Dimensional analysis of nerve models

General equations for (i) a uniform patch of nerve membrane, (ii) a continuous (unmyelinated) axon and (iii) a noded (myelinated) axon are analyzed using dimensional analysis. The original dimensioned equations are transformed to dimensionless equations. These equations contain dimensionless constants called similarity parameters which are functions of the physical constants or parameters of the system. (The similarity parameters are analogous to such quantities as the Reynolds number and Mach number used in fluid dynamics.) There is one similarity parameter for each of cases (i) and (ii), and four for case (iii). All dimensioned systems having the same values of all the similarity parameters form a similarity class.Once a quantity such as threshold stimulus or conduction velocity is computed for one member of any similarity class, the same quantity can be easily computed for any other member of the same class, by a simple formula containing the physical constants of the system, called a generating equation, or by an even simpler expression of proportionality, called a scaling relation.     

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.     

A note on the asymptotic reduction of the Hodgkin-Huxley equations for nerve impulses

The (standard) FitzHugh reduction of the Hodgkin-Huxley equations for the propagation of nerve impulses ignores the dynamics of the activation gates. This assumption is invalid and leads to an over-estimation of the wave speed by a factor of 5 and the wrong dependence of wave speed on sodium channel conductance. The error occurs because a non-dimensional parameter, which is assumed to be small in the FitzHugh reduction, is in fact large (≈18). We analyse the Hodgkin-Huxley equations for propagating nerve impulses in the limit that this non-dimensional parameter is large, and show that the analytical results are consistent with numerical simulations of the Hodgkin-Huxley equations.     

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.     

Electrical Characteristics of Insect Mechanoreceptors

External direct coupled recordings from the neurons of the mechanosensory hairs of insects show nerve impulses and graded slow potentials in response to deformation of the hair. These slow potentials or receptor potentials are negative going, vary directly with the magnitude of the stimulus, and show no overshoot when returning to baseline. The impulses have an initial positive phase which varies in size directly with the amplitude of the receptor potential. The receptor potential is related to the generator potential for the impulse in that it must attain some critical level before impulses are produced, and the frequency of impulses varies directly with amplitude of the receptor potential. The receptor potential does not return to the baseline after each impulse. In some receptors static deformation of the hair will maintain the receptor potential. It appears likely that both the receptor potential and the variation in size of the impulses are caused by a change in conductance of the cell membrane at the receptor site, and that the receptor potential originates at a site which is not invaded by the propagated impulses.     

Conduction in bundles of demyelinated nerve fibers: computer simulation

This study presents a model of action potential propagation in bundles of myelinated nerve fibers. The model combines the single-cable formulation of Goldman and Albus (1967) with a basic representation of the ephaptic interaction among the fibers. We analyze first the behavior of the conduction velocity (CV) under the change of the various conductance parameters and temperature. The main parameter influencing the CV is the fast sodium conductance, and the dependence of CV on the temperature is linear up to 30 degrees C. The increase of myelin thickness above its normal value (5 microm) gives a slight increase in CV. The CV of the single fiber decreases monotonically with the disruption of myelin, but the breakdown is abrupt. There is always conduction until the thickness is larger than 2% of its original value, at which with at this point a sharp transition of CV to zero occurs. Also, the increase of temperature can block conduction. At 5% of the original thickness there is still spike propagation, but an increase of 2 degrees C causes conduction block. These results are consistent with clinical observations. Computer simulations are performed to show how the CV is affected by local damage to the myelin sheath, temperature alterations, and increased ephaptic coupling (i.e., coupling of electrical origin due to the electric neutrality of all the nerve) in the case of fiber bundles. The ephaptic interaction is included in the model. Synchronous impulse transmission and the formation of "condensed" pulse states are found. Electric impulses with a delay of 0.5 ms are presented to the system, and the numerical results show that, for increasing coupling, the impulses tend to adjust their speed and become synchronized. Other interesting phenomena are that spurious spikes are likely to be generated when ephaptic interaction is raised and that damaged axons suffering conduction block can be brought into conduction by the normal functioning fibers surrounding them. This is seen also in the case of a large number of fibers (N=500). When all the fibers are stimulated simultaneously, the conduction velocity is found to be strongly dependent on the level of ephaptic coupling and a sensible reduction is observed with respect to the propagation along an isolated axon even for low coupling level. As in the case of three fibers, spikes tend to lock and form collective impulses that propagate slowly in the nerve. On the other hand, if only 10% of fibers are stimulated by an external input, the conduction velocity is only 2% less than that along a single axon. We found a threshold value for the ephaptic coupling such that for lower values it is impossible to recruit the damaged fibers into conduction, for values of the coupling equal to this threshold only one fiber can be restored by the nondamaged fibers, and for values larger than the threshold an increasing number of fibers can return to normal functioning. We get values of the ephaptic coupling such that 25% of axons can be damaged without change of the collective conduction.     

Proton Hopping: A Proposed Mechanism for Myelinated Axon Nerve Impulses

Myelinated axon nerve impulses travel 100 times more rapidly than impulses in non‐myelinated axons. Increased speed is currently believed to be due to ‘hopping’ or ‘saltatory propagation’ along the axon, but the mechanism by which impulses flow has never been adequately explained. We have used modeling approaches to simulate a role for proton hopping in the space between the plasma membrane and myelin sheath as the mechanism of nerve action‐potential flow.     

Temperature Characteristics of Excitation in Space-Clamped Squid Axons

Temperature characteristics of excitability in the squid giant axon were measured for the space-clamped axon with the double sucrose gap technique. Threshold strength-duration curves were obtained for square wave current pulses from 10 µsec to 10 msec and at temperatures from 5°C to 35°C. The threshold change of potential, at which an action potential separated from a subthreshold response, averaged 17 mv at 20°C with a Q¹⁰ of 1.15. The average threshold current density at rheobase was 12 µa/cm² at 20°C with a Q¹⁰ of 2.35 compared to 2.3 obtained previously. At short times the threshold charge was 1.5·10^-8 coul/cm². This was relatively independent of temperature and occasionally showed a minimum in the temperature range. At intermediate times and all temperatures the threshold currents were less than for both the single time constant model and the two factor excitation process as developed by Hill. FitzHugh has made computer investigations of the effect of temperature on the excitation of the squid axon membrane as represented by the Hodgkin-Huxley equations. These are in general in good agreement with our experimental results.     

The Information Capacity of Nerve Cells Using a Frequency Code

Approximate equations are derived for the amount of information a nerve cell or group of nerve cells can transmit about a stimulus of a given duration using a frequency code (i.e., assuming the mean frequency of nerve impulses measures the intensity of a maintained stimulus). The equations take into account the variability of successive interspike intervals, and any serial correlations between successive intervals, but do not require detailed assumptions about the mechanism of impulse initiation. The errors involved in using these approximations are evaluated for neurons which discharge either completely regularly, completely at random (Poisson process) or show a particular type of intermediate variability (gamma distribution model). The errors become negligibly small as the stimulus duration or the number of functionally similar nerve cells increases. The conditions for applying these equations to experimental data are discussed. The application of these equations should help considerably in eliminating the enormous discrepancies between some earlier estimates for the information processing capabilities of single nerve cells and systems of nerve cells.     

Influence of TRPV1 on diabetes-induced alterations in thermal pain sensitivity

Animals detect environmental changes through sensory neural mechanisms that enable them to differentiate the quality, intensity and temporal characteristics of stimuli. The 'doctrine of specific nervous energies' postulates that the different sensory modalities experienced by humans result of the activation of specific nervous pathways. Identification of functional classes of sensory receptors provided scientific support to the concept that somatosensory modalities (touch, pain, temperature, kinesthesis) are subserved by separate populations of sensory receptor neurons specialized in detecting innocuous and injurious stimuli of different quality (mechanical forces, temperature, chemical compounds). The identification of receptor proteins activated by different physicochemical stimuli, in particular ion channels of the Transient Receptor Potential (TRP) superfamily, has put forward the concept that specificity of peripheral sensory receptor neurons is determined by their expression of a particular "molecular sensor" that confers to each functional type its selectivity to respond with a discharge of nerve impulses to stimuli of a given quality. Nonetheless, recent experimental data suggest that the various molecular sensors proposed as specific transducer molecules for stimuli of different quality are not as neatly associated with the distinct functional types of sensory receptors as originally proposed. First, many ion channel molecules initially associated to the transduction of only one particular form of energy are also activated by stimuli of different quality, implying a limited degree of specificity in their transducing capacities. Second, molecular sensors associated with a stimulus quality and hence to a sensory receptor type and ultimately to a sensory modality may be concomitantly expressed in sensory receptor neurons functionally defined as specific for another stimulus quality. Finally, activation of voltage gated channels involved primarily in nerve impulse generation can also influence the gating of transducing channels, dramatically modifying their activation profile. Thus, we propose that the capacity exhibited by the different functional types of somatosensory receptor neurons to preferentially detect and encode specific stimuli into a discharge of nerve impulses, appears to result of a characteristic combinatorial expression of different ion channels in each neuronal type that finally determines their transduction and impulse firing properties. Transduction channels don't operate in isolation and their cellular context should also be taken into consideration to fully understand their function. Moreover, the inhomogeneous distribution of transduction and voltage-gated channels at soma, axonal branches and peripheral endings of primary sensory neurons influences the characteristics of the propagated impulse discharge that encodes the properties of the stimulus. Alteration of this concerted operation of ion channels in pathological conditions may underlie the changes in excitability accompanying peripheral sensory neuron injuries.     

Peripheral and central nervous responses evoked by small water movements in a cephalopod

Potentials were recorded from the epidermal head lines and from the CNS of young cuttlefish, Sepia officinalis, in response to weak water movements. 1. Within the test range 0.5-400 Hz a sinusoidal water movement elicits up to 4 components of response if the electrode is placed on a headline: (i) a positive phasic ON response; (ii) a tonic frequency-following microphonic response; (iii) a slow negative OFF response; and (iv) compound nerve impulses. 2. The amplitude of both the ON wave and the microphonic potential depends on stimulus frequency, stimulus amplitude and stimulus rise time. Frequencies around 100 Hz and short rise times are most effective in eliciting strong potentials. The minimal threshold was 0.06 microns peak-to-peak water displacement at 100 Hz (18.8 microns/s as velocity). 3. Change of direction of tangential sphere movement (parallel vs. across the head lines) has only a small effect on the microphonic and the summed nerve potentials. 4. Frequency and/or amplitude modulations of a carrier stimulus elicit responses at the onset and offset of the modulation and marked changes in the tonic microphonic response. 5. Evoked potentials can be recorded from the brain while stimulating the epidermal lines with weak water movements. The brain potentials differ in several aspects from the potentials of the head lines and show little or no onset or offset wave at the transitions of a frequency and amplitude modulation.     

Low temperature painful stimulus alters brain wave pattern of transcutaneous electrical stimulus

The somatosensory evoked response recorded from the scalp over the somatosensory cortex was used to examine the interaction between painful cold and transcutaneous electrical stimuli delivered concomitantly. When a painful cold stimulus was applied to the palmar receptive field of the median nerve while that nerve was being stimulated with electrical pulses at the wrist, there was an augmentation of an early component of the somatosensory evoked response manifested by an increase in the amplitude of a wave segment in comparison with room temperature controls. This augmentation depended on there being normal conduction of nerve impulses in both the population of small and large peripheral nerve fibers as compared to a state in which conduction was blocked selectively by a local anesthetic or a pressure cuff in those small and large fibers, respectively. The augmentation was not found to be characteristic of an arousal phenomenon, but was localized to the somatosensory cortex. This might represent the effects of a non-specific thalamortical projection system on a specific one.     

Inhibition in the eye of Limulus

In the compound lateral eye of Limulus each ommatidium functions as a single receptor unit in the discharge of impulses in the optic nerve. Impulses originate in the eccentric cell of each ommatidium and are conducted in its axon, which runs without interruption through an extensive plexus of nerve fibers to become a fiber of the optic nerve. The plexus makes interconnections among the ommatidia, but its exact organization is not understood. The ability of an ommatidium to discharge impulses in the axon of its eccentric cell is reduced by illumination of other ommatidia in its neighborhood: the threshold to light is raised, the number of impulses discharged in response to a suprathreshold flash of light is diminished, and the frequency with which impulses are discharged during steady illumination is decreased. Also, the activity that can be elicited under certain conditions when an ommatidium is in darkness can be inhibited similarly. There is no evidence for the spread of excitatory influences in the eye of Limulus. The inhibitory influence exerted upon an ommatidium that is discharging impulses at a steady rate begins, shortly after the onset of the illumination on neighboring ommatidia, with a sudden deep minimum in the frequency of discharge. After partial recovery, the frequency is maintained at a depressed level until the illumination on the neighboring receptors is turned off, following which there is prompt, though not instantaneous recovery to the original frequency. The inhibition is exerted directly upon the sensitive structure within the ommatidium: it has been observed when the impulses were recorded by a microelectrode thrust into an ommatidium, as well as when they were recorded more proximally in single fibers dissected from the optic nerve. Receptor units of the eye often inhibit one another mutually. This has been observed by recording the activity of two optic nerve fibers simultaneously. The mediation of the inhibitory influence appears to depend upon the integrity of nervous interconnections in the plexus: cutting the lateral connections to an ommatidium abolishes the inhibition exerted upon it. The nature of the influence that is mediated by the plexus and the mechanism whereby it exerts its inhibitory action on the receptor units are not known. The depression of the frequency of the discharge of nerve impulses from an ommatidium increases approximately linearly with the logarithm of the intensity of illumination on receptors in its vicinity. Inhibition of the discharge from an ommatidium is greater the larger the area of the eye illuminated in its vicinity. However, equal increments of area become less effective as the total area is increased. The response of an ommatidium is most effectively inhibited by the illumination of ommatidia that are close to it; the effectiveness diminishes with increasing distance, but may extend for several millimeters. Illumination of a fixed region of the eye at constant intensity produces a depression of the frequency of discharge of impulses from a nearby ommatidium that is approximately constant, irrespective of the level of excitation of the ommatidium. The inhibitory interaction in the eye of Limulus is an integrative process that is important in determining the patterns of nervous activity in the visual system. It is analogous to the inhibitory component of the interaction that takes place in the vertebrate retina. Inhibitory interaction results in the exaggeration of differences in sensory activity from different regions of the eye illuminated at different intensities, thus enhancing visual contrast.     

Traveling Wave Solutions of a Nerve Conduction Equation

We consider a pair of differential equations whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh-Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse-shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.     

Generator processes of repetitive activity in a pacinian corpuscle

Response patterns resulting from repetitive mechanical stimulation of the corpuscle depend on (1) the time course of recovery of the generator potential, on (2) the recovery of critical firing height, and on (3) the stimulus strength/generator potential function. By either augmenting stimulus frequency at constant strength, or by reducing strength at constant frequency, a sequence of propagated potentials is turned into a pattern of alternating regenerative and generator responses. In such a pattern an extra impulse can be set up whenever an extra stimulus produces a generator potential of enough amplitude to reach the firing height of the corresponding period. The new requirements of firing height introduced by the refractory trail of the extra impulse determine resetting of periodicity and appearance of a "compensatory pause." The decay time of the single generator potential is independent of stimulus duration. This is interpreted as a factor determining receptor adaptation. Upon repetitive stimulation at intervals above ½ decay time of the single generator potential, a compound generator potential is built up which shows no spontaneous decline. However, in spite of being considerably greater than the firing height for single impulses, the constant level of depolarization of the compound generator potential is unable to produce propagated potentials. A hypothesis is brought forward which considers the generator potential to arise from membrane units with fluctuating excitability scattered over the non-myelinated nerve ending.     

蜚蠊单个棘—钟形感器冲动发放的特性

本文分析了蜚镰后胸足单个棘一钟形感器对机械位移刺激的反应模式以及冲动发放的特性.结果证明该感器是一种适应较慢的相位性触觉感受器,对触刺激有相当稳定的反应.     

A mathematical model of a myelinated nerve fiber for analysis of impulse conduction mechanisms during the relative refractory period

It was shown by means of a mathematical model of a myelinated nerve fiber (Frankenhaeuser — Huxley) that an increase in threshold and decrease in the amplitude of the action potential (AP) during the relative refractory period are due mainly to sodium inactivation. The contribution of increased potassium permeability to these changes is small, for the chief component of the outgoing ionic current in the node of Ranvier is not the potassium current, but the leak current. Given the ratio between these currents the increase in threshold and graduation of the action potential in the node membrane are less marked than in the membrane of the squid giant axon. At the beginning of the relative refractory period the AP evoked by strong stimulation is conducted only to the next node. Later in the refractory period impulses are conducted incrementally, and the threshold for the spreading impulse is higher than the threshold for spike excitation in the stimulated node. Delay in impulse conduction between refractory nodes leads to the formation of a retrograde depolarization wave. The reasons for differences in the mechanisms of impulse conduction along unmyelinated and myelinated refractory fibers are discussed.Vishnevskii Institute of Surgery, Academy of Medical Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 4, No. 2, pp. 201–207, March–April, 1972.     

Refractoriness of sural nerve in humans

High-frequency stimulation of peripheral nerve bundles is frequently used in clinical tests and physiologic experiments to study presynaptic and postsynaptic effects. To understand the postsynaptic effects, it is important to ensure that each pulse in the train is equally effective in stimulating the presynaptic nerve bundle; however, the optimal interpulse interval (IPI) and the stimulus intensity at which each pulse is equally effective in stimulating the same number of axons are not known. The magnitude of the compound action potential produced by each pulse in a train was tested on the sural nerve of 4 healthy human subjects. The stimulus train (2-4 pulses) was applied to the sural nerve at the lateral malleolus, and neural responses were recorded from just below the knee. With 2-pulse trains, families of curves between IPIs (1-6 ms) and normalized amplitudes of the second response were plotted for different stimulus intensities. Visual inspection of the data showed that the curves fell into 2 groups: with stimulus intensities <2.5x perception threshold (Th), the test response appeared partially at longer IPIs, whereas with stimulus intensities >=3x Th, partial recovery of the test response was earlier. The interval for complete recovery was statistically the same for low- and high-intensity stimulation. With more than 2 pulses in a stimulus train (IPI = 5 ms), the amplitude of the compound action potential (CAP) was not affected significantly. These results are important in understanding both the presynaptic and postsynaptic responses when presynaptic axon bundles are stimulated at high frequencies.     

The standard Hodgkin-Huxley model and squid axons in reduced external Ca++ fail to accommodate to slowly rising currents.

Accommodation may be defined as an increase in the threshold of an excitable membrane when the membrane is subjected to a sustained subthreshold depolarizing stimulus. Some excitable membranes show accommodation in response to currents which rise linearly at a very slow rate. In this report we point out a theoretical and an experimental counterexample, i.e., a nerve model and an axon which do not accommodate. The nerve model is the standard Hodgkin-Huxley axon, which Hodgkin and Huxley expected not to be excited by a very slowly rising current. This expectation is often quoted as fact, in spite of contrary calculations which we confirm. We have found that squid axons in seawater with reduced divalent cation concentration also do not accommodate to slowly rising currents.