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

Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction-diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction-diffusion system is Baer and Rinzel's continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon "complete accommodation."

Breakdown of accommodation in nerve: a possible role for persistent sodium current

Background

Accommodation and breakdown of accommodation are important elements of information processing in nerve fibers, as they determine how nerve fibers react to natural slowly changing stimuli or electrical stimulation. The aim of the present study was to elucidate the biophysical mechanism of breakdown of accommodation, which at present is unknown.

Results

A model of a space-clamped motor nerve fiber was developed. It was found that this new model could reproduce breakdown of accommodation when it included a low-threshold, rapidly activating, persistent sodium current. However, the phenomenon was not reproduced when the persistent sodium current did not have fast activation kinetics or a low activation threshold.

Conclusion

The present modeling study suggests that persistent, low-threshold, rapidly activating sodium currents have a key role in breakdown of accommodation, and that breakdown of accommodation can be used as a tool for studying persistent sodium current under normal and pathological conditions.

Analysis of the Optimal Channel Density of the Squid Giant Axon Using a Reparameterized Hodgkin–Huxley Model

A reparameterized Hodgkin–Huxley-type model is developed that improves the 1952 model's fit to the biological action potential. In addition to altering Na+inactivation and K+activation kinetics, a voltage-dependent gating-current mechanism has been added to the model. The resulting improved model fits the experimental trace nearly exactly over the rising phase, and it has a propagation velocity that is within 3% of the experimentally measured value of 21.2 m/s (at 18.5°C). Having eliminated most inaccuracies associated with the velocity-dependent rising phase of the action potential, the model is used to test Hodgkin's maximum velocity hypothesis, which asserts that channel density has evolved to maximize conduction velocity. In fact the predicted optimal channel density is more than twice as high as the actual squid channel density. When the available capacitance is reduced to approximate more modern serial Na+-channel models, the optimal channel density is 4 times the actual value. We suggest that, although Hodgkin's maximum velocity hypothesis is acceptable as a first approximation, the microscopic optimization perspective of natural selection will not explain the channel density of the squid unless other constraints are taken into account, for example, the metabolic costs of velocity.

Threshold for repetitive activity for a slow stimulus ramp: a memory effect and its dependence on fluctuations

We have obtained new insights into the behavior of a class of excitable systems when a stimulus, or parameter, is slowly tuned through a threshold value. Such systems do not accommodate no matter how slowly a stimulus ramp is applied, and the stimulus value at onset of repetitive activity shows a curious, nonmonotonic dependence on ramp speed. (Jakobsson, E. and R. Guttman. Biophys. J. 1980. 31:293-298.) demonstrated this for squid axon and for the Hodgkin-Huxley (HH) model. Furthermore, they showed theoretically that for moderately slow ramps the threshold increases as the ramp speed decreases, but for much slower ramp speeds threshold decreases as the ramp speed decreases. This latter feature was found surprising and it was suggested that the HH model, and squid axon in low calcium, exhibits reverse accommodation. We have found that reverse accommodation reflects the influence of persistent random fluctuations, and is a feature of all such excitable systems. We have derived an analytic condition which yields an approximation for threshold in the case of a slow ramp when the effect of fluctuations are negligible. This condition predicts, and numerical calculations confirm, that the onset of oscillations occurs beyond the critical stimulus value which is predicted by treating the stimulus intensity as a static parameter, i.e., the dynamic aspect of a ramp leads to a delay in the onset. The condition further demonstrates a memory effect, i.e., firing threshold is dependent on the initial state of the system. For very slow ramps then, fluctuations diminish both the delay and memory effects. We characterize the class of excitable systems for which these behaviors are expected, and we illustrate the phenomena for the HH model and for a model of cAMP-receptor dynamics in Dictyostelium discoideum.