From squid to mammals with the HH model through the Nav channels' half-activation-voltage parameter

Abstract

The model family analyzed in this work stems from the classical Hodgkin-Huxley model (HHM). for a single-compartment (space-clamp) and continuous variation of the voltage-gated sodium channels (Nav) half-activation-voltage parameter ΔV1/2, which controls the window of sodium-influx currents. Unlike the baseline HHM, its parametric extension exhibits a richer multitude of dynamic regimes, such as multiple fixed points (FP's), bi- and multistability (coexistence of FP's and/or periodic orbits). Such diversity correlates with a number of functional properties of excitable neural tissue, such as the capacity or not to evoke an action potential (AP) from the resting state, by applying a minimal absolute rheobase current amplitude. The utility of the HHM rooted in the giant squid for the descriptions of the mammalian nervous system is of topical interest. We conclude that the model's fundamental principles are still valid (up to using appropriate parameter values) for warmer-blooded species, without a pressing need for a substantial revision of the mathematical formulation. We demonstrate clearly that the continuous variation of the ΔV1/2 parameter comes close to being equivalent with recent HHM 'optimizations'. The neural dynamics phenomena described here are nontrivial. The model family analyzed in this work contains the classical HHM as a special case. The validity and applicability of the HHM to mammalian neurons can be achieved by picking the appropriate ΔV1/2 parameter in a significantly broad range of values. For such large variations, in contrast to the classical HHM, the h and n gates' dynamics may be uncoupled - i.e. the n gates may no longer be considered in mere linear correspondence to the h gates. ΔV1/2 variation leads to a multitude of dynamic regimes - e.g. models with either 1 fixed point (FP) or with 3 FP's. These may also coexist with stable and/or unstable periodic orbits. Hence, depending on the initial conditions, the system may behave as either purely excitable or as an oscillator. ΔV1/2 variation leads to significant changes in the metabolic efficiency of an action potential (AP). Lower ΔV1/2 values yield a larger range of AP response frequencies, and hence provide for more flexible neural coding. Such lower values also contribute to faster AP conduction velocities along neural fibers of otherwise comparable-diameter. The 3 FP case brings about an absolute rheobase current. In comparison in the classical HHM the rheobase current is only relative - i.e. excitability is lost after a finite amount of elapsed stimulation time. Lower ΔV1/2 values translate in lower threshold currents from the resting state.