypeptides co-fractionate with neurofilaments in discontinuous sucrose gradient centrifugation and on gel filtration chromatography on Sepharose 4B. The protein-phosphate bond behaves like a phospho-ester, and labeled phospho-serine is identified in an acid hydrolysate of the protein. The generality of this phenomenon in various species and its possible physiological significance are discussed.     

Effects of narcotics on the giant axon of the squid

—Levorphanol (10^-3 M) reversibly blocked conduction in the giant axon of the squid and axons from the walking legs of spider crab and lobster. Similar concentrations of levallorphan and dextrorphan blocked conduction in the squid giant axon. Under the same experimental condition morphine caused an approximately 40 per cent decrease in spike height. Levorphanol did not affect the resting potential or resistance of the squid axon. Spermidine, spermine and dinitrophenol had little or no direct effect on the action potential nor did they alter the potency of levorphanol. Concentrations of levorphanol as low as 5 × 10^-5 M blocked repetitive or spontaneous activity in the squid axon induced by decreasing the divalent cations in the medium. After exposure to tritiated levorphanol, the axoplasm and envelope of the squid axon accumulated up to 500 per cent of the concentration of tritium found in the external medium, dependent on time of exposure, and other variables. At pH 6 the levels of penetration were 33-50% of those found at pH 8, which correlates with our observation that levorphanol is about 33 % as potent in blocking the action potential at pH 6. The penetrability of levorphanol was not affected by spermidine, dinitrophenol or cottonmouth moccasin venom. Levorphanol did not alter the penetration of [C¹⁴]acetylcholine nor did it render the squid axon sensitive to it. The block of axonal conduction by compounds of the morphine series is discussed both as to possible mechanisms and significance.     

Transient polarization currents in the squid giant axon.

It has been repeatedly noted that the change of conformation of the molecules that serve as the ion-selective channels for sodium and potassium conductance in the nerve membrane will be accompanied by a change in the dipole moment of the molecule. This time-dependent change of dipole moment will produce transient currents in the membrane. The canonical form for these currents is determined with conventional statistical mechanics formalism. It is pointed out that the voltage dependence of the conductance channel conductance determines the free energy of the system to within a factor that is an unknown function of the voltage. Since the dipole currents do not depend on this unknown function, they are completely determined 0y the observed properties of the conductance system. The predicted properties of these dipole currents, their time constants and strengths, are calculated. By using the observed properties of gating currents, the density of the sodium channels is computed. The predicted properties of the dipole currents are found to compare satisfactorily with the observed properties of gating currents.     

Electrical phenomena in nerve; squid giant axon

The action of a number of agents, which may be classified as "stabilizers" and "unstabilizers" on the electrical oscillations and after-potentials in the squid giant axon has been examined. The effects on the spike, "positive overshoot," and "potassium potential" were also observed, but where possible concentrations were employed which left these phenomena unaltered. Veratrine augmented the oscillations and the negative after-potential, particularly with repetitive stimulation. Yohimbine caused a small long lasting positive after-potential and depressed the oscillations, effects also enhanced with repetitive activity. Cocaine and procaine suppressed the oscillations and the negative after-potential but DDT was completely inert. An elevation in the medium calcium depressed the oscillations and the naturally occurring negative after-potential; negative after-potentials induced with veratrine were increased by calcium. A decrease in the potassium augmented the oscillations and the negative after-potential. A hypothesis is presented in which these effects are interpreted in terms of potassium concentration at the fiber surface as regulated by a labile permeability and metabolism. This is discussed in relation to the available evidence for these factors. It is a pleasure to acknowledge the author's indebtedness to Dr. D. E. S. Brown, Director, and to his staff at the Bermuda Biological Station for Research for the cooperation and special facilities provided during the initiation of this work. Dr. T. Baylor of Princeton University very kindly provided the camera and film used in Bermuda.     

Pressure dependence of the potassium currents of squid giant axon

Summary The effect of pressure upon the delayed, K, voltage-clamp currents of giant axons from the squidLoligo vulgaris was studied in axons treated with 300nm TTX to block the early, Na, currents. The effect of TTX remained unaltered by pressure. The major change produced by pressures up to 62 MPa is a slowing down of the rising phase of the K currents by a time scaling factor which depends on pressure according to an apparent activation volume, V, of 31 cm³/mole at 15°C; V increased to about 42 cm³/mole at 5°C.Pressure slightly increased the magnitude, but did not produce any obvious major change in the voltage dependence, of the steady-state K conductance estimated from the current jump at the end of step depolarizations of small amplitude (to membrane potentials,E, 20 mV) and relatively short duration. At higher depolarizations, pressure produced a more substantial increase of the late membrane conductance, associated with an apparent enhancement of a slow component of the K conductance which could not be described within the framework of the Hodgkin-Huxley (HH)n ⁴ kinetic scheme.The apparent V values that characterize the pressure dependence of the early component of the K conductance are very close to those that describe the effect of pressure on Na activation kinetics, and it is conceivable that they are related to activation volumes involved in the isomerization of the normal K channels. The enhancement of the slow component of membrane conductance by pressure implies either a large increase in the conductance of the ionic channels that are responsible for it or a strong relative hastening of their turn-on kinetics.     

Transport of Na+ inside the giant axon of squid

The transport mechanism of Na ions within the nerve cell was studied by measuring the radioactivity distribution profile of 22Na that had been intracellularly injected into the giant axon. Specifically, we tested whether or not the movement of Na ions is coupled with the process of "fast axonal transport." Results of our measurements indicate that the intracellular transport of Na+ and the fast axonal transport are two independent processes. Very few Na ions are irreversibly sequestered into the axoplasmic vesicles involved in axonal transport. The movement of Na+ inside the axon can be modeled by a one-dimension diffusion. The effective diffusion coefficient of the intracellular Na+ was determined in this study.     

Transport of Na+ inside the giant axon of squid

The transport mechanism of Na ions within the nerve cell was studied by measuring the radioactivity distribution profile of²²Na that had been intracellularly injected into the giant axon. Specifically, we tested whether or not the movement of Na ions is coupled with the process of “fast axonal transport.” Results of our measurements indicate that the intracellular transport of Na⁺ and the fast axonal transport are two independent processes. Very few Na ions are irreversibly sequestered into the axoplasmic vesicles involved in axonal transport. The movement of Na⁺ inside the axon can be modeled by a one-dimension diffusion. The effective diffusion coefficient of the intracellular Na⁺ was determined in this study.     

Anion conductances of the giant axon of squid Sepioteuthis.

Anion conductances of giant axons of squid, Sepioteuthis, were measured. The axons were internally perfused with a 100-mM tetraethylammonium-phosphate solution and immersed in a 100-mM Ca-salt solution (or Mg-salt solution) containing 0.3 microns tetrodotoxin. The external anion composition was changed. The membrane currents had a large amount of outward rectification due to anion influx across Cl- channels of the membrane (Inoue, 1985). The amount of outward rectification depended on the species of anion used and was strongly influenced by temperature and internal pH. In contrast to the anion conductances themselves, the conductance relative to Cl- (gA/gCl) was found to be quite stable against changes in the membrane potential, temperature, and pH. It is therefore suggested that each gA/gCl is an intrinsic quantity of the Cl- channel of the squid axon membrane. The sequence and values of gA/gCl obtained in this study were NO3- (1.80) greater than I- (1.40) greater than Br- (1.07) greater than Cl- (1.00) greater than MeSO3- (0.46) greater than H2PO2- (0.33) greater than CH3COO- (0.29) greater than SO4(2-) (0.06).