Kinetic Theory Model for Ion Movement through Biological Membranes: I. Field-Dependent Conductances in the Presence of Solution Symmetry

A model for ion movement through specialized sites in the plasma membrane is presented and analyzed using techniques from nonequilibrium kinetic theory. It is assumed that ions traversing these specialized regions interact with membrane molecules through central conservative forces. The membrane molecules are approximated as massive spherical scattering centers so that ionic fractional energy losses per collision are much less than one. Equations for steady-state membrane ionic currents and conductances as functions of externally applied electric field strength are derived and numerically analyzed, under the restriction of identical solutions on each size of the membrane and constant electric fields within the membrane. The analysis is carried through for a number of idealized ion-membrane molecule central force interactions. For any interaction leading to a velocity-dependent ion-membrane molecule collision frequency, the membrane chord conductance is a function of the externally applied electric field. Interactions leading to a collision frequency that is an increasing (decreasing) function of ionic velocity are characterized by chord conductances that are decreasing (increasing) functions of field strength. For ion-neutral molecule interactions, the conductance is such a rapidly decreasing function of field strength that the slope conductance becomes negative for all field strengths above a certain value.     

Kinetic Theory Model for Ion Movement through Biological Membranes: III. Steady-State Electrical Properties with Solution Asymmetry

An electrodiffusion model for plasma membrane ion transport, which takes into account the influence of high electric field strengths and ion-membrane molecule interactions, is presented and analyzed. A generalized Nernst-Planck equation for steady-state situations is derived which has electric field-dependent mobility and diffusion coefficients. Under the assumption of a constant electric field within the membrane, this equation is integrated to give a more general form of the Goldman equation. Based on this equation numerical computations of ionic chord conductance as a function of applied electric field strength were carried out for several permeant ion concentration ratios. The model is capable of yielding significantly larger rectification ratios than is the Goldman equation. Further, high field asymptotes to the current vs. electric field strength curve do not generally intersect at the origin.     

Kinetic Theory Model for Ion Movement through Biological Membranes: II. Interionic Selectivity

The equation presented in the previous paper for steady-state membrane ionic current as a function of externally applied electric field strength is numerically analyzed to determine the influence of ionic and membrane molecule parameters on current densities. The model displays selectivity between different ions. A selectivity coefficient S_i, defined as the ratio of current carried by an ionic species i at a given field strength to the current carried by a reference species at the same field strength, has the following properties: (a) S_i is a function of electric field strength except for ion-membrane molecule interactions yielding velocity independent collision frequencies; (b) for ion-membrane molecule interactions characterized by a collision frequency that is a decreasing (increasing) function of increasing ionic velocity, ions whose S_i > 1 (<1) at zero field strength will show maxima (minima) (minima[maxima]) in their S_i vs. electric field strength curves.     

Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction

Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh–Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.     

Biological effects of oscillating electric fields: Role of voltage-sensitive ion channels

An alternating component of potential across the membrane of an excitable cell may change the membrane conductance by interacting with the voltagesensing charged groups of the protein macromolecules that form voltage-sensitive ion channels. Because the probability that a voltage sensor is in a given state is a highly nonlinear function of the applied electric field, the average occupancy of a particular state will change in an oscillating electric field of sufficient magnitude. This “rectification” at the level of the voltage sensors could result in conformational changes (gating) that would modify channel conductance. A simplified two-state model is examined where the relaxation time of the voltage sensor is assumed to be considerably faster than the fastest changes of ionic conductance. Significant changes in the occupancy of voltage sensor states in response to an applied oscillating electric field are predicted by the model.     

Current-Voltage Curves of Porous Membranes in the Presence of Pore-Blocking Ions: I. Narrow Pores Containing No More Than One Moving Ion

We propose a physical model for voltage-dependent conductance changes of excitable cell membranes. It is based on competition of uni- and bivalent ions for chains of stable sites extending through the membrane. These one-dimensional pathways (pores) have different profiles of chemical potential for the two ionic species so that bivalent ions can block the passage of univalent ions at large membrane potentials. We treat the special case that each pore is either empty or, because of electrostatic repulsion, contains no more than one uni- or bivalent ion at a time. A system of linear differential equations describes the time-dependent probabilities of the various possible pore states. The states are limited by transition rate constants involving the profile of the chemical potential, the membrane voltage, the ionic concentrations in the adjacent baths, and electrostatic interactions between the ions. The steady-state solutions (Kirchhoff-Hill theorem) yield expressions for the relationship between the small signal conductance of univalent ions and the concentration of these ions in the external bathing medium (a saturation curve) and for the ionic currents and the steady-state current-voltage curve (N-shaped). From the latter curve we compute the shift of theshold potential caused by concentration changes of the external bathing medium. The model yields a number of predictions which can be tested experimentally.     

Kinetics of Excitable Membranes : Voltage amplification in a diffusion regime

An understanding of the properties of excitable membranes requires the calculation of ion flow through the membrane, including the effects of nonuniformity in the transverse membrane properties (mobilities, fixed charge, electric field). Permeability is apparently controlled at the external interface. Two factors may be involved here: the statistical blocking of pores by divalent cations, and activation energy. Only the former is included in the present treatment. When the total transmembrane voltage is varied, a redistribution in ionic concentration occurs. This can cause a change in boundary (zeta) potential, large in comparison with the applied voltage change—"voltage amplification." The result is a steep change in membrane conductance. The calculated flow curves are compared with experimental results. The Appendix gives an outline of the numerical method used for solving the boundary value problem with several diffusible ions, across a nonuniform regime.     

BIOMEMBRANE EXCITABILITY STUDIED WITHIN A WIDE-BAND FREQUENCY OF AN INTERACTING EXOGENOUS ELECTRIC FIELD

There does exist increasing experimental and theoretical evidence that supports the existence of a coupling between exogenous electromagnetic fields and ion channels located within the membrane of excitable cells. One of the most tantalizing consequences of such interactions points to a resonant-like behavior of this class of electrical non-linear systems leading to an optimized information transfer along excitable membranes. Herein, we present novel evidence showing that action potentials may occur in biomembranes within even the subthreshold excitation range, provided that concomitantly with the depolarizing stimulus, an exogenous low-amplitude oscillatory electric field of proper frequency (centered around 10 kHz) interacts with the biomembrane. As we present it, this phenomenon may be explained if one takes into consideration the resonant-like electrical properties of the linearized, small-signal impedance of the simplified, equivalent electrical representation of the studied biomembrane.     

A Test of the Theory of the Steady-State Properties of an Ion Exchange Membrane with Mobile Sites and Dissociated Counterions

An experimental model system, formally equivalent to a liquid ion exchange membrane having completely dissociated sites and counterions, has been devised in order to test the steady-state properties recently deduced theoretically for such a membrane by Conti and Eisenman, (1966). In this system we have obtained quantitative experimental confirmation of the following theoretical expectations. (a) The current-voltage relationship is nonlinear and exhibits finite limiting currents with strong applied fields. (b) The mobile sites rearrange within the “membrane” under applied electric field to give a linear concentration profile and a logarithmic electric potential profile in the steady state. We have also extended the theory to consider the instantaneous conductance in the steady state. Theory and experiment indicate that in a mobile site membrane the instantaneous conductance in the steady state is not given by the chord conductance of the steady-state current-voltage relationship, in contrast to the situation in a fixed site membrane. This finding suggests a way of testing whether ions permeate across an unknown membrane by a fixed site or a dissociated mobile site mechanism.     

Kinetics of the Opening and Closing of Individual Excitability-Inducing Material Channels in a Lipid Bilayer

The kinetics of the opening and closing of individual ion-conducting channels in lipid bilayers doped with small amounts of excitability-inducing material (EIM) are determined from discrete fluctuations in ionic current. The kinetics for the approach to steady-state conductance during voltage clamp are determined for lipid bilayers containing many EIM channels. The two sets of measurements are found to be consistent, verifying that the voltage-dependent conductance of the many-channel EIM system arises from the opening and closing of individual EIM channels. The opening and closing of the channels are Poisson processes. Transition rates for these processes vary exponentially with applied potential, implying that the energy difference between the open and closed states of an EIM channel is linearly proportional to the transmembrane electric field. A model incorporating the above properties of the EIM channels predicts the observed voltage dependence of ionic conductance and conductance relaxation time, which are also characteristic of natural electrically excitable membranes.     

Effect of 3-phenylindole on lipophilic ion and carrier-mediated ion transport across bilayer lipid membranes

The physical effects of 3-phenylindole, an antimicrobial compound which interacts with phospholipids, on ion transport across phosphatidylcholine-cholesterol bilayers have been investigated using three lipophilic ions and one ion-carrier complex. It was found that 3-phenylindole increased membrane electrical conductance of positively charged membrane probes and decreased electrical conductance of negatively charged probes. The enhancement of conductance detected by nonactin-K+ complex and tetraphenylarsonium+ was several orders of magnitude, whereas the suppression of conductance due to tetraphenylborate- and dipicrylamine- was less than a factor of ten. Presence of 3-phenylindole in aqueous phase slightly decreased adsorption of tetraphenylborate- and dipicrylamine- at the membrane surface. From the voltage dependence of the steady-state conductance it was shown that 3-phenylindole induced kinetic limitation of membrane transport of potassium mediated by nonactin. No such limitation was found in the case of tetraphenylarsonium+ transport. These results are shown to be consistent with the present concept of ion diffusion in membranes and the assumption that 3-phenylindole decreases the electric potential in the membrane interior. The asymmetry of the effect of 3-phenylindole on the magnitude of conductance changes for positively and negatively charged membrane permeable ions is also discussed as a reflection of the discreteness of both the absorbed 3-phenylindole and lipid dipoles.     

Quantitative description of end plate current and its reconstruction

An attempt was made to quantify the postsynaptic current based on the experimental data of the voltage clamp method. The conductance change in postsynaptic membrane was derived from the postsynaptic conductance of voltage clamped postsynaptic membrane, then it's temporal characteristics and dependence on the clamped voltage have been quantified. The temporal characteristics was found to be explained by the introduction of two schematic operators, active and inactive. This idea was applied to a simple electrical circuit model of the postsynaptic cells. Besides the change in postsynaptic potentials of normal synapse in excitable state was culculated as its application.     

The Numerical Solution of the Time-Dependent Nernst-Planck Equations

Calculations are reported of the time-dependent Nernst-Planck equations for a thin permeable membrane between electrolytic solutions. Charge neutrality is assumed for the time-dependent case. The response of such a membrane system to step current input is measured in terms of the time and space changes in concentration, electrical potential, and effective conductance. The report also includes discussion of boundary effects that occur when charge neutrality does not hold in the steady-state case.     

Electrophorus electricus as a model system for the study of membrane excitability

The stunning sensations produced by electric fish, particularly the electric eel, Electrophorus electricus, have fascinated scientists for centuries. Within the last 50 years, however, electric cells of Electrophorus have provided a unique model system that is both specialized and appropriate for the study of excitable cell membrane electrophysiology and biochemistry. Electric tissue generates whole animal electrical discharges by means of membrane potentials that are remarkably similar to those of mammalian neurons, myocytes and secretory cells. Electrocytes express ion channels, ATPases and signal transduction proteins common to these other excitable cells. Action potentials of electrocytes represent the specialized end function of electric tissue whereas other excitable cells use membrane potential changes to trigger sophisticated cellular processes, such as myofilament cross-bridging for contraction, or exocytosis for secretion. Because electric tissue lacks these functions and the proteins associated with them, it provides a highly specialized membrane model system. This review examines the basic mechanisms involved in the generation of the electrical discharge of the electric eel and the membrane proteins involved. The valuable contributions that electric tissue continues to make toward the understanding of excitable cell physiology and biochemistry are summarized, particularly those studies using electrocytes as a model system for the study of the regulation of membrane excitability by second messengers and signal transduction pathways.     

Analysis of an excitable dendritic spine with an activity-dependent stem conductance

 Dendritic spines are the major target for excitatory synaptic inputs in the vertebrate brain. They are tiny evaginations of the dendritic surface consisting of a bulbous head and a tenuous stem. Spines are considered to be an important locus for plastic changes underlying memory and learning processes. The findings that synaptic morphology may be activity-dependent and that spine head membrane may be endowed with voltage-dependent (excitable) channels is the motivation for this study. We first explore the dynamics, when an excitable, yet morphologically fixed spine receives a constant current input. Two parameter Andronov–Hopf bifurcation diagrams are constructed showing stability boundaries between oscillations and steady-states. We show how these boundaries can change as a function of both the spine stem conductance and the conductance load of the attached dendrite. Building on this reference case an idealized model for an activity-dependent spine is formulated and analyzed. Specifically we examine the possibility that the spine stem resistance, the tunable “synaptic weight” parameter identified by Rall and Rinzel, is activity-dependent. In the model the spine stem conductance depends (slowly) on the local electrical interactions between the spine head and the dendritic cable; parameter regimes are found for bursting, steady states, continuous spiking, and more complex oscillatory behavior. We find that conductance load of the dendrite strongly influences the burst pattern as well as other dynamics. When the spine head membrane potential exhibits relaxation oscillations a simple model is formulated that captures the dynamical features of the full model. Received: 10 January 1997/Revised version: 25 March 1997     

Theory of electromechanical effects in nerve

The electromechanical transduction mechanisms operating in nerve membranes are considered theoretically. For mechanical-to-electrical transduction (mechanical generator potentials), a model is proposed in which the surface charge on the membrane mediates stress-induced changes in the intramembrane electric field, thus opening transmembrane ion conductance channels or reducing the ion selectivity of the membrane via leak conductance pathways. For electrical-to-mechanical transduction (axon diameter change with excitation), an investigation into two well-known electrostatic properties of dielectrics, electrostriction and piezoelectricity, in the context of the nerve membrane is undertaken which predicts a few percent change in axon dimensions for voltage- and space-clamped axons.     

Membrane excitability and dissipative instabilities

Summary Electrical excitation is interpreted in terms of a cooperative structural transition of membrane protomers coupled with the translocation of a permeant molecule in a non-equilibrium environment. Equations for flow of permeant and for membrane conformation are derived for the simple case of a single non-charged permeant. On the basis of a few simple physical assumptions, the theory predicts several important properties of electrically excitable membranes: the steepness of the relation between membrane conductance and potential, the presence of a negative conductance, and the occurrence of instabilities following rapid perturbations of membrane environment, giving rise to some simple cases of action potentials. Several experimental tests of the membrane with its changes of electrical properties are proposed. From a thermodynamic point of view, an electrically excitable membrane, in its resting state, lies beyond a dissipative instability and consequently is in a non-equilibrium state but with stable organization, a dissipative structure of Prigogine. Membrane excitation following a small perturbation of the environment would correspond to a jump from such an organization to another stable organization but close to thermodynamic equilibrium. It is shown how the cooperative molecular properties of the membrane are amplified by energy dissipation at the macroscopic level.     

The role of passive calcium influx through the cell membrane in galvanotaxis

Passive calcium influx is one of the theories to explain the cathodal galvanotaxis of cells that utilize the electric field to guide their motion. When exposed to an electric field, the intracellular fluid becomes polarized, leading to positive charge accumulation on the cathodal side and negative charge accumulation on the anodal side. The negative charge on the anodal side attracts extracellular calcium ions, increasing the anodal calcium concentration, which is supposed to decrease the mobile properties of this side. Unfortunately, this model does not capture the Ca²⁺ dynamics after its presentation to the intracellular fluid. The ions cannot permanently accumulate on the anodal side because that would build a potential drop across the cytoplasm leading to an ionic current, which would carry positive ions (not only Ca²⁺) from the anodal to the cathodal part through the cytoplasm. If the cytoplasmic conductance for Ca²⁺ is low enough compared to the membrane conductance, the theory could correctly predict the actual behavior. If the ions move through the cytoplasm at a faster rate, compensating for the passive influx, this theory may fail. This paper contains a discussion of the regimes of validity for this theory.     

Inactivation of monazomycin-induced voltage-dependent conductance in thin lipid membranes. I. Inactivation produced by long chain quaternary ammonium ions

The voltage-dependent conductance induced in thin lipid membranes by monazomycin undergoes inactivation upon the introduction of quaternary ammonium ions (QA) having a long alkyl chain (e.g. dodecyltrimethylammonium [C12]) to the side containing monazomycin. That is, in response to a step of voltage the conductance rises to a peak and then falls to a much lower steady-state value. We demonstrate that the basis of this phenomenon is the ability of QA to pass through the stimulated membrane and bind to the opposite surface. As a consequence, the surface potential on that side becomes more positive, thus reducing the voltage across the membrane proper and turning off the monazomycin-induced conductance. Because the flux of QA through the membrane increases linearly with conductance, we believe that these ions pass through the monazomycin channels. QA permeability increases with alkyl chain length; remarkably, in spite of its much larger size, C12 is about 150 times more permeant than K+. It appears, therefore, that there is a hydrophobic region of the cahnnel that favors the alkyl chain; we propose that this region is formed by the hydrophobic faces of the monazomycin channels in lipid bilayers to QA inactivation of potassium channels in the squid giant azon, and suggest that there may be a common structural feature for the two channels. It is possible that some of the inactivation phenomena in excitable cells may arise from local field changes not measurable by the recording electrodes.     

Effect of conditioning polarization of soma of giant neurons of mollusks on the mechanism of action-potential generation

To ascertain the properties of an excitable membrane of the soma of giant neurons of mollusks, experiments were carried out to study the effect of conditioning shift of the membrane potential on the mechanism of action-potential generation. The effect of conditioning was assessed from changes in the action-potential curve and its first derivative, as well as from the curve of transmembrane currents under voltage clamp conditions. It was found that a change in membrane potential evokes at least two reactions which have opposite effects on the mechanism of generation of action potentials. These reactions evidently have different time characteristics. One of these does not differ notably from the reaction recorded for other excitable structures, and is manifested in the activation (with hyperpolarization) or inactivation (with depolarization) of the mechanism generating action potentials. The other reaction contributes either to an increase (with depolarization) or a decrease (with hyperpolarization) in the efficiency of this mechanism. Conditioning polarization also has a marked effect on the system responsible for repolarization of the membrane during generation of action potentials. This effect is manifested in a change in the reaction of this system to tetraethylammonium ions. The specific membrane systems sustaining excitability and reacting to changes in the strength of the membrane's electrical field were found to be very inert. After a shift in the potential to a given stable level a rearrangement, lasting sometimes tens of seconds, takes place in the membrane.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 2, No. 1, pp. 91–99, January–February, 1970.