Criteria of bistability of the cylindrical dendrite with the variable negative slope of the N-shaped current-voltage membrane characteristic

The criteria for hystheresis in the input current-voltage relation of a cylindrical dendrite, i.e. cable bistability, were studied earlier in case of the constant negative slope of the N-shaped membrane current-voltage characteristic. For a membrane with a variable negative slope of the current-voltage characteristic, only sufficient conditions of dendritic bistability were formulated: [equation: see text], where df/dV/h is the negative slope of the membrane current-voltage characteristic at zero current point, h; X is the electrotonic length of the dendrite. We propose to use as the necessary condition of bistability the above equation but with the maximal value of the negative slope df/dV/max instead of df/dV/h. Calculations illustrate that this necessary condition, with acceptable accuracy, can be used as the necessary and sufficient condition of the cable bistability when the N-shaped current-voltage characteristic of the membrane is arbitrary.     

Analysis of the effects of cesium ions on potassium channel currents in biological membranes

Cesium ions block potassium channels in biological membranes in a voltage dependent manner. For example, external cesium blocks inward current with little or no effect on outward current. Consequently, it produces a characteristic N-shaped current-voltage relationship. We have modeled this result by single file diffusion of ions in a narrow channel spanning the membrane with a special blocking site in the channel for cesium ions. The model enables us to make detailed comparisons of the effects of cesium on potassium channels in different types of biological membranes.     

Theoretical analysis of the firing pattern of a neuron with stationary N-shaped current voltage characteristic of its dendritic membrane

To confirm the hypothesis of the N-shaped current-voltage characteristic curve of slow ionic currents of the dendritic membrane the role of processes taking place on that membrane in organization of the firing pattern of the nerve cells was examined. On a mathematical model dependence of discharge frequency on strength of depolarizing current can be divided into two ranges. The second range of sharply increased steepness of the curve of discharge frequency versus current can be explained by transitions of the distal parts of the dendrites from resting potential to persistent depolarization. The possibility of hysteresis of the frequency-current curve is postulated and the spontaneous discharge of neurons as an off-response to strong depolarization is explained.     

Electrical bistability in a neuron model with monostable dendritic and axosomatic membranes

In a two-compartment mathematical model, we studied the reason for and conditions of manifestation of electrical bistability in a neuron composed of monostable parts. One compartment of the model simulated the dendrites; their membrane was monostable at high depolarization and characterized by an N-shaped steady current-voltage (I–V) characteristic endowed by inward synaptic current through voltage-dependent channels sensitive to N-methyl-D-aspartate (NMDA). Another compartment simulated the axosomatic region with a positively sloped linearizedI–V characteristic of the membrane monostable at the resting membrane potential. For the whole cell, bistability was obvious at a subcritical intensity of NMDA activation; the reason was the current directed from the more depolarized dendritic region into the somatic region, and the necessary condition was that the above somatopetal core current must exceed the net inward transmembrane current (the latter was the sum of the inward synaptic and outward passive extrasynaptic currents) of the dendritic compartment. This relation essentially depended on the size of the dendrites. Neirofiziologiya/Neurophysiology, Vol. 32, No. 2, pp. 98–101, March–April, 2000.     

Electrical Bistability of the Motoneuronal Dendritic Membrane Determined by Non-inactivating Inward Currents: a Model Study

We investigated features of the spatial pattern of electrical bistable states of dendrites using a computer model of an abducens motoneuron with the dendritic branching reconstructed in detail. The dendritic membrane has an N-shaped current-voltage relation (I-V curve) determined mainly by the presence of L-type calcium channels. Such channels, according to indirect experimental data, are present in the dendrites of these cells together with glutamatergic NMDA-type channels also capable of determining electrical bistability of the membrane and the corresponding specific patterns of electrical activity generated by such neurons. For our model, we obtained steady-state local I-V curves and transferred spatial distribution maps of the membrane potential difference (surface density of transmembrane currents), as well as increments of the axial dendritic current, to three-dimensional images of the reconstructed branching dendrites. The latter increments determine the contribution of a dendritic site in general axial current delivering the charge to the trigger zone of a neuron. The simulation results showed that incorporation of non-inactivating calcium channels into dendritic membrane leads to the origination of a pattern of spatial distribution of bistable electrical states in the dendrites, which were not described earlier. Such features are most important under conditions of a stable state of high depolarization of the relevant parts of the dendrites. In this case, the respective feature was the existence of a zone of maximum density of the inward transmembrane current, which covers areas of first-order branching of all dendrites. Since the greatest relative contribution to the total current belongs to the inward calcium current, the above zone of first branchings can be considered a “hot spot” zone characterized by increased entry of Ca²⁺. This may have important functional consequences for local intracellular calcium signaling.     

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.     

The Ventral Photoreceptor Cells of Limulus : III. A voltage-clamp study

In the dark, the ventral photoreceptor of Limulus exhibits time-variant currents under voltage-clamp conditions; that is, if the membrane potential of the cell is clamped to a depolarized value there is an initial large outward current which slowly declines to a steady level. The current-voltage relation of the cell in the dark is nonlinear. The only ion tested which has any effect on the current-voltage relation is potassium; high potassium shifts the reversal potential towards zero and introduces a negative slope-conductance region. When the cell is illuminated under voltage-clamp conditions, an additional current, the light-induced current, flows across the cell membrane. The time course of this current mimics the time course of the light response (receptor potential) in the unclamped cell; namely, an initial transient phase is followed by a steady-state phase. The amplitude of the peak transient current can be as large as 60 times the amplitude of the steady-state current, while in the unclamped cell the amplitude of the peak transient voltage never exceeds 4 times the amplitude of the steady-state voltage. The current-voltage relations of the additional light-induced current obtained for different instants of time are also nonlinear, but differ from the current-voltage relations of the dark current. The ions tested which have the greatest effect on the light-induced current are sodium and calcium; low sodium decreases the current, while low calcium increases the current. The data strongly support the hypothesis that two systems of electric current exist in the membrane. Thus the total ionic current which flows in the membrane is accounted for as the sum of a dark current and a light-induced current.     

Effect of natural convection on the current-voltage characteristic of a DC discharge in neon at intermediate pressures

The parameters of the positive column of a glow discharge in neon are calculated with allowance for the induced hydrodynamic motion. It is shown that natural convection in the pressure range of ∼0.1 atm significantly affects the profiles of the parameters of the positive column and its current-voltage characteristic. The convection arising at large deposited energies improves heat removal, due to which the temperature in the central region of the discharge becomes lower than that calculated from the heat conduction equation. As a result, the current-voltage characteristic is shifted. With allowance for convection, the current-voltage characteristic changes at currents much lower than the critical current at which a transition into the constricted state is observed. This change is uniquely related to the Rayleigh number in the discharge. Thus, a simplified analysis of thermal conduction and diffusion, even with detailed account of kinetic processes occurring in the positive column, does not allow one to accurately calculate the current-voltage characteristic and other discharge parameters at intermediate gas pressures.     

Epilepsy in kcnj10 Morphant Zebrafish Assessed with a Novel Method for Long-Term EEG Recordings

We aimed to develop and validate a reliable method for stable long-term recordings of EEG activity in zebrafish, which is less prone to artifacts than current invasive techniques. EEG activity was recorded with a blunt electrolyte-filled glass pipette placed on the zebrafish head mimicking surface EEG technology in man. In addition, paralysis of agarose-embedded fish using D-tubocurarine excluded movement artifacts associated with epileptic activity. This non-invasive recording technique allowed recordings for up to one hour and produced less artifacts than impaling the zebrafish optic tectum with a patch pipette. Paralyzed fish survived, and normal heartbeat could be monitored for over 1h. Our technique allowed the demonstration of specific epileptic activity in kcnj10a morphant fish (a model for EAST syndrome) closely resembling epileptic activity induced by pentylenetetrazol. This new method documented that seizures in the zebrafish EAST model were ameliorated by pentobarbitone, but not diazepam, validating its usefulness. In conclusion, non-invasive recordings in paralyzed EAST syndrome zebrafish proved stable, reliable and robust, showing qualitatively similar frequency spectra to those obtained from pentylenetetrazol-treated fish. This technique may prove particularly useful in zebrafish epilepsy models that show infrequent or conditional seizure activity.     

Remote Modulation of Current Transfer from Dendritic Glutamatergic Synapses by GABA-ergic Synapses of the Somatic Zone of Motoneurons: a Simulation Study

Until now, information concerning spatial interaction of postsynaptic excitation and inhibition in neuronal dendrites remains rather limited. In model experiments, we studied spatial effects of tonic co-activation of GABA-ergic synapses situated on the soma and axon hillock of a motoneuron and dendritic glutamatergic synapses with receptors sensitive or insensitive to N-methyl-D-aspartate. We analyzed distribution maps of the transmembrane potentials and excitatory currents transferred toward the soma over the reconstructed dendritic arborization of a rat abducens motoneuron (three-dimensional reconstruction). In the motoneuron, isolated tonic excitation of glutamatergic synapses induced two stable states of low (downstate) or high (upstate) spatially heterogeneous dendritic depolarization, which decayed with unequal rates along different dendritic paths. In this case, the local steady-state current-voltage relation of the dendritic membrane became N-shaped due to a limb of the negative slope within a certain voltage range. The upstate corresponding to plateau potentials associated with stereotyped motor activity patterns was analyzed in detail. In this state, most proximal dendritic sites were the main sources of the excitatory current reaching the soma, while the contribution from distal sites was negligible. Co-activation of GABA-synapses located at the soma and axon hillock reduced this depolarization and shifted the main excitatory current source from a perisomatic location to the middle, structurally more complex, region of the dendritic arborization. The more remote dendritic region having a greater membrane area and receiving a greater number of synaptic contacts became directly involved in the supply of the trigger zone by the excitatory current. We suggest that a special, not described earlier, operational mechanism of postsynaptic inhibition is manifested in the above spatial effects of activation of strategically located inhibitory synapses, and that the list of known crucial inhibitory mechanisms (namely hyperpolarization and shunting of the postsynaptic membrane) must be expanded.     

A Theoretical Study on the Sucrose Gap Technique as Applied to Multicellular Muscle Preparations: III. Methodical Errors in the Determination of Inward Currents

The analysis of errors associated with saline-sucrose interdiffusion in sucrose gap experiments on multicellular muscle preparations described in two previous papers (Lammel, E., 1981, Biophys. J., 36:533-553, 555-573) is extended to the determination of current-voltage relations that contain an activated inward current component. The membrane current-voltage (i_t-V_m) relation used in the computations was N-shaped and consisted of two components, an outward (background) current (i_bg) with properties of anomalous (inward-going) membrane rectification, and an inward current (i_s) resembling the slow inward current of cardiac muscle. Reconstruction of current-voltage relations, which simulate those determined experimentally, indicates that in the potential range in which the total membrane current (i_t) is outward, i_t is measured too high, whereas it is measured too low in the range of net inward current. Reversal potentials of the inward and outward components are both shifted to more negative values, that of the inward current being more affected. Simulation of the experimental approach to evaluate i_s as the difference between i_t and i_bg shows that errors that produce values too high for i_bg are partly compensated by errors that lead to values of the net inward component that are too low. The basic features of the distorting effects analyzed are independent of different assumptions made on the selectivity of the slow inward current channels. They are related to currents emerging from the sucrose compartment (local circuit as well as externally applied currents).     

Two purified fractions of alamethicin have different conductance properties

The properties of two purified alamethicin fractions, Fraction 4 and Fraction 6, have been studied in phosphatidylethanolamine (PE) membranes and phosphatidylserine (PS) membranes. Membranes doped with Fraction 4 show well-defined single channel conductance (mean lifetime about 20 ms). The autocorrelation function of the current fluctuations has one relaxation time of the same order as the mean lifetime of the single channels, and the current response to a voltage pulse follows an exponential with only one time constant. The conductance of a membrane doped with Fraction 6 has a voltage-independent part and a current-voltage curve with a slope that is half the slope of the Fraction 4 current-voltage curve. In the presence of Fraction 6, PS membranes and PE membranes both have symmetrical current-voltage curves even with Fraction 6 added to only one side. We did not detect any well-defined single channel levels in the presence of Fraction 6, and autocorrelation analysis of the current due to Fraction 6 gave two characteristic correlation times: a fast time (about 5 ms) and a slow time (about 50 ms). High current level kinetics of Fraction 6 also show two time constants. A possible explanation for the differences between the two fractions is that Fraction 6 monomers have a lower dipole moment than those of Fraction 4. The difference in channel stability can be explained by a lowered tendency of the monomers to line up parallel to the field. The negative branch and voltage-independent conductance can be explained by lowered energy of insertion of monomers into the membrane, and lowered energy of interaction between the monomers and the electric field.     

Phase resetting of the rhythmic activity of embryonic heart cell aggregates. Experiment and theory.

Injection of a current pulse of brief duration into an aggregate of spontaneously beating chick embryonic heart cells resets the phase of the activity by either advancing or delaying the time of occurrence of the spontaneous beat subsequent to current injection. This effect depends upon the polarity, amplitude, and duration of the current pulse, as well as on the time of injection of the pulse. The transition from prolongation to shortening of the interbeat interval appears experimentally to be discontinuous for some stimulus conditions. These observations are analyzed by numerical investigation of a model of the ionic currents that underlie spontaneous activity in these preparations. The model consists of: Ix, which underlies the repolarization phase of the action potential, IK2, a time-dependent potassium ion pacemaker current, Ibg, a background or time-independent current, and INa, an inward sodium ion current that underlies the upstroke of the action potential. The steady state amplitude of the sum of these currents is an N-shaped function of potential. Slight shifts in the position of this current-voltage relation along the current axis can produce either one, two, or three intersections with the voltage axis. The number of these equilibrium points and the voltage dependence of INa contribute to apparent discontinuities of phase resetting. A current-voltage relation with three equilibrium points has a saddle point in the pacemaker voltage range. Certain combinations of current-pulse parameters and timing of injection can shift the state point near this saddle point and lead to an interbeat interval that is unbounded . Activation of INa is steeply voltage dependent. This results in apparently discontinuous phase resetting behavior for sufficiently large pulse amplitudes regardless of the number of equilibrium points. However, phase resetting is fundamentally a continuous function of the time of pulse injection for these conditions. These results demonstrate the ionic basis of phase resetting and provide a framework for topological analysis of this phenomenon in chick embryonic heart cell aggregates.     

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

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

The Voltage Dependence of the Cardiac Membrane Conductance

Solutions have been computed for the point polarization of a sheet-like membrane obeying the equations used previously (Noble, 1960, 1962) to reproduce the Purkinje fiber action potential. It was found that, in spite of the gross non-linearity of the membrane current-voltage relations, the relations between total polarizing current and displacement of membrane potential at various distances from the polarizing electrode are remarkably linear. It is therefore concluded that Johnson and Tille's (1960, 1961) results showing linear polarizing current-voltage relations obtained by passing current through the membrane from a microelectrode during the plateau of the rabbit ventricular action potential do not conflict with the Hodgkin-Huxley theory of electrical activity.     

Sodium ions as blocking agents and charge carriers in the potassium channel of the squid giant axon

Instantaneous K channel current-voltage (I-V) relations were determined by using internally perfused squid axons. When K was the only internal cation, the I-V relation was linear for outward currents at membrane potentials up to +240 mV inside. With 25-200 mM Na plus 300 mM K in the internal solution, an N-shaped I-V curve was seen. Voltage-dependent blocking of the K channels by Na produces a region of negative slope in the I-V plot (F. Bezanilla and C. M. Armstrong. 1972. J. Gen Physiol, 60: 588). At higher voltages (greater than or equal to 160 mV) we observed a second region of increasing current and a decrease in the fraction of the K conductance blocked by Na. Internal tetraethylammonium (TEA) ions blocked currents over the whole voltage range. In a second series of experiments with K-free, Na-containing internal solutions, the I-V curve turned sharply upward about +160 mV. The current at high voltages increased with increasing internal Na concentration was largely blocked by internal TEA. These data suggest that the K channel becomes substantially more permeable to Na at high voltages. This change is apparently responsible for the relief, at high transmembrane voltages, of the blocking effect seen in axons perfused with Na plus K mixtures. Each time a Na ion passed through, vacating the blocking site, the channel would transiently allow K ions to pass through freely.     

The N-Shaped Current-Potential Characteristic in Frog Skin: II. Kinetic Behavior during Ramp Voltage Clamp

Previous step voltage-clamp measurements on frog skin showed the presence of an N-shaped current-potential (I-V) relation in excitable skin. However, the collection and reconstruction of I-V data using discrete step changes of skin potential was tedious because of the long refractory period (up to 1 min) in frog skin. A direct and rapid (5 msec) method for recording the N-shaped I-V characteristic in real time is presented. Ramp functions are used as the command to the clamp system instead of a step function. Consequently the skin potential is forced to change in a linear manner (as commanded) and the skin current can be recorded as a continuous function of the controlled change of skin potential. With the ramp clamp, a low-resistance membrane state ( 10 Omega . cm(2)) resembling a breakdown phenomenon was observed at high skin potential ( 300 mv). Entry into the low resistance state resulted in a collapse of the N-shaped I-V relation to a nearly linear function. The utility of the ramp measurement is demonstrated by predicting (1) that the maximum rate of rise of the spike occurs at a voltage corresponding to the valley (local minimum) in the N-shaped I-V curve, (2) that the rate of rise of the spike increases with increasing clamp currents, (3) the voltage peak of the spike, and (4) the time course of the rising phase of the spike.     

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.     

Ion Channel Density Regulates Switches between Regular and Fast Spiking in Soma but Not in Axons

The threshold firing frequency of a neuron is a characterizing feature of its dynamical behaviour, in turn determining its role in the oscillatory activity of the brain. Two main types of dynamics have been identified in brain neurons. Type 1 dynamics (regular spiking) shows a continuous relationship between frequency and stimulation current (f-I_stim) and, thus, an arbitrarily low frequency at threshold current; Type 2 (fast spiking) shows a discontinuous f-I_stim relationship and a minimum threshold frequency. In a previous study of a hippocampal neuron model, we demonstrated that its dynamics could be of both Type 1 and Type 2, depending on ion channel density. In the present study we analyse the effect of varying channel density on threshold firing frequency on two well-studied axon membranes, namely the frog myelinated axon and the squid giant axon. Moreover, we analyse the hippocampal neuron model in more detail. The models are all based on voltage-clamp studies, thus comprising experimentally measurable parameters. The choice of analysing effects of channel density modifications is due to their physiological and pharmacological relevance. We show, using bifurcation analysis, that both axon models display exclusively Type 2 dynamics, independently of ion channel density. Nevertheless, both models have a region in the channel-density plane characterized by an N-shaped steady-state current-voltage relationship (a prerequisite for Type 1 dynamics and associated with this type of dynamics in the hippocampal model). In summary, our results suggest that the hippocampal soma and the two axon membranes represent two distinct kinds of membranes; membranes with a channel-density dependent switching between Type 1 and 2 dynamics, and membranes with a channel-density independent dynamics. The difference between the two membrane types suggests functional differences, compatible with a more flexible role of the soma membrane than that of the axon membrane.     

Voltage- and time-dependent action of histrionicotoxin on the endplate current of the frog muscle

Histrionicotoxin, a toxin isolated from skin secretions of a Colombian arrow poison frog, Dendrobates histrionicus, decreased the amplitude and time-course of the endplate current, and altered the voltage dependence of the half-decay time. In addition, the toxin produced a characteristic nonlinearity in the current-voltage relationship of the endplate current when 3-s voltage conditioning steps were used. Reduction in time of the conditioning steps to 10 ms made the current-voltage relationship linear. The decrease in peak amplitude of the endplate current (epc) produced by histrionicotoxin measured during long hyperpolarizing conditioning steps was fitted by a single exponential function. The calculated rate constants ranged from 0.03 to 0.14 s-1 and varied with membrane potential at hyperpolarizing levels. The voltage- and time-dependent action of histrionicotoxin does not require an initial activation of receptors by acetylcholine (ACh). The characteristic of the current-voltage relationship can be accounted for by the observed voltage and time dependency of the attenuation of the endplate current amplitude in the presence of histrionicotoxin during long conditioning steps. These effects of histrionicotoxin on the peak amplitude, and on the voltage and time dependence of the epc were concentration-dependent and slowly reversible upon washing out the toxin. Thus, the voltage- and time-dependent action of histrionicotoxin at the endplate is related to an increase in the affinity between the toxin and the ACh receptor-ionic channel complex. This increase in affinity is postulated to be due to a conformational change of the macromolecule in the presence of histrionicotoxin which is demonstrated to be relatively slow, i.e., on the order of tens of seconds.