Sodium inactivation in nerve fibers

A number of models proposed to account for the sodium conductance changes are shown to fall into two classes. The Hodgkin-Huxley (HH) model falls into a class (I) in which the conductance depends on two or more independent variables controlled by independent processes. The Mullins, Hoyt, and Goldman models fall into class II in which conductance depends directly on one variable only, a variable which is controlled by two or more coupled processes. The HH and Hoyt models are used as specific examples of the two classes. It is shown that, contrary to a recently published report, the results from double experiments can be equally well accounted for by both models. It is also shown that steady-state conditioning, or “inactivation,” curves, obtained at more than one test potential, can be used to distinguish the two models. The HH equations predict that such curves should be shifted, by very small amounts, in the hyperpolarizing direction when more depolarizing test potentials are used, while the Hoyt model predicts that they should be shifted in the depolarizing direction, by quite appreciable amounts. Several pieces of published experimental information are used as tests of these predictions, and give tentative support to the class II model. Further experiments are necessary before a definite conclusion can be reached.     

Determining the contributions of divisive and subtractive feedback in the Hodgkin-Huxley model

The Hodgkin-Huxley (HH) model is the basis for numerous neural models. There are two negative feedback processes in the HH model that regulate rhythmic spiking. The first is an outward current with an activation variable n that has an opposite influence to the excitatory inward current and therefore provides subtractive negative feedback. The other is the inactivation of an inward current with an inactivation variable h that reduces the amount of positive feedback and therefore provides divisive feedback. Rhythmic spiking can be obtained with either negative feedback process, so we ask what is gained by having two feedback processes. We also ask how the different negative feedback processes contribute to spiking. We show that having two negative feedback processes makes the HH model more robust to changes in applied currents and conductance densities than models that possess only one negative feedback variable. We also show that the contributions made by the subtractive and divisive feedback variables are not static, but depend on time scales and conductance values. In particular, they contribute differently to the dynamics in Type I versus Type II neurons.     

Comparison of ion current noise predicted from different models of the sodium channel gating mechanism in nerve membrane

Summary The theoretical power density spectrumS(f) of ion current noise is calculated from several models of the sodium channel gating mechanism in nerve membrane. Sodium ion noise experimental data from the frog node of Ranvier [Conti, F.,et al. (1976),J. Physiol. (London) 262:699] is used as a test of the theoretical results. The motivation for recent modeling has been evidence for a coupling between sodium activation and inactivation from voltage clamp data. The two processes are independent of one another in the Hodgkin and Huxley (HH) model [Hodgkin, A.L., Huxley, A.F. (1952),J. Physiol. (London) 117:500] The noise data is consistent with HH, as noted by Contiet al. (1976). The theoretical results given here appear to indicate that only one case of coupling models is also consistent with the noise data.     

Transmission in the squid giant synapse: a model based on voltage clamp studies

1. Voltage clamp studies were performed in squid giant synapse after blockage of the voltage-dependent sodium and potassium conductances. 2. Presynaptic depolarization under these conditions demonstrates the presence of voltage-dependent calcium conductance change for the duration of the voltage step, and a tail current at the break of the pulse. 2. This calcium current triggers a postsynaptic response which can be measured directly at the postsynaptic fiber. 4. These voltage clamp experiments have allowed the development of a mathematical model that describes the kinetics of the calcium current and the relationship between calcium current and transmitter release.     

Selective modification of sodium channel gating in lobster axons by 2, 4, 6-trinitrophenol: Evidence for two inactivation mechanisms

Trinitrophernol (TNP) selectively alters the sodium conductance system of lobster giant axons as measured in current clamp and voltage clamp experiments using the double sucrose gap technique. TNP has no measurable effect on potassium currents but reversibly prolongs the time-course of sodium currents during maintained depolarizations over the full voltage range of observable currents. Action potential durations are increased also. Tm of the Hodgkin-Huxley model is not markedly altered during activation of the sodium conductance but is prolonged during removal of activation by repolarization, as observed in sodium tail experiments. The sodium inactivation versus voltage curve is shifted in the hyperpolarizing direction as is the inactivation time constant curve, measured with conditioning voltage steps. This shift speeds the kinetics of inactivation over part of the same voltage range in which sodium currents are prolonged, a contradiction incompatible with the Hodgkin-Huxley model. These results are interpreted as support for a hypothesis of two inactivation processes, one proceeding directly from the resting state and the other coupled to the active state of sodium conductance.     

The Physical Interpretation of Mathematical Models for Sodium Permeability Changes in Excitable Membranes

This paper deals with the physical interpretation of existing mathematical models which describe the transient sodium conductance changes in excitable membranes. It is shown that there are clear limitations to the specificity of inferences which may be drawn about physical mechanism from the behavior of abstract models. Within these limitations, it is shown that a pronounced inactivation shift is not necessarily evidence for coupling between the events responsible for the rise and inactivation of the sodium conductance, but that the inactivation shift may be associated with an event whose rate explicitly depends on the rate of continuous voltage change or magnitude of instantaneous voltage change.     

Vestibular integrator neurons have quadratic functions due to voltage dependent conductances

The nonlinear properties of the dendrites of the prepositus hypoglossi nucleus (PHN) neurons are essential for the operation of the vestibular neural integrator that converts a head velocity signal to one that controls eye position. A novel system of frequency probing, namely quadratic sinusoidal analysis (QSA), was used to decode the intrinsic nonlinear behavior of these neurons under voltage clamp conditions. Voltage clamp currents were measured at harmonic and interactive frequencies using specific nonoverlapping stimulation frequencies. Eigenanalysis of the QSA matrix reduces it to a remarkably compact processing unit, composed of just one or two dominant components (eigenvalues). The QSA matrix of rat PHN neurons provides signatures of the voltage dependent conductances for their particular dendritic and somatic distributions. An important part of the nonlinear response is due to the persistent sodium conductance (g_NaP), which is likely to be essential for sustained effects needed for a neural integrator. It was found that responses in the range of 10 mV peak to peak could be well described by quadratic nonlinearities suggesting that effects of higher degree nonlinearities would add only marginal improvement. Therefore, the quadratic response is likely to sufficiently capture most of the nonlinear behavior of neuronal systems except for extremely large synaptic inputs. Thus, neurons have two distinct linear and quadratic functions, which shows that piecewise linear?+?quadratic analysis is much more complete than just piecewise linear analysis; in addition quadratic analysis can be done at a single holding potential. Furthermore, the nonlinear neuronal responses contain more frequencies over a wider frequency band than the input signal. As a consequence, they convert limited amplitude and bandwidth input signals to wider bandwidth and more complex output responses. Finally, simulations at subthreshold membrane potentials with realistic PHN neuron models suggest that the quadratic functions are fundamentally dominated by active dendritic structures and persistent sodium conductances.     

Voltage-clamp measurement of visually-evoked conductances with whole-cell patch recordings in primary visual cortex

Whole cell patch recordings have been realized in the primary visual cortex of the anesthetized and paralyzed cat, in order to better characterize input resistance and time constant of visual cortical cells in vivo. Measurements of conductance changes evoked by visual stimulation were derived from voltage clamp recordings achieved in continuous mode at two or more different subtreshold holding potentials. They show that the magnitude of the conductance increase can reach up to 300% of the mean conductance at rest. The observation of similar changes for the preferred and antagonist responses, when flashing ON and OFF, a test stimulus in pure ON and OFF subfields supports the hypothesis of a role for shunting inhibition in the spatial organization of simple receptive fields.     

Effects of physostigmine on the voltage dependent ionic conductances of skeletal muscle fibres

The voltage dependent ionic conductances were studied by analysing the phase plane trajectories of action potentials evoked by electrical stimulation of the sartorius muscles of the frog (Rana esculenta). The delayed outward potassium current was measured also under voltage clamp conditions on muscle fibres of either the frog (Rana esculenta) or Xenopus laevis. On analysing the effect of physostigmine decreasing the peak amplitude, the rate of both the rising and falling phases of the action potentials, it was revealed that the alkaloid at a concentration of 1 mmol/l reduced significantly both the delayed potassium conductance and the outward ionic current values during the action potentials. The inhibition of sodium conductance and inward ionic current was less expressed. The maximum value of delayed potassium conductance measured under voltage clamp conditions was decreased by 1 mmol/l physostigmine. The time constant determined from the development of delayed potassium conductance was increased at a given membrane potential. The voltage vs. n relationship describing the membrane potential dependence of the delayed rectifier was not influenced by physostigmine. It has been concluded that physostigmine changes the time course of the action potentials by decreasing the value of both voltage dependent ionic conductances and by slowing down their kinetics. It is discussed that results obtained from the phase plane analysis of complex pharmacological effects can only be accepted with some restrictions.     

Potassium current activated by intracellular sodium in quail trigeminal ganglion neurons

Whole-cell voltage clamp and single-channel recordings were performed on cultured trigeminal ganglion neurons from quail embryos in order to study a sodium-activated potassium current (KNa). When KNa was activated by a step depolarization in voltage clamp, there was a proportionality between KNa and INa at all voltages between the threshold of INa and ENa. Single-channel recordings indicated that KNa could be activated already by 12 mM intracellular sodium and was almost fully activated at 50 mM sodium. 100 mM lithium, 100 mM choline, or 5 microM calcium did not activate KNa. The relationship between the probability for the channel to be open (Po) vs. the sodium concentration and the relationship of KNa open time-distributions vs. the sodium concentration suggest that two to three sodium ions bind cooperatively before KNa channels open. KNa channels were sensitive to depolarization; at 12 mM sodium, a 42-mV depolarization caused an e-fold increase in Po. Under physiological conditions, the conductance of the KNa channel was 50 pS. This conductance increased to 174 pS when the intra- and extracellular potassium concentrations were 75 and 150 mM, respectively.     

Light scattering spectroscopy of the squid axon membrane

Light scattering studies on the giant squid axon were done using the technique of optical mixing spectroscopy. This experimental approach is based on the use of laser light to detect the fluctuations of membrane macromolecules which are associated with conductance fluctuations. The light scattering spectra were similar to the Lorentzian-like behavior of conductance fluctuations, possibly reflecting an underlying conformational change in the specific membrane sites responsible for the potassium ion conductance. The amplitude of the spectra measured, increased when the membrane was depolarized and decreased on hyperpolarization. The spectra were fit to the sum of two terms, a (1/fcomponent and a simple Lorentzian term. Spectra from deteriorating axons did not show sensitivity to membrane potential changes. It is shown theoretically that fluctuations due to the voltage-dependent variable, n, of the Hodgkin-Huxley formalism are identical to the voltage fluctuations. The derived power spectrum is that of a second order system, capable of showing resonance peaking only if the voltage dependence of the potassium rate constants is included in the analysis. The lack of resonance peaking in the observed light scattering spectra, indicates that the data are best described by a damped second order system.     

Voltage-Dependent Conductance Induced in Thin Lipid Membranes by Monazomycin

When present in micromolar amounts on one side of phospholipid bilayer membranes, monazomycin (a positively charged, polyene-like antibiotic) induces dramatic voltage-dependent conductance effects. Voltage clamp records are very similar in shape to those obtained from the potassium conductance system of the squid axon. The steady-state conductance is proportional to the 5th power of the monazomycin concentration and increases exponentially with positive voltage (monazomycin side positive); there is an e-fold change in conductance per 4–6 mv. The major current-carrying ions are univalent cations. For a lipid having no net charge, steady-state conductance increases linearly with KCl (or NaCl) concentration and is unaffected by Ca⁺⁺ or Mg⁺⁺. The current-voltage characteristic which is normally monotonic in symmetrical salt solutions is converted by a salt gradient to one with a negative slope-conductance region, although the conductance-voltage characteristic is unaffected. A membrane treated with both monazomycin and the polyene antibiotic nystatin (which alone creates anion-selective channels) displays bistability in the presence of a salt gradient. Thus monazomycin and nystatin channels can exist in parallel. We believe that many monazomycin monomers (within the membrane) cooperate to form a multimolecular conductance channel; the voltage control of conductance arises from the electric field driving monazomycin molecules at the membrane surface into the membrane and thus affecting the number of channels that are formed.     

Axon voltage-clamp simulations. III. Postsynaptic region.

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

The voltage-dependent step of the chloride transporter of Valonia utricularis encounters a Nernst-Planck and not an Eyring type of potential energy barrier

Voltage-clamp experiments were performed on cells of the giant marine alga Valonia utricularis to study the voltage dependence of the previously postulated chloride transporter (Wang, J., G. Wehner, R. Benz, and U. Zimmermann. 1991. Biophys. J. 59:235-248). Only one exponential current relaxation (apart from the capacitive spike) could be resolved up to a clamp voltage of ~120 mV within the time resolution of our experimental instrumentation (100 μs). This means that the rate constants of the heterogeneous complexation, k_R (association) and k_D (dissociation), were too fast to be resolved. Therefore, the “Läuger” model for carrier-mediated ion transport with equilibrium heterogeneous surface reaction was used to fit the experimental results. The voltage dependence of the initial membrane conductance was used for the evaluation of the voltage dependence of the translocation rate constant of the complexed carriers, k_AS. The initial conductance was found to be independent on the clamp voltage, which means that the translocation rate constant k_AS is a linear function of the applied voltage and that the voltage dependence of the translocation of charged carriers through the plasmalemma could be explained by a square-type Nernst-Planck barrier. The movement of the complexed form of the carrier through the membrane may be explained by a diffusion process rather than by simple first-order kinetic jump across an Eyring-type potential well. The current relaxation after a voltage clamp was studied as a function of the external chloride concentration. The results allowed an estimation of the stability constant, K, of the heterogeneous complexation reaction and a calculation of the translocation rate constants of the free and the complexed carriers, k_s and k_AS, respectively.     

Reduction of stochastic conductance-based neuron models with time-scales separation

We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.     

Sodium currents in the giant axon of the crab Carcinus maenas

Measurements were made of the kinetics and steady-state properties of the sodium conductance changes in the giant axon of the crab Carcinus maenas. The conductance measurements were made in the presence of small concentrations of tetrodotoxin and as much electrical compensation as possible in order to minimize errors caused by the series resistance. After an initial delay of 10-150 microsec, the conductance increase during depolarizing voltage clamp pulses followed the Hodgkin-Huxley kinetics. Values of the time constant for the activation of the sodium conductance lay on a bell-shaped curve with a maximum under 180 microsec at -40 mV (at 18 degrees C). Values of the time constant for the inactivation of the sodium conductance were also fitted using a bell-shaped curve with a maximum under 7 msec at -70 mV. The effects of membrane potential on the fraction of Na channels available for activation studied using double pulse protocols suggest that hyperpolarizing potentials more negative than -100 mV lock a fraction of the Na channels in a closed conformation.     

Limitations of the dual voltage clamp method in assaying conductance and kinetics of gap junction channels.

The electrical properties of gap junctions in cell pairs are usually studied by means of the dual voltage clamp method. The voltage across the junctional channels, however, cannot be controlled adequately due to an artificial resistance and a natural resistance, both connected in series with the gap junction. The access resistances to the cell interior of the recording pipettes make up the artificial resistance. The natural resistance consists of the cytoplasmic access resistances to the tightly packed gap junction channels in both cells. A mathematical model was constructed to calculate the actual voltage across each gap junction channel. The stochastic open-close kinetics of the individual channels were incorporated into this model. It is concluded that even in the ideal case of complete compensation of pipette series resistance, the number of channels comprised in the gap junction may be largely underestimated. Furthermore, normalized steady-state junctional conductance may be largely overestimated, so that transjunctional voltage dependence is easily masked. The model is used to discuss conclusions drawn from dual voltage clamp experiments and offers alternative explanations for various experimental observations.     

Measurement,control, and modeling of submerged acetic acid fermentation

For control and optimization of large scale bioprocesses, mathematical models are needed to describe transient growth and/or product formation. Such models can only be developed from reliable experimental data. A computerized experimental system was applied to submerged acetic acid fermentation with industrial Acetobacter strains in order to obtain quantitatively reproducible long-term data. Automated repeated-batch fermentations were carried out over a period of one year. It was found that consideration of substrate, product, and biomass concentrations alone was not sufficient to describe transient culture conditions. At least one more internal parameter must be taken into account. A delay-time model was developed which takes into consideration the variable concentration of an internal component of the cells, the ribonucleic acid. This model was used to simulate the acetic acid fermentation. The simulation results agreed well with the experimental data. Thus, the validity of the model assumptions could be confirmed. The model was capable of simulating the lag-phase of growth as well as lysis of microorganisms due to product inhibition.