Edge of Chaos Theory Sheds Light Into the All-to-None Phenomenon in Neurons—Part I: On the Fundamental Role of the Sodium Ion Channel

The Edge of Chaos Principle lies at the origin of emergent phenomena in physical systems. Recurring to powerful concepts from the theory, establishing its rules, it is possible to identify regions of the parameter space of a system, endowing the latter with a high degree of excitability, which can then manifest itself vividly, as complexity emerges across the respective physical medium upon apparently-harmless changes to the environmental conditions. In this manuscript, the first of a two-paper contribution, the Edge of Chaos Principle is invoked to explain the mechanisms, underlying the development of recently-reported yet-unexplained oscillations across a biological cell, consisting of a voltage-driven sodium ion channel, including leakage effects. Our in-depth investigation of the Local Activity and Edge of Chaos of the biological cell, based upon the rigorous mathematical description, first proposed by Hodgkin and Huxley in 1952, identifies the subcritical Hopf, which emerges across its physical medium as it enters one of its two Edge of Chaos domains, giving birth to the aforementioned oscillations, whose unstable nature is thus revealed. The existence of an unstable limit-cycle attractor in the state space of the Hodgkin-Huxley neuron model is crucially important for the All-to-None dynamical phenomenon, which distinctively characterises the evolution of the membrane capacitance voltage under DC synaptic current sweep. Thus the discovery of the capability of the sodium ion channel to generate unstable oscillations on its own, highlights the fundamental role of this biological memristor in the mechanisms behind emergence and extinction of an Action Potential across a neuronal axon. As the cell, studied in this manuscript, is unable to sustain stable oscillations, the second companion paper shall demonstrate how the insertion of a membrane capacitance across the sodium ion channel, reduced to a first-order system by neglecting the dynamics of the fast activation gate variable, is necessary and sufficient to endow the original biological cell with the capability to undergo also a supercritical Hopf bifurcation, besides the subcritical one, which allows the observation of the entire life cycle of a neuronal spike under DC synaptic current sweep, including, especially, the All-to-None phenomenon spawned out of a fold or saddle-node limit cycle bifurcation.