Locally active memristor (LAM), which has an ability to amplify fluctuations, is a natural component for constructing artificial neuron circuits. This paper proposes a novel third-order neuron circuit by paralleling two LAMs and a capacitor. Firstly, two parallel LAMs are equivalently modeled as a second-order LAM to facilitate theoretical analysis. Regarding the third-order neuron, the parameter design and operating condition are obtained by calculating its small signal impedance functions poles or Jacobin matrixs eigenvalues. It is demonstrated that the neuron exhibits various neuromorphic behaviors, including periodic spiking, chaos and burst-number adaptation. Due to the two different LAMs, the proposed thirdorder system has multiple equilibrium points, leading to the generation of coexisting attractors. Interestingly, the generated chaotic attractor does not revolve around a single unstable equilibrium point, but is located between two unstable equilibrium points. Furthermore, the emergence of oscillating behaviors is dependent on the distance between the two unstable equilibrium points. Detailed theoretical and simulation analysis are presented to investigate the neuron dynamics and provide an explanation for the observed neuromorphic behaviors. Finally, physical circuit implementation of the neuron is constructed based on the memristor emulator, which also demonstrates the practicability of the proposed neuron model and the correctness of the theoretical analysis.

The symmetry-breaking phenomenon, appearing, under suitable conditions, when identical reaction cells, quiet on their own, are let interact via diffusion processes, is dubbedTuring Instability. Its local form exposes the local destabilization, which allows two multistable cells lose stability at one of its locallyasymptotically-stableoperating points. While the globalTuring Instabilityand its mechanisms have been recently explained in (Ascoli et al., 2022), its local form and an experimental demonstration of these complex effects on a physical memristive medium have not been reported yet. This paper investigates a local form ofTuring Instabilityin a two-cell array, when one of the possiblelocally asymptotically-stableandlocally-activestatic solutions loses stability, when let interact with an identical reaction cell via diffusion processes, resulting in the emergence of two different static solutions after transients fade away. In order to study its mechanisms, this paper first introduces a current-controlled Chua Corsage Memristor (CCM), and demonstrates the operating point destabilization in a single current-controlled CCM-based cell. Adding a dissipative resistor and a capacitor to the current-controlled CCM, preliminarily poised on anedge of chaosoperating point, gives birth to two unstable circuits, inducing a local quiescent bi-stability and a local oscillation, respectively. The mechanisms behind a local form ofTuring Instability, appearing in a current-controlled CCM-based two-cell array, have been elucidated, and the bifurcation, spawning symmetry-breaking effects, locally, across the cellular network, has been identified. Both numerical and experimental results confirm the correctness of the theoretical analysis.

Inspired by recent research reported in [1], this paper investigates the bio-inspired bifurcation patterns of a simple memristive neuron on edge of chaos. The adopted memristive neuron, comprising a DC current source, a current-controlled locally active memristor, and a capacitor, successfully reproduces the bifurcation cascade patterns observed in the Hodgkin-Huxley (H-H) neuron model, including fold limit cycle bifurcation (FLCB), subcritical Hopf bifurcation (SUB-HB), and supercritical Hopf bifurcation (SUP-HB). Through attraction basin analysis and pulse-based initial state regulation, we verify the coexistence phenomenon of stable and unstable limit cycles induced by FLCB, as well as the bistable behaviors triggered by SUB-HB. Furthermore, taking resistively coupled memristive neurons as an example, we explore the influence of the dynamics of individual neurons on the bifurcation patterns of coupled networks, where two neurons have identical parameters but different initial states. The results demonstrate that the three bifurcation modes also emerge in memristive coupled networks, and their evolutionary patterns are closely related to the dynamic behaviors of individual neurons. Finally, hardware experiments successfully reproduce the bifurcation cascade phenomenon thereby validating the correctness of theoretical analysis and simulation results.