This paper presents a rigorous and comprehensive nonlinear circuit-theoretic foundation for the memristive Hodgkin-Huxley Axon Circuit model. We show that the Hodgkin-Huxley Axon comprises a potassium ion-channel memristor and a sodium ion-channel memristor, along with some mundane circuit elements. From this new perspective, many hitherto unresolved anomalous phenomena and paradoxes reported in the literature are explained and clarified. The yet unknown nonlinear dynamical mechanisms which give birth to the action potentials remain hidden within the memristors, and the race is on for uncovering the ultimate truth.

By exploiting the new concepts of CA characteristic functions and their associated attractor time-τ maps, a complete characterization of the long-term time-asymptotic behaviors of all 256 one-dimensional CA rules are achieved via a single "probing" random input signal. In particular, the graphs of the time-1 maps of the 256 CA rules represent, in some sense, the generalized Green's functions for Cellular Automata. The asymptotic dynamical evolution on any CA attractor, or invariant orbit, of 206 (out of 256) CA rules can be predicted precisely, by inspection. In particular, a total of 112 CA rules are shown to obey a generalized Bernoulli στ-shift rule, which involves the shifting of any binary string on an attractor, or invariant orbit, either to the left, or to the right, by up to 3 pixels, and followed possibly by a complementation of the resulting bit string. The most intriguing result reported in this paper is the discovery that the four Turing-universal rules [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], and only these rules, exhibit a 1/f power spectrum.

This paper shows the action potential (spikes) generated from the Hodgkin–Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.