Some Relations between Action Potential and Resting Potential of the Lobster Giant Axon

Experiments were performed to determine the quantitative relation existing between action potential and resting potential of the lobster giant axon. Resting potential changes were induced by either increasing the external potassium concentration or by reducing the external calcium concentration. For either treatment the action potential amplitude is proportional to the logarithm of the resting potential minus a constant. This constant is equivalent to the minimum resting potential at which a propagated spike is possible, and is larger for depolarization in low calcium than in high potassium. Thus the change in action potential per unit change in resting potential is greater in low external calcium than in high external potassium. Analog computer solutions to the Hodgkin-Huxley equations for squid axon membrane potentials show that, if the initial conditions are properly specified, the action potential is proportional to the logarithm of the potassium potential minus a constant. The experimental results and the analog computations suggest that reducing external calcium produces changes in the invertebrate axon that cannot be accounted for solely on the basis of alterations in the potassium potential.     

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.     

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.     

Intermittency in the Hodgkin-Huxley model

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.     

A model for axonal propagation incorporating both radial and axial ionic transport.

We present an axonal model that explicitly includes ionic diffusion in the intracellular, periaxonal, and extracellular spaces and that incorporates a Hodgkin-Huxley membrane, extended with potassium channel inactivation and active ion transport. Although ionic concentration changes may not be significant in the time course of one action potential, they are important when considering the long-term behavior (seconds to minutes) of an axon. We demonstrate this point with simulations of transected axons where ions are moving between the intra- and extracellular spaces through an opening that is sealing with time. The model predicts that sealing must occur within a critical time interval after the initial injury to prevent the entire axon from becoming permanently depolarized. This critical time interval becomes considerably shorter when active ion transport is disabled. Furthermore, the model can be used to study the effects of sodium and potassium channel inactivation; e.g., sodium inactivation must be almost complete (within 0.02%) to obtain simulation results that are realistic.     

A model of motor and sensory axon activation in the median nerve using surface electrical stimulation

Surface electrical stimulation has the potential to be a powerful and non-invasive treatment for a variety of medical conditions but currently it is difficult to obtain consistent evoked responses. A viable clinical system must be able to adapt to variations in individuals to produce repeatable results. To more fully study the effect of these variations without performing exhaustive testing on human subjects, a system of computer models was created to predict motor and sensory axon activation in the median nerve due to surface electrical stimulation at the elbow. An anatomically-based finite element model of the arm was built to accurately predict voltages resulting from surface electrical stimulation. In addition, two axon models were developed based on previously published models to incorporate physiological differences between sensory and motor axons. This resulted in axon models that could reproduce experimental results for conduction velocity, strength-duration curves and activation threshold. Differences in experimentally obtained action potential shape between the motor and sensory axons were reflected in the models. The models predicted a lower threshold for sensory axons than motor axons of the same diameter, allowing a range of sensory axons to be activated before any motor axons. This system of models will be a useful tool for development of surface electrical stimulation as a method to target specific neural functions.     

Action Potential Initiation in the Hodgkin-Huxley Model

A recent paper of B. Naundorf et al. described an intriguing negative correlation between variability of the onset potential at which an action potential occurs (the onset span) and the rapidity of action potential initiation (the onset rapidity). This correlation was demonstrated in numerical simulations of the Hodgkin-Huxley model. Due to this antagonism, it is argued that Hodgkin-Huxley-type models are unable to explain action potential initiation observed in cortical neurons in vivo or in vitro. Here we apply a method from theoretical physics to derive an analytical characterization of this problem. We analytically compute the probability distribution of onset potentials and analytically derive the inverse relationship between onset span and onset rapidity. We find that the relationship between onset span and onset rapidity depends on the level of synaptic background activity. Hence we are able to elucidate the regions of parameter space for which the Hodgkin-Huxley model is able to accurately describe the behavior of this system.     

The frequency response function and sinusoidal threshold properties of the Hodgkin-Huxley model of action potential encoding

The behaviour of the space-clamped Hodgkin-Huxley model has been studied using bandlimited white noise (0–50 Hz) as the input membrane current and taking the output as a point process in time given by the peaks of the action potentials. The frequency response and coherence functions were measured by use of the Fourier transform and digital filtering of the spike train. The results obtained are in good agreement with those already published for the simple integrator and leaky integrator models of neuronal encoding, as well as the earlier studies on the response of the Hodgkin-Huxley model to steady currents. In addition, the threshold of the model to sinusoidal membrane currents has been measured as a function of frequency over the range of 0.1–100 Hz. This shows a relatively constant level up to 2 Hz and then a clear minimum at 60 Hz, in agreement with measured thresholds of squid axons. These results are discussed in terms of the possible contributions of action potential encoding mechanisms to the frequency responses and sinusoidal thresholds which have been measured for rapidly adapting receptors.     

Analysis of Travelling Waves Associated with the Modelling of Aerosolised Skin Grafts

A previous model developed by the authors investigates the growth patterns of keratinocyte cell colonies after they have been applied to a burn site using a spray technique. In this paper, we investigate a simplified one-dimensional version of the model. This model yields travelling wave solutions and we analyse the behaviour of the travelling waves. Approximations for the rate of healing and maximum values for both the active healing and the healed cell densities are obtained. PACS 92B05     

Temporally coherent organization and instabilities in squid giant axons

Instabilities and dynamic structure of the modified Hodgkin-Huxley equations (Adelman & FitzHugh, 1975) for sensitized axons were studied as a function of the sodium concentration in the external medium surrounding the axon. At the same time electrophysiological activities in squid giant axons were experimentally observed to confirm the results of the numerical calculation. It was found that the resting state of the axon was thermodynamically equivalent to a thermodynamic structure of an asymptotically stable equilibrium point. The state of spontaneous repetitive firing of action potentials corresponds to the dissipative structure with a stable limit cycle. The temporally coherent organization is realized through instability of the equilibrium point.     

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.     

A fully coupled transient excited state model for the sodium channel

Summary The axon membrane is simulated by standard Hodgkin-Huxley leakage and potassium channels plus a coupled transient excited state kinetic scheme for the sodium channel. This scheme for the sodium channel is as proposed previously by the author. Simultations are presented showing the form of the action potential, threshold behavior, accommodation, and repetitive firing. It is seen that the form of the individual action potential, its all-or-none nature, and its refractory period are well simulated by this model, as they are by the standard Hodgkin-Huxley model. However, the model differs markedly from the Hodgkin-Huxley model with respect to repetitive firing and accommodation to stimulating currents of slowly rising intensity, in ways that are anomn to be related to those features of the sodium inactivation which are anomalous to the H-H model. The tendency for repetitive firing is highly dependent on that parameter which primarily determintes the existence of the inactivation shift in voltage clamp experiments, in such a way that the more pronounced the inactivation shift, the less the tendency for repetitive firing,. The tendency for accommodation is highly dependent on that parameter which primarily determines the “τ_c − τ_h” separation, in such a way that the greater the separation the greater the tendency for the membrane to accommodate without firing action potentials to a slowly rising current.     

Global bifurcations in the FitzHugh-Nagumo equations for nerve wave propagation

The FitzHugh-Nagumo equations for action potential propagation along nerve axons and the corresponding ordinary differential equations for travelling waves are solved numerically. Above a critical value, a constant bias current can drive a wave-front solution. At the critical value, a global bifurcation occurs. As a result, the wave front switches into a pulse.Based on a thesis by one of the authors (H. F.).     

A study of the effects of externally applied sodium-ions and detection of spatial non-uniformity of the squid axon membrane under internal perfusion

Electrical properties of the axon membrane were examined under internal perfusion of squid giant axons with a dilute solution of NaF or CsF. The rate of propagation of the action potential was markedly enhanced when NaCl was added to the external CaCl₂ solution. The membrane conductance both at rest and during the action potential was increased with increasing Na-concentration in the external medium. In the perfusion zone of these axons, the action potentials in different parts of the membrane were found to terminate in a more-or-less spatially random and temporally irregular fashion. When the electric field outside the axon membrane was examined with hyperfine glass-pipette electrodes, small rectangular potential changes of uniform amplitude were observed. The small potential changes, which resemble those obtained by Bean in EIM-treated lipid bilayer, were interpreted as indicating spatial non-uniformity of the axon membrane during excitation. The importance of long-range electric interaction between different parts of the axon membrane is emphasized.     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.