Impulses and Physiological States in Theoretical Models of Nerve Membrane

Van der Pol''s equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting “BVP model” has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer''s theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a “physiological state diagram,” with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.     

Mechanical wave propagation on nerve axons

The mechanical waves which may occur in or on a nerve axon are considered. It is shown that under rather general assumptions mechanical waves with phase velocities corresponding to nerve impulse propagation velocities belong to the natural modes of the axon membrane. This means that the mechanical disturbances which are connected with almost all models of nerve impulse propagation are readily supported by the axon, suggesting a possibility of interaction between electrical and mechanical events in the axon.     

Superposition of impulse activity in a rapidly-adapting afferent unit model

Mechanosensory afferent units consist of a parent axon, the peripheral axonal arborization, and the branch terminal mechanoreceptors. The present work uses a mathematical model to describe the contribution of a given number of rapidly-adapting mechanoreceptors to the impulse pattern of their parent axon. In the model impulses initiated by any driven mechanoreceptor instantaneously propagate orthodromically and antidromically. The model also incorporates the axonal absolute refractory period as well as ortho-and antidromically elicited recovery cycles. In separate computations, periodic or random (Poisson process) trains of short-duration stimuli at constant amplitude are delivered to a given number (N=2–30) of co-innervated mechanoreceptors. The superposition of component impulse trains always departs from the theoretical ideal (Poisson process). Such departures are attributable to: (i) the number of driven mechanoreceptors, when N is small, (ii) axonal absolute refractory period, during maximal amplitude stimulation, and (iii) antidromic recovery cycles as well as absolute refractoriness, during submaximal-amplitude stimulation. Computations reveal that this instantaneous reset model results in the elimination of information extracted by driven mechanoreceptors. Model predictions with Poisson stimulation at varied amplitudes are compared to G-hair afferent unit responses to analogous stimulation. Qualitatively opposite results with respect to parent axonal impulse patterns imply that the axonal arborization is not simply a substrate for impulse propagation from branch terminals to parent axon.     

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.     

Effect of doublet impulse sequences in the crayfish claw opener muscles and the computer-simulated neuromuscular synapse

The effects of doublet impulse sequences of the excitatory motor axon on the movement of the claw opener muscles in the crayfish were examined. The excitatory motor axon was stimulated electrically with various patterns of doublet impulse sequences generated by a digital computer. Doublet impulse sequences of stimulation produced a larger sustained movement than an uniform impulse sequences at the same mean rate of stimulation. The movement was largest when the interval between the impulses of a doublet was about 5 ms. This interval generated a movement amplitude 25% greater than that for the uniform impulse sequence. A simple model was formulated to stimulate the neuromuscular synapse of the claw opener muscle. The relationship between stimulation sequences with alternating long and short intervals and responses (firing probabilities) of the neuromuscular synapse at the same mean rate was investigated. The responses was classified into two typical types which are noneffective Type I and effective Type II to the absolute refractory period (ARP). The characteristics which are larger responses with short intervals in Type I and reduction of responses in the ARP region of Type II formed a plateau peak of the experimental results. By incorporating the reduction of end-plate potential (EPP) as a property of nonlinear rule for temporal summation into the model, it was shown that Type I response is maximal with a plateau peak at short interval, agreeing well with the experimental results from the claw opener muscles.     

A mode control model of a neuron's axon and dendrites

The ability of a neuron network to process information depends upon the ability of the individual neurons to transport impulses and to control the signal transport process in other neurons. The transport process for the action potential seen at the axon depends upon the excitable characteristic of the neural membrane. Propagation of signals in the dendrites, where synaptic imputs are most likely processed, is not clearly understood. Extracellular recordings of dendritic systems indicate that the dendrites are partially excitable and can conduct spikes. Further, electrical stimulation of the reticular formation or specific thalamic nuclei suggest that the conduction process can be modified in the dendrites of cortical cells.A Mode Control model is described which demonstrates many of the observed transport and control properties of dendrite and axon membrane. The model is based upon a simple extension of Fitzhugh's BVP model. Lateral transport over the membrane has been introduced by applying Kirchhoff's laws. Reinterpreting the variables, the influence of membrane potential, pH, and calcium ions can be identified. Modification of the voltage-current characteristic of the membrane model can change the axon model to a dendrite model. The dendrite model possesses a diffusion equation mode, a wave equation mode and a pulse mode. Signals are transferred in the wave and pulse mode and blocked in the diffusion mode. The dendrite's mode is controlled by the resting depolarization level. Experimental evidence tends to confirm these phenomena.The work described in this paper was performed while attending the University of California, Berkeley, under a National Institutes of Health Traineeship.     

Digital Computer Solutions for Excitation and Propagation of the Nerve Impulse

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.     

Asymmetry of motor impulses and neuromuscular synapses produced in crayfish claws by unilateral immobilization

Summary The fast axon supplying the closer muscle in crayfish (Procambarus clarkii) normally fires few impulses and generates large excitatory postsynaptic potentials (EPSPs) that fatigue rapidly with repeated stimulation. When the dactyl of one claw is immobilized in the closed position, impulse production in the fast axon decreases on the immobilized side and increases on the contralateral side. On the immobilized side, EPSPs become larger but more readily depressed with repeated stimulation, while converse changes occur on the contralateral side.In order to establish whether the smaller number of impulses on the immobilized side was responsible for the changes in EPSPs, extra impulses were generated in the fast axon of immobilized claws by implanting electrodes in the claw. Raising the impulse production to equal or exceed that of the contralateral side did not prevent the changes in EPSPs produced by immobilization. Thus, it is probable that changes in the level of synaptic input to central parts of the fast closer excitor neuron are mainly responsible for altered physiological properties of peripheral synapses, rather than the fast axon's impulse traffic per se.     

Effect of reduced impulse activity on the development of identified motor terminals in Drosophila larvae

In Drosophila larvae, motoneurons show distinctive differences in the size of their synaptic boutons; that is, axon 1 has type Ib ("big" boutons) terminals and axon 2 has type Is ("small" boutons) terminals on muscle fibers 6 and 7. To determine whether axon 1 develops large boutons due to its high impulse activity, we reduced impulse activity and examined the motor terminals formed by axon 1. The number of functional Na(+) channels was reduced either with the nap(ts) mutation or by adding tetrodotoxin (TTX) to the media (0.1 microg/g). In both cases, the rate of locomotion was decreased by approximately 40%, presumably reflecting a decrease in impulse activity. Locomotor activity was restored to above wild-type (Canton-S) levels when nap(ts) was combined with a duplication of para, the Na(+)-channel gene. Lucifer yellow was injected into the axon 1 motor terminals, and we measured motor terminal area, length, the number of branches, and the number and width of synaptic boutons. Although all parameters were smaller in nap(ts) and TTX-treated larvae compared to wild-type, most of these differences were not significant when the differences in muscle fiber size were factored out. Only bouton width was significantly smaller in both different nap(ts) and TTX-treated larvae: boutons were about 20% smaller in nap(ts) and TTX-treated larvae, and 20% larger in nap(ts); Dp para(+) compared to wild-type. In addition, terminal area was significantly smaller in nap(ts) compared to wild-type. Bouton size at Ib terminals with reduced impulse activity was similar to that normally seen at Is terminals. Thus, differences in impulse activity play a major role in the differentiation of bouton size at Drosophila motor terminals.     

ELECTRIC IMPEDANCE OF THE SQUID GIANT AXON DURING ACTIVITY

Alternating current impedance measurements have been made over a wide frequency range on the giant axon from the stellar nerve of the squid, Loligo pealii, during the passage of a nerve impulse. The transverse impedance was measured between narrow electrodes on either side of the axon with a Wheatstone bridge having an amplifier and cathode ray oscillograph for detector. When the bridge was balanced, the resting axon gave a narrow line on the oscillograph screen as a sweep circuit moved the spot across. As an impulse passed between impedance electrodes after the axon had been stimulated at one end, the oscillograph line first broadened into a band, indicating a bridge unbalance, and then narrowed down to balance during recovery. From measurements made during the passage of the impulse and appropriate analysis, it was found that the membrane phase angle was unchanged, the membrane capacity decreased about 2 per cent, while the membrane conductance fell from a resting value of 1000 ohm cm.² to an average of 25 ohm cm.² The onset of the resistance change occurs somewhat after the start of the monophasic action potential, but coincides quite closely with the point of inflection on the rising phase, where the membrane current reverses in direction, corresponding to a decrease in the membrane electromotive force. This E.M.F. and the conductance are closely associated properties of the membrane, and their sudden changes constitute, or are due to, the activity which is responsible for the all-or-none law and the initiation and propagation of the nerve impulse. These results correspond to those previously found for Nitella and lead us to expect similar phenomena in other nerve fibers.     

Conduction in demyelinated axons—A simplified model

This paper is concerned with conduction of the nervous impulse in a myelinated axon and the effect of demyelination on conduction characteristics. A model of nerve conduction called the “gunpowder fuse” model is presented which accurately predicts conduction velocities in both myelinated and unmyelinated nerves. The effect on conduction velocity in this model by reducing myelin thickness is examined by utilizing basic data and building and equivalent circuit. The result is a curve relating reduced conduction velocity to reduced myelin thickness. A similar analysis and resultant curve is derived from a saltatory conduction model. Supported in part by National Multiple Sclerosis Society Research Grant No. 516 and Air Force Grant AFOSR 669-67.     

Response characteristics of the BVP neuron model to periodic pulse inputs.

The characteristics of the BVP neuron model response to periodic pulse stimuli are investigated. Temporal patterns of the output of the model are analyzed as a function of the stimulus intensity and period. The BVP model exhibits the same chaotic behavior, and a Cantor function-like graph of the response frequency (mean firing rate) as in electrophysiological experiments. This shows that the BVP model describes the complicated response characteristics of the neuron at least qualitatively.     

Further study of soma,dendrite, and axon excitation in single neurons

The present investigation continues a previous study in which the soma-dendrite system of sensory neurons was excited by stretch deformation of the peripheral dendrite portions. Recording was done with intracellular leads which were inserted into the cell soma while the neuron was activated orthodromically or antidromically. The analysis was also extended to axon conduction. Crayfish, Procambarus alleni (Faxon) and Orconectes virilis (Hagen), were used. 1. The size and time course of action potentials recorded from the soma-dendrite complex vary greatly with the level of the cell's membrane potential. The latter can be changed over a wide range by stretch deformation which sets up a "generator potential" in the distal portions of the dendrites. If a cell is at its resting unstretched equilibrium potential, antidromic stimulation through the axon causes an impulse which normally overshoots the resting potential and decays into an afternegativity of 15 to 20 msec. duration. The postspike negativity is not followed by an appreciable hyperpolarization (positive) phase. If the membrane potential is reduced to a new steady level a postspike positivity appears and increases linearly over a depolarization range of 12 to 20 mv. in various cells. At those levels the firing threshold of the cell for orthodromic discharges is generally reached. 2. The safety factor for conduction between axon and cell soma is reduced under three unrelated conditions, (a) During the recovery period (2 to 3 msec.) immediately following an impulse which has conducted fully over the cell soma, a second impulse may be delayed, may invade the soma partially, or may be blocked completely. (b) If progressive depolarization is produced by stretch, it leads to a reduction of impulse height and eventually to complete block of antidromic soma invasion, resembling cathodal block, (c) In some cells, when the normal membrane potential is within several millivolts of the relaxed resting state, an antidromic impulse may be blocked and may set up within the soma a local potential only. The local potential can sum with a second one or it may sum with potential changes set up in the dendrites, leading to complete invasion of the soma. Such antidromic invasion block can always be relieved by appropriate stretch which shifts the membrane potential out of the "blocking range" nearer to the soma firing level. During the afterpositivity of an impulse in a stretched cell the membrane potential may fall below or near the blocking range. During that period another impulse may be delayed or blocked. 3. Information regarding activity and conduction in dendrites has been obtained indirectly, mainly by analyzing the generator action under various conditions of stretch. The following conclusions have been reached: The large dendrite branches have similar properties to the cell body from which they arise and carry the same kind of impulses. In the finer distal filaments of even lightly depolarized dendrites, however, no axon type all-or-none conduction occurs since the generator potential persists to a varying degree during antidromic invasion of the cell. With the membrane potential at its resting level the dendrite terminals contribute to the prolonged impulse afternegativity of the soma. 4. Action potentials in impaled axons and in cell bodies have been compared. It is thought that normally the over-all duration of axon impulses is shorter. Local activity during reduction of the safety margin for conduction was studied. 5. An analysis was made of high frequency grouped discharges which occasionally arise in cells. They differ in many essential aspects from the regular discharges set up by the generator action. It is proposed that grouped discharges occur only when invasion of dendrites is not synchronous, due to a delay in excitation spread between soma and dendrites. Each impulse in a group is assumed to be caused by an impulse in at least one of the large dendrite branches. Depolarization of dendrites abolishes the grouped activity by facilitating invasion of the large dendrite branches.     

An integrated blood volume pulse biofeedback system for migraine treatment

A blood volume pulse (BVP) biofeedback system is described that integrates BVP amplitude to provide a signal appropriate for auditory feedback. In comparison to binary BVP feedback methods, this integrated system offers the advantages of continuous feedback and increased scoring ease. The validity of this system was established by correlating the integrated BVP output with trough-to-peak measurements of the raw BVP signal during unassisted relaxation and temporal BVP biofeedback with eight migraine headache patients. Within-subject correlations of the integrated and raw BVP outputs ranged from .82 to .98 (X=.95). Although the integrated method admits unwanted BVP changes in rate, correlation analyses showed this confound factor to be small. Increments in biofeedback training effects were observed during the treatment course. Substantive migraine relief was achieved by the end of treatment and therapeutic gains were maintained at 1-year follow-up. In conclusion, it appears that this method successfully presents continuous auditory feedback from an integrated BVP signal resulting in therapeutic benefits to migraineurs.     

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.     

Firing properties of the soma and axon of the abdominal stretch receptor neurons in the crayfish (Astacus leptodactylus)

Action potentials (APs) and impulse responses in the soma and axon of the rapidly and slowly adapting (SA) abdominal stretch receptor neurons of the crayfish (Astacus leptodactylus) were recorded with single microelectrode current-clamp technique. Impulse frequency response to constant current injection was almost constant in the SA neuron while the response decayed completely in the rapidly adapting (RA) neuron. Mean impulse frequency responses to current stimulations were similar in the receptor neuron pairs. In the RA neuron additional current steps evoked additional impulses while a sudden drop in the current amplitude caused adaptation. Impulse duration was dependent on the rate of rise when current ramps were used. Adaptation was facilitated when calculated receptor current was used. Exposing the neuron to 3 mmol/l TEA or scorpion venom resulted in partly elongated impulse responses. SA neuron could continuously convert the current input into impulse frequency irrespective of previous stimulation conditions. Exposing the SA neuron to 3 mmol/l TEA or 1 mmol/l Lidocaine reduced impulse duration to large current stimulations. The SA neuron fired spontaneously if it was exposed to 5-10 mmol/l Lidocaine or 10(-2) mg/ml Leiurus quinquestriatus venom. The action potential (AP) amplitudes in the RA soma, RA axon, SA soma, and SA axon were significantly different between components of all pairs. Duration of the AP in the axon of the RA neuron was significantly shorter than those in the RA soma, SA soma, and SA axon. Diameter of the RA axon was larger than that of the SA axon. Non-adapting impulse responses were promptly observed only in the SA axons. The results indicate that the RA neuron is a sort of rate receptor transducing the rapid length changes in the receptor muscle while the SA neuron is capable of transducing the maintained length changes in the receptor muscle. The differences in firing properties mainly originate from the differences in the active and passive properties of the receptor neurons.     

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.     

动态神经元的网络模型——Ⅲ.电路实现

在动态神经元网络数学模型的基础上,利用模拟有源器件与数学开关电路组成的硬件系统来达到模拟生物神经元的目的。模拟的结果表明：这个硬件具有神经元脉冲发放的动态过程,系统中每部分的输出信号分别对应于突触后电位、感受器电位、始段分级电位和轴突上脉冲发放等波形,与生物实验资料相似,是一个比计算机仿真更接近实际的连续模型,它将为小型神经元网络的动态特性分析提供了更直观,更可靠的手段。     

Extra spike formation in sensory neurons and the disruption of afferent spike patterning

The peculiar pseudounipolar geometry of primary sensory neurons can lead to ectopic generation of "extra spikes" in the region of the dorsal root ganglion potentially disrupting the fidelity of afferent signaling. We have used an explicit model of myelinated vertebrate sensory neurons to investigate the location and mechanism of extra spike formation, and its consequences for distortion of afferent impulse patterning. Extra spikes originate in the initial segment axon under conditions in which the soma spike becomes delayed and broadened. The broadened soma spike then re-excites membrane it has just passed over, initiating an extra spike which propagates outwards into the main conducting axon. Extra spike formation depends on cell geometry, electrical excitability, and the recent history of impulse activity. Extra spikes add to the impulse barrage traveling toward the spinal cord, but they also travel antidromically in the peripheral nerve colliding with and occluding normal orthodromic spikes. As a result there is no net increase in afferent spike number. However, extra spikes render firing more staccato by increasing the number of short and long interspike intervals in the train at the expense of intermediate intervals. There may also be more complex changes in the pattern of afferent spike trains, and hence in afferent signaling.     

Reliable, responsive pacemaking and pattern generation with minimal cell numbers: the crustacean cardiac ganglion

Investigations of the electrophysiology of crustacean cardiac ganglia over the last half-century are reviewed for their contributions to elucidating the cellular mechanisms and interactions by which a small (as few as nine cells) neuronal network accomplishes extremely reliable, rhythmical, patterned activation of muscular activity-in this case, beating of the neurogenic heart. This ganglion is thus a model for pacemaking and central pattern generation. Favorable anatomy has permitted voltage- and space-clamp analyses of voltage-dependent ionic currents that endow each neuron with the intrinsic ability to respond with rhythmical, patterned impulse activity to nonpatterned stimulation. The crustacean soma and initial axon segment do not support impulse generation but integrate input from stretch-sensitive dendrites and electrotonic and chemically mediated synapses on axonal processes in neuropils. The soma and initial axon produce a depolarization-activated, calcium-mediated, sustained potential, the "driver potential," so-called because it drives a train of impulses at the "trigger zone" of the axon. Extreme reliability results from redundancy and the electrotonic coupling and synaptic interaction among all the neurons. Complex modulation by central nervous system inputs and by neurohormones to adjust heart pumping to physiological demands has long been demonstrated, but much remains to be learned about the cellular and molecular mechanisms of action. The continuing relevance of the crustacean cardiac ganglion as a relatively simple model for pacemaking and central pattern generation is confirmed by the rapidly widening documentation of intrinsic potentials such as plateau potentials in neurons of all major animal groups. The suite of ionic currents (a slowly inactivating calcium current and various potassium currents, with variations) observed for the crustacean cardiac ganglion have been implicated in or proven to underlie a majority of the intrinsic potentials of neurons involved in pattern generation.