Subthreshold Behavior and Phenomenological Impedance of the Squid Giant Axon

Abstract

The oscillatory behavior of the cephalopod giant axons in response to an applied current has been established by previous investigators. In the study reported here the relationship between the familiar "RC" electrotonic response and the oscillatory behavior is examined experimentally and shown to be dependent on the membrane potential. Computations based on the threecurrent system which was inferred from electrical measurements by Hodgkin and Huxley yield subthreshold responses in good agreement with experimental data. The point which is developed explicitly is that since the three currents, in general, have nonzero resting values and two currents, the "Na" system and the "K" system, are controlled by voltage-dependent time-variant conductances, the subthreshold behavior of the squid axon in the small-signal range can be looked upon as arising from phenomenological inductance or capacitance. The total phenomenological impedance as a function of membrane potential is derived by linearizing the empirically fitted equations which describe the time-variant conductances. At the resting potential the impedance consists of three structures in parallel, namely, two series RL elements and one series RC element. The true membrane capacitance acts in parallel with the phenomenological elements, to give a total impedance which is, in effect, a parallel R, L, C system with a "natural frequency" of oscillation. At relatively hyperpolarized levels the impedance "degenerates" to an RC system. The subthreshold behavior of the axonal membrane, exemplified by the classical "electrotonus" of the myelinated nerve (Hermann, 1905; Hodgkin and Rushton, 1946;Davis and Lorente de N6, 1947) and the "local response" and oscillatory behavior of the giant axon in the cephalopods, Loligo and Sepia (Hodgkin, 1938;Arvanitaki, 1939 Arvanitaki, , 1941Brink et al., 1946), have been studied by many investigators. In recent years, perhaps due to the advent of the microelectrode technique with its emphasis on large amplitude recordings, electrophysiologists have shown only moderate interest in the subthreshold be-497

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“…As already pointed out in the BIntroduction,^linear-ized equations of conductance-based models are known to provide a good approximation of the neural response in the subthreshold regime (Mauro et al 1970). However, as the transmembrane potential approaches threshold, nonlinearities become more prominent, thereby questioning the validity of the linearity assumption.…”

“…We want to test this hypothesis by comparing the threshold predictions of the Hodgkin and Huxley (HH) model of the giant squid axon to its linearized version. The HH model is chosen because its linear behavior is well known (Mauro et al 1970;Koch 1984) and because it shows a subthreshold resonance. The predictions for the HH model are made using the softcell package software (Weiss 2000).…”

“…Fig. 19b in Mauro et al 1970). For the linear model, threshold is assumed to be inversely proportional to the maximal amplitude of the model_s response to a unitary input.…”

“…The method of conductance-based model linearization is well known and has been described in details in several studies (Mauro et al 1970;Brunel et al 2003). Consider a single compartment neuron with n voltage-gated ion channels (Fig.…”

“…It may also, in return, provide a basis to develop more realistic conductance-based models of the human AN. Linearization of conductance-based models is valid in the subthreshold domain, where the variations of the transmembrane potential remain small (Mauro et al 1970). Different subthreshold behaviors can be obtained depending on the characteristics of the neuron.…”