The electric activities of neurons are dependent on the complex electrophysiological condition in neuronal system, and it indicates that the complex distribution of electromagnetic field could be detected in the neuronal system. According to the Maxwell electromagnetic induction theorem, the dynamical behavior in electric activity in each neuron can be changed due to the effect of internal bioelectricity of nervous system (e.g., fluctuation of ion concentration inside and outside of cell). As a result, internal fluctuation of electromagnetic field is established and the effect of magnetic flux across the membrane should be considered during the emergence of collective electrical activities and signals propagation among a large set of neurons. In this paper, the variable for magnetic flow is proposed to improve the previous Hindmarsh–Rose neuron model; thus, a four-variable neuron model is designed to describe the effect of electromagnetic induction on neuronal activities. Within the new neuron model, the effect of magnetic flow on membrane potential is described by imposing additive memristive current on the membrane variable, and the memristive current is dependent on the variation of magnetic flow. The dynamics of this modified model is discussed, and multiple modes of electric activities can be observed by changing the initial state, which indicates memory effect of neuronal system. Furthermore, a practical circuit is designed for this improved neuron model, and this model could be suitable for further investigation of electromagnetic radiation on biological neuronal system.

Model of electrical activity in a neuron under magnetic flow effect

By utilizing a memristor to substitute a coupling resistor in the realization circuit of a three-dimensional chaotic system having one saddle and two stable node-foci, a novel memristive hyperchaotic system with coexisting infinitely many hidden attractors is presented. The memristive system does not display any equilibrium, but can exhibit hyperchaotic, chaotic, and periodic dynamics as well as transient hyperchaos. In particular, the phenomenon of extreme multistability with hidden oscillation is revealed and the coexistence of infinitely many hidden attractors is observed. The results illustrate that the long term dynamical behavior closely depends on the memristor initial condition thus leading to the emergence of hidden extreme multistability in the memristive hyperchaotic system. Additionally, hardware experiments and PSIM circuit simulations are performed to verify numerical simulations.

Hidden extreme multistability in memristive hyperchaotic system

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.

Multiple attractors can be found in many nonlinear dynamical system with multistability. Recently, experimental attractors with two stable saddle-foci were reported to find in a non-ideal active voltage-controlled memristor based Chua's circuit. In this paper we focus on the multiple attractors found in the proposed memristive Chua's circuit. Concretely, by numerical simulations of mathematical model, hardware circuit experiments and PSIM circuit simulations, multiple attractors with different initial states are revealed and with the dimensionless system equations, complex dynamics with different initial conditions are further discussed. Theoretical derivation results indicate that the normalized memristive Chua's system has two stable nonzero saddle-foci in globally adjusting normalized parameter region and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.

Multiple attractors in a non-ideal active voltage-controlled memristor based Chua's circuit

Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria

Bursting, an important communication activity in biological neurons and endocrine cells, has been widely found in fast-slow dynamical systems In this paper, a modified second-order generalized memristor, memristive diode bridge cascaded with LC network, is presented and its fingerprints of the pinched hysteresis loops are analyzed By replacing the parallel resistor with the modified generalized memristor, a novel memristive Wien-bridge oscillator is constructed and its mathematical model is established, from which the dynamical behaviors of symmetric chaotic and periodic bursting oscillations are observed and the corresponding bifurcation mechanisms are explained Based on a hardware realization circuit, experimental observations are performed, which verify the numerical simulations

Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator

When a sinusoidal voltage stimulus is applied, memristive Chua’s circuit becomes a non-autonomous periodically forced nonlinear circuit. By utilizing theoretical formulations, simulations and experimental verifications, the complex transient dynamics of the periodically forced memristive Chua’s circuit is investigated in this paper. It can be found that the equilibrium point of the circuit switches between a line equilibrium and no equilibrium with the time evolutions, and the circuit exhibits period, chaos and also hyperchaos in a parameter range of the stimulus frequency. Moreover, some abundant interesting nonlinear phenomena including transient chaos, transient hyperchaos and chaotic beats are revealed numerically and verified experimentally.

Complex transient dynamics in periodically forced memristive Chua’s circuit

Hidden attractors were reported to find in a normalised Chua's equation. To find the hidden attractors in a practical Chua's circuit, a modified Chua's diode is presented and its equivalent realisation circuit is designed. On the basis of proposed modified Chua's diode, a new Chua's circuit is rebuilt and hidden attractors under special initial conditions are captured by numerical simulations and circuit simulations.

Pan Jiang

Papers

Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator

Complex transient dynamics in periodically forced memristive Chua’s circuit

Hidden attractors in a practical Chua's circuit based on a modified Chua's diode