This tutorial clarifies the axiomatic definition of (v(α); i(β)) circuit elements via a lookup table dubbed an A-pad, of admissible (v; i) signals measured via Gedanken probing circuits. The (v(α); i(β)) elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element, characterized by a state-dependent Ohm's law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency ω and tends to a straight line as ω ~ ∞, for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used hence-forth as a moniker for memristive devices.

The Fourth Element.

From a pedagogical point of view, the memristor is defined in this tutorial as any 2-terminal device obeying a state-dependent Ohm’s law. This tutorial also shows that from an experimental point of view, the memristor can be defined as any 2-terminal device that exhibits the fingerprints of “pinched” hysteresis loops in the v–i plane. It also shows that memristors endowed with a continuum of equilibrium states can be used as non-volatile analog memories. This tutorial shows that memristors span a much broader vista of complex phenomena and potential applications in many fields, including neurobiology. In particular, this tutorial presents toy memristors that can mimic the classic habituation and LTP learning phenomena. It also shows that sodium and potassium ion-channel memristors are the key to generating the action potential in the Hodgkin-Huxley equations, and that they are the key to resolving several unresolved anomalies associated with the Hodgkin-Huxley equations. This tutorial ends with an amazing new result derived from the new principle of local activity, which uncovers a minuscule life-enabling Goldilocks zone, dubbed the edge of chaos, where complex phenomena, including creativity and intelligence, may emerge. From an information processing perspective, this tutorial shows that synapses are locally-passive memristors, and that neurons are made of locally-active memristors.

Memristor, Hodgkin-Huxley, and Edge of Chaos.

A new 3-D chaotic dynamical system with a peanut-shaped closed curve of equilibrium points is introduced in this work. Since the new chaotic system has infinite number of rest points, the new chaotic model exhibits hidden attractors. A detailed dynamic analysis of the new chaotic model using bifurcation diagrams and entropy analysis is described. The new nonlinear plant shows multi-stability and coexisting convergent attractors. A circuit model using MultiSim of the new 3-D chaotic model is designed for engineering applications. The new multi-stable chaotic system is simulated on a field-programmable gate array (FPGA) by applying two numerical methods, showing results in good agreement with numerical simulations. Consequently, we utilize the properties of our chaotic system in designing a new cipher colour image mechanism. Experimental results demonstrate the efficiency of the presented encryption mechanism, whose outcomes suggest promising applications for our chaotic system in various cryptographic applications.

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A 3-D Multi-Stable System With a Peanut-Shaped Equilibrium Curve: Circuit Design, FPGA Realization, and an Application to Image Encryption

The study of dynamics on artificial neurons and neuronal networks is of great significance to understand brain functions and develop neuromorphic systems. Recently, memristive neuron and neural network models offer great potential in the investigation of neurodynamics. Many chaotic dynamics including chaos, transient chaos, hyperchaos, coexisting attractors, multistability, and extreme multistability have been researched based on the memristive neurons and neural networks. In this review, we firstly introduce the basic definition of chaotic dynamics and review several traditional artificial neuron and neural network models. Then we categorize memristive neuron and neural network models with different biological function mechanisms into five types: memristive autapse neuron, memristive synapse-coupled bi-neuron network, memristive synaptic weight neural network, neuron under electromagnetic radiation, and neural network under electromagnetic radiation. The modeling mechanisms of each type are explained and described in detail. Furthermore, the pioneer works and some recent important papers related to those types are introduced. Finally, some open problems in this field are presented to further explore future work.

Review on chaotic dynamics of memristive neuron and neural network

This exposition shows that the potassium ion-channels and the sodium ion-channels that are distributed over the entire length of the axons of our neurons are in fact locally-active memristors. In particular, they exhibit all of the fingerprints of memristors, including the characteristic pinched hysteresis Lissajous figures in the voltage-current plane, whose loop areas shrink as the frequency of the periodic excitation signal increases. Moreover, the pinched hysteresis loops for the potassium ion-channel memristor, and the sodium ion-channel memristor, from the Hodgkin-Huxley axon circuit model are unique for each periodic excitation signal. An in-depth circuit-theoretic analysis and characterizations of these two classic biological memristors are presented via their small-signal memristive equivalent circuits, their frequency response, and their Nyquist plots. Just as the Hodgkin-Huxley circuit model has stood the test of time, its constituent potassium ion-channel and sodium ion-channel memristors are destined to be classic examples of locally-active memristors in future textbooks on circuit theory and bio-physics.

Brains Are Made of Memristors.

A bistable nonvolatile locally-active memristor and its complex dynamics

S-type locally-active memristor (LAM) has a great potential for brain- inspired neuromorphic computing, where the S-type LAM-based oscillator is a fundamental building block. Concerning the S-type LAM, this paper constructs a material-independent model in simple mathematical expression, which can be relatively easily analyzed. By biasing the memristor into the locally- active region, and connecting it with a capacitor, a second-order oscillator can be built. The small-signal equivalent circuit of the memristor and its frequency response are applied to determine the period oscillation frequency range and compensation capacitance. Hopf bifurcation theory is used to analyze oscillation mechanism of the second-order circuit and appropriate capacitance. By adding an extra inductor into the second-order oscillator, a novel third-order chaotic circuit is developed, where a saddle-focus is derived to create chaos. Its dynamic characteristics are investigated via Lyapunov exponents, bifurcation diagrams, dynamic route map, and so on. The local activities of the single memristor, second-order oscillator, and third-order chaotic circuit are verified through the mathematical analysis. Finally, physical circuit realizations of the S- type LAM-based oscillators, including the memristor emulator, are presented. Both simulation and experimental results demonstrate the practicability of the proposed mathematical model and the validity of the theoretical analysis.

S-Type Locally Active Memristor-Based Periodic and Chaotic Oscillators

A number of important applications would benefit from the introduction of locally-active memristors, which is defined to be any memristor that exhibits negative differential memristance for at leas...

https://www.worldscientific.com/doi/pdf/10.1142/S0218127419300301

Switching Characteristics of a Locally-Active Memristor with Binary Memories

This paper designs a simple mathematical model for the voltage-controlled locally-active memristor (LAM), which undoubtedly simplifies the theoretical analysis. Owing to the N-shaped outline of the DC V -I curve, this LAM is called as N-shaped LAM. The small-signal analysis method is exploited to obtain the equivalent circuit of the N-shaped LAM when it is biased into the locally-active region. The calculation results show that the LAM can be equivalent to a conductance with a parasitic capacitor, which makes for a deep insight into the mechanisms at the origin of LAM based oscillator. The theoretical analysis reveals that the N-shaped LAM along with an inductor and a battery can generate oscillation, which is confirmed by the numerical simulation. In addition, observe from simulation results that as the inductance increases, the oscillation behavior of the system deviates from the sinusoidal signal and tends to a relaxation oscillation. Finally, the physical circuit realization of the N-shaped LAM based oscillator including the memristor emulator is proposed. The good agreement between the experimental results and simulation results further demonstrates the correctness and feasibility of the theoretical design and analysis.

/pdf/modeling-simplification-and-dynamic-behavior-of-n-shaped-hvg1nj91m4.pdf

Modeling Simplification and Dynamic Behavior of N-Shaped Locally-Active Memristor Based Oscillator

This paper presents a chaotic circuit based on a nonvolatile locally active memristor model, with non-volatility and local activity verified by the power-off plot and the DC V-I plot, respectively. It is shown that the memristor-based circuit has no equilibrium with appropriate parameter values and can exhibit three hidden coexisting heterogeneous attractors including point attractors, periodic attractors, and chaotic attractors. As is well known, for a hidden attractor, its attraction basin does not intersect with any small neighborhood of any unstable equilibrium. However, it is found that some attractors of this circuit can be excited from an unstable equilibrium in the locally active region of the memristor, meaning that its basin of attraction intersects with neighborhoods of an unstable equilibrium of the locally active memristor. Furthermore, with another set of parameter values, the circuit possesses three equilibria and can generate self-excited chaotic attractors. Theoretical and simulated analyses both demonstrate that the local activity and an unstable equilibrium of the memristor are two reasons for generating hidden attractors by the circuit. This chaotic circuit is implemented in a digital signal processing circuit experiment to verify the theoretical analysis and numerical simulations.

Yujiao Dong

Papers

A bistable nonvolatile locally-active memristor and its complex dynamics

S-Type Locally Active Memristor-Based Periodic and Chaotic Oscillators

Switching Characteristics of a Locally-Active Memristor with Binary Memories

Modeling Simplification and Dynamic Behavior of N-Shaped Locally-Active Memristor Based Oscillator

Coexisting hidden and self-excited attractors in a locally active memristor-based circuit