Subthreshold Oscillatory Responses of the Hodgkin-Huxley Cable Model for the Squid Giant Axon

Abstract

The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.

The h Channel Mediates Location Dependence and Plasticity of Intrinsic Phase Response in Rat Hippocampal Neurons

“…Phenomenological inductances (alternately called resonating conductances) are contributed by conductances that actively oppose changes in membrane voltage (Cole, 1968;Sabah and Leibovic, 1969;Hutcheon and Yarom, 2000). The functional expression of a phenomenological inductance is not only dependent on the kinetics and voltage dependence of the mediating ion channel, but also on the passive properties of the membrane they reside in (Table 1).…”

“…It was later demonstrated that these inductive reactances as well as their capacitive counterparts emerge because of the presence of voltage-dependent, time-variant conductances (Mauro, 1961;Cole, 1968;Sabah and Leibovic, 1969;Mauro et al, 1970), and such reactances have been variously named as "anomalous" (Cole, 1949) or "phenomenological" reactances (Mauro et al, 1970;Koch, 1984). Importantly, whereas excitatory conductance changes contribute to a capacitive reactance, conductance changes that aid in membrane recovery yield an inductive reactance (Cole, 1968;Sabah and Leibovic, 1969).…”

The role of negative conductances in neuronal subthreshold properties and synaptic integration

“…This approach relies on linearizing voltage-dependent cable conductances around the membrane potential V R . The method was first employed by Sabah and Leibovic (1969) and further worked on by Koch (1984), and it elucidates certain general principles of quasi-active cables. This approach allowed classifying I NaP as a regenerative current, which produces a positive feedback and amplifies membrane potential changes, boosting and broadening excitatory postsynaptic potentials (EPSPs).…”

Untitled

“…The response of the system for small signal stimulus is calculable, and compares well with experiment [9,10].…”

“…Time, milliseconds m = ma{v)] h = h"(v = 0), n = na{v = 0), (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and the resulting wave form for v and m is a step instead of a pulse, as shown in Fig. 1.4.…”