Hodgkin-Huxley Theory Research Articles

Dynamic Analysis of Neuron Models

Based on the Hodgkin–Huxley theory, this paper establishes several nonlinear system models, analyzes the models’ stability, and studies the conditions for repetitive discharge of neuronal membrane potential. Our dynamic analysis showed that the main channel currents (the fast transient sodium current, the potassium delayed rectifier current, and the fixed leak current) of a neuron determine its dynamic properties and that the GHK formula will greatly widen the stimulation current range of the repetitive discharge condition compared with the Nernst equation. The model including the change in ion concentration will lead to spreading depression (SD)-like depolarization, and the inclusion of a Na-K pump will weaken the current stimulation effect by decreasing the extracellular K accumulation. The results indicate that the Hodgkin–Huxley model is suitable for describing the response to initial stimuli, but due to changes in ion concentration, it is not suitable for describing the response to long-term stimuli.

Key factors dominating the neural coding preference to 1/f signal

Experiments have demonstrated that cortical and sensory neurons prefer to response to signals with characteristics of long-term correlation or 1/f noise feature better than signals with no correlation like white-noise-type [1]. In order to study the underlying mechanism, we built up a cortical neuronal model [2] based on Hodgkin-Huxley theory to study the correlations between neuron kinetics and signal statistics. Interestingly, we observed that (see Figure 1a,b) white-noise-type signal (cutoff frequency >10000Hz) is hard to induce action potentials unless signals with very strong intensity while 1/f signal and low-pass filtered white noise type signal (cutoff frequency <1000Hz) can easily induce action potentials with high firing rate at low signal intensity (quantified by signal standard deviation STD), see Figure ​Figure1a.1a. Moreover, the half-height duration of action potentials is also varying with more sensitivity to 1/f signal or filtered white noise than pure white noise (Figure ​(Figure1b).1b). In addition, neural firing rate and spike duration are more sensitive to 1/f signal than filtered white noise. This is a clear positive evidence of neuronal coding preference to signals with long-term correlations. Figure 1 Interactions among neuron kinetics and signal statistics. a) Firing rate vs signal STD for white noise, filtered white noise and 1/f noise. b) Half-height duration of action potentials vs signal STD for white noise, filtered white noise and 1/f noise. ... To further reveal the key factors dominating the preference of neuronal dynamics to colored and white noise, we systematically varied the values of Na+ and K+ channel time constants and channel rate constants. Figure ​Figure1c1c showed that the firing rate doesn't change much as a function of sodium activation time constant (τm) for a given 1/f signal (STD = 1). However, the firing rate decrease dramatically with an increase of sodium inactivation time constant (τh) or potassium activation time constant (τn) for this signal. For very large τh and τn, neuronal response dynamics start to be saturate for all signals, see Figure ​Figure1d.1d. These results indicate that both sodium inactivation time constant (τh) and potassium activation time constant (τn) may be the key factors dominating neural coding preference to signals with different correlated statistic features. In sum, this study demonstrated that ionic channel time constant of sodium inactivation and potassium activation may be the dominating factors accounting for the neuronal spiking sensitivity favorable to signals with temporal correlations.

Penetration of Action Potentials During Collision in the Median and Lateral Giant Axons of Invertebrates

The collisions of two simultaneously generated impulses in the giant axons of both earthworms and lobster propagating in orthodromic and antidromic direction were investigated. The experiments have been performed on the extracted ventral cords of Lumbricus terrestris and the abdominal ventral cord of lobster, Homarus americanus, by using external stimulation and recording. The collision of two nerve impulses of orthodromic and antidromic propagation didn't result in the annihilation of the two signals contrary to the common notion that is based on the existence of a refractory period in the well-known Hodgkin-Huxley theory. However, the results are in agreement with the electromechanical soliton theory for nerve pulse propagation as suggested by Heimburg and Jackson (Proc. Natl. Acad. Sci. USA 102, 9790 (2005)).

Lipid Ion Channels and the Role of Proteins

In the absence of proteins, synthetic lipid membranes can display quantized conduction events for ions that are virtually indistinguishable from those of protein channels. The phenomenological similarities between typical conductances are striking: they are of equal order and show similar lifetime distributions and current histograms. They can include conduction bursts, flickering, and multistep conductance. Lipid channels can be gated by voltage and blocked by drugs. They respond to changes in lateral membrane tension and temperature. Thus, they behave like voltage-gated, temperature-gated, and mechano-sensitive protein channels, or like receptors. The similarity between lipid and protein channels poses an important problem for the interpretation of protein channel data. For example, the Hodgkin-Huxley theory for nerve pulse conduction requires a selective mechanism for the conduction of sodium and potassium ions. To this end, the lipid membrane must act both as a capacitor and as an insulator. Nonselective ion conductance by mechanisms other than the gated protein channels challenges the proposed mechanism for pulse propagation. Nevertheless, textbooks rarely describe the properties of the lipid membrane surrounding the proteins in their discussions of membrane models. These similarities lead to important questions: Do these similarities in lipid and protein channels result from a common mechanism, or are these similarities fortuitous? What distinguishes protein channels from lipid channels, if anything? In this Account, we document experimental and theoretical findings that show the similarity between lipid and protein channels. We discuss important cases where protein channel function strongly correlates with the properties of the lipid. Based on statistical thermodynamics simulations, we discuss how such correlations could come about. We suggest that proteins can act as catalysts for lipid channel formation and that this hypothesis can explain some of the unexplained correlations between protein and lipid membrane function.

Properties of cortical axons for energy efficient cortical action potentials

Brains are extremely energy demanding. Recent studies [1] have demonstrated that AP generated in mammalian neurons are more energy efficient than squid giant axon [2]. In an companion study, we have showed that a warm body temperature is a critical factor responsible for the high efficient cortical action potentials in endotherm brain neurons than those in ectotherm neurons. In this study, we investigate that besides warm body temperature, what are the principal factors affecting energy cost in action potential generation. Since the energy cost is directly related to Na+ and K+ channels, we build up a cortical neuronal model based on Hodgkin-Huxley theory [3] and investigate the effect of changes in Na+ and K+ distribution density on the axon energetics. Figure ​Figure11 shows that there exist a pair of optimal Na+ and K+ conductance values, leading to the maximal energy efficiency. Below the level of the optimal Na+ conductance, the action potential could not be generated, while above this level, the energy cost will increase gradually. For a fixed Na+ conductance, a lower K+ conductance results in an unstable spiking dynamics, and a higher K+ conductance results in a high energy cost. Therefore, once a neuron developed a certain amount of Na+ channel density, there exist an optimal K+ channel density, where the spike dynamics is stable while the energy cost is minimized. Figure 1 Spike energy efficiency is dominated by values of Na+ and K+ channel conductance. Then we studied the energy efficiency for action potential initiation and propagation in unmyelinated and myelinated axons. Comparing to a point model, Na+ entry ratio (SER) increases for action potential initiation in the unmyelinated axon. This is because that an additional amount of INa is required to flow into neighborhood axon compartment when AP is propagated on the axon. This axial Na+ current results in a rapid rising phase of membrane potential, triggering a large amount of local Na+ influx for local action potential generation. Therefore, this axial Na+ current saves energy for action potential generation in the neighborhood compartment during AP propagation. Compared to the unmyelinated axon, spike propagation in a myelinated axon saves energy extensively for AP propagation. In summing, the energy cost for action potential initiation and propagation in cortical axons is not only ionic channel dependent, but also axonal property dependent.

A comparision of Hodgkin-Huxley and soliton neural theories

Abstract. Hodgkin and Huxley were the pioneers to abstract biological neuron as an electric circuit and nerve signal as the voltage impulse. The Hodgkin-Huxley theory (Hodgkin and Huxley, 1952) has set the direction and defined the goals for much of the ensuing research in biophysics. However, in 2005, T. Heimburg and A. D. Jackson, biophysicists from Copenhagen proposed a new neural theory called Soliton theory (Heimburg and Jackson, 2005). In this theory, the nerve conduction is proposed as a density wave. In this paper, Hodgkin-Huxley and Soliton theories are described and a theoretical comparison has been carried out throughout the analysis of the theories and models.

Equations for the membrane potential

The problem of an electric signal propagating over a thin layer modeling the cell membrane is considered in the electrostatic approximation. At both sides of the membrane there are bulk conductors with different properties. Various types of boundary conditions are examined. Solutions are obtained for a cylindrical fiber and planar layer and compared with the relationships used in the classical Hodgkin-Huxley theory. It is shown that agreement with the cable theory is obtained only if the system is fully symmetrical and the near-membrane layers are thin.

Cortical Action Potential Backpropagation Explains Spike Threshold Variability and Rapid-Onset Kinetics

Neocortical action potential responses in vivo are characterized by considerable threshold variability, and thus timing and rate variability, even under seemingly identical conditions. This finding suggests that cortical ensembles are required for accurate sensorimotor integration and processing. Intracellularly, trial-to-trial variability results not only from variation in synaptic activities, but also in the transformation of these into patterns of action potentials. Through simultaneous axonal and somatic recordings and computational simulations, we demonstrate that the initiation of action potentials in the axon initial segment followed by backpropagation of these spikes throughout the neuron results in a distortion of the relationship between the timing of synaptic and action potential events. In addition, this backpropagation also results in an unusually high rate of rise of membrane potential at the foot of the action potential. The distortion of the relationship between the amplitude time course of synaptic inputs and action potential output caused by spike backpropagation results in the appearance of high spike threshold variability at the level of the soma. At the point of spike initiation, the axon initial segment, threshold variability is considerably less. Our results indicate that spike generation in cortical neurons is largely as expected by Hodgkin-Huxley theory and is more precise than previously thought.

Spikes too kinky in the cortex?

The Hodgkin–Huxley theory that explains the mechanism of how neurons fire forms the cornerstone of computational neuroscience. But something it hasn't predicted is happening in the brain cortex. Neurons in the brain communicate via electric pulses or action potentials of a fraction of a volt, that last about a thousandth of a second. In 1952, Alan Hodgkin and Andrew Huxley received a share in the Nobel prize for their theory of action potential generation, developed from work on the squid giant axon. Ever since it has been tacitly assumed that nerve impulses are generated in much the same way in all animals from slugs to humans. Now a study of the cortical neurons of higher animals suggests that this assumption may need to be revised. Key features of cortical action potential initiation depart from predictions of the Hodgkin–Huxley theory, as the neurons are much more tailored for fast information processing than was assumed.

Unique features of action potential initiation in cortical neurons

Neurons process and encode information by generating sequences of action potentials. For all spiking neurons, the encoding of single-neuron computations into sequences of spikes is biophysically determined by the cell's action-potential-generating mechanism. It has recently been discovered that apparently minor modifications of this mechanism can qualitatively change the nature of neuronal encoding. Here we quantitatively analyse the dynamics of action potential initiation in cortical neurons in vivo, in vitro and in computational models. Unexpectedly, key features of the initiation dynamics of cortical neuron action potentials--their rapid initiation and variable onset potential--are outside the range of behaviours described by the classical Hodgkin-Huxley theory. We propose a new model based on the cooperative activation of sodium channels that reproduces the observed dynamics of action potential initiation. This new model predicts that Hodgkin-Huxley-type dynamics of action potential initiation can be induced by artificially decreasing the effective density of sodium channels. In vitro experiments confirm this prediction, supporting the hypothesis that cooperative sodium channel activation underlies the dynamics of action potential initiation in cortical neurons.

A kinetic model for voltage-gated ion channels in cell membranes based on the path integral method

A kinetic model of cell membrane ion channels is proposed based on the path integral method. From the Pauli-type master equations valid on a macroscopic time scale, we derive a first-order differential equation or the kinetic equation which governs temporal evolution of the channel system along the paths of extreme probability. Using known parameters for the batrachotoxin (BTX)-modified sodium channels in squid giant axon, the time dependence of the channel activation and the voltage dependence of the corresponding time constants ( τ ) are examined numerically. It is found that the channel activation relaxes to the steady (or equilibrium)-state values for a given membrane potential and the corresponding time constant reaches a maximum at a certain potential and thereafter decreases in magnitude as the membrane potential increases. A qualitative comparison between these results and the results of Hodgkin–Huxley theory, path probability method and thermodynamic models as well as the cut-open axon technique is presented. Good agreement is achieved.

AN ANALYTIC PICTURE OF NEURON OSCILLATIONS

Current induced oscillations of a space clamped neuron action potential demonstrates a bifurcation scenario originally encapsulated by the four-dimensional Hodgkin–Huxley equations. These oscillations were subsequently described by the two-dimensional FitzHugh–Nagumo Equations in close agreement with the Hodgkin–Huxley theory. It is shown that the FitzHugh–Nagumo equations can to close approximation be reduced to a generalized van der Pol oscillator externally driven by the current. The current functions as an external constant force driving the action potential. As a consequence approximate analytic expressions are derived which predict the bifurcation scenario, the amplitudes of the oscillations and the oscillation periods in terms of the current and the physiological constants of the FitzHugh–Nagumo model. A second reduction permits explicit analytic solution and results in a spiking model which can be multiply coupled and extended to include the dynamics of phase locking, entrainment and chaos characteristic of time-dependent synaptic inputs.

Potassium Channels and the Atomic Basis of Selective Ion Conduction

All living cells are surrounded by a thin, approximately 40 A thick lipid bilayer called the cell membrane. The cell membrane holds the contents of a cell in one place so that the chemistry of life can occur, but it is a barrier to the movement of certain essential ingredients including the ions Na, K, Ca, and Cl. The barrier to ion flow across the membrane—known as the dielectric barrier—can be understood at an intuitive level: the interior of the cell membrane comprises an oily substance and ions are more stable in water than in oil. The energetic preference of an ion for water arises from the electric field around the ion and its interaction with neighboring molecules. Water is an electrically polarizable substance, which means that its molecules rearrange in an ion’s electric field so that negative oxygen atoms point in the direction of cations and positive hydrogen atoms point toward anions. These electrically stabilizing interactions are much weaker in a less polarizable substance such as oil. Thus, an ion will tend to stay in the water on either side of a cell membrane rather than enter and cross the membrane. Yet numerous cellular processes, ranging from electrolyte transport across epithelia to electrical signal production in neurons, depend on the flow of ions across the membrane. To mediate the flow, specific protein catalysts known as ion channels exist in the cell membrane. Ion channels exhibit the following three essential properties: (1) they conduct ions rapidly; (2) many ion channels are highly selective, which means only certain ion species flow while others are excluded; (3) their function is regulated by processes known as gating, that is, ion conduction is turned on and off in response to specific environmental stimuli. Fig. 1 summarizes these properties. The modern history of ion channels began in 1952 when Hodgkin and Huxley published their seminal papers on the theory of the action potential in the squid giant axon [1–4]. A fundamental element of their theory was that the axon membrane undergoes changes in its permeability to Na and K ions. The Hodgkin–Huxley theory did not address the mechanism by which changes in the membrane perme-

The Hodgkin-Huxley theory of the action potential.

The Hodgkin-Huxley (H-H) theory of theaction potential, formulated 50 years ago,remains one of the great success stories inbiology, and ranks among the most signif-icant conceptual breakthroughs in neuro-science. Together with the artificial neuralnetworks of McCulloch and Pitts, the quan-tal theory of Katz, and the cable theory ofRall, all developed at around the same time,the H-H theory provided the foundationfor modern computational neuroscience.The theory was the culmination of anintense experimental and theoretical col-laboration between Hodgkin and Huxley,from 1938 to the publication in 1952 oftheir five landmark papers in the

Chaos in Ionic Transport Through Membranes

The chaotic nature of ionic current through nuclear filters has been detected and analyzed by means of Liapunov exponents and bifurcation diagrams. The "crowd model" describing the nonlinear behavior of single channel ionic current, based on the family of logistic maps, has been formulated and applied to analysis of the patch clamp recordings. Some characteristic features of the system read out from the bifurcation diagrams of the two-parameter logistic map, and also from the extended Hodgkin–Huxley theory have been presented. The relation between the logistic difference schemes (and, at the same time, the class of logistic differential equations) and the logistic maps as well as some possibilities of using the crowd model to biological membranes have also been shown.

The Axon: Structure, Function, and Pathophysiology

The Axon: Structure, Function, and Pathophysiology. S.G. Waxman, J.D. Kocsis, P.K. Stys (Editors). New York, Oxford University Press, 1995. Pages: 714. Price: $175. ISBN: 0-1950-8293-1. This is a good book for the casual reader and for the reader wishing to obtain an accurate and substantial survey of axon physiology and pathophysiology. Its 34 chapters are up-to-date and contain an excellent bibliography of original literature and review articles. Although the book is multi-authored, it is sufficiently cross-referenced to be used as a comprehensive textbook. Chapter 1 is an historical overview by Sir Andrew Huxley, which provides an entertaining and informative account of the work leading up to the Hodgkin-Huxley theory of membrane potentials. The remaining chapters are divided into two major sections, the first one covering the normal axon and the second one covering the principles of axonal pathophysiology. The section on the normal axon begins with two chapters describing...

Entrainment in nerve by a ferroelectric model: Bifurcation and limit cycles

The excitation in nerve that is self-organized in a dissipative structure with the resting membrane potential (an equilibrium structure) occurs on an equilibrium space for the cusp catastrophe. The space is given by a nonlinear state equation η 3 + aη + b = 0 deduced from a chemical network model which is applied to Leuchtag's ferroelectric hypothesis for Na channels, where −η corresponds to the membrane potential, a and b are control parameters related to the dipole-dipole and dipole-ion interactions, respectively. A phase transition of the membrane organized in a region, a < 0 ( T < T c ), can be determined by a parameter which describes the difference from equilibrium. When the membrane in a self-oscillation is disturbed by a periodical Na current with the natural frequency of the membrane or near one, a stable limit cycle of the potential arises through an entrainment. With modified Zeeman's formulas for the movements of a and b in the equation, the transitions are calculated to arise at two points (lowest s 1 and highest s h limits) discontinuously, so s h which is the subcritical point differs from the result by the modified Hodgkin-Huxley theory. This seems to show a characteristic of the catastrophe.

Computation of long-distance propagation of impulses elicited by Poisson-process stimulation

1. The purpose of this work was to determine whether computed temporally coded axonal information generated by Poisson process stimulation were modified during long-distance propagation, as originally suggested by S. A. George. Propagated impulses were computed with the use of the Hodgkin-Huxley equations and cable theory to simulate excitation and current spread in 100-microns-diam unmyelinated axons, whose total length was 8.1 cm (25 lambda) or 101.4 cm (312.5 lambda). Differential equations were solved numerically, with the use of trapezoidal integration over small, constant electrotonic and temporal steps (0.125 lambda and 1.0 microsecond, respectively). 2. Using dual-pulse stimulation, we confirmed that for interstimulus intervals between 5 and 11 ms, the conduction velocity of the second of a short-interval pair of impulses was slower than that of the first impulse. Further, with sufficiently long propagation distance, the second impulse's conduction velocity increased steadily and eventually approached that of the first impulse. This effect caused a spatially varying interspike interval: as propagation proceeded, the interspike interval increased and eventually approached stabilization. 3. With Poisson stimulation, the peak amplitude of propagating action potentials varied with interspike interval durations between 5 and 11 ms. Such amplitude attenuation was caused by the incomplete relaxation of parameters n (macroscopic K-conductance activation) and h (macroscopic Na-conductance inactivation) during the interspike period. 4. The stochastic properties of the impulse train became less Poisson-like with propagation distance. In cases of propagation over 99.4 cm, the impulse trains developed marked periodicities in Interevent Interval Distribution and Expectation Density function because of the axially modulated transformation of interspike intervals. 5. Despite these changes in impulse train parameters, the arithmetic value of the mean interspike interval did not change as a function of propagation distance. This work showed that in theory, whereas the pattern of Poisson-like impulse codes was modified during long-distance propagation, their mean rate was conserved.

Open-channel block of Na+ channels by intracellular Mg2+

1. Macroscopic and single-channel currents through several types of cloned rat brain Na+ channels, expressed in Xenopus oocytes, were measured using the patch-clamp technique. 2. For all cloned channel types and for endogenous Na+ channels in chromaffin cells, intracellular Mg2+ blocks outward currents in a voltage-dependent manner similar to that in rat brain type II Na+ channel (Pusch et al. 1989). 3. A sodium-channel mutant ('cZ-2') with long single-channel open times was used to examine the voltage-dependent reduction of single-channel outward current amplitudes by intracellular Mg2+. This reduction could be described by a simple blocking mechanism with half-maximal blockage at 0 mV in 1.8 mM intracellular Mg2+ and a voltage-dependence of e-fold per 39 mV (in approximately 125 mM [Na]i); this corresponds to a binding-site at an electrical distance of 0.32 from the inside of the membrane. 4. At low Mg2+ concentrations and high voltages, the open-channel current variance is significantly elevated with respect to zero [Mg]i. This indicates that Mg2+ acts as a fast blocker rather than gradually decreasing current, e.g. by screening of surface charges. Analysis of the open-channel variance yielded estimates of the block and unblock rate constants, which are of the order of 2.10(8) M-1 S-1 and 3.6.10(5) S-1 at 0 mV for the mutant cZ-2. 5. A quantitative analysis of tail-currents of wild-type II channels showed that the apparent affinity for intracellular Mg2+ strongly depends on [Na]i. This effect could be explained in terms of a multi-ion pore model. 6. Simulated action potentials, calculated on the basis of the Hodgkin-Huxley theory, are significantly reduced in their amplitude and delayed in their onset by postulating Mg2+ block at physiological levels of [Mg]i.

The interaction of ionic currents mediating single spike activity in retinal amacrine cells of the tiger salamander

We investigated the ionic interactions responsible for the characteristic nonrepetitive spike activity of amacrine cells. First we measured 4 pharmacologically separable ionic components: a voltage-gated, transient inward sodium current, a voltage-gated, sustained inward calcium current, a calcium-gated, sustained outward potassium current, and a voltage-gated, transient outward potassium current. The measurements provided the time course and magnitudes of the underlying conductances as functions of voltage. Each current was simulated following conventional Hodgkin-Huxley theory. A composite of the simulated currents was analytically reassembled to generate an approximation of the voltage response to a current step. By artificially varying the magnitude and kinetics of the different conductances in the simulation, we determined the range of values that supported the nonrepetitive spike-like response. Amacrine cells tend to remain refractory following an initial spike because (1) the entire activation range for potassium is located at positive potentials with respect to sodium inactivation, so sodium inactivation is never fully extinguished, and (2) the fully activated sodium conductance is of insufficient magnitude to subsequently reach threshold, given this residual inactivation. Shifting the sodium inactivation range by 10 mV, or increasing sodium conductance by 5 times, leads to a more repetitive form of activity. Changes in the magnitude, time course, or activation range of the potassium conductance cannot alter these conditions.