The assembly of ionic currents in a thalamic neuron. III. The seven-dimensional model

We replace our earlier three-dimensional ha-model of a thalamic neuron (Rose & Hindmarsh, Proc. R. Soc. Lond. B 237, 289-312 (1989b)) by a seven-dimensional ha-model. The stability and state diagrams for this seven-dimensional model are shown to be similar to those of the three-dimensional system with which it is compared. Two examples that illustrate how the seven-dimensional model can be related to experimental recordings are then discussed in detail. In each case we show the state and stability diagrams and the time courses of the different ionic currents during the burst response. We discuss how the various components of the state diagram could be determined experimentally. These experiments are illustrated by using our model to simulate the results of actual experiments. Finally the state and stability diagrams are transferred to a current-voltage diagram. This shows the advantage of the state diagram compared with the current-voltage diagram for representing the dynamics of the firing of a thalamic neuron.     

The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model

We have previously discussed qualitative models for bursting and thalamic neurons that were obtained by modifying a simple two-dimensional model for repetitive firing. In this paper we report the results of making a similar sequence of modifications to a more elaborate six-dimensional model of repetitive firing which is based on the Hodgkin-Huxley equations. To do this we first reduce the six-dimensional model to a two-dimensional model that resembles our original two-dimensional qualitative model. This is achieved by defining a new variable, which we call q. We then add a subthreshold inward current and a subthreshold outward current having a variable, z, that changes slowly. This gives a three-dimensional (v,q,z) model of the Hodgkin-Huxley type, which we refer to as the z-model. Depending on the choice of parameter values this model resembles our previous models of bursting and thalamic neurons. At each stage in the development of these models we return to the corresponding seven-dimensional model to confirm that we can obtain similar solutions by using the complete system of equations. The analysis of the three-dimensional model involves a state diagram and a stability diagram. The state diagram shows the projection of the phase path from v,q,z space into the v,z plane, together with the projections of the curves z = 0 and v = q = 0. The stability of the points on the curve v = q = 0, which we call the v, q nullcurve, is determined by the stability diagram. Taken together the state and stability diagrams show how to assemble the ionic currents to produce a given firing pattern.     

A model of a thalamic neuron

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. Hyperpolarization from rest is followed by post-inhibitory rebound. The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x-y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.     

Analysis of bursting in a thalamic neuron model

We extend a quantitative model for low-voltage, slow-wave excitability based on the T-type calcium current (Wang et al. 1991) by juxtaposing it with a Hodgkin-Huxley-like model for fast sodium spiking in the high voltage regime to account for the distinct firing modes of thalamic neurons. We employ bifurcation analysis to illustrate the stimulus-response behavior of the full model under both voltage regimes. The model neuron shows continuous sodium spiking when depolarized sufficiently from rest. Depending on the parameters of calcium current inactivation, there are two types of low-voltage responses to a hyperpolarizing current step: a single rebound low threshold spike (LTS) upon release of the step and periodic LTSs. Bursting is seen as sodium spikes ride the LTS crest. In both cases, we analyze the LTS burst response by projecting its trajectory into a fast/slow phase plane. We also use phase plane methods to show that a potassium A-current shifts the threshold for sodium spikes, reducing the number of fast sodium spikes in an LTS burst. It can also annihilate periodic bursting. We extend the previous work of Rose and Hindmarsh (1989a–c) for a thalamic neuron and propose a simpler model for thalamic activity. We consider burst modulation by using a neuromodulator-dependent potassium leakage conductance as a control parameter. These results correspond with experiments showing that the application of certain neurotransmitters can switch firing modes. Received: 18 July 1993/Accepted in revised form: 22 January 1994     

Role of ionic currents in the physiological response to angiotensin II.

Studies using conventional and patch-clamp microelectrode techniques demonstrate that in a number of cell types angiotensin II (AII) causes reversible changes in transmembrane ionic currents, and that these effects can be mimicked by various membrane-associated and cytosolic messengers. AII modulates the current amplitude of ion channels, as well as their activation threshold and their open/closed time probability. Stimulatory and inhibitory effects on ion channel activity are a fundamental feature of the development of AII actions on target organs.     

The role of ionic currents in establishing developmental pattern

This paper reviews a theory of pattern establishment and pattern restoration by endogenous ionic currents. These currents are supposed to be generated by a certain separation of ion leaks and ion pumps in cell membranes. In so far as these currents act back to further this separation, they would be part of a regenerative process that initially establishes positional values. Later in development, particularly in epimorphic regeneration, when positional values are restored or extended, these currents are supposed to leak through sites of discontinuity in such values and thus trigger growth. This paper also reviews the factual evidence for this view: evidence that developmental currents are, indeed, very widespread; evidence in a few cases, particularly in Cecropia follicles and in wounded cavy skin, that they can generate substantial voltage differences or gradients; evidence that comparable artificial fields can move charged macromolecules along cell membranes and polarize cell growth; and direct evidence in a few case, particularly fucoid eggs, Cecropia follicles and regenerating amphibian limbs, that ion currents do, in fact, act back to direct or further development. The paper also presents a particular theory, based upon ionic currents, of the reversal of thyroid cell polarity by serum.     

The course of ionic transmembrane currents during cardiac action potentials.

The course of the total transmembrane ionic current (Ii) during a natural action potential (AP) was reconstructed from a family of current traces recorded for single voltage clamp depolarization steps to various levels. The experiments were performed on 9 papillary cat muscles driven at 0.5 per second in oxygenated 31 degrees C Tyrode. Under varying experimental conditions very good agreement was found between the resulting Ii curve and another indicator of Ii, the first time derivative of the AP (dV/dt). Furthermore, the coefficient needed to adjust dV/dt to reconstructed Ii may serve as an indicator of the membrane capacity. The results suggest the validity of the employed approximation and, in general, the adequacy of the sucrose gap technique applied to cardiac muscle.     

Effects of 4-aminopyridine on potassium currents in a molluscan neuron

The effects of 4-aminopyridine (4-AP) on the delayed K+ current and on the Ca2+-activated K+ current of the Aplysia pacemaker neurons R-15 and L-6 were studied. The delayed outward K+ current was measured in Ca2+- free artificial seawater (ASW) containing tetrodotoxin (TTX), using brief depolarizing clamp pulses. External (and internal) 4-AP blocks the delayed K+ current in a dose-dependent manner but does not block the leakage current. Our results show that one 4-AP molecule combines with a single receptor site and that the block is voltage dependent with an apparent dissociation constant (K4-AP) of approximately 0.8 mM at 0 mV. K4-AP increases e-fold for a 32-mV change in potential, which is consistent with the block occurring approximately 0.8 of the distance through the membrane electrical field. The 4-AP block appears to depend upon stimulus frequency as well as upon voltage. The greater speed of onset of the block produced by internal 4-AP relative to when it is used externally suggests that 4-AP acts from inside the cell. The Ca2+-activated K+ current was measured in Ca2+-free ASW containing TTX, using internal Ca2+-ion injection to directly activate the K+ conductance. Low external 4-AP concentrations (less than 2 mM) have no effect on the Ca2+-activated K+ current, but concentrations of 5 mM or greater increase the K+ current. Internal 4-AP has the same effect. The opposing effects of 4-AP on the two components of the K+ current can be seen in measurements of the total outward K+ current at different membrane potentials in normal ASW and during the repolarizing phase of the action potential.     

Zinc-induced changes in ionic currents of cardiomyocytes

The whole-cell voltage-clamp technique was applied to isolated ventricular myocytes to investigate the effects of extracellular and intracellular zinc application on L-type Ca²⁺ channel currents (I _Ca). Extracellular zinc exposure at micromolar concentration induced a reversible (with washout of ZnCl₂) reduction (30%) of I _Ca with no change in current-voltage relationship. On the other hand, an increase of intracellular free-zinc concentration, [Zn²⁺]_i, from normal (less than 1 nM) to approx 7 nM with 10 μM Zn-pyrithione exposure caused an inhibition of 33±6% in the peak of the I _Ca and altered the voltage dependency of L-type Ca²⁺ channels with a 10-mV left shift and a hump at around −40 mV in its current-voltage relation. In contrast, N,N,N′,N′-tetrakis(2-pyridylmethyl)ethylenediamine (TPEN) strongly inhibited the I _Ca (42±2%), with only a small but detectable outward shift of the holding current measured at the end of the pulses. Zn-pyrithione and TPEN caused a reproducible decrease of the I _Ca. Interestingly, TPEN application, without Zn-pyrithione pretreatment, inhibited the I _Ca (35±2%) with no change in voltage dependency. Taken together, the results suggest that both extracellular and intracellular zinc increases under pathological conditions in cardiomyocytes can alter the I _Ca, but their effects are not in the same order and same manner. One should consider these possible side effects when it is suggested to be vital to cardiovascular cell integrity and functions.     

One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death II: transition state metastability

This is the second of two papers in which we study a mathematical model of cytoskeleton-induced neuron death. Recent evidence indicates that aggravated assembly or destruction of the cytoskeleton can trigger programmed death in neurons, by mechanisms as yet poorly understood. In our model, assembly control of the neuronal cytoskeleton interacts with both cellular stress levels and cytosolic free radical concentrations to trigger neurodegeneration. This trigger mechanism is further modulated by a diffusible toxic factor released from dying neurons. In the companion report we established that the model relates the observed general patterns of neuron decline to specific scales of cytoskeleton reorganization and cell-cell interaction strength. In this paper we study the transit of neurons through states intermediate between initial viability and cell death in our model. We find that the stochastic flow of neuron fate, from viability to cell death, self-organizes into two distinct temporal phases. There is a rapid relaxation of the initial neuron population to a more disordered phase that is long-lived, or metastable, with respect to the time scales of change in single cells. Strikingly, cellular egress from this metastable phase follows the one-hit kinetic pattern of exponential decline now established as a principal hallmark of cell death in neurodegenerative disorders. Intermediate state metastability may therefore be an important element in the systems biology of one-hit neurodegeneration.     

Sodium ionic and gating currents in mammalian cells

Ionic and gating currents from voltage-gated sodium channels were recorded in mouse neuroblastoma cells using the path-clamp technique. Displacement currents were measured from whole-cell recordings. The gating charge displaced during step depolarizations increased with the applied membrane potential and reached saturating levels above 20 mV Prolonged large depolarizations produced partial immobilization of the gating charge, and only about one third of the displaced charge was quickly reversed upon return to negative holding potentials. The activation and inactivation properties of macroscopic sodium currents were characterized by voltage-clamp analysis of large outside-out patches and the single-channel conductance was estimated from nonstationary noise analysis. The general properties of the sodium channels in mouse neuroblastoma cells are very similar to those previously reported for various preparations of invertebrate and vertebrate nerve cells.Offprint requests to: O. Moran     

Study of ionic currents across a model membrane channel using Brownian dynamics.

Brownian dynamics simulations have been carried out to study ionic currents flowing across a model membrane channel under various conditions. The model channel we use has a cylindrical transmembrane segment that is joined to a catenary vestibule at each side. Two cylindrical reservoirs connected to the channel contain a fixed number of sodium and chloride ions. Under a driving force of 100 mV, the channel is virtually impermeable to sodium ions, owing to the repulsive dielectric force presented to ions by the vestibular wall. When two rings of dipoles, with their negative poles facing the pore lumen, are placed just above and below the constricted channel segment, sodium ions cross the channel. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength does not increase the channel conductance further. When only those ions that acquire a kinetic energy large enough to surmount a barrier are allowed to enter the narrow transmembrane segment, the channel conductance decreases monotonically with the barrier height. This barrier represents those interactions between an ion, water molecules, and the protein wall in the transmembrane segment that are not treated explicitly in the simulation. The conductance obtained from simulations closely matches that obtained from ACh channels when a step potential barrier of 2-3 kTr is placed at the channel neck. The current-voltage relationship obtained with symmetrical solutions is ohmic in the absence of a barrier. The current-voltage curve becomes nonlinear when the 3 kTr barrier is in place. With asymmetrical solutions, the relationship approximates the Goldman equation, with the reversal potential close to that predicted by the Nernst equation. The conductance first increases linearly with concentration and then begins to rise at a slower rate with higher ionic concentration. We discuss the implications of these findings for the transport of ions across the membrane and the structure of ion channels.     

The manganese-stabilizing protein is required for photosystem II assembly/stability and photoautotrophy in higher plants

Interfering RNA was used to suppress the expression of two genes that encode the manganese-stabilizing protein of photosystem II in Arabidopsis thaliana, MSP-1 (encoded by psbO-1, At5g66570), and MSP-2 (encoded by psbO-2, At3g50820). A phenotypic series of transgenic plants was recovered that expressed high, intermediate, and low amounts of these two manganese-stabilizing proteins. Chlorophyll fluorescence induction and decay analyses were performed. Decreasing amounts of expressed protein led to the progressive loss of variable fluorescence and a marked decrease in the fluorescence quantum yield (F(v)/F(m)) in both the absence and the presence of dichloromethylurea. This result indicated that the amount of functional photosystem II reaction centers was compromised in the plants that exhibited intermediate and low amounts of the manganese-stabilizing proteins. An analysis of the decay of the variable fluorescence in the presence of dichlorophenyldimethylurea indicated that charge recombination between Q ((A-)) and the S(2) state of the oxygen-evolving complex was seriously retarded in the plants that expressed low amounts of the manganese stabilizing proteins. This may have indicated a stabilization of the S(2) state in the absence of the extrinsic component. Immunological analysis of the photosystem II protein complement indicated that significant losses of the CP47, CP43, and D1 proteins occurred upon the loss of the manganese-stabilizing proteins. This indicated that these extrinsic proteins were required for photosystem II core assembly/stability. Additionally, although the quantity of the 24-kDa extrinsic protein was only modestly affected by the loss of the manganese-stabilizing proteins, the 17-kDa extrinsic protein dramatically decreased. The control proteins ribulose bisphosphate carboxylase and cytochrome f were not affected by the loss of the manganese-stabilizing proteins; the photosystem I PsaB protein, however, was significantly reduced in the low expressing transgenic plants. Finally, it was determined that the transgenic plants that expressed low amounts of the manganese-stabilizing proteins could not grow photoautotrophically.     

Interactive responses of a thalamic neuron to formalin induced lasting pain in behaving mice

Thalamocortical (TC) neurons are known to relay incoming sensory information to the cortex via firing in tonic or burst mode. However, it is still unclear how respective firing modes of a single thalamic relay neuron contribute to pain perception under consciousness. Some studies report that bursting could increase pain in hyperalgesic conditions while others suggest the contrary. However, since previous studies were done under either neuropathic pain conditions or often under anesthesia, the mechanism of thalamic pain modulation under awake conditions is not well understood. We therefore characterized the thalamic firing patterns of behaving mice in response to nociceptive pain induced by inflammation. Our results demonstrated that nociceptive pain responses were positively correlated with tonic firing and negatively correlated with burst firing of individual TC neurons. Furthermore, burst properties such as intra-burst-interval (IntraBI) also turned out to be reliably correlated with the changes of nociceptive pain responses. In addition, brain stimulation experiments revealed that only bursts with specific bursting patterns could significantly abolish behavioral nociceptive responses. The results indicate that specific patterns of bursting activity in thalamocortical relay neurons play a critical role in controlling long-lasting inflammatory pain in awake and behaving mice.     

Charges, currents, and potentials in ionic channels of one conformation.

Flux through an open ionic channel is analyzed with Poisson-Nernst-Planck (PNP) theory. The channel protein is described as an unchanging but nonuniform distribution of permanent charge, the charge distribution observed (in principle) in x-ray diffraction. Appropriate boundary conditions are derived and presented in some generality. Three kinds of charge are present: (a) permanent charge on the atoms of the protein, the charge independent of the electric field; (b) free or mobile charge, carried by ions in the pore as they flux through the channel; and (c) induced (sometimes called polarization) charge, in the pore and protein, created by the electric field, zero when the electric field is zero. The permanent charge produces an offset in potential, a built-in Donnan potential at both ends of the channel pore. The system is completely solved for bathing solutions of two ions. Graphs describe the distribution of potential, concentration, free (i.e., mobile) and induced charge, and the potential energy associated with the concentration of charge, as well as the unidirectional flux as a function of concentration of ions in the bath, for a distribution of permanent charge that is uniform. The model shows surprising complexity, exhibiting some (but not all) of the properties usually attributed to single filing and exchange diffusion. The complexity arises because the arrangement of free and induced charge, and thus of potential and potential energy, varies, sometimes substantially, as conditions change, even though the channel structure and conformation (of permanent charge) is strictly constant. Energy barriers and wells, and the concomitant binding sites and binding phenomena, are outputs of the PNP theory: they are computed, not assumed. They vary in size and location as experimental conditions change, while the conformation of permanent charge remains constant, thus giving the model much of its interesting behavior.     

Selenium-induced alterations in ionic currents of rat cardiomyocytes

In the present study, rats were treated with sodium selenite (5 micromol/kg body weight/day, ip) for 4 weeks and the parameters of contractile activity, action potential, L-type Ca2+-current (ICaL), as well as transient outward (Ito), inward rectifier (IK1), and steady state (Iss) K+-currents were investigated. Sodium selenite treatment increased rat blood glucose level and lowered plasma insulin level, significantly. This treatment also caused slightly prolongation in action potential with no significant effects on spontaneous contraction parameters and intracellular Ca2+ transients of the heart preparations. These effects were associated with marked alterations in the kinetics of both ICaL and Ito including a significant slowing in both inactivation time constants of ICaL and a significant shift to negative potential at half-inactivation of these channels without any change in the current density. Also, there was a significantly faster inactivation of Ito and no shift in half-inactivation of this channel without any change in its current density. Consequently, there was a approximately 50% increase in total charges carried by Ca2+ current and approximately 50% decrease in total charges carried by K+ currents of the treated rat cardiomyocytes. Additionally we observed a significant inhibition in IK1 density in treated rat cardiomyocytes. Oxidized glutathione level was significantly increased (70%) while the observed decrease in reduced glutathione was much less. Since a shift in redox state of regulatory proteins is related with cell dysfunction, selenium-induced increase in blood glucose and decrease in plasma insulin may correlate these alterations. These alterations, in the kinetics of the channels and in IK1 density, might lead to proarrhythmic effect of chronic selenium supplementation.