Memristor neurons and their coupling networks based on Edge of Chaos Kernel

Chua's theory of local activity shows that local activity is the origin of complexity, and the complexity can only occur on or near the stable locally active domain, referred to as Edge of Chaos (EOC). Very recently, for the voltage-controlled locally active memristors, a new physical concept dubbed “Edge of Chaos Kernel” (EOCK), which consists of a series combination between a negative resistance and a negative inductance and exhibits the EOC phenomenon of being stable yet potentially unstable, was defined and applied to the Hodgkin-Huxley neural circuit model. When an EOCK is coupled to a passive environment, its stability is disrupted, resulting in the emergence of action potentials, chaos and various complex phenomena. This paper proposes the dual version of the EOCK called as R-C EOCK, which consists of the parallel combination between a negative resistance and a negative capacitance. We show that the actual NbO memristor manufactured by NaMLab essentially belongs to a current-controlled locally active memristor which contains a R-C EOCK and gives the signature of its EOCK and EOC. We construct a second-order neuron based on the NbO memristor when connected in parallel with a passive capacitor, and further prove that only memristors endowed with an EOCK can generate action potential. On this basis, we construct a minimum cellular neural network with only 7 components based on two NbO memristor neurons and a passive coupling resistor, in which neuromorphic behaviors of static and dynamic pattern formation may emerge if and only if the single neuron has an EOCK and is poised on the EOC. The analysis in this paper explains the dynamic mechanism of Smale's paradox, in which two mathematically dead neurons coupled by a passive environment may become alive, under the same or different input current excitation, which are more in line with the actual biological neural networks.