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.     

A gradient model of cardiac pacemaker myocytes

We have formulated a spatial-gradient model of action potential heterogeneity within the rabbit sinoatrial node (SAN), based on cell-specific ionic models of electrical activity from its central and peripheral regions. The ionic models are derived from a generic cell model, incorporating five background and exchange currents, and seven time-dependent currents based on three- or four-state Markov schemes. State transition rates are given by non-linear sigmoid functions of membrane potential.

By appropriate selection of parameters, the generic model is able to accurately reproduce a wide range of action potential waveforms observed experimentally. Specifically, the model can fit recordings from central and peripheral regions of the SAN with RMS errors of 0.3987 and 0.7628 mV, respectively. Using a custom least squares parameter optimisation routine, we have constructed a spatially-varying gradient model that exhibits a smooth transition in action potential characteristics from the central to the peripheral region, whilst ensuring individual membrane currents remain physiologically accurate. Smooth transition action potential characteristics include maximum diastolic potential, overshoot potential, upstroke velocity, action potential duration and cycle length. The gradient model is suitable for developing higher dimensional models of the right atrium, in which action potential heterogeneity within nodal tissue may be readily incorporated.     

A theory for the membrane potential of living cells

We give an explicit formula for the membrane potential of cells in terms of the intracellular and extracellular ionic concentrations, and derive equations for the ionic currents that flow through channels, exchangers and electrogenic pumps. We demonstrate that the work done by the pumps equals the change in potential energy of the cell, plus the energy lost in downhill ionic fluxes through the channels and exchangers. The theory is illustrated in a simple model of spontaneously active cells in the cardiac pacemaker. The model predicts the experimentally observed intracellular ionic concentration of potassium, calcium and sodium. Likewise, the shapes of the simulated action potential and five membrane currents are in good agreement with experiment. We do not see any drift in the values of the concentrations in a long time simulation, and we obtain the same asymptotic values when starting from the full equilibrium situation with equal intracellular and extracellular ionic concentrations. Received: 9 December 1998 / Revised version: 30 August 1999 / Accepted: 15 October 1999     

Series resistance compensation for whole-cell patch-clamp studies using a membrane state estimator

Whole-cell patch-clamp techniques are widely used to measure membrane currents from isolated cells. While suitable for a broad range of ionic currents, the series resistance (R(s)) of the recording pipette limits the bandwidth of the whole-cell configuration, making it difficult to measure rapid ionic currents. To increase bandwidth, it is necessary to compensate for R(s). Most methods of R(s) compensation become unstable at high bandwidth, making them hard to use. We describe a novel method of R(s) compensation that overcomes the stability limitations of standard designs. This method uses a state estimator, implemented with analog computation, to compute the membrane potential, V(m), which is then used in a feedback loop to implement a voltage clamp; we refer to this as state estimator R(s) compensation. To demonstrate the utility of this approach, we built an amplifier incorporating state estimator R(s) compensation. In benchtop tests, our amplifier showed significantly higher bandwidths and improved stability when compared with a commercially available amplifier. We demonstrated that state estimator R(s) compensation works well in practice by recording voltage-gated Na(+) currents under voltage-clamp conditions from dissociated neonatal rat sympathetic neurons. We conclude that state estimator R(s) compensation should make it easier to measure large rapid ionic currents with whole-cell patch-clamp techniques.     

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.     

Multiple-state model of ionic channel in reproducing the time course of electrophysiological events.

BACKGROUND: The predictions of the Hodgkin-Huxley model do not accurately fit all the measurements of voltage-clamp currents, gating charge and single-channel currents. There are many quantitative differences between the predicted and measured characteristics of the sodium and potassium channels. For example, the two-state gate model has exponential onset kinetics, whereas the sodium and potassium conductances show S-shaped activation and the sodium conductance shows an exponential inactivation. In this paper we shall examine a more general channel model that can more faithfully represent the measured properties of ionic channels in the membrane of the excitable cell. METHODS: The model is based on the generalisation of the notion of a channel with a discrete set of states. Each state has state attributes such as the state conductance, state ionic current and state gating charge. These variables can have quite different waveforms in time, in contrast with a two-state gate channel model, in which all have the same waveforms. RESULTS: The kinetics of all variables are equivalent: gating and ionic currents give equivalent information about channel kinetics; both the equilibrium values of the current and the time constants are functions of membrane potential. The results are in almost perfect concordance with the experimental data regarding the characteristics of nerve impulse. CONCLUSIONS: The expected values of the gating charge and the ionic conductance are weighted sums of the state occupancy probabilities, but the weights differ: for the expected value of the gating charge the weights are the state gating charges and for the expected value of the ionic conductance the weights are the state conductances. Since these weights are different, the expected values of the gating charge and the ionic conductance will differ.     

The shaping of intrinsic membrane potential oscillations: positive/negative feedback,ionic resonance/amplification,nonlinearities and time scales

The generation of intrinsic subthreshold (membrane potential) oscillations (STOs) in neuronal models requires the interaction between two processes: a relatively fast positive feedback that favors changes in voltage and a slower negative feedback that opposes these changes. These are provided by the so-called resonant and amplifying gating variables associated to the participating ionic currents. We investigate both the biophysical and dynamic mechanisms of generation of STOs and how their attributes (frequency and amplitude) depend on the model parameters for biophysical (conductance-based) models having qualitatively different types of resonant currents (activating and inactivating) and an amplifying current. Combinations of the same types of ionic currents (same models) in different parameter regimes give rise to different types of nonlinearities in the voltage equation: quasi-linear, parabolic-like and cubic-like. On the other hand, combinations of different types of ionic currents (different models) may give rise to the same type of nonlinearities. We examine how the attributes of the resulting STOs depend on the combined effect of these resonant and amplifying ionic processes, operating at different effective time scales, and the various types of nonlinearities. We find that, while some STO properties and attribute dependencies on the model parameters are determined by the specific combinations of ionic currents (biophysical properties), and are different for models with different such combinations, others are determined by the type of nonlinearities and are common for models with different types of ionic currents. Our results highlight the richness of STO behavior in single cells as the result of the various ways in which resonant and amplifying currents interact and affect the generation and termination of STOs as control parameters change. We make predictions that can be tested experimentally and are expected to contribute to the understanding of how rhythmic activity in neuronal networks emerge from the interplay of the intrinsic properties of the participating neurons and the network connectivity.     

Modeling the leech heartbeat elemental oscillator I. Interactions of intrinsic and synaptic currents

We have developed a biophysical model of a pair of reciprocally inhibitory interneurons comprising an elemental heartbeat oscillator of the leech. We incorporate various intrinsic and synaptic ionic currents based on voltage-clamp data. Synaptic transmission between the interneurons consists of both a graded and a spike-mediated component. By using maximal conductances as parameters, we have constructed a canonical model whose activity appears close to the real neurons. Oscillations in the model arise from interactions between synaptic and intrinsic currents. The inhibitory synaptic currents hyperpolarize the cell, resulting in activation of a hyperpolarization-activated inward currentI _h and the removal of inactivation from regenerative inward currents. These inward currents depolarize the cell to produce spiking and inhibit the opposite cell. Spike-mediated IPSPs in the inhibited neuron cause inactivation of low-threshold Ca⁺⁺ currents that are responsible for generating the graded synaptic inhibition in the opposite cell. Thus, although the model cells can potentially generate large graded IPSPs, synaptic inhibition during canonical oscillations is dominated by the spike-mediated component.     

Spiking resonances in models with the same slow resonant and fast amplifying currents but different subthreshold dynamic properties

The generation of spiking resonances in neurons (preferred spiking responses to oscillatory inputs) requires the interplay of the intrinsic ionic currents that operate at the subthreshold voltage level and the spiking mechanisms. Combinations of the same types of ionic currents in different parameter regimes may give rise to different types of nonlinearities in the voltage equation (e.g., parabolic- and cubic-like), generating subthreshold (membrane potential) oscillations patterns with different properties. These nonlinearities are not apparent in the model equations, but can be uncovered by plotting the voltage nullclines in the phase-plane diagram. We investigate the spiking resonant properties of conductance-based models that are biophysically equivalent at the subthreshold level (same ionic currents), but dynamically different (parabolic- and cubic-like voltage nullclines). As a case study we consider a model having a persistent sodium and a hyperpolarization-activated (h-) currents, which exhibits subthreshold resonance in the theta frequency band. We unfold the concept of spiking resonance into evoked and output spiking resonance. The former focuses on the input frequencies that are able to generate spikes, while the latter focuses on the output spiking frequencies regardless of the input frequency that generated these spikes. A cell can exhibit one or both types of resonances. We also measure spiking phasonance, which is an extension of subthreshold phasonance (zero-phase-shift response to oscillatory inputs) to the spiking regime. The subthreshold resonant properties of both types of models are communicated to the spiking regime for low enough input amplitudes as the voltage response for the subthreshold resonant frequency band raises above threshold. For higher input amplitudes evoked spiking resonance is no longer present in these models, but output spiking resonance is present primarily in the parabolic-like model due to a cycle skipping mechanism (involving mixed-mode oscillations), while the cubic-like model shows a better 1:1 entrainment. We use dynamical systems tools to explain the underlying mechanisms and the mechanistic differences between the resonance types. Our results demonstrate that the effective time scales that operate at the subthreshold regime to generate intrinsic subthreshold oscillations, mixed-mode oscillations and subthreshold resonance do not necessarily determine the existence of a preferred spiking response to oscillatory inputs in the same frequency band. The results discussed in this paper highlight both the complexity of the suprathreshold responses to oscillatory inputs in neurons having resonant and amplifying currents with different time scales and the fact that the identity of the participating ionic currents is not enough to predict the resulting patterns, but additional dynamic information, captured by the geometric properties of the phase-space diagram, is needed.     

Parameter estimation in cardiac ionic models

We examine the problem of parameter estimation in mathematical models of excitable cell cardiac electrical activity using the well-known Beeler–Reuter (1977) ionic equations for the ventricular action potential. The estimation problem can be regarded as equivalent to the accurate reconstruction of ionic current kinetics and amplitudes in an excitable cell model, given only action potential experimental data. We show that in the Beeler–Reuter case, all ionic currents may be reasonably reconstructed using an experimental design consisting of action potential recordings perturbed by pseudo-random injection currents.

The Beeler–Reuter model was parameterised into 63 parameters completely defining all membrane current amplitudes and kinetics. Total membrane current was fitted to model-generated experimental data using a ‘data-clamp’ protocol. The experimental data consisted of a default action-potential waveform and an optional series of perturbed waveforms generated by current injections. Local parameter identifiability was ascertained from the reciprocal condition value (1/λ) of the Hessian at the known solution. When fitting to a single action potential waveform, the model was found to be over-determined, having a 1/λ value of 3.6e−14. This value improved slightly to 1.4e−10 when an additional 2 perturbed waveforms were included in the fitting process, suggesting that the additional data did not overly improve the identifiability problem. The additional data, however, did allow the accurate reconstruction of all ionic currents. This indicates that by appropriate experimental design, it may be possible to infer the properties of underlying membrane currents from observation of transmembrane potential waveforms perturbed by pseudo-random currents.     

Pharmacological and kinetic analysis of K channel gating currents

We have measured gating currents from the squid giant axon using solutions that preserve functional K channels and with experimental conditions that minimize Na channel contributions to these currents. Two pharmacological agents were used to identify a component of gating current that is associated with K channels. Low concentrations of internal Zn2+ that considerably slow K channel ionic currents with no effect on Na channel currents altered the component of gating current associated with K channels. At low concentrations (10-50 microM) the small, organic, dipolar molecule phloretin has several reported specific effects on K channels: it reduces K channel conductance, shifts the relationship between channel conductance and membrane voltage (Vm) to more positive potentials, and reduces the voltage dependence of the conductance-Vm relation. The K channel gating charge movements were altered in an analogous manner by 10 microM phloretin. We also measured the dominant time constants of the K channel ionic and gating currents. These time constants were similar over part of the accessible voltage range, but at potentials between -40 and 0 mV the gating current time constants were two to three times faster than the corresponding ionic current values. These features of K channel function can be reproduced by a simple kinetic model in which the channel is considered to consist of two, two-state, nonidentical subunits.     

Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model

Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.     

Membrane interactions and metal ion effects on bilayer permeation of the lipophilic ion modulator DP-109

DP-109, a lipophilic bivalent metal ion modulator currently under preclinical development for neurodegenerative disorders, was designed to have membrane-associated activity, thereby restricting its action to the vicinity of cell membranes. We describe the application of a colorimetric phospholipid/polydiacetylene (PDA) biomimetic membrane assay in elucidating DP-109 membrane interactions and penetration into lipid bilayers. In this membrane model, visible quantifiable color changes were monitored in studying membrane interactions. The colorimetric data identified a biphasic concentration-dependent interaction, with a break point around the critical micelle concentration (CMC) of DP-109. The kinetics and colorimetric dose-response profile of DP-109 indicate that the compound inserts into the lipid bilayers rather than being localized at the bilayer surface. Analysis of interactions of DP-109 with phospholipid/PDA vesicles in which ionic gradients were imposed indicates that membrane activity of DP-109 is strongly affected by electrochemical gradients imposed by K+ and Zn2+. The ionic gradient effects suggest that the insertion of DP-109 into the membrane may depend on the membrane potential.     

Estimation of ionic channel surface density from relaxation ionic and gating currents

It is shown that excitable membrane surface density of channels can be estimated from ionic and gating current relaxations. The gating currents are determined thermodynamically from a multistate kinetic model. The parameters of the kinetic model are derived from ionic current relaxations. The assumptions regarding the gating process made here are the same as those made in fluctuation analysis previously regarded as the only method that may yield channel density from membrane currents.     

On the relation between displacement currents and activation of the sodium conductance in the squid giant axon.

The early time course of the current passing across the membrane in squid giant axons in which the ionic currents have been blocked reveals substantial asymmetries during and after the application of hyperpolarizing and depolarizing voltage-clamp pulses of identical size. Since the integral of the 'on' and 'off' current transients is zero, these currents must result from charge movements confined to the membrane and, therefore, they are nonlinear displacement currents. The steady state rearrangement of the charges as a consequence of sudden displacements of the membrane potential is consistent with a Boltzmann distribution of charges between two states characterized by different energy levels. Following changes in membrane potential the charges undergo a first order transition between these states. The relaxation time constant for the transition at a given temperature is a function of membrane potential. We propose that these displacement currents arise from a redistribution of the charges involved in the sodium gating system.     

Determinants of time-dependent membrane conductance. The nonrole of classical ion-membrane molecule interactions

We have examined the steady-state and time-dependent electrical properties of a model membrane system. The model assumes that the directed velocity and energy of ions moving through the membrane are determined by the applied electric field, ionic diffusion forces, and central elastic collisions between ions and membrane molecules. A simple analysis of the steady-state electrical properties of the model yields results identical with ones obtained previously using a more complex analysis procedure. The time-dependent conductance changes of the model in response to a step change in electric field strength when there is solution symmetry display three qualitative patterns dependent on the nature of the ion-membrane molecule interaction. One of the patterns of conductance change is quite similar to that observed in the sodium conductance system of a number of excitable tissues: an initial conductance rise to a maximum (activation) followed by a decay to a final steady-state value (inactivation). However, the correspondence between the time-dependent model behavior and known experimental behavior of excitable systems is only qualitative. We conclude that the classical ion-membrane molecule interactions we consider are not involved in determining time-dependent conductance processes in the excitable systems for which comparison is possible.     

Interaction of antitumor α-lactalbumin—oleic acid complexes with artificial and natural membranes

The specific complexes of human α-lactalbumin (α-LA) with oleic acid (OA), HAMLET and LA-OA-17 (OA-complexes), possess cytotoxic activity against tumor cells but the mechanism of their cell penetration remains unclear. To explore the molecular mechanisms underlying interaction of the OA-complexes with the cell membrane, their interactions with small unilamellar dipalmitoylphosphatidylcholine (DPPC) vesicles and electroexcitable plasma membrane of internodal native and perfused cells of the green alga Chara corallina have been studied. The fractionation (Sephadex G-200) of mixtures of the OA-complexes with the vesicles shows that OA-binding increases the affinity of α-LA to DPPC vesicles. Calcium association decreases protein affinity to the vesicles; the effect being less pronounced for LA-OA-17. The voltage clamp technique studies show that LA-OA-17, HAMLET, and their constituents produce different modifying effects on the plasmalemmal ionic channels of the Chara corallina cells. The irreversible binding of OA-complexes to the plasmalemma is accompanied by changes in the activation-inactivation kinetics of developing integral transmembrane currents, suppression of the Ca²⁺ current and Ca²⁺-activated Cl⁻ current, and by increase in the nonspecific K⁺ leakage currents. The latter reflects development of nonselective permeability of the plasma membrane. The HAMLET-induced effects on the plasmalemmal currents are less pronounced and potentiated by LA-OA-17. The control experiments with OA and intact α-LA show their qualitatively different and much less pronounced effects on the transmembrane ionic currents. Thus, the modification of α-LA by OA results in an increase in the protein association with the model lipid bilayer and in drastic irreversible changes in permeability of several types of the plasmalemmal ionic channels.     

Heuristics for the Hodgkin–Huxley system

Hodgkin and Huxley (HH) discovered that voltages control ionic currents in nerve membranes. This led them to describe electrical activity in a neuronal membrane patch in terms of an electronic circuit whose characteristics were determined using empirical data. Due to the complexity of this model, a variety of heuristics, including relaxation oscillator circuits and integrate-and-fire models, have been used to investigate activity in neurons, and these simpler models have been successful in suggesting experiments and explaining observations. Connections between most of the simpler models had not been made clear until recently. Shown here are connections between these heuristics and the full HH model. In particular, we study a new model (Type III circuit): It includes the van der Pol-based models; it can be approximated by a simple integrate-and-fire model; and it creates voltages and currents that correspond, respectively, to the h and V components of the HH system.