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.     

The Effect of Membrane Parameters on the Properties of the Nerve Impulse

The effect of varying membrane capacitance, conductance, and rate constants on the properties of the nerve impulse is considered in terms of the degree of regeneration in the Hodgkin-Huxley model for the squid giant axon. It is shown through computer simulation that reducing regeneration generally increases the duration of the action potential and decreases its amplitude, rate of rise, and conduction velocity. The threshold becomes much less sharp and the amplitude of the response of a patch of membrane grades with stimulus strength. A second stimulus, applied shortly after a first stimulus, considerably perturbs the membrane potential from its original time-course. Under certain conditions, the nerve signal can propagate with a small decrement.     

A calculation of the magnetic field of a nerve action potential.

The magnetic field outside an isolated axon is calculated using transmembrane potential data to specify the boundary conditions to a solution of Laplace's equation. It is shown that the contribution to the magnetic field from the current inside the membrane is two orders of magnitude larger than that from the external current. The contribution from current within the membrane is negligible. Comparisons are made between waveforms calculated for a crayfish lateral axon and those measured for a frog sciatic nerve. This calculation suggests that the magnetic field measured outside nerves can be used to determine their internal current without puncturing the nerve membrane.     

Semiconductor theory of ion transport in thin lipid membranes: I. Potential and field distributions

In the theory as presented in this paper and the following one, we shall attempt to apply the semiconductor principles and methods to the study of ion transport in thin lipid membranes. Detailed formulations are given on the potential energy barriers at the interfaces, voltage drops in the polar and non-polar regions, and potential and field distributions in the diffuse double layer and within a charged membrane. These results will be used mainly as the boundary conditions for the solution of ion flow as to be given in the following paper. The analysis clearly indicates that the ion transport is interface-limited and is profoundly influenced by the presence of surface charges. An explanation of Na⁺ extrusion in nerve membrane is given based on the field distribution analysis. The theory also suggests that the “membrane potential” depends mainly on surface charges but not necessarily on ion permeation through the membrane.     

Towards a molecular theory of the nerve membrane

Ion transport through the nerve membrane is considered in terms of barrier-limited fluxes calculated from absolute reaction rate theory. Equations are developed to describe the conformational transitions of an enzyme embedded in the membrane to provide a low-energy transport site. The enzyme transitions are controlled by binding and hydrolytic release of an acetylcholine-like molecule, which in turn depends on ion association with a single negative charge on the enzyme. Simulation of the equations gives good agreement with typical experimental voltage clamps and action potentials. A steady-state negative resistance is found in isoosmolar potassium, and the model shows excitation by an acetylcholine pulse under conditions mimicking the postsynaptic membrane. The implications of the model for development of a molecular theory of the nerve membrane are considered.     

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.     

Random currents through nerve membranes

The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f ^3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.     

Transmembrane electrical potential of excitable membranes: a pore analysis influence of surface charges and surface dipoles

A treatment is proposed in order to establish the general expression of the zero-current transmembrane potential of excitable membranes. The membrane model considered here is that of a hydrocarbon layer which is impermeable to ions and which represents the lipid bilayer matrix. In this matrix are incorporated ionic channels. The ion transport process through the channels is described by the absolute-rate theory applied to pores which are seen as chains of potential energy maxima and minima. Only one of the energy barriers corresponds to the gate step, and it is strongly dependent on the transmembrane potential. The kinetic equation is related to the zero current, to electrostatic boundary conditions and to the Gouy-Chapman equation for the aqueous diffuse layer.     

A theory of ion permeation through membranes with fixed neutral sites

Summary Some model membranes and biological membranes behave as if ion permeation were controlled by fixed neutral sites, i.e., by groups that are polar but lack net charge. By solving the boundary conditions and Nernst-Planck flux equations, this paper derives the expected properties of four types of membranes with fixed neutral sites: model 1, a membrane thick enough that microscopic electroneutrality is obeyed; model 2, same as model 1 but with a free-solution shunt in parallel; model 3, a membrane thin enough that microscopic electroneutrality is violated; and model 4, same as model 3 but with a free-solution shunt in parallel. The conductance-concentration relation and the current-voltage relation in symmetrical solutions are approximately linear for all four models. Partial ionic conductances are independent of each other for a thin membrane but not for a thick membrane. Sets of permeability ratios derived from conductances, dilution potentials, or biionic potentials agree with each other in a thin membrane but not in a thick membrane. The current-voltage relation in asymmetrical single-salt solutions is linear for a thick membrane but nonlinear for a thin membrane. Examples of potential and concentration profiles in a thin membrane are calculated to illustrate the meaning of space charge and the electroneutrality condition. The experimentally determined properties (by A. Cass, A. Finkelstein & V. Krespi) of thin lipid membranes containing “pores” of the anion-selective antibiotic nystatin are in reasonable agreement with model 3. Tests are suggested for deciding if a membrane of unknown structure has neutral sites, whether it is thick or thin, and whether the sites are fixed or mobile.     

A Note on the Local Current Associated with the Rising Phase of a Propagating Impulse in Nonmyelinated Nerve Fibers

To extend our recent paper dealing with the cable properties and the conduction velocity of nonmyelinated nerve fibers (Bull. Math. Biol. 64, 1069; 2002), the behavior of the local current associated with the rising phase of a propagating action potential is discussed. It is shown that the process of charging the membrane capacity by means of the local current plays a crucial role in determining the velocity of nerve conduction. The symmetry of the local current with respect to the boundary between the resting and active regions of the nerve fiber is emphasized. It is noted that there are several simple quantitative rules governing the intensities of the capacitive, resistive and total membrane currents observed during the rising phase of an action potential.     

A network thermodynamic method for numerical solution of the Nernst-Planck and Poisson equation system with application to ionic transport through membranes

Simple techniques of network thermodynamics are used to obtain the numerical solution of the Nernst-Planck and Poisson equation system. A network model for a particular physical situation, namely ionic transport through a thin membrane with simultaneous diffusion, convection and electric current, is proposed. Concentration and electric field profiles across the membrane, as well as diffusion potential, have been simulated using the electric circuit simulation program, SPICE. The method is quite general and extremely efficient, permitting treatments of multi-ion systems whatever the boundary and experimental conditions may be.     

Poisson-Boltzmann Theory for Membranes with Mobile Charged Lipids and the pH-Dependent Interaction of a DNA Molecule with a Membrane

We consider a planar stiff model membrane consisting of mobile surface groups whose state of charge depends on the pH and the ionic composition of the adjacent electrolyte solution. To calculate the mean-field interaction potential between a charged object and such a model membrane, one needs to solve a Poisson-Boltzmann boundary value problem. We here derive and discuss the boundary condition at the membrane surface, a condition that is generally appropriate for biological membranes where two charge-regulating mechanisms are present at the same time: the pH-dependent chemical charge regulation and a regulation through the in-plane mobility of the surface groups. As an application of this general formalism, we consider the specific example of a single DNA molecule, approximated by a cylinder with smeared-out surface charges, interacting with such a model membrane. We study the effect that the two competing charge-regulating mechanisms have on the DNA/membrane interaction and the distribution of surface ions in the plane of the membrane. We find that, at short DNA-membrane distances, membrane fluidity can have a considerable impact on the DNA adsorption behavior and can lead to such counterintuitive phenomena as the adsorption of a negatively charged DNA onto a (on average) negatively charged membrane.     

Diffusion approximation of the neuronal model with synaptic reversal potentials

The stochastic neuronal model with reversal potentials is approximated. For the model with constant postsynaptic potential amplitudes, a deterministic approximation is the only one which can be applied. The diffusion approximations are performed under the conditions of random postsynaptic potential amplitudes. New diffusion models of nerve membrane potential are devised in this way. These new models are more convenient for an analytical treatment than the original model with discontinuous trajectories.     

An Approach to the Current-Voltage Characteristics of Nerve Membranbs Based on Adsorption Phenomena

Using Stern's double-layer adsorption model for the density of cations in the membrane pores, a quantitative approach to the stationary current-voltage characteristic of nerve membranes is developed. The interaction of mobile cations with the negative fixed charges, located inside the membrane, constitutes a resistance for the current through the membrane. The stepwise increase in the resistance for the hyperpolarization is ascribed to a stronger interaction accompanying a depletion of the adsorbed cations from the interior. Thermodynamic treatment of flows and forces is adapted to the situation, to give a current voltage relation amenable to experimental check. The value of the resting potential thus obtained gives a deviation from Nernst equation applied to the ion for which the membrane is mainly permeable. The effect of the membrane double-layer potential on the potential range in which the transition from low to high resistance takes place, is explicitly incorporated. Finally, a comparison of the theory with the experimental results for the squid axon and frog nerve fibers is made.     

Action potential refractory period in axonal demyelination: a computer simulation

Axonal demyelination leads to an increase in the refractory period for propagation of the action potential. Computer simulations were used to investigate the mechanism by which changes in the passive properties of the internodal membrane increase the refractory period. The properties of the voltage dependent ion channels can be altered to restore conduction in demyeliated nerve fibers. The ability of these alterations to decrease the refractory period of demyelinated model nerve fibers was compared. The model nerve fiber contained six nodes. The action potential was stimulated at node one and propagated to node six. The internode between nodes three and four was demyelinated in a graded manner. The absolute refractory period for propagation of the action potential through the demyelinated internode increased as the number of myelin wraps was reduced to less than 25% of the normal value. The increase in refractory period was found to be due to a reduction in the rate or repolarization of the action potential at node three. The delay in repolarization reduced the rate of recovery of inactivated Na channels and slowed the closing of K channels. The rate of repolarization of node three was reduced by the conduction delay for the depolarization of node four caused by demyelination of the preceeding internode. In these simulations the increase in refractory period due to demyelination was eliminated by slowing the onset of Na channel inactivation. A small reduction of the K conductance also decreased the refractory period. However, larger reductions eliminated this effect.     

Ionic mechanism of 5-hydroxytryptamine induced hyperpolarization and inhibitory junctional potential in body wall muscle cells of Hirudo medicinalis.

Five-hydroxytryptamine (5-HT) causes a hyperpolarization and increased conductance of the leech body wall muscle cell membrane. If 5-HT is applied in the absence of the Cl minus ion, the response appears as a depolarization, whereas if 5-HT is applied in the absence of the K+ ion, the response is a hyperpolarization. In both cases, the conductance of the muscle cell membrane is increased. Stimulation of the peripheral nerve to the body wall muscle produces a complex junctional potential in muscle cells. Exposing the muscle to d-tubocurarine (d-TC) eliminates the excitatory component (EJP) of the complex potential. The inhibitory potential (IJP) that remains has an equilibrium potential at approximately 65 mV. Furthermore, this IJP appears as a depolarization when the nerve is stimulated in the presence of d-TC and low CL minus, whereas this is not the case if the nerve is stimulated in the presence of d-TC and low K+. The drugs BOL-148 and cyproheptadine block the IJP's in the body wall muscle. These data are interpreted as indicating that 5-HT acts on leech body wall muscle cells by increasing the conductance to the Cl minus ion and that the IJP's caused by nerve stimulation are probably the result of 5-HT release at nerve terminals. As a final point, it has been shown that the inhibition by 5-HT of the spontaneous EJP's that occur on the leech body wall muscle results from an inhibition of central neurons and not from any direct effect on the muscle cell or on peripheral synapses.     

Electrical correlates of secretion in endocrine and exocrine cells

Many types of secretory cells including neurons and cells of endocrine and exocrine glands show changes in electrical potential and resistance when secretion is stimulated. These electrical correlates result from the movement of ions across the cell membrane through specific ion-selective channels. In neurons and certain endocrine cells (such as pancreatic beta cells and certain cells of the anterior pituitary), these channels are voltage dependent and open transiently upon depolarization leading to action potentials. Thus some endocrine cells are electrically excitable, a property previously held to occur only in nerve and muscle. In other nonexcitable endocrine and exocrine cells (such as the pancreas and parotid), ion channels are responsive to either occupancy of specific membrane receptors or changes in intracellular metabolites and second messengers. Ion fluxes through these latter channels also lead to changes in the electrical potential and resistance, but these changes are generally more sustained and action potentials are not seen. The entry of Ca2+ through both voltage-dependent and voltage-independent ion channels plays a major role in the activation of secretion via exocytosis.     

A study of the change in conductivity of the neuronal membrane caused by a generator potential using a method of computer modeling

Injection of cAMP induces in snail neurons generator potential, which is related to an increase of sodium and decrease of potassium permeability of the neuron outer membrane. A model is proposed which takes into account cAMP diffusion inside the neuron from the injection place and interaction of these molecules with the intercellular system controlling permeability of the outer membrane. Resulting impulse generation induces calcium ions current through the outer membrane. The model also considers calcium diffusion toward cAMP and its effect on the rate of the enzyme work destroying cAMP. Agreement between the calculations of ionic current I(t) and the experiment permits determination of the model parameters and calculation of the observed change of time distribution of nerve impulses when calcium input is significant.     

The biomagnetic signature of a crushed axon. A comparison of theory and experiment.

The response of a crayfish medial giant axon to a nerve crush is examined with a biomagnetic current probe. The experimental data is interpreted with a theoretical model that incorporates both radial and axial ionic transport and membrane kinetics similar to those in the Hodgkin/Huxley model. Our experiments show that the effects of the crush are manifested statically as an elevation of the resting potential and dynamically as a reduction in the amplitude of the action current and potential, and are observable up to 10 mm from the crush. In addition, the normally biphasic action current becomes monophasic near the crush. The model reflects these observations accurately, and based on the experimental data, it predicts that the crush seals with a time constant of 45 s. The injury current density entering the axon through the crush is calculated to be initially on the order of 0.1 mA/mm2 and may last until the crush seals or until the concentration gradients between the intra- and extracellular spaces equilibrate.     

The role of the adenine nucleotide translocator in oxidative phosphorylation. A theoretical investigation on the basis of a comprehensive rate law of the translocator

A minimum model of adenine nucleotide exchange through the inner membrane of mitochondria is presented. The model is based on a sequential mechanism, which presumes ternary complexes formed by binding of metabolites from both sides of the membrane. The model explains the asymmetric kinetics of ADP-ATP exchange as a consequence of its electrogenic character. In energized mitochondria, a part of the membrane potential suppresses the binding of extramitochondrial ATP in competition with ADP. The remaining part of the potential difference inhibits the back exchange of internal ADP for external ATP. The assumption of particular energy-dependent conformational states of the translocator is not necessary. The model is not only compatible with the kinetic properties reported in the literature about the adenine nucleotide exchange, but it also correctly describes the response of mitochondrial respiration to the extramitochondrial ATP/ADP ratio under different conditions. The model computations reveal that the translocation step requires some loss of free energy as driving force. The size of the driving force depends on the flux rate as well as on the extra- and intramitochondrial ATP/ADP quotients. By both quotients the translocator controls the export of ATP formed by oxidative phosphorylation in mitochondria.