Simulation study on effects of channel noise on differential conduction at an axon branch.

Effects of membrane channel noise (random opening and closing of ion channels) are studied on spike conduction at a branching point on an axon. Computer simulation is done on the basis of a stochastic version of the Hodgkin-Huxley cable model, into which the channel noise is incorporated. It is shown that the channel noise makes conduction of spikes into daughter branches random; spikes randomly succeed or fail in conduction into daughter branches. The conduction is then randomly differential even though the forms and properties of daughter branches are the same. The randomness is considerable when the radius of an axon is small (approximately 1 microns).     

Theory of Threshold Fluctuations in Nerves: I. Relationships between Electrical Noise and Fluctuations in Axon Firing

Relations describing threshold fluctuation phenomena in nerves are derived by calculating the approximate response of the Hodgkin-Huxley (HH) axon to electrical noise. We use FitzHugh's reduced phase space approximation and describe the dynamics of a noisy nerve by a two-dimensional brownian motion. The theory predicts the functional form and parametric dependence of the relation between probability of firing and stimulus strength. Expressions are also obtained for the firing probability as a function of stimulus duration and for the distribution of latency times as a function of stimulus strength.     

Squid Axon Membrane Response to White Noise Stimulation

The current from a white noise generator was applied as a stimulus to a space-clamped squid axon in double sucrose gap. The membrane current and the voltage response of the membrane were then amplified, recorded on magnetic tape, and the stimulus was cross-correlated with the response. With subthreshold stimuli, a cross-correlation function resembling that obtained from a resonant parallel circuit is obtained. As the intensity of the input noise is increased, the cross-correlation function resembles that obtained from a less damped oscillatory circuit. When the noise intensity is further increased so that an appreciable frequency of action potentials is observed, an additional component appears in the experimental cross-correlogram. The subthreshold cross-correlogram is analyzed theoretically in terms of the linearized Hodgkin-Huxley equations. The subthreshold axon approximates a parallel resonant circuit. The circuit parameters are temperature dependent, with resonant frequency varying from approximately 100 Hz at 10°C to approximately 250 Hz at 20°C. The Q₁₀ of the resonant frequency is equal to 1.9. These values are in agreement with values found previously for subthreshold oscillations following a single action potential.     

Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.     

Modelling of biological encoding mechanisms as a system with distributed parameters]

The encoder region in receptors and neurons is represented by the inhomogeneous origin of the axon. The axon diameter and the excitability are in fact space-dependent. For the analysis the soma is described by a system with concentrated parameters followed by an inhomogeneous axon. The membrane properties are approximated by the slightly modified Hodgkin-Huxley equations. The assumption that the space-dependence of the excitability originates in variations of the conductance value for sodium ions accounts for a number of experimental results. The influence of other membrane parameters upon the mechanism of impulse generation and transmission has also been analysed.     

Two stable steady states in the Hodgkin-Huxley axons.

Two stable steady states were found in the numerical solution of the Hodgkin-Huxley equations for the intact squid axon bathed in potassium-rich sea water with an externally applied inward current. Under the conditions the two stable steady-states exist, the Hodgkin-Huxley equations have a complex bifurcation structure including, in addition to the two stable steady-states, a stable limit cycle, two unstable equilibrium points, and one asymptotically stable equilibrium point. It was also concluded that two stable steady states can appear in the Hodgkin-Huxley axons when the leak current is comparable to the currents through the Na and K channels.     

Spike initiation and propagation on axons with slow inward currents

We investigate spike initiation and propagation in a model axon that has a slow regenerative conductance as well as the usual Hodgkin-Huxley type sodium and potassium conductances. We study the role of slow conductance in producing repetitive firing, compute the dispersion relation for an axon with an additional slow conductance, and show that under appropriate conditions such an axon can produce a traveling zone of secondary spike initiation. This study illustrates some of the complex dynamics shown by excitable membranes with fast and slow conductances.     

Gating molecule interactions in K+ channels. A consistent explanation for cole-moore shifts and potential ramp experiments

The enhanced induction period of potassium channel currents in squid giant axon induced by hyperpolarizing prepulses (the Cole-Moore shift) is observed and analyzed for a range of depolarizing step potentials. The induction periods produced when the axon is voltage clamped with ascending potential ramps are also analyzed since both sets of experiments are incompatible with fourth-power dependence proposed by Hodgkin and Huxley for the squid K⁺ channels. When the Hodgkin-Huxley equations are modified to include the effects of interactions between gating molecules within individual channels, both the Cole-Moore and ascending potential ramp data are described with a fourth-power dependence. The constant interaction parameters which provide a consistent fit for all the data are based on the geometrical arrangement of gating molecules within the channel and the total interaction energy which stabilizes the four gating molecules in their closed configuration. A tetrahedral gating molecule geometry and an interaction energy of only 471 cal/mole provide optimal fits of all the data; the modified equations retain the ability to describe data presently described by the Hodgkin-Huxley equations in the depolarizing regime.     

Excitation properties of the squid axon membrane and model systems with current stimulation. Statistical evaluation and comparison.

The space-clamped squid axon membrane and two versions of the Hodgkin-Huxley model (the original, and a strongly adapting version) are subjected to a first order dynamic analysis. Stable, repetitive firing is induced by phase-locking nerve impulses to sinusoidal currents. The entrained impulses are then pulse position modulated by additional, small amplitude perturbation sinusoidal currents with respect to which the frequencies response of impulse density functions are measured. (Impulse density is defined as the number of impulses per unit time of an ensemble of membranes with each membrane subject to the same stimulus). Two categories of dynamic response are observed: one shows clear indications of a corner frequency, the other has the corner frequency obscured by dynamics associated with first order conductance perturbations in the interspike interval. The axon membrane responds with first order perturbations whereas the unmodified Hodgkin-Huxley model does not. Quantitative dynamic signatures suggest that the relaxation times of axonal recovery excitation variables are twice as long as those of the corresponding model variables. A number of other quantitative differences between axon and models, including the values of threshold stimuli are also observed.     

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 leading edge approximation to the nerve axon problem

A negative resistance piece-wise linear model, and one which is the sum of two sine terms are used to solve the nerve axon problem for leading edge waveshape, pulse velocity, maximum rate of rise, and rise time for the “Hodgkin-Huxley axon.” The results are compared analytically and numerically to experiment and the calculated results of Hodgkin and Huxley, a fuse model, and a cubic solution.     

Nonlinear cable equations for axons. II. Computations and experiments with external current electrodes

We have investigated the steady-state potential and current distributions resulting from current injection into a close-fitting channel into which a squid axon is placed. Hybrid computer solutions of the cable equations, using the Hodgkin-Huxley equations to give the membrane current density, were in good agreement with experimental observations. A much better fit was obtained when the Hodgkin-Huxley leakage conductance was reduced fivefold.     

Inferences about membrane properties from electrical noise measurements

Four sources of electrical noise in biological membranes, each with a different physical basis, are discussed; the analysis of each type of noise potentially yields a different sort of information about membrane properties. (a) From the thermal noise spectrum, the passive membrane impedance may be obtained, so that thermal noise measurements are essentially equivalent to the type of since wave analysis carried out by Cole and Curtis. (b) If adequately high frequency measurements could be made, the shot noise spectrum should give information about the average motion of a single ion within the membrane. (c) The number of charge carriers and single ion mobilities within the membrane can possibly be inferred from measurements of noise with a 1/f spectrum. Available data indicate, for example, that increases in axon membrane conductance are not achieved by modulations in the mobility of ions within the membrane. (d) Fluctuations arising from the mechanisms normally responsible for membrane conductance changes can produce a type of electrical noise. Analysis of such conductance fluctuations provides a way to assess the validity of various microscopic models for the behavior of individual channels. Two different probabilistic interpretations of the Hodgkin-Huxley equations are investigated here and shown to yield different predictions about the spectrum of conductance fluctuations; thus, appropriate noise measurements may serve to eliminate certain classes of microscopic models for membrane conductance changes. Further, it is shown how the analysis of conductance fluctuations can, in some circumstances, provide an estimate of the conductance of a single channel.     

Poisson process stimulation of an excitable membrane cable model.

The convergence of multiple inputs within a single-neuronal substrate is a common design feature of both peripheral and central nervous systems. Typically, the result of such convergence impinges upon an intracellularly contiguous axon, where it is encoded into a train of action potentials. The simplest representation of the result of convergence of multiple inputs is a Poisson process; a general representation of axonal excitability is the Hodgkin-Huxley/cable theory formalism. The present work addressed multiple input convergence upon an axon by applying Poisson process stimulation to the Hodgkin-Huxley axonal cable. The results showed that both absolute and relative refractory periods yielded in the axonal output a random but non-Poisson process. While smaller amplitude stimuli elicited a type of short-interval conditioning, larger amplitude stimuli elicited impulse trains approaching Poisson criteria except for the effects of refractoriness. These results were obtained for stimulus trains consisting of pulses of constant amplitude and constant or variable durations. By contrast, with or without stimulus pulse shape variability, the post-impulse conditional probability for impulse initiation in the steady-state was a Poisson-like process. For stimulus variability consisting of randomly smaller amplitudes or randomly longer durations, mean impulse frequency was attenuated or potentiated, respectively. Limitations and implications of these computations are discussed.     

Slow Changes of Potassium Permeability in the Squid Giant Axon

A slow potassium inactivation i.e. decrease of conductance when the inside of the membrane is made more positive with respect to the outside, has been observed for the squid axon. The conductance-potential curve is sigmoid shaped, and the ratio between maximum and minimum potassium conductance is at least 3. The time constant for the change of potassium conductance with potential is independent of the concentration of potassium in the external solution, but dependent upon potential and temperature. At 9 degrees C and at the normal sea water resting potential, the time constant is 11 sec. For lower temperature or more depolarizing potentials, the time constant is greater. The inactivation can be described by modifying the Hodgkin-Huxley equation for potassium current, using one additional parameter. The modified equation is similar in form to the Hodgkin-Huxley equation for sodium current, suggesting that the mechanism for the passive transport of potassium through the axon membrane is similar to that for sodium.     

Bifurcation of periodic solutions of Hodgkin-Huxley model for the squid giant axon

The Hodgkin-Huxley model of the space-clamped squid giant axon is shown to admit unstable periodic solutions for current stimuli less than the stimulus at which the rest state becomes linearly unstable. The periodic solutions are demonstrated both by bifurcation theory and by numerical integration. The presence of subcritical unstable oscillations explains the discontinuous behaviour of the amplitude of the repetitive response as a function of current stimulus     

Repetitive activity of a branched Hodgkin-Huxley axon with multiple encoding sites

Afferent activity in a receptor afferent fiber with several encoding sites is generally believed to represent the activity of the fastest pacemaker that resets all more slowly encoding sites. Alternatively, some impulse mixing as well as some nonlinear summation of receptor current to a single encoder have been considered. In this article the repetitive firing activity of a Hodgkin-Huxley axon consisting of two branches that join into a single stem axon was investigated. The model axon was stimulated by constant-current injection into either the right or the left or both branches. It was found that the model axon generated an (infinite) train of action potentials if the input current was large enough. The discharge frequency found was constant, and on combined stimulation of both branches with different current, the site of impulse initiation was always in the branch receiving the higher input current, excluding a simple impulse mixing. On the other hand, the combined stimulation of both branches evoked repetitive firing with a higher frequency than expected by the pacemaker-resetting hypothesis. Moreover, a stimulus that is subthreshold for repetitive firing if injected into one branch yields repetitive firing when it is injected into both branches, a behavior inconsistent with impulse mixing and pacemaker resetting. On the other hand, current injection into one branch allowed repetitive activity only within a rather limited range of firing frequencies. Using distributed current injection into both branches, however, allowed many more different firing frequencies. Such behavior is inconsistent with both pacemaker resetting and (nonlinear) input current summation. Consequently, the repetitive firing behavior of a branched Hodgkin-Huxley axon with multiple encoding sites appears to be more complex than postulated in the simple hypotheses.     

Potassium-ion conduction noise in squid axon membrane.

Spectral analysis (1-1000 Hz) of spontaneous fluctuations of potential and current in small areas of squid (Loligo pealei) axon shows two forms of noise: f-1 noise occurs in both excitable and inexcitable axons with an intensity which depends upon the driving force for potassium ions. The other noise has a spectral form corresponding to a relaxation process, i.e. its asymptotic behavior at low frequencies is constant, and at high frequencies it declines with a slope of -2. This latter noise occurs only in excitable axons and was identified in spectra by (1) its disappearance after reduction of K+ current by internal perfusion with solutions containing tetraethylammonium (TEA+), Cs+ or reduced [Ki+] and (2) its insensitivity to block of Na+ conduction and active transport. The transition frequency of relaxation spectra are also voltage and temperature dependent and relate to the kinetics of K+-conduction in the Hodgkin-Huxley formulation. These data strongly suggest that the relaxation noise component arises from the kinetic properties of K+ channels. The f-1 noise is attributed to restricted diffusion in conducting K+ channels and/or leakage pathways. In addition, an induced K+ conduction noise associated with the binding of TEA+ and triethyldecylammonium ion to membrane sites is described. Measurement of the induced noise may provide an alternative means of characterizing the kinetics of interaction of these molecules with the membrane and also suggests that these and other pharmacological agents may not be useful in identifying noise components related to the sodium conduction mechanism which, in these experiments, appears to be much lower in intensity than either the normal K conduction or induced noise components.