Regeneration restores some of the altered electrical properties of axotomized bullfrog B-cells

In bullfrog B-type sympathetic neurones axon injury produces substantial changes in somal membrane properties. These include a shortening of action potential afterhyperpolarization (AHP) and an increase in action potential (AP) duration. In the present experiments we compared two injury situations: nerve crush, which was followed by regeneration, and nerve cut, after which regeneration to the original target was prevented, to investigate whether these electrophysiological changes were related to axon regeneration. Both crush and cut injuries produced a similar maximum decrease in AHP duration (to 33 and 30%) by 14 days after axotomy. After nerve crush, AHP duration recovered to within control values by 42 days, while after cut it remained depressed. AHP amplitude decreased to the same extent after nerve crush or cut (to 62 and 58%), but the rate of decrease was slower following crush when compared with cut, and following both types of injury it still remained depressed at 42 and 49 days. Changes in AP duration also took longer to occur following nerve crush, reaching maximal values at 35-42 days, at which time AHP duration had returned to within the normal range. The early reduction in AHP duration and its rapid recovery in regenerating neurones suggests that the current underlying this membrane property is regulated by events associated with axon outgrowth and peripheral reconnection. In contrast, changes in AHP amplitude and AP repolarization appeared to be independent of the occurrence of axon regeneration and remained abnormal at 49 days despite the recovery of AHP duration. These results imply that the electrophysiological changes seen in B-cells following injury are differentially regulated during subsequent regeneration.     

The magnetic field of a single axon. A comparison of theory and experiment.

The magnetic field and the transmembrane action potential of a single nerve axon were measured simultaneously. The volume conductor model was used to calculate the magnetic field from the measured action potential, allowing comparison of the model predictions with the experimental data. After analyzing the experiment for all systematic errors, we conclude that the shape of the magnetic field can be accurately predicted from the transmembrane potential and, more importantly, the shape of the transmembrane potential can be calculated from the magnetic field. The data are used to determine ri, the internal resistance per unit length of the axon, to be 19.3 +/- 1.9 k omega mm-1, implying a value for the internal conductivity of 1.44 +/- 0.33 omega -1 m-1. Magnetic measurements are compared with standard bioelectric techniques for studying nerve axons.     

An Analysis of the Membrane Potential along a Clamped Squid Axon

A partially depolarized squid axon membrane is assumed to have a quasi-steady state negative resistance, the membrane potential is clamped at one point, and a distribution of potential along the axon is obtained from the cable equation. Nominal experimental values of -2 ohm cm² for the membrane and 6 ohm cm² for the internal and external current electrodes and the axoplasm and sea water between them are used for illustration. The potential and current may be uniform for an axon and electrode length less than 1.2 mm. For a long axon the potential varies as the cosine of the distance within 0.8 mm of the control point. Beyond this the potential variation is exponential and the entire pattern is about 5 mm long. The average current density out to 0.3 mm from the control point is within 10 per cent of the potential clamp value. These distributions are stable for control amplifications of about unity and more.     

A new double-chamber model of ion channels. Beyond the Hodgkin and Huxley model

This paper proposes a new double-chamber model (DCM) of ion channels. The model ion channel consists of a series of three pores alternating with two chambers. The chambers are net negatively charged. The chamber's electric charge originates from dissociated amino acid side chains and is pH dependent. The chamber's net negative charge is compensated by cations present inside the chamber and in a diffuse electric layer outside the chamber. The pore's permeability is constant independent of time. One pore of the sodium channel and one of the potassium channel is a voltage-sensing pore. Due to the channel's structure, ions flow through the pores and chambers in a time-dependent manner. The model reproduces experimental voltage clamp and action potential data. The current flowing through a single sodium channel is less then one femtoampere. The DCM is considerably simpler then the Hodgkin and Huxley model (HHM) used to describe the electrophysiological properties of an axon. Unlike the HHM, the DCM can explain refractoriness, anode break excitation, accommodation and the effect of pH and temperature on the channels without additional parameters. In the DCM, the axon membrane shows repetitive activity depending on the channel density, sodium to potassium channel ratio and external potassium concentration. In the DCM, the action potential starts from 'hot spot areas' of higher channel densities and a higher sodium to potassium channel ratio, and then propagates through the whole axon.     

Propagation of action potentials in squid giant axons. Repetitive firing at regions of membrane inhomogeneities

Effects of reduction in potassium conductance on impulse conduction were studied in squid giant axons. Internal perfusion of axons with tetraethylammonium (TEA) ions reduces G K and causes the duration of action potential to be increased up to 300 ms. This prolongation of action potentials does not change their conduction velocity. The shape of these propagating action potentials is similar to membrane action potentials in TEA. Axons with regions of differing membrane potassium conductances are obtained by perfusing the axon trunk and one of its two main branches with TEA after the second branch has been filled with normal perfusing solution. Although the latter is initially free of TEA, this ion diffuses in slowly. Up until a large amount of TEA has diffused into the second branch, action potentials in the two branches have very different durations. During this period, membrane regions with prolonged action potentials are a source of depolarizing current for the other, and repetitive activity may be initiated at transitional regions. After a single stimulus in either axon region, interactions between action potentials of different durations usually led to rebound, or a short burst, of action potentials. Complex interactions between two axon regions whose action potentials have different durations resembles electric activity recorded during some cardiac arrhythmias.     

A model for the polarization of neurons by extrinsically applied electric fields.

A model is presented for the subthreshold polarization of a neuron by an applied electric field. It gives insight into how morphological features of a neuron affect its polarizability. The neuronal model consists of one or more extensively branched dendritic trees, a lumped somatic impedance, and a myelinated axon with nodes of Ranvier. The dendritic trees branch according to the 3/2-power rule of Rall, so that each tree has an equivalent cylinder representation. Equations for the membrane potential at the soma and at the nodes of Ranvier, given an arbitrary specified external potential, are derived. The solutions determine the contributions made by the dendritic tree and the axon to the net polarization at the soma. In the case of a spatially constant electric field, both the magnitude and sign of the polarization depend on simple combinations of parameters describing the neuron. One important combination is given by the ratio of internal resistances for longitudinal current spread along the dendritic tree trunk and along the axon. A second is given by the ratio between the DC space constant for the dendritic tree trunk and the distance between nodes of Ranvier in the axon. A third is given by the product of the electric field and the space constant for the trunk of the dendritic tree. When a neuron with a straight axon is subjected to a constant field, the membrane potential decays exponentially with distance from the soma. Thus, the soma seems to be a likely site for action potential initiation when the field is strong enough to elicit suprathreshold polarization. In a simple example, the way in which orientation of the various parts of the neuron affects its polarization is examined. When an axon with a bend is subjected to a spatially constant field, polarization is focused at the bend, and this is another likely site for action potential initiation.     

C-Fiber Recovery Cycle Supernormality Depends on Ion Concentration and Ion Channel Permeability

Following each action potential, C-fiber nociceptors undergo cyclical changes in excitability, including a period of superexcitability, before recovering their basal excitability state. The increase in superexcitability during this recovery cycle depends upon their immediate firing history of the axon, but also determines the instantaneous firing frequency that encodes pain intensity. To explore the mechanistic underpinnings of the recovery cycle phenomenon a biophysical model of a C-fiber has been developed. The model represents the spatial extent of the axon including its passive properties as well as ion channels and the Na/K-ATPase ion pump. Ionic concentrations were represented inside and outside the membrane. The model was able to replicate the typical transitions in excitability from subnormal to supernormal observed empirically following a conducted action potential. In the model, supernormality depended on the degree of conduction slowing which in turn depends upon the frequency of stimulation, in accordance with experimental findings. In particular, we show that activity-dependent conduction slowing is produced by the accumulation of intraaxonal sodium. We further show that the supernormal phase results from a reduced potassium current K_dr as a result of accumulation of periaxonal potassium in concert with a reduced influx of sodium through Na_v1.7 relative to Na_v1.8 current. This theoretical prediction was supported by data from an in vitro preparation of small rat dorsal root ganglion somata showing a reduction in the magnitude of tetrodotoxin-sensitive relative to tetrodotoxin -resistant whole cell current. Furthermore, our studies provide support for the role of depolarization in supernormality, as previously suggested, but we suggest that the basic mechanism depends on changes in ionic concentrations inside and outside the axon. The understanding of the mechanisms underlying repetitive discharges in recovery cycles may provide insight into mechanisms of spontaneous activity, which recently has been shown to correlate to a perceived level of pain.     

Current-Voltage Relations in the Lobster Giant Axon Membrane Under Voltage Clamp Conditions

The sucrose-gap method introduced by Stämpfli provides a means for the application of a voltage clamp to the lobster giant axon, which responds to a variety of different experimental procedures in ways quite similar to those reported for the squid axon and frog node. This is particularly true for the behavior of the peak initial current. However, the steady state current shows some differences. It has a variable slope conductance less than that of the peak initial current. The magnitude of the steady state slope conductance is related to the length of the repolarization phase of the action potential, which does not have an undershoot in the lobster. The steady state outward current is maintained for as long as 100 msec.; this is in contrast to a decline of about 50 per cent in the squid axon. Lowering the external calcium concentration produces shifts in the current-voltage relations qualitatively similar to those obtained from the squid axon. On the basis of the data available, there is no reason to doubt that the Hodgkin and Huxley analysis for the squid giant axon in sea water can be applied to the lobster giant axon.     

On the role of subthreshold dynamics in neuronal signaling

The role of subthreshold dynamics in neuronal signaling is examined using periodic pulse train stimulation of the Fitzhugh-Nagumo (FN) model of nerve membrane excitability and results from the squid giant axon as an experimental data base. For a broad range of stimulus conditions the first pulse in a pulse train elicited an action potential, whereas all subsequent pulses elicited subthreshold responses, both in the axon and in the FN model. These results are not well described by the Hodgkin and Huxley 1952 model. Various different patterns of subthreshold responses, including chaotic dynamics, can be observed in both systems-the FN model and the axon-depending upon stimulus conditions. For some conditions action potentials are occasionally interspersed among the subthreshold events with randomly occurring interspike intervals. The randomness is directly attributable to the underlying subthreshold chaos-deterministic chaos-rather than to a stochastic noise source. We conclude that this mechanism may contribute to multimodal interspike interval histograms which have been observed from individual neurons throughout the nervous system.     

Pacemaker activity of the rabbit sinoatrial node. A comparison of mathematical models.

In the past decade, three mathematical models describing the pacemaker activity of the rabbit sinoatrial node have been developed: the Bristow-Clark model, the Irisawa-Noma model, and the Noble-Noble model. In a comparative study it is demonstrated that these models, as well as subsequent modifications, all have several drawbacks. A more accurate model, describing the pacemaker activity of a single pacemaker cell isolated from the rabbit sinoatrial node, was constructed. Model equations, including equations for the T-type calcium current, are based on experimental data from voltage clamp experiments on single cells that were published during the last few years. In contrast to the other models, only a small amount of background current contributes to the overall electrical charge flow. The action potential parameters of the model cell, its responses to voltage clamp steps and its current-voltage relationships have been computed. The model is used to discuss the relative contribution of membrane current components to the slow diastolic depolarization phase of the action potential.     

Action currents, internodal potentials, and extracellular records of myelinated mammalian nerve fibers derived from node potentials.

The potential distribution within the internodal axon of mammalian nerve fibers is derived by applying known node potential waveforms to the ends of an equivalent circuit model of the internode. The complete spatial/temporal profile of action potentials synthesized from the internodal profiles is used to compute the node current waveforn, and the extracellular action potential around fibers captured within a tubular electrode. For amphibia, the results agreed with empirical values. For mammals, the amplitude of the node currents plotted against conduction velocity was fitted by a straight line. The extracellular potential waveform depended on the location of the nodes within the tube. For tubes of length from 2 to 8 internodes, extracellular wave amplitude (mammals) was about one-third of the product of peak node current and tube resistance (center to ends). The extracellular potentials developed by longitudinal and radial currents in an anisotropic medium (fiber bundle) are compared.     

Block of sodium channels by internal mono- and divalent guanidinium analogues. Modulation by sodium ion concentration.

We have investigated the block of squid axon sodium channels by mono- and divalent guanidinium analogues. The action of these compounds on steady state sodium currents was independent of the presence or absence of the normal inactivation process. Block by both mono- and divalent analogues was voltage-dependent, but was a steeper function of potential for divalent molecules. The voltage-dependence could not, in general, be reproduced by a simple model based on Boltzmann's equation. Inhibition of steady state currents by guanidinium ions with 50 mM internal sodium was reasonably well described by a 1:1 drug/channel binding function. Increasing the internal sodium ion concentration increased both the degree and voltage-dependence of current inhibition. This is in sharp contrast to the decrease in inactivation caused by internal sodium. Changes in the external sodium concentration had very little effect on drug block. These results are consistent with a model of the sodium channel as a multi-ion pore. Only a small increase in block can be produced by increased internal sodium in a three-barrier two-site model, but a four-barrier three-site model can reproduce these experimental findings. The implications of these results for physical models of inactivation are discussed.     

Neuromagnetic thresholds

Estimates are made of the effects which a steady magnetic field may have on the action potential. These are divided into two classes: Current distortion and inductive effects. Employing relations of the Hall Effect, it is found that for distortion of the action-potential current-pattern of the squid axon, magnetic field strengths of the order of Gauss are required. Changes in the current pattern due to inductive effects similarly imply that a magnetic field $\text{B ? (2·6/α)}$ kgauss distributed over ? 1 cm of axon would measurably perturb current flow in the action potential. The phenomenological coefficient α characterizes the asymmetry of the current pattern. For the case of a cylindrically symmetric response, α ? 0, in which case inductive perturbation of the action potential is nullified. Finite inductive effects for the case of the cylindrically asymmetric action potential are found also to be dependent on the orientation of the magnetic field with respect to the action potential. The effects of the extent of the applied magnetic field are also discussed.     

Parameter estimation in cardiac ionic models

We examine the problem of parameter estimation in mathematical models of excitable cell cardiac electrical activity using the well-known Beeler–Reuter (1977) ionic equations for the ventricular action potential. The estimation problem can be regarded as equivalent to the accurate reconstruction of ionic current kinetics and amplitudes in an excitable cell model, given only action potential experimental data. We show that in the Beeler–Reuter case, all ionic currents may be reasonably reconstructed using an experimental design consisting of action potential recordings perturbed by pseudo-random injection currents.

The Beeler–Reuter model was parameterised into 63 parameters completely defining all membrane current amplitudes and kinetics. Total membrane current was fitted to model-generated experimental data using a ‘data-clamp’ protocol. The experimental data consisted of a default action-potential waveform and an optional series of perturbed waveforms generated by current injections. Local parameter identifiability was ascertained from the reciprocal condition value (1/λ) of the Hessian at the known solution. When fitting to a single action potential waveform, the model was found to be over-determined, having a 1/λ value of 3.6e−14. This value improved slightly to 1.4e−10 when an additional 2 perturbed waveforms were included in the fitting process, suggesting that the additional data did not overly improve the identifiability problem. The additional data, however, did allow the accurate reconstruction of all ionic currents. This indicates that by appropriate experimental design, it may be possible to infer the properties of underlying membrane currents from observation of transmembrane potential waveforms perturbed by pseudo-random currents.     

Subtle Paranodal Injury Slows Impulse Conduction in a Mathematical Model of Myelinated Axons

This study explores in detail the functional consequences of subtle retraction and detachment of myelin around the nodes of Ranvier following mild-to-moderate crush or stretch mediated injury. An equivalent electrical circuit model for a series of equally spaced nodes of Ranvier was created incorporating extracellular and axonal resistances, paranodal resistances, nodal capacitances, time varying sodium and potassium currents, and realistic resting and threshold membrane potentials in a myelinated axon segment of 21 successive nodes. Differential equations describing membrane potentials at each nodal region were solved numerically. Subtle injury was simulated by increasing the width of exposed nodal membrane in nodes 8 through 20 of the model. Such injury diminishes action potential amplitude and slows conduction velocity from 19.1 m/sec in the normal region to 7.8 m/sec in the crushed region. Detachment of paranodal myelin, exposing juxtaparanodal potassium channels, decreases conduction velocity further to 6.6 m/sec, an effect that is partially reversible with potassium ion channel blockade. Conduction velocity decreases as node width increases or as paranodal resistance falls. The calculated changes in conduction velocity with subtle paranodal injury agree with experimental observations. Nodes of Ranvier are highly effective but somewhat fragile devices for increasing nerve conduction velocity and decreasing reaction time in vertebrate animals. Their fundamental design limitation is that even small mechanical retractions of myelin from very narrow nodes or slight loosening of paranodal myelin, which are difficult to notice at the light microscopic level of observation, can cause large changes in myelinated nerve conduction velocity.     

An autoregulation model in the system of squid axonal channels

A model that describes the interactions in the system of squid axon channels is proposed. The variables of the model are channel concentration and membrane voltage. The model represents the simplest mathematical formulation of the data on axon membrane, its components, and their role in the generation and propagation of action potential.     

Analytical reconstruction of the neuronal input current from spike train data

The time course of the current driving action potential generation at a neuron investigated experimentally is in general not measurable directly. In this paper an indirect method is introduced that allows estimation of this unknown current time course using only spike train data. Assuming the leaky integrator model as valid for the action potential encoding site of the investigated neuron, the unknown input current is obtained by determining (analytically) a current time course that upon injection into the leaky integrator model evokes action potential sequences identical to those observed experimentally. Applications of this current-reconstruction procedure to neuronal output data obtained from a leaky integrator model showed that the procedure allows a good estimation of the underlying input current even if the membrane time constant of the investigated neuron is not known exactly. Additionally, an application of current reconstruction to experimental data obtained from a cat muscle spindle primary afferent subject to repeated -stimuli is demonstrated.     

Demonstration of two stable potential states in the squid giant axon under tetraethylammonium chloride

1. Intracellular injection of tetraethylammonium chloride (TEA) into a giant axon of the squid prolongs the duration of the action potential without changing the resting potential (Fig. 3). The prolongation is sometimes 100-fold or more. 2. The action potential of a giant axon treated with TEA has an initial peak followed by a plateau (Fig. 3). The membrane resistance during the plateau is practically normal (Fig. 4). Near the end of the action potential, there is an apparent increase in the membrane resistance (Fig. 5D and Fig. 6, right). 3. The phenomenon of abolition of action potentials was demonstrated in the squid giant axon treated with TEA (Fig. 7). Following an action potential abolished in its early phase, there is no refractoriness (Fig. 8). 4. By the method of voltage clamp, the voltage-current relation was investigated on normal squid axons as well as on axons treated with TEA (Figs. 9 and 10). 5. The presence of stable states of the membrane was demonstrated by clamping the membrane potential with two voltage steps (Fig. 11). Experimental evidence was presented showing that, in an "unstable" state, the membrane conductance is not uniquely determined by the membrane potential. 6. The effect of low sodium water was investigated in the axon treated with TEA (Fig. 12). 7. The similarity between the action potential of a squid axon under TEA and that of the vertebrate cardiac muscle was stressed. The experimental results were interpreted as supporting the view that there are two stable states in the membrane. Initiation and abolition of an action potential were explained as transitions between the two states.     

Excitation of squid giant axons below 0 degree C.

The excitation of the squid giant axon that had been perfused intra- and extracellularly with solutions containing a high concentration of glycerol could be observed below 0 degree C. The action potential could be elicited at normal strengths of electrical stimuli. The time-course of the action potential was slowed, whereas the resting potential and the amplitude of the action potential changed only slightly. The membrane current under the voltage clamp at -6.3 degrees C was about 100-fold slower than that in normal sea water at 8.7 degrees C because of the large viscosity of glycerol solutions and the low temperature. The Q10 values of the magnitude and the time-course of the membrane current were 2.3 and 1/4.0, respectively.