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This paper deduces that the potassium and calcium ion currents in the two-dimensional (2D) Morris–Lecar neuron model can be respectively characterized by a passive memristor and a locally active memristor. Then a memristive Morris–Lecar neuron model with an equivalent circuit scheme is first constructed. The equilibrium trajectory and its stability are analyzed, which displays fold and Hopf bifurcations with proper model parameters. Bursting behavior and mixed-mode oscillations with after-depolarizing potential for low-frequency current stimulus are numerically disclosed. Besides, the bifurcation mechanisms for bursting behavior and mixed-mode oscillations are theoretically deduced. Furthermore, the firing patterns and antimonotonicity phenomenon for high-frequency current stimulus are numerically discovered. This study provides a new concept for building memristive bionic circuits from neuron models with well-defined ion channel currents. • A memristive Morris–Lecar neuron model with an equivalent circuit scheme is newly constructed. • Dynamical effects of current stimuli with high-frequency and low-frequency are explored. • Bifurcation mechanisms periodic bursting behavior and mixed-mode oscillations are deduced. [ABSTRACT FROM AUTHOR]

The diversity of firing patterns and their bifurcation mechanisms of a neuron circuit are crucial for exploiting bionic applications. This paper constructs an N-type locally active memristor (LAM)-based FitzHugh–Nagumo (FHN) circuit by replacing the tunnel-diode in the original FHN circuit. Numerical simulations and hardware measurements reveal that the N-type LAM-based FHN circuit can reproduce rich neuromorphic firing patterns of quasi-periodic/periodic bursting behaviors and chaotic/periodic spiking behaviors. The bursting and spiking behaviors are triggered by a low-frequency stimulus and a high-frequency one, respectively. Besides, the fold and Hopf bifurcation sets are depicted. Then, the bifurcation mechanisms for the quasi-periodic and periodic bursting behaviors via Hopf/Hopf and Hopf/fold bifurcations are theoretically deduced by time-domain waveform and equilibrium trajectory. The numerical results and hardware experiments exhibit the effectiveness of the proposed memristive FHN circuit in reproducing abundant firing patterns of bursting and spiking behaviors. • An N-type locally active memristor-based FHN circuit is first built by replace the tunnel-diode. • Fast–slow dynamics versus to high-frequency and low-frequency stimuli are explored. • Bifurcation mechanisms for quasi-periodic and periodic bursting behaviors are deduced. • Hardware experiments are executed to validate the generation of spiking and bursting behaviors. [ABSTRACT FROM AUTHOR]

The diversity of spiking activity of a bionic circuit is a vital footstone in developing spike-based applications. The bionic circuit constructed by membrane theory frequently employs an invariable capacitor to characterize the electrophysiological behaviors of the neuron membrane. Actually, the thickness and medium property of a neuron membrane are regulated by its membrane potential, which leads to the invariable capacitor suffering from inaccuracy in expressing the regulating process. To solve this issue, a memcapacitive emulator with controllable capacitance is deployed to characterize the neuron membrane in this paper. Then, a memcapacitive and memristive emulator-based (MC-MR-emulator-based) bionic circuit is first built, which involves only a memcapacitive emulator, a locally active memristive emulator, a DC voltage source, and an externally applied current stimulus. Numerical explorations display that the MC-MR-emulator-based bionic circuit can generate rich bifurcation behaviors, e.g., period-doubling bifurcation, tangent bifurcation, and crisis scenario, related to the current stimulus, memristive emulator parameters, and memcapacitive emulator parameters. These bifurcation behaviors lead to that the MC-MR-emulator-based bionic circuit can produce abundant periodic and chaotic spiking activities. In analog experiments, memcapacitor and memristor emulators are utilized. PCB-based hardware experimental results well address the validity of the numerical explorations and further exhibit the effectiveness of the MC-MR-emulator-based bionic circuit in generating spiking activities. [ABSTRACT FROM AUTHOR]

The famous Hodgkin-Huxley circuit contains two time-varying resistors to describe the electrophysiological characteristics of sodium and potassium ion channels. But the time-varying resistors are expressed by mixed exponential equations, which are unavailable to directly implement in analog circuit and hinders hardware application of the Hodgkin-Huxley circuit. To hit this issue, a simplified memristive Hodgkin-Huxley (m-HH) circuit is proposed, which only involves two locally active memristors (LAMs) to depict the sodium and potassium ion channels, a capacitor to describe the neuron membrane, a DC current to represent external stimulus, and two DC voltages to express the reversal potentials. MATLAB-based numerical simulations and analog circuit-based hardware measurements display that the simplified m-HH circuit can generate memristor parameter- and DC current-related periodic spiking behaviors with different periodicities and chaotic spiking behavior. This delights that electrophysiological characteristics of ion channels and external stimulus can be employed for regulating these periodic and chaotic spiking behaviors. It is interesting that the simplified m-HH circuit can generate frequency self-adaptation and coexisting firing patterns. The inter-spike interval bifurcation diagram shows that the frequency of the periodic spiking behavior increases with the increase of externally applied DC current. Besides, the hybrid parameter bifurcation diagram displays that the simplified m-HH circuit can generate memristor initial state-related coexisting firing patterns of period-6 with chaos, period-3 with chaos, and period-3 with period-8. These offer us the unique insight into explaining some biological firing patterns. The most significance is that the analog hardware circuit is feasible in reproducing these periodic and chaotic spiking behaviors and benefits for developing spiking-based neuromorphic hardware. • A simplified Hodgkin-Huxley circuit with memristive sodium and potassium ion channels is built. • Periodic/chaotic spiking behaviors, frequency self-adaptation, and coexisting firing patterns are numerically disclosed. • PCB-based analog circuit is manually made and hardware experiments are performed to verify the numerical results. [ABSTRACT FROM AUTHOR]

As a connecting synapse, memristor is considered as a feasible device for representing the magnetic induction between two coupled neurons to enable the information exchange. However, the effect of memristor as a coupling synapse between two map-based neurons on synchronous activity has not received sufficient attention. To this end, an extensible locally active memristor is proposed, and its various characteristics are demonstrated by several existing methods. Based on the proposed memristor and two map-based neurons, a memristor synapse-coupled neuron model is constructed to investigate the memristor-induced synchronous firing activities. Taking the coupling strength and memristor synapse initial state as controllable variables, several analysis methods, such as phase diagram, normalized mean synchronization error, similarity function, and phase difference, are used to clearly reveal various synchronization phenomena triggered by the introduced memristor, including complete synchronization, lag synchronization, and phase synchronization, as well as the accompanying spiking/bursting firing behaviors. In particular, the generation and transition of synchronization depend on the initial states of the memristor synapse-coupled neuron model, resulting in the coexistence of firing modes. The correctness of these synchronous firing activities is validated by digital hardware experiments. Numerical and hardware results show that the memristor, as a coupling synapse, provides a flexible and reliable control scheme for the synchronization between neurons. • An extensible locally active memristor is presented and its characteristics are demonstrated. • A memristor synapse-coupled neuron model is constructed based on two map-based neurons. • The memristor-induced synchronous firing activities are revealed by several analysis methods. • Digital hardware experiments validate the correctness of these synchronous firing activities. [ABSTRACT FROM AUTHOR]

Memristor, especially the locally active one, is widely used in emulating the synapse-inspired activities in neural networks, in which the connection between memristor properties and neuronal electrical activities requires further investigation. In this paper, a bi-neuron model is constructed by bi-directionally connecting a locally active memristor between the membrane potentials of two deterministic 3D HR neuron models. Complex spiking/bursting firing activities and their transitions are disclosed under different memristor control parameters. The bifurcation mechanisms of the revealed synchronous and asynchronous firing activities are explored inside each individual neuron by taking the externally input state variables as modulation parameters. These results demonstrate the crucial effect of the locally active memristor synapse on firing activities of its coupled network. FPGA-based digital circuit experiment is finally performed to verify the numerical results. This research is beneficial to understanding, control and application of firing activities in neural networks. • An LSCN model is proposed by two 3D HR models via LAM memristor coupling. • The connection between memristor features and neural electrical activities is investigated. • Bifurcation mechanism of neural electrical activities is interpreted. [ABSTRACT FROM AUTHOR]

Firing activities provide the potential possibility for achieving bio-brain functionality with high energy-efficient and high-speed information processing performance. This inspires the design of bionic circuits to generate firing activities and develop brain-like applications. In this paper, a dual mem-elements based cellular neural network (CNN) cell is constructed to produce bionic firing activities, in which a non-ideal memcapacitor and an N-type locally active memristor are employed to emulate the functions of the neuronal membrane. The proposed CNN cell has an excitation-dependent equilibrium trajectory and stability. Numerical analysis shows that the dual mem-elements based CNN cell has abundant dynamical behaviors of forward/reverse period-doubling bifurcation routes, chaos crisis, tangent bifurcation, and bubbles with the change of model parameters of the CNN cell, memcapacitor, and exciting source. As a result, the rich firing patterns' transition can be observed from the two-dimensional dynamics evolution. The analog circuit of the proposed CNN cell is designed, and then a PCB-based hardware circuit is implemented. The experimental results certify the accuracy of the theoretical and numerical analysis. • A dual mem-elements based CNN cell is newly proposed by employing memristor and memcapacitor. • Bionic firing activities of the periodic/chaotic spiking behaviors are numerically disclosed. • Mem-element emulators based CNN circuit is manually made and hardware experiments are executed. [ABSTRACT FROM AUTHOR]

To characterize neuronal firing activities and elucidate its bifurcation mechanisms, a memristor-based FitzHugh-Nagumo (FHN) circuit is designed based on the FHN circuit architecture combined with a first-order memristive simulator, and its normalized system with periodic and quasi-periodic bursting oscillations is established. With the change of externally applied excitation, the memristor-based FHN system has the time-varying equilibrium point where the number, position and stability evolve slowly over time. In an evolution period of the time-domain waveform, different fold and/or Hopf bifurcations are triggered, resulting in periodic or quasi-periodic bursting oscillations. To explain the intrinsic bifurcation mechanisms, the fold and Hopf bifurcation sets are built and the transitions between the resting and spiking states are demonstrated, thus identifying the Hopf/fold and Hopf/Hopf bursting oscillations. Finally, based on the circuit simulation model, analog circuit simulations and hardware circuit measurements are developed for the memristor-based FHN circuit to confirm MATLAB numerical simulations. In addition, it is worth noting that the proposed circuit is a simple non-autonomous memristive neuron circuit that is particularly easy to physically implement. • A memristor-based FitzHugh-Nagumo circuit with periodic/quasi-periodic bursting oscillations is designed. • The time-varying equilibrium points with stability evolutions for some parameter settings are analyzed. • Bifurcations to periodic and quasi-periodic bursting oscillations are theoretically demonstrated. • Analog circuit simulations and hardware circuit measurements are developed for the proposed circuit. [ABSTRACT FROM AUTHOR]

In this paper, a general configuration for nonlinear circuit employing current-controlled nonlinearity is proposed, which has a simple topology and only contains three linear energy storage components, one nonlinear element, and one multi-port network that can be made by linear resistor. Under this configuration, we design two new five-element chaotic circuits including the first third-order one employing a current-controlled anti-parallel connection diode pair and the second fourth-order one employing a current-controlled memristor emulator. The analyses show that the third-order chaotic circuit is dual to the canonical Chua's circuit, and the extended fourth-order one can be regarded as a dual canonical memristive Chua's circuit. The basic analyses including the model description, equilibrium point and stability are presented. Several rich dynamical properties of the two chaotic circuits are investigated using numerical simulations and certified by both Pspice circuit simulations and analog circuit measurements. • A general configuration for nonlinear circuits employing current-controlled nonlinearity is proposed. • Two novel chaotic circuits with current-controlled Chua's diode and memristor emulator are presented. • The presented configuration provides a guidance for the extension of dual Chua's circuit family. [ABSTRACT FROM AUTHOR]

Electromagnetic induction plays a vital role in impacting the neuron dynamics, since the electromagnetic induction is triggered alongside with the complex interaction of membrane potential and various ions transport. In other words, the electromagnetic induction is reversely subjected to the membrane potential fluctuation and ions distribution. Flux-controlled memristor described by flux and voltage has been employed to restrict the electromagnetic induction effect. This paper presents a flux-controlled memristor with sinusoidal mem-conductance function and hyperbolic tangent function modulated input. The memristor can availably reflect the non-uniform ions distribution inside and outside neuron membrane. Then, an improved memristive FitzHugh-Nagumo (mFHN) neuron model is built to explore the dynamical effect of the memristive electromagnetic induction. Numerical simulations and theoretical analysis are performed, which reveal that the mFHN neuron model possesses no equilibrium point and can generate abundant hidden dynamics. Interestingly, hidden coexisting behavior is triggered by memristor initial-offset boosting, which evokes the emergence of infinitely hidden coexisting firing patterns. Besides, bifurcation mechanism of the memristor initial-offset boosting behavior is theoretically analyzed. Furthermore, PowerSIM-based (PSIM-based) analog circuit simulations and microcontroller unit based (MCU-based) digital experiments are executed, the captured results perfectly verify the correctness of numerical results and the feasibility of analog/digital hardware experiments. • An improved mFHN neuron model having no equilibrium point is built. • Hidden dynamics and memristor initial-offset boosting behavior with its bifurcation mechanism is explored. • PSIM-based analog circuit simulations and MCU-based digital experiments are executed. [ABSTRACT FROM AUTHOR]

Recently, discrete memristor maps can be directly constructed using discrete memristors. However, some discrete memristors with reciprocal polynomial memristances cannot be directly used to generate mapping models. To achieve an available memristive map, a simple and effective implementation scheme is proposed to construct a two-dimensional (2-D) sine-bounded memristive map (SBMM). The proposed map has line invariant point and its stability is extremely related to parameters and memristor initial state. The parameter-dependent complex dynamics, coexisting attractors with riddled attraction basin, and initial state-induced dynamics effect are disclosed using several numerical measures. SBMM has complex dynamics distributions with initial state effects and can generate chaotic and hyperchaotic attractors with riddled attraction basins. These chaotic/hyperchaotic sequences have excellent performance and can be applied to generate pseudorandom binary number generators (PBNGs). Besides, FPGA hardware implementation is performed, and the numerical results are verified by experiments. • A simple implementation scheme is proposed to construct a 2-D sine-bounded memristive map. • Complex dynamics, coexisting attractors, and initial state-induced dynamics effect are disclosed. • Pseudorandom binary number generator application and FPGA implementation are performed. [ABSTRACT FROM AUTHOR]

Eleven years have passed since the discovery of the first chaotic hidden attractor in Chua system with piecewise-linear nonlinearity. Experimental observation of such attractor is still a hard work as the attraction basin of such hidden attractor does not intersect with any system equilibrium point and is far away from the origin. The key technical difficulty is the accurate configuration of non-zero initial values for dynamic elements. In this Letter, offset control is applied to shift the attraction basin and mapping the non-zero initial values to zero ones. Consequently, the attraction basin is intersect with origin, which allows us to experimentally observe such hidden attractor via pre-discharge all dynamic elements. • Graphical structure of attraction basin of hidden Chua's attractor is investigated. • Offset control is applied for effective observation of hidden attractor in experiment. • The observating strategy of hidden oscillations can be extended to high-dimensional nonlinear systems. [ABSTRACT FROM AUTHOR]