A fully coupled transient excited state model for the sodium channel

Summary The axon membrane is simulated by standard Hodgkin-Huxley leakage and potassium channels plus a coupled transient excited state kinetic scheme for the sodium channel. This scheme for the sodium channel is as proposed previously by the author. Simultations are presented showing the form of the action potential, threshold behavior, accommodation, and repetitive firing. It is seen that the form of the individual action potential, its all-or-none nature, and its refractory period are well simulated by this model, as they are by the standard Hodgkin-Huxley model. However, the model differs markedly from the Hodgkin-Huxley model with respect to repetitive firing and accommodation to stimulating currents of slowly rising intensity, in ways that are anomn to be related to those features of the sodium inactivation which are anomalous to the H-H model. The tendency for repetitive firing is highly dependent on that parameter which primarily determintes the existence of the inactivation shift in voltage clamp experiments, in such a way that the more pronounced the inactivation shift, the less the tendency for repetitive firing,. The tendency for accommodation is highly dependent on that parameter which primarily determines the “τ_c − τ_h” separation, in such a way that the greater the separation the greater the tendency for the membrane to accommodate without firing action potentials to a slowly rising current.     

A transient excited state model for sodium permeability changes in excitable membranes.

In this paper we explore the properties of a mathematical model for the passive sodium permeability system of excitable membranes. This model is distinguished by the explicit inclusion of a rate constant which depends not on instantaneous voltage, but on rate of voltage change. Actually, the model is a rather modest modification of the Hodgkin-Huxley model, but displays some behaviors which the H-H model does not. Among these behaviors are a pronounced inactivation shift (for certain parameter values), a difference between inactivation time constant as measured by turning off a sodium current under sustained depolarization and as measured by double pulse experiments, skip runs under sustained current stimulation, and accommodation to slowing rising currents.     

Structural stability,alternate descriptions and information

The inactivation properties of a model of the nerve membrane are examined. The inactivation kinetics are closely first order and may be characterized by Hodgkin-Huxley (H-H) parameters h_∞ and τ_h which depend on potential in agreement with experiments. Some differences from the H-H equations are identified. The forms predicted for τ_h variation with hyper-polarization and change of external [K⁺] agree with available data. While the inactivation time delay predicted by the model is too small to be detected experimentally, there are grounds for expecting that it may be larger in other tissues, as observed in Myxicola giant axons. The variation of the delay with test potential is predicted to be exponential. Although the model is coupled in the sense defined by Hoyt, it gives rise to an inactivation shift of negligible magnitude. However, introducing a simple variability in one physical parameter leads to the observed form of both the peak transient current voltage relation and the inactivation shift. Inactivation shift thus does not unambiguously indicate coupling; that it results from parametric heterogeneity may be a better hypothesis, and is readily testable. The inactivation shift dependence on current ratio, from experimental data, can be used to correct for the effects of parametric heterogeneity and obtain the value of a previously predicted fundamental parameter of excitable membranes. It is suggested that the effects of parametric heterogeneity must be considered in interpreting experiments and designing models for excitable systems.     

Native Gating Behavior of Ion Channels in Neurons with Null-Deviation Modeling

Computational modeling has emerged as an indispensable approach to resolve and predict the intricate interplay among the many ion channels underlying neuronal excitability. However, simulation results using the classic formula-based Hodgkin-Huxley (H-H) model or the superior Markov kinetic model of ion channels often deviate significantly from native cellular signals despite using carefully measured parameters. Here we found that the filters of patch-clamp amplifier not only delayed the signals, but also introduced ringing, and that the residual series resistance in experiments altered the command voltages, which had never been fully eliminated by improving the amplifier itself. To remove all the above errors, a virtual device with the parameters exactly same to that of amplifier was introduced into Markov kinetic modeling so as to establish a null-deviation model. We demonstrate that our novel null-deviation approach fully restores the native gating-kinetics of ion-channels with the data recorded at any condition, and predicts spike waveform and firing patterns clearly distinctive from those without correction.     

A conversational program for analytical modelling of action potentials in nerve and muscle.

A conversational program is described which is being used for exploratory modelling of digestive tract electrical activity. The structure of the mathematical model is based on the Hodgkin-Huxley equations for nerve axons and the Noble equations for Purkinje fibres of the heart. The parameters of the models are changed conversationally to investigate the effects of frequency, amplitude and waveshape. In this way it is intended to extend the models to include nervous control. The program is also being used to correlate the behaviour of H-H type equations with the simpler dynamics of Van der Pol's equation which is the basis of an oscillator-array model for the stomach and small intestines.     

A fully coupled transient excited state model for the sodium channel

Summary The behavior under voltage clamp conditions of a coupled kinetic scheme for the sodium channel is examined. The scheme is given diagrammatically by: tano Numerical simulations are presented which show that this model fits the voltage clamp data which are well described by the Hodgkin-Huxley equations, but also gives the sorts of behavior anomalous to the Hodgkin-Huxley model which have been seen experimentally. Further, straightforward changes in parameter values are shown to be capable of mimicking the ways in which some axonal preparations differ from others. Detailed, but admittedly heuristic, arguments are presented for the propositions that: 1) the model is minimal; i.e. no simpler kinetic model will fit the array of data simulated, and: 2) the transient excited state is necessary; i.e. no model of comparable simplicity with pure voltage dependent kinetics will fit the array of data simulated.     

Frequency entrainment of squid axon membrane

Summary Sinusoidally varying stimulating currents were applied to space-clamped squid giant axon membranes in a double sucrose gap apparatus. Stimulus parameters varied were peak-to-peak current amplitude, frequency, and DC offset bias. In response to these stimuli, the membranes produced action potentials in varying patterns, according to variation of input stimulus parameters. For some stimulus parameters the output patterns were stable and obviously periodic with the periods being simple multiples of the input period; for other stimulus parameters no obvious periodicity was manifest in the output. The experimental results were compared with simulations using a computer model which was modified in several ways from the Hodgkin-Huxley model to make it more representative of our preparation. The model takes into account K⁺ accumulation in the periaxonal space, features of Na⁺ inactivation which are anomalous to the Hodgkin-Huxley model, sucrose gap hyperpolarization current, and membrane current noise. Many aspects of the experiments are successfully simulated but some are not, possibly because some very slow process present in the preparation is not included in the model.     

The fluorescence detected guanidine hydrochloride equilibrium denaturation of wild-type staphylococcal nuclease does not fit a three-state unfolding model

A three-state equilibrium unfolding of a protein can be difficult to detect if two of the states fail to differ in some easily measurable way. It has been unclear whether staphylococcal nuclease unfolds in a two-state fashion, with only the native and denatured states significantly populated at equilibrium, or in a three-state manner, with a well-populated intermediate. Since equilibrium unfolding experiments are commonly used to determine protein stability and the course of denaturation are followed by changes in the fluorescence which has difficulty in distinguishing various states, this is a potential problem for many proteins. Over the course of twenty years we have performed more than one hundred guanidine hydrochloride equilibrium denaturations of wild-type staphylococcal nuclease; to our knowledge, a number of denaturations unrivaled in any other protein system. A careful examination of the data from these experiments shows no sign of the behavior predicted by a three-state unfolding model. Specifically, a three-state unfolding should introduce a slight, but characteristic, non-linearity to the plot of stability versus denaturant concentration. The average residuals from this large number of repeated experiments do not show the predicted behavior, casting considerable doubt on the likelihood of a three-state unfolding for the wild-type protein. The methods used for analysis here could be applied to other protein systems to distinguish a two-state from a three-state denaturation.     

Co-operative response of chemically excitable membrane III. Three-state model

The general three-state model is formulated first, which is the direct extension of the unified two-state model previously formulated (Kijima & Kijima, 1978). In this model, each protomer in a symmetrically interacting system (oligomers or lattices) can take three states, S, R and Q, where S and R states are the same as in the two-state model and Q state is another state either corresponding to a different open-state of ionophore from R open-state or corresponding to another closed state of ionophore. The model has no restriction on the value of Hill coefficient at the midpoint of the dose-response curves in contrast to two-state models. It is applied on GABA sensitive inhibitory synapse of crayfish muscle to account for anomalous behaviour of the membrane in I^? solution.The simplified versions of the above general three-state model are also formulated (simplified three-state model), in which it is assumed that R and Q state are equivalent in regard to the nearest neighbor interaction. By this assumption, R and Q state are collectively treated as state A and mathematical formula obtained on Ising model are applicable on this model. This model is applied on the insect sugar receptor which was shown to be incompatible with the two-state models (Kijima & Kijima, 1980). Further simplification of the above simplified model results in two convenient models: three-state KNF model and three-state MWC model, which have minimum parameters but sufficient to account for most experiments. They give plausible physico-chemical base on the “classical model” in which the existence of both inactive and active ligand-receptor complex is assumed.     

Two and three dimensional reductions of the Hodgkin-Huxley system: separation of time scales and bifurcation schemes

We study two different two-dimensional reductions of the Hodgkin-Huxley equations. We show that they display the same qualitative bifurcation scheme as the original equations but overestimate the current range where periodic emission occurs. This is essentially due to the assumption that the evolution of the sodium activation variable m is instantaneous with respect to the dynamics of the variables h and n, an hypothesis that breaks down at high values of the injected current. To prove this point we compare the current-amplitude relation, the current-frequency relation, and the shapes of individual spikes for the two reduced models to the results obtained for the original Hodgkin-Huxley model and for a three dimensional model with instantaneous sodium activation. We show that a more satisfying agreement with the original Hodgkin-Huxley equations is obtained if we modify the evolution equation for the potential by incorporating the prominent features of the dynamics of m.     

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.     

Simulation of non-enzymatic template-directed synthesis of oligonucleotides and peptides

A simulation model is proposed for the template- and sequence-directed (TSD) condensation of two trideoxyribonucleotide 3'-phosphate molecules into a hexameric template with palindromic sequence studied experimentally by von Kiedrowski (1986;Angew. Int. Ed. Engl.25, 932--935). The model simulates reasonably well the kinetics of synthesis of both the template, and the pyrophosphate product which is not directly involved in the autocatalytic reaction. It offers quantitative approximation of the different rate constants of the processes involved in the reaction. The model simulates and gives predictions for the influence of factors such as the initial concentrations of the trimers and the template, and gives predictions for the effect of temperature on the dynamics of the autocatalytic reaction. The model also simulates well the production rate of a different self-replicating system (coiled coil peptide) used in the experiments of Lee et al. (1997;Nature390, 591--594). Comparing the different rate constants, it seems that chain elongation occurs at higher rates in the peptide system (at 23 degrees C) than in the nucleotide one (at 0 degrees C), but that the relative contribution of template-directed synthesis is significantly larger with the nucleotides.     

Oscillation and coding in a formal neural network considered as a guide for plausible simulations of the insect olfactory system

For the analysis of coding mechanisms in the insect olfactory system, a fully connected network of synchronously updated McCulloch and Pitts neurons (MC-P type) was developed [Quenet, B., Horn, D., 2003. The dynamic neural filter: a binary model of spatio-temporal coding. Neural Comput. 15 (2), 309-329]. Considering the update time as an intrinsic clock, this "Dynamic Neural Filter" (DNF), which maps regions of input space into spatio-temporal sequences of neuronal activity, is able to produce exact binary codes extracted from the synchronized activities recorded at the level of projection neurons (PN) in the locust antennal lobe (AL) in response to different odors [Wehr, M., Laurent, G., 1996. Odor encoding by temporal sequences of firing in oscillating neural assemblies. Nature 384, 162-166]. Here, in a first step, we separate the populations of PN and local inhibitory neurons (LN) and use the DNF as a guide for simulations based on biological plausible neurons (Hodgkin-Huxley: H-H type). We show that a parsimonious network of 10 H-H neurons generates action potentials whose timing represents the required codes. In a second step, we construct a new type of DNF in order to study the population dynamics when different delays are taken into account. We find synaptic matrices which lead to both the emergence of robust oscillations and spatio-temporal patterns, using a formal criterion, based on a Normalized Euclidian Distance (NED), in order to measure the use of the temporal dimension as a coding dimension by the DNF. Similarly to biological PN, the activity of excitatory neurons in the model can be both phase-locked to different cycles of oscillations which remind local field potential (LFP), and nevertheless exhibit dynamic behavior complex enough to be the basis of spatio-temporal codes.     

Modeling of the adrenergic response of the human IKs current (hKCNQ1/hKCNE1) stably expressed in HEK-293 cells

Stable coexpression of human (h)KCNQ1 and hKCNE1 in human embryonic kidney (HEK)-293 cells reconstitutes a nativelike slowly activating delayed rectifier K+ current (HEK-I(Ks)), allowing beta-adrenergic modulation of the current by stimulation of endogenous receptors in the host cell line. HEK-I(Ks) was enhanced two- to fourfold by isoproterenol (EC50 = 13 nM), forskolin (10 microM), or 8-(4-chlorophenylthio)adenosine 3',5'-cyclic monophosphate (50 microM), indicating an intact cAMP-dependent ion channel-regulating pathway analogous to the PKA-dependent regulation observed in native cardiac myocytes. Activation kinetics of HEK-I(Ks) were accurately fit with a novel modified second-order Hodgkin-Huxley (H-H) gating model incorporating a fast and a slow gate, each independent of each other in scale and adrenergic response, or a "heterodimer" model. Macroscopically, beta-adrenergic enhancement shifted the current activation threshold to more negative potentials and accelerated activation kinetics while leaving deactivation kinetics relatively unaffected. Modeling of the current response using the H-H model indicated that observed changes in gating could be explained by modulation of the opening rate of the fast gate. Under control conditions at nearly physiological temperatures (35 degrees C), rate-dependent accumulation of HEK-I(Ks) was observed only at pulse frequencies exceeding 3 Hz. Rate-dependent accumulation of I(Ks) at high pulsing rate had two phases, an initial staircaselike effect followed by a slower, incremental accumulation phase. These phases are readily interpreted in the context of a heterodimeric H-H model with two independent gates with differing closing rates. In the presence of isoproterenol after normalizing for its tonic effects, rate-dependent accumulation of HEK-I(Ks) appeared at lower pulse frequencies and was slightly enhanced (approximately 25%) over control.     

The singularly perturbed Hodgkin-Huxley equations as a tool for the analysis of repetitive nerve activity

A qualitative analysis of the Hodgkin-Huxley model (Hodgkin and Huxley 1952), which closely mimics the ionic processes at a real nerve membrane, is performed by means of a singular perturbation theory. This was achieved by introducing a perturbation parameter that, if decreased, speeds up the fast variables of the Hodgkin-Huxley equations (membrane potential and sodium activation), whereas it does not affect the slow variables (sodium inactivation and potassium activation). In the most extreme case, if the perturbation parameter is set to zero, the original four-dimensional system degenerates to a system with only two differential equations. This degenerate system is easier to analyze and much more intuitive than the original Hodgkin-Huxley equations. It shows, like the original model, an infinite train of action potentials if stimulated by an input current in a suitable range. Additionally, explanations for the increased sensitivity to depolarizing current steps that precedes an action potential can be found by analysis of the degenerate system. Using the theory of Mishchenko and Rozov (1980) it is shown that the degenerate system does not only represent a simplification of the original Hodgkin-Huxley equations but also gives a valid approximation of the original model at least for stimulating currents that are constant within a suitable range.     

Design and analysis of arm-in-cage experiments: inference for three-state progressive disease models with common periodic observation times

Summary .   We develop statistical methods for designing and analyzing arm-in-cage experiments used to test the efficacy of insect repellents and other topical treatments. In these experiments, a controlled amount of the treatment is applied to a volunteer's forearm, which then is exposed to the insects by being placed into a special cage. Arms are not kept in the cages continuously, but rather placed there periodically for a brief period of time, during which it is noted whether an insect lands (but does not bite) or (lands and) bites. Efficacy of a repellent can be described using a progressive three-state model in which the first two states represent varying degrees of protection (no landing and landing without biting) and the third state occurs once protection is completely lost (biting). Because subjects within a treatment group follow the same cage visit schedule, transition times between states are interval censored into one of several fixed intervals. We develop an approach that uses a mixture of nonparametric and parametric techniques for estimating the parameters of interest when sojourn times are dependent. Design considerations for arm-in-cage experiments are addressed and the proposed methods are illustrated on data from a recent arm-in-cage experiment as well as simulated data.     

How is information transmitted in a nerve?

In the last 15 years, a debate has emerged about the validity of the famous Hodgkin-Huxley model for nerve impulse. Mechanical models have been proposed. This note reviews the experimental properties of the nerve impulse and discusses the proposed alternatives. The experimental data, which rule out some of the alternative suggestions, show that while the Hodgkin-Huxley model may not be complete, it nevertheless includes essential features that should not be overlooked in the attempts made to improve, or supersede, it.     

Hodgkin-Huxley Axon: Increased Modulation and Linearity of Response to Constant Current Stimulus

Repetitive response patterns resembling those of tonic receptors were obtained by increasing the potassium system time constant in the Hodgkin-Huxley (H-H) equations. The increase in time constant varied with membrane potential. Calculated spike frequencies varied linearly with the magnitude of the constant current stimulus; in addition, minimum frequencies were greatly reduced, and the frequency range increased. Modification of the maximum ionic conductances, membrane capacitance, and rate constant voltage dependence was found to vary the minimum frequency, current at that frequency, slope, and over-all modulation of the modified responses.     

Comparaison des modeles de neurone de Hodgkin-Huxley et de Nelson

Le modèle de Nelson peut-être considéré comme une approximation du modèle de Hodgkin-Huxley. Moins précis, il est plus maniable. Le modèle de Nelson peut également être considéré comme une généralisation du modèle de Hodgkin-Huxley. En effet, il introduit des liaisons synaptiques localisées ou diffusantes, et un processus de facilitation. Le mécanisme des liaisons synaptiques ne se traduit pas facilement dans le langage de Hodgkin-Huxley. Par contre, le processus de facilitation s'interprète facilement. Nelson's model can be taken as an approximation of Hodgkin-Huxley's model. Its precision is lesser, but it is more usable. Nelson's model can also be taken as a generalization of Hodgkin-Huxley's one; for it introduces localized or diffusing synaptic connexions and a facilitating process. The mechanism of synaptic connexions cannot be easily translated into Hodgkin-Huxley's language. On the contrary, the facilitating process is easily interpreted.