Electronic measurement of the intracellular concentration and net flux of sodium in the squid axon

Analytical solutions of the Frankenhaeuser-Huxley equations I: minimal model for backpropagation of action potentials in sparsely excitable dendrites

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na(+) channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na(+) ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na(+) and K(+). In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Phi) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na(+) activation (micro) and a state-dependent inactivation (eta) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na(+) channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.

Action of external divalent ion reduction on sodium movement in the squid giant axon

Voltage clamp measurements of the sodium potential have been made on the resting squid giant axon to study the effect of variations in external divalent ion concentration upon net sodium flux. From these measurements the intracellular sodium concentration and the net sodium inflow were calculated using the Nernst relation and constant activity coefficients. While an axon bathed in artificial sea water shows a slow increase in internal sodium concentration, the rate of sodium accumulation is increased about two times by reducing external calcium and magnesium concentrations to 0.1 times their normal values. The mean inward net sodium flux increases from a mean control value of 97 pmole/cm² sec. to 186 pmole/cm² sec. in low divalent solution. Associated with these effects of external divalent ion reduction are a marked decrease in action potential amplitude, little or no change in resting potential, and a shift along the voltage axis of the curve relating peak sodium conductance to membrane potential similar to that obtained by Frankenhaeuser and Hodgkin (1957). These results implicate divalent ions in long term (minutes to hours) sodium permeability.

EFFECTS OF REPLACEMENT OF EXTERNAL SODIUM CHLORIDE WITH SUCROSE ON MEMBRANE CURRENTS OF THE SQUID GIANT AXON

It was observed that a reduction of the sodium chloride concentration in the external solution bathing a squid giant axon by replacement with sucrose resulted in marked decreases in the peak inward and steady-state outward currents through the axon membrane following a step decrease in membrane potential. These effects are quantitatively acounted for by the increase in series resistance resulting from the decreased conductivity of the sea water and the assumption that the sodium current obeys a relation of the form I = k(1)C(1) - k(2)C(2) where C(1), C(2) are internal and external ion activities and k(1), k(2) are independent of concentration. It is concluded that the potassium ion current is independent of the sodium concentration. That the inward current is carried by sodium ions has been confirmed. The electrical potential (or barrier height) profile in the membrane which drives sodium ions appears to be independent of sodium ion concentration or current. A specific effect of the sucrose on hyperpolarizing currents was observed and noted but not investigated in detail.

EFFECT OF DETERGENT ON ELECTRICAL PROPERTIES OF SQUID AXON MEMBRANE

The effects of detergents on squid giant axon action and resting potentials as well as membrane conductances in the voltage clamp have been studied. Anionic detergents (sodium lauryl sulfate, 0.1 to 1.0 mM; dimethyl benzene sulfonate, 1 to 20 mM, pH 7.6) cause a temporary increase and a later decrease of action potential height and the value of the resting potential. Cationic detergent (cetyl trimethyl ammonium chloride, 6 x 10^-5M or more, pH 7.6) generally brings about immediate and irreversible decreases in the action and resting potentials. Non-ionic detergent (tween 80, 0.1 M, pH 7.6) causes a slight reversible reduction of action potential height without affecting the value of the resting potential. Both anionic and cationic detergents generally decrease the sodium and potassium conductances irreversibly. The effect of non-ionic detergent is to decrease the sodium conductance reversibly, leaving the potassium conductance almost unchanged.

Action potential propagation and threshold parameters in inhomogeneous regions of squid axons

The squid giant axon was used as a model system in which to determine the independent contributions of membrane excitability and diameter changes to threshold parameters and propagation of action potentials in inhomogeneous regions. The membrane excitability of a segment of an axon was altered by changes in the bathing solution, while its effective electrical diameter was increased by the insertion of a low-resistance axial wire. In computer simulations of these experiments, similar alterations were made in the membrane's conductance and axon's diameter. The inflexions in the shapes of action potentials propagating into a region with abrupt decreases in axial resistance become more pronounced when the interval between impulses was shortened. At short intervals, propagation of the second impulse failed. In contrast, reduction of membrane excitability produced inflexion-free changes in action potential shape and allowed a close-following second impulse to pass through the inhomogeneity. A combined decrease in membrane excitability and increase in diameter of the same region exaggerated the changes in action potential shape characteristic of the diameter increase alone. Threshold parameters were obtained from 'strength-duration' excitability relationships measured by injection of current at different points along the axon. When only the membrane excitability was reduced, threshold characteristics changed smoothly from one region of the nerve to another. In contrast, lowering the internal resistance or increasing the diameter in one region of a nerve lowered the time constant of excitation and the threshold for brief (relative to rheobasic) current stimuli in the small-diameter region near the transition while raising them in the larger-diameter region.

Diffusion of sodium in axoplasm of myelinated nerve fibre. Potential clamp analysis

The diffusion of sodium and lithium in the axoplasm of myelinated fibres of Xenopus laevis was investigated. The sodium current associated with pulse steps was measured and the internal sodium concentration was calculated from these measurements with the constant field equation. The internal sodium concentration was increased either (i) by massive pulse trains or (ii) by applying a sodium containing solution to a cut end of the fibre. The time course of the concentration change was measured. Equations were solved numerically for a model axon where sodium diffused freely in the axoplasm. It was found that the experimental results were satisfactorily predicted by this model axon. The calculated diffusion coefficient for sodium in axoplasm seemed to be somewhat lower than that in water. Similar experiments were performed with lithium. These results agreed with the predictions based on unrestricted diffusion in the axoplasm. It is concluded that no major diffusion barriers were present within the axoplasm.

Axon voltage-clamp simulations. III. Postsynaptic region

This is the third in a series of four papers in which we present the numerical simulations of the application of the voltage clamp technique to excitable cells. In this paper we discuss the problem of voltage clamping a region of a cylindrical cell using microelectrodes for current injection and voltage recording. A recently developed technique (Llinás et al., 1974) of internal application of oil drops to electrically insulate a short length of the postsynaptic region of the squid giant synapse is evaluated by simulation of the voltage clamp of an excitable cylindrical cell of finite length with variable placement of the current and voltage electrodes. Our results show that ENa can be determined quite accurately with feasible oil gap lengths but that the determination of the reversal potential for the synaptic conductance, ES, can be considerably in error. The error in the determination of ES dependp, and especially the membrane resistance at the time the synaptic conductance occurs. It is shown that the application of tetraethylammonium chloride to block the active potassium conductance very significantly reduces the error in the determination of ES. In addition we discuss the effects of cable length and electrode position on the apparent amplitude and time course of the syn aptic conductance change. These results are particularly relevant to the application of the voltage clamp technique to cells with nonsomatic synapses. The method of simulation presented here provides a tool for evaluation of voltage clamp analysis of synaptic transmission for any cell with known membrane parameters and geometry.

The temperature dependence of the movement of sodium ions associated with nerve impulses

1. The movement of sodium ions across the membrane of the squid giant axon was measured by the use of radioactive tracers. Unidirectional fluxes were measured at rest and when the nerve was stimulated. The difference was considered the extra flux association with nerve impulses.2. The extra influx in intact axons at room temperature was 5.5 p-mole/cm(2). impulse. At 6 degrees C the extra influx was 6.5 p-mole/cm(2). impulse giving a Q(10) of 1/1.2.3. In perfused axons a Q(10) of 1/1.6 was obtained for the extra sodium influx in bracketed experiments on individual axons.4. The Q(10) of the extra sodium efflux associated with nerve impulses was found to be 1/1.2 in intact axons.5. Hodgkin & Huxley had predicted a much larger temperature dependence for the extra fluxes. If this difference between prediction and experiment does not result from some experimental error, then the class of models for the ion fluxes suggested by Hodgkin & Huxley may be inapplicable.

Synaptic current at the squid giant synapse

Transmission in the giant synapse of squid was studied by measuring synaptic currents in the voltage-clamped postsynaptic giant axon. These currents varied linearly with the axon's membrane potential, and showed an intercept on the voltage axis at, or near, the sodium equilibrium potential. The intercept shifted in seawater containing less sodium by even more than the shift in the sodium equilibrium potential. It is concluded that the transmitter at this synapse causes a significant change in the sodium conductance only.

Cation movements in perfused giant axons

1. The net movements of Na and K into resting squid giant axons perfused with a K(2)SO(4) artificial axoplasm have been determined. Na enters at a rate of 131 +/- 25 p-mole/cm(2).sec. The K leakage from the fibre is 942 +/- 566 p-mole/cm(2).sec; clearly this latter figure shows how large is the uncertainty in the exact value for the potassium leakage.2. The net entry of sodium associated with activity in such fibres has been measured and is 5.7 +/- 0.7 p-mole/cm(2).impulse.