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.     

Ion fluxes in giant excised cardiac membrane patches detected and quantified with ion-selective microelectrodes

We have used ion-selective electrodes (ISEs) to quantify ion fluxes across giant membrane patches by measuring and simulating ion gradients on both membrane sides. Experimental conditions are selected with low concentrations of the ions detected on the membrane side being monitored. For detection from the cytoplasmic (bath) side, the patch pipette is oscillated laterally in front of an ISE. For detection on the extracellular (pipette) side, ISEs are fabricated from flexible quartz capillary tubing (tip diameters, 2-3 microns), and an ISE is positioned carefully within the patch pipette with the tip at a controlled distance from the mouth of the patch pipette. Transport activity is then manipulated by solution changes on the cytoplasmic side. Ion fluxes can be quantified by simulating the ion gradients with appropriate diffusion models. For extracellular (intrapatch pipette) recordings, ion diffusion coefficients can be determined from the time courses of concentration changes. The sensitivity and utility of the methods are demonstrated with cardiac membrane patches by measuring (a) potassium fluxes via ion channels, valinomycin, and Na/K pumps; (b) calcium fluxes mediated by Na/Ca exchangers; (c) sodium fluxes mediated by gramicidin and Na/K pumps; and (d) proton fluxes mediated by an unknown electrogenic mechanism. The potassium flux-to-current ratio for the Na/K pump is approximately twice that determined for potassium channels and valinomycin, as expected for a 3Na/2K pump stoichiometery (i.e., 2K/charge moved). For valinomycin-mediated potassium currents and gramicidin-mediated sodium currents, the ion fluxes calculated from diffusion models are typically 10-15% smaller than expected from the membrane currents. As presently implemented, the ISE methods allow reliable detection of calcium and proton fluxes equivalent to monovalent cation currents <1 pA in magnitude, and they allow detection of sodium and potassium fluxes equivalent to <5 pA currents. The capability to monitor ion fluxes, independent of membrane currents, should facilitate studies of both electrogenic and electroneutral ion-coupled transporters in giant patches.     

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     

Modeling of discrete currents of single ion channels of cell membranes using synthetic nanometer pores in polyethylene terephthalate films

Study of the conductivity of single supernarrow pores (1–15 nm in diameter) formed in thin membranes (10–12 μm in thickness) from polyethylene terephthalate (PETP) has revealed discrete changes in the currents passing through such pores when applied from an external source of potential difference of 200–1000 mV. Based on several characteristics, such discrete currents (discrete conductivity changes) appeared to be identical to the so-called currents of single ionic channels in cell membranes. The supernarrow pores whose properties are described in the present work were obtained by alkaline etching of tracks in thin PETP membranes (a variant of the so-called nuclear filters). On the walls of the pores, carboxyl groups, i.e., negative fixed charges, and their compensating counterion (cation) layer are formed. Upon setting the potential difference onto the PETP membrane, this cation layer is able to transfer current, through a process called surface conductance. In the case of nanometer-sized diameters of the pores, such surface conductance can turn out to be dominating. We have shown that these discrete changes of currents passing through the nanometer pores are associated with metastability of their surface conductance. In the highly cation-selective channels in the cell membranes, there should inevitably exist an area with dominating cation surface conductance and, hence, conductance metastability. Therefore, a new explanation is proposed of the characteristic discreteness of the currents of single cation-specific ionic channels in cell membranes. Such an explanation does not rule out the existence of any other traditional explanation of the discreteness of ion channel currents.     

Discerning ionic currents and their kinetics from input impedance data

The excitable nature of a biological cell is manifested in the many voltage gated ion channels that perforate its membrane. The forms of the associated ionic currents, and in particular the functions that govern their kinetics, permit one to distinguish, electrophysiologically, between various cell types. We show, in the context of FitzHugh-Nagumo and Morris-Lecar models and without recourse to voltage or space clamping, that such currents and kinetics may be stably inferred from a cell’s voltage response to a specified input current.     

Simulations of the cardiac action potential based on the Hodgkin-Huxley kinetics with the use of Microsoft Excel spreadsheets

The purpose of this study was to develop a method to simulate the cardiac action potential using a Microsoft Excel spreadsheet. The mathematical model contained voltage-gated ionic currents that were modeled using either Beeler-Reuter (B-R) or Luo-Rudy (L-R) phase 1 kinetics. The simulation protocol involves the use of in-cell formulas directly typed into a spreadsheet. The capability of spreadsheet iteration was used in these simulations. It does not require any prior knowledge of computer programming, although the use of the macro language can speed up the calculation. The normal configuration of the cardiac ventricular action potential can be well simulated in the B-R model that is defined by four individual ionic currents, each representing the diffusion of ions through channels in the membrane. The contribution of Na+ inward current to the rate of depolarization is reproduced in this model. After removal of Na+ current from the model, a constant current stimulus elicits an oscillatory change in membrane potential. In the L-R phase 1 model where six types of ionic currents were defined, the effect of extracellular K+ concentration on changes both in the time course of repolarization and in the time-independent K+ current can be demonstrated, when the solutions are implemented in Excel. Using the simulation protocols described here, the users can readily study and graphically display the underlying properties of ionic currents to see how changes in these properties determine the behavior of the heart cell. The method employed in these simulation protocols may also be extended or modified to other biological simulation programs.     

Permeation and interaction of monovalent cations with the cGMP-gated channel of cone photoreceptors

We measured the ion selectivity of cGMP-dependent currents in detached membrane patches from the outer segment of cone photoreceptors isolated from the retina of striped bass. In inside-out patches excised from either single or twin cones the amplitude of these currents, under symmetric ionic solutions, changed with the concentration of cGMP with a dependence described by a Hill equation with average values, at +80 mV, of Km = 42.6 microM and n = 2.49. In the absence of divalent cations, and under symmetric ionic solutions, the I-V curves of the currents were linear over the range of -80 to +80 mV. The addition of Ca altered the form of the I-V curve to a new function well described by an empirical equation that also describes the I-V curve of the photocurrent measured in intact photoreceptors. The monovalent cation permeability sequence of the cGMP-gated channels in the absence of divalent ions was PK > PNa = PLi = PRb > PCs (1.11 > 1.0 = 0.99 = 0.96 > 0.82). The conductance selectivity sequence at +80 mV was GNa = GK > GRb > GCs > GLi (1.0 = 0.99 > 0.88 > 0.74 > 0.60). The organic cations tetramethylammonium (TMA) and arginine partially blocked the current, but the larger ion, arginine, was permeant, whereas the smaller ion, TMA, was not. The amplitude of the outward current through the channels increased with the concentration of monovalent cations on the cytoplasmic membrane surface, up to a saturating value. The increase was well described by the adsorption isotherm of a single ion binding site within the channel with average binding constants, at +80 mV, of 104 mM for Na and 37.6 mM for Li. By assuming that the ion channel contains a single ion binding site in an energy trough separated from each membrane surface by an energy barrier, and using Eyring rate theory, we simulated I-V curves that fit the experimental data measured under ionic concentration gradients. From this fit we conclude that the binding site interacts with one ion at a time and that the energy barriers are asymmetrically located within the membrane thickness. Comparison of the quantitative features of ion permeation and interaction between the cGMP-gated channels of rod and cone photoreceptors reveals that the ion binding sites are profoundly different in the two types of channels. This molecular difference may be particularly important in explaining the differences in the transduction signal of each receptor type.     

Mechanosensitive ion channels

The article concentrates on the concepts of mechanosensitive ion channels that are present in practically all cells of an organism. Considered are kinetic scheme and activation principles of mechanic-sensitive ion channels. The forces affecting those channels are discussed in detail. The qualities of the channels in lipid monolayer, bilayer and real cell membrane are under consideration. Discussed are various models that analyze possibilities of channel opening depending on the membrane tension. Under discussion are the data received from studying single channels, currents in whole-cell configuration and cloned channels built into bilayer, liposomes and membrane blebs. Problems of transmitting mechanic energy to the channel through the bilayer and through the cytoskeleton are investigated. Inhibitors and activators of mechanosensitive ion channels are mentioned and their effects are considered. The functional classification of mechanosensitive ion channels is given. Described are cation SACs, potassium SACs, Ca(2+)-sensitive and Ca(2+)-insensitive SACs, anion SACs, nonselective SACs and SICs. It is proved that mechanosensitive ion channels can produce considerable currents enough to change the cell electrogenesis.     

Open-state substructure of single chloride channels from Torpedo electroplax

Chloride channels from Torpedo californica electroplax were inserted into planar phospholipid membranes, and single-channel currents were studied at high time-resolution. The open channel fluctuates rapidly between three substates, with conductances of 18.5, 9.4 and 0 pS in 150 mM Cl-. Under various ionic conditions the three substates are always equally spaced in conductance; at various voltages leading to different probabilities of observing the three substates, the substate frequencies are always binomially distributed. The conclusion emerges that the conducting of unit of Cl- channel is composed of two identical Cl- diffusion pathways, each with a voltage-dependent gate.     

Membrane currents in the oocyte of the toad Bufo arenarum

The amphibian oocyte cell model is widely used for heterologous expression of ionic channels and receptors. Little is known, however, about the physiology of oocyte cell models other than Xenopus laevis. In this study, the two-electrode voltage clamp technique was used to assess the most common electrical patterns of oocytes of the South American toad Bufo arenarum. Basal membrane resistance, resting potential, and ionic currents were determined in this cell model. The oocyte transmembrane resistance was 0.35 M(Omega), and the resting potential in normal saline was about -33 mV with a range between -20 mV and -50 mV. This is, to our knowledge, the first attempt to begin an understanding of the ion transport mechanisms of Bufo arenarum oocytes. This cell model may provide a viable alternative to the expression of ion channels, in particular those endogenously observed in Xenopus laevis oocytes.     

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.     

Ion channels: does each subunit do something on its own?

The advent of the patch-clamp technique 25 years ago revolutionized the study of ion channels. This method also made it possible to measure the kinetic behavior of single protein molecules. The low-noise recordings of ionic currents through single channels, coupled with other cutting-edge technologies, have revealed a rich complexity of functional states that are not readily explained by simple allosteric protein models such as the popular concerted model and the sequential model. Although these models can each account for elements of ion channel function, we propose that variations or extensions of the lesser-known general allosteric model provide a more promising framework for explaining the intricate behaviors of ion channels.     

A Continuum Neuronal Model for the Instigation and Propagation of Cortical Spreading Depression

Cortical spreading depression (CSD) waves can occur in the cortices of various brain structures and are associated with the spread of depression of the electroencephalogram signal. In this paper, we present a continuum neuronal model for the instigation and spreading of CSD. Our model assumes that the brain-cell microenvironment can be treated as a porous medium consisting of extra- and intracellular compartments. The main mechanisms in our model for the transport of ions into and out of neurons are cross-membrane ionic currents and (active) pumps, coupled with diffusion in the extracellular space. To demonstrate the applicability of our model, we have carried out extensive numerical simulations under different initial conditions and inclusion of various mechanisms. Our results show that CSD waves can be instigated by injecting cross-membrane ionic currents or by applying KCl in the extracellular space. Furthermore, the estimated speeds of CSD waves are within the experimentally observed range. Effects of specific ion channels, background ion concentrations, extracellular volume fractions, and cell swelling on the propagation speed of CSD are also investigated.     

M5 mesoscopic and macroscopic models for mesenchymal motion

In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.Dedicated to K.P. Hadeler, a great scientist, teacher, and friend.     

Expression and canalicular localization of two isoforms of the ClC-3 chloride channel from rat hepatocytes

The molecular identities of functional chloride channels in hepatocytes are largely unknown. We examined the ClC-3 chloride channel in rat hepatocytes and found that mRNA for two different isoforms is present. A short form is identical to the previously reported sequence for rat ClC-3, and a long form contains a 176-bp insertion immediately upstream of the translation initiation site. This predicts a 58-amino acid NH(2) terminal insertion. Both long and short form mRNA was expressed in diverse tissues of the rat. Transient transfection of the long form in CHO-K1 cells resulted in currents with an I(-) > B(-) > Cl(-) selectivity sequence, outward rectification, and inactivation at positive voltages. Short form currents had identical ionic selectivity but displayed a more extreme outward rectification and showed no voltage-dependent inactivation. Immunofluorescence and immunoblots localized native ClC-3 preferentially but not exclusively to the canalicular membrane. We have therefore identified a new isoform of rat ClC-3 and shown that expression of both isoforms produces functional channels. In hepatocytes, ClC-3 is located in association with the canalicular membrane.     

Transfection of the CD20 cell surface molecule into ectopic cell types generates a Ca2+ conductance found constitutively in B lymphocytes

CD20 is a plasma membrane phosphoprotein expressed exclusively by B lymphocytes. mAb binding to CD20 alters cell cycle progression and differentiation, indicating that CD20 plays an essential role in B lymphocyte function. Whole-cell patch clamp and fluorescence microscopy measurements of plasma membrane ionic conductance and cytosolic-free Ca2+ activity, respectively, were used to directly examine CD20 function. Transfection of human T and mouse pre-B lymphoblastoid cell lines with CD20 cDNA and subsequent stable expression of CD20 specifically increased transmembrane Ca2+ conductance. Transfection of CD20 cDNA and subsequent expression of CD20 in nonlymphoid cells (human K562 erythroleukemia cells and mouse NIH-3T3 fibroblasts) also induced the expression of an identical transmembrane Ca2+ conductance. The binding of a CD20-specific mAb to CD20+ lymphoblastoid cells also enhanced the transmembrane Ca2+ conductance. The mAb-enhanced Ca2+ currents had the same conductance characteristics as the CD20- associated Ca2+ currents in CD20 cDNA-transfected cells. C20 is structurally similar to several ion channels; each CD20 monomer possesses four membrane spanning domains, and both the amino and carboxy termini reside within the cytoplasm. Biochemical cross-linking of cell-surface molecules with subsequent immunoprecipitation analysis of CD20 suggests that CD20 may be present as a multimeric oligomer within the membrane, as occurs with several known membrane channels. Taken together, these findings indicate that CD20 directly regulates transmembrane Ca2+ conductance in B lymphocytes, and suggest that multimeric complexes of CD20 may form Ca2+ conductive ion channels in the plasma membrane of B lymphoid cells.     

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.     

Identifying kinetic gating mechanisms for ion channels by using two-dimensional distributions of simulated dwell times.

Ion channels are integral membrane proteins that regulate ionic flux through cell membranes by opening and closing (or gating) their pores. The gating can be monitored by observing step changes in the current flowing through single channels. Analysis of the durations of the open and closed intervals and of the correlations among the interval durations can give insight into the gating mechanism. Although it is well known that the correlation information can be essential to distinguish among possible gating mechanisms, it has been difficult to use this information because it has not been possible to correct the predicted correlations for the distortion of the single-channel data because of filtering and noise. To overcome this limitation we present a method based on a comparison of simulated and experimental two-dimensional dwell-time distributions constructed by analysing simulated and experimental single-channel currents in an identical manner. The simulated currents incorporate the true effects of filtering and noise, the two-dimensional distributions retain the correlation information, and the identical analysis allows direct maximum-likelihood comparison of the simulated and experimental two-dimensional distributions. We show that the two-dimensional simulation method has a greatly increased ability to distinguish among models, compared with methods that use one-dimensional distributions.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.