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.     

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.     

Noise effects on spike propagation in the stochastic Hodgkin-Huxley models

Effects of membrane current noise on spike propagation along a nerve fiber are studied. Additive current noise and channel noise are considered by using stochastic versions of the Hodgkin-Huxley model. The results of computer simulation show that the membrane noise causes considerable variation of the propagation time of a spike (thus changes in interspike intervals) for a small unmyelinated fiber of radius 0.1 approximately 1 micron.     

Modeling the interactions between stimulation and physiologically induced APs in a mammalian nerve fiber: dependence on frequency and fiber diameter

Electrical stimulation of nerve fibers is used as a therapeutic tool to treat neurophysiological disorders. Despite efforts to model the effects of stimulation, its underlying mechanisms remain unclear. Current mechanistic models quantify the effects that the electrical field produces near the fiber but do not capture interactions between action potentials (APs) initiated by stimulus and APs initiated by underlying physiological activity. In this study, we aim to quantify the effects of stimulation frequency and fiber diameter on AP interactions involving collisions and loss of excitability. We constructed a mechanistic model of a myelinated nerve fiber receiving two inputs: the underlying physiological activity at the terminal end of the fiber, and an external stimulus applied to the middle of the fiber. We define conduction reliability as the percentage of physiological APs that make it to the somatic end of the nerve fiber. At low input frequencies, conduction reliability is greater than 95% and decreases with increasing frequency due to an increase in AP interactions. Conduction reliability is less sensitive to fiber diameter and only decreases slightly with increasing fiber diameter. Finally, both the number and type of AP interactions significantly vary with both input frequencies and fiber diameter. Modeling the interactions between APs initiated by stimulus and APs initiated by underlying physiological activity in a nerve fiber opens opportunities towards understanding mechanisms of electrical stimulation therapies.     

On reinforcement and interference between stimuli

It is shown that the current “two-factor” theory of nerve excitation can account for sustained inhibition or enhancement by a sequence of stimulus pulses, and for the decrease in the reinforcement period with each successive pulse of the train.     

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.     

Effect of stimulus (postsynaptic current) shape on fibre excitation.

Effects of variation of the stimulus pulse shape on the excitation of a nonmyelinated nerve fibre were studied using a mathematical model based on the Hodgkin-Huxley equations. Efficiency of smoothly changing pulses was compared with that of rectangular pulses. For pulses shorter than the time to excitation, the rate of the stimulus rise did not determine the ability of a smoothly changing pulse to excite the fibre. For a given stimulus duration, the main factor was the pulse area or the charge delivered by the pulse. The strength-duration curve for smoothly changing pulses was a nonmonotonic function, in contrast to the curve for rectangular pulses. The dependence of latency on changes in the pulse area was non-linear. It would be nonmonotonic when the pulse area variation were due to the stimulus duration or the stimulus rise duration. More that one propagating intracellular action potential (IAP) could arise upon fibre activation by a long smoothly changing threshold stimulus. Upon activation of relatively short fibres the IAP could arise not at the site of the smoothly changing stimulus injection. The rectangular pulses of long duration were more efficient than the corresponding smoothly changing ones. Irrespective of the shape, the pulses whose duration at the foot is 1-2 ms, are more suitable for a prolonged threshold fibre activation.     

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.     

Poisson process stimulation of an excitable membrane cable model.

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

Effect of Low Sodium, Tetrodotoxin, and Temperature Variation upon Excitation

The lowering of external sodium raised both the constant quantity threshold, Q_o, and the rheobase, I_o, in both real space-clamped squid axons and the theoretical axon as computed on the basis of the standard Hodgkin-Huxley equations. In both real and theoretical axons the minimum intensity for excitability for short pulses, which occurs at about 15°C, was still present when low sodium replaced seawater. Low sodium did not affect the temperature dependence of the strength-duration relationship in the range, 5° to 25°C. The excitability of tetrodotoxin-treated real axons was found to be more temperature-dependent than that of normal real axons. Also the data on dosage-response to TTX of real axons fit the dose-response relationship of a hypothetical system in which one TTX ion binds reversibly to its receptor to produce a fraction of the inhibitory effect, the curve being identical to a simple adsorption isotherm. The Hodgkin-Huxley equations describe the broad outline of events occurring during excitation quite well.     

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye.     

Response characteristics of neurons in the medial geniculate body of the little brown bat to simple and temporally-patterned sounds

We examined the auditory response properties of neurons in the medial geniculate body of unanesthetized little brown bats (Myotis lucifugus). The units' selectivities to stimulus frequency, amplitude and duration were not significantly different from those of neurons in the inferior colliculus (Condon et al. 1994), which provides the primary excitatory input to the medial geniculate body, or in the auditory cortex (Condon et al. 1997) which receives primary input from the medial geniculate body. However, in response to trains of unmodulated tone pulses, the upper cutoff frequency for time-locked discharges (64 ± 46.9 pulses per second or pps) and the mean number of spikes per pulse (19.2 ± 12.2 pps), were intermediate to those for the inferior colliculus and auditory cortex. Further, in response to amplitude-modulated pulse trains, medial geniculate body units displayed a degree of response facilitation that was intermediate to that of the inferior colliculus and auditory cortex inferior colliculus: 1.32 ± 0.33; medial geniculate body: 1.75 ± 0.26; auditory cortex: 2.52 ± 0.96, P < 0.01). These data suggest that the representation of isolated tone pulses is not significantly altered along the colliculo-thalamo-cortical axis, but that the fidelity of representation of temporally patterned signals progressively degrades along this axis. The degradation in response fidelity allows the system to better extract the salient feature in complex amplitude-modulated signals. Accepted: 9 January 1999     

Inhibition in branched afferent neurons of the bullfrog tongue

Summary Single fibers of the bullfrog glossopharyngeal nerve give rise to several peripheral branches, each innervating separate fungiform papillae on the dorsal surface of the tongue. Extracellular electrodes were used to stimulate and record simultaneously from several papillae and from the central branch.Minor changes in centrally recorded neural output were caused by collision of action potentials originating in separate branches of a common fiber.Following an antidromic or orthodromic action potential in any branch, a series of excitability changes occured in that branch. Normal excitability was regained within 5 msec of an action potential, but was followed by a secondary decrease in excitability, which reached a minimum approximately 50 msec after the spike, and returned to normal within 200–400 msec after the spike. Subthreshold stimuli caused no depression, while doubling the stimulus strength above threshold did not enhance depression. After several spikes, both amplitude and duration of depression increased. Depression could be evoked even after the gustatory receptors were surgically removed.Post-stimulus depression in fiber branches is suggested as one source of gustatory adaptation, and may also contribute to interference between stimulating substances.The authors are particularly grateful for assistance and advice from Dr. Douglas Junge, of the School of Dentistry and Department of Physiology at the University of California, Los Angeles. The reported work was supported by NIDR Contract No. 69-2227 to Dr. Junge, and was carried out while one of the authors (JAM) held a PHS postdoctoral traineeship with the Department of Zoology, U.C.L.A., and the other (MSB) held a NIH predoctoral traineeship with the Department of Anatomy, U.C.L.A. Draughts of the paper have been read and criticized by Dr. Junge and Dr. J. P. Leader, of Auckland University.     

Using interspike intervals to quantify noise effects on spike trains in temperature encoding neurons

This paper examines how noise interacts with the non-linear dynamical mechanisms of neuronal stimulus. We study the spike trains generated by a minimal Hodgkin-Huxley type model of a cold receptor neuron. The distributions of interspike intervals(ISIs) of purely deterministic simulations exhibit considerable differences compared to the noisy ones. We quantify the effect of noise using ISI return plots and the ISI-distance recently proposed by Kreuz et al. (J Neurosci Meth, 165:151–161, 2007). It is shown that the spike trains of a cold receptor neuron are more strongly affected by noise for low temperatures than for high temperatures. This trend is also observed in both regimes of cold receptors: tonic firing(which occurs for low and high temperatures) and bursting (which occurs for intermediate temperatures).     

Dendritic sodium spikes endow neurons with inverse firing rate response to correlated synaptic activity

Many neurons possess dendrites enriched with sodium channels and are capable of generating action potentials. However, the role of dendritic sodium spikes remain unclear. Here, we study computational models of neurons to investigate the functional effects of dendritic spikes. In agreement with previous studies, we found that point neurons or neurons with passive dendrites increase their somatic firing rate in response to the correlation of synaptic bombardment for a wide range of input conditions, i.e. input firing rates, synaptic conductances, or refractory periods. However, neurons with active dendrites show the opposite behavior: for a wide range of conditions the firing rate decreases as a function of correlation. We found this property in three types of models of dendritic excitability: a Hodgkin-Huxley model of dendritic spikes, a model with integrate and fire dendrites, and a discrete-state dendritic model. We conclude that fast dendritic spikes confer much broader computational properties to neurons, sometimes opposite to that of point neurons.     

Statistical characteristics of climbing fiber spikes necessary for efficient cerebellar learning

 Mean firing rates (MFRs), with analogue values, have thus far been used as information carriers of neurons in most brain theories of learning. However, the neurons transmit the signal by spikes, which are discrete events. The climbing fibers (CFs), which are known to be essential for cerebellar motor learning, fire at the ultra-low firing rates (around 1 Hz), and it is not yet understood theoretically how high-frequency information can be conveyed and how learning of smooth and fast movements can be achieved. Here we address whether cerebellar learning can be achieved by CF spikes instead of conventional MFR in an eye movement task, such as the ocular following response (OFR), and an arm movement task. There are two major afferents into cerebellar Purkinje cells: parallel fiber (PF) and CF, and the synaptic weights between PFs and Purkinje cells have been shown to be modulated by the stimulation of both types of fiber. The modulation of the synaptic weights is regulated by the cerebellar synaptic plasticity. In this study we simulated cerebellar learning using CF signals as spikes instead of conventional MFR. To generate the spikes we used the following four spike generation models: (1) a Poisson model in which the spike interval probability follows a Poisson distribution, (2) a gamma model in which the spike interval probability follows the gamma distribution, (3) a max model in which a spike is generated when a synaptic input reaches maximum, and (4) a threshold model in which a spike is generated when the input crosses a certain small threshold. We found that, in an OFR task with a constant visual velocity, learning was successful with stochastic models, such as Poisson and gamma models, but not in the deterministic models, such as max and threshold models. In an OFR with a stepwise velocity change and an arm movement task, learning could be achieved only in the Poisson model. In addition, for efficient cerebellar learning, the distribution of CF spike-occurrence time after stimulus onset must capture at least the first, second and third moments of the temporal distribution of error signals. Received: 28 January 2000 / Accepted in revised form: 2 August 2000     

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.     

A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance that allows calculation of both the speed and shape of the traveling pulses, in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses that is valid with good accuracy in a wide range of the parameters.     

Manifestations of dynamic coding of the amplitude-modulated sounds on the level of auditory nerve fibres

Average firing rate of the auditory nerve fiber as function of the level of the tone with the frequency equal to characteristic frequency of the fibers, can be defined as an input-output characteristic. It is known that the steepening of the input-output characteristic of the real auditory nerve fiber is more, and the width is less than the spontaneous activity of the fiber. The latter characterizes fiber's ability to generate spikes, if the stimulus is absent. However it is known, that the real auditory nerve fibers with low spontaneous activity reproduce amplitude modulation of the signals much better, than the fibers with high spontaneous activity. From the results of simulation experiments, it follows that the dynamic properties of the auditory nerve fibers, providing fine tuning or adaptation of a fiber threshold under the stimulus level but not the static input-output characteristics, are the reason of fibers reproduction of stimuli amplitude modulations. However the auditory nerve fibers with high spontaneous activity due to abrupt input-output characteristic are capable to reproduce modulations of sounds whose levels are lower than a threshold of the fiber, if a weak signal adds to a weak broadband noise. This is a phenomenon of stochastic resonance found in the reactions of auditory nerve fibers.