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Neurons can exhibit abundant electrical activities due to physical effects of various electrophysiology environments. The electromagnetic induction flows can be triggered by changes in neuron membrane potential, which can be equivalent to a memristor applying on membrane potential. To imitate the electromagnetic induction effects, we propose a three-variable memristor-based Wilson neuron model. Using several kinetic analysis methods, the memristor parameter- and initial condition-related electrical activities are explored intensively. It is revealed that the memristive Wilson neuron model can display rich electrical activities, including the asymmetric coexisting electrical activities and antimonotonicity phenomenon. Finally, using off-the-shelf discrete components, an analog circuit on a hardware level is implemented to verify the numerically simulated coexisting electrical activities. Studying these rich electrical activities in neurons can build the groundwork to widen the neuron-based engineering applications.

Investigating the chaotic dynamics in neural networks holds significant importance in elucidating brain-like neural activities and guiding brain-like learning. The multi-scroll chaos, due to its intricate topological structure, has garnered interest in the study of brain-like chaotic neural networks. Previous researches have primarily focused on ordinary multi-scroll attractors, while there has been little research on symmetric multi-scroll attractors. Symmetric attractors are typically more diverse and have more flexible evolutionary and higher stability which may lead to more stable system responses. The purpose of this paper is to investigate the symmetric multi-scroll phenomenon generated under the influence of the memristor controlled by multi-level-logic pulse in Hopfield Neural Network (HNN). Firstly, a memristive HNN capable of generating multi-scroll is proposed, serving as the foundation for studying the influence of multi-level-logic pulse. Through theoretical and numerical analysis, the dynamic behavior of the proposed memristive HNN is examined and simulation results reveal the emergence of multi-scroll attractors and initial offset coexisting behavior. Subsequently, a multi-level-logic pulse is introduced into the memristor to simulate one of its parameters. The experimental results reveal that the introduction of multi-level-logic pulse expands the original multi-scroll structure into a symmetric structure. Furthermore, it enlarges the chaotic parameter range of the system, which holds significant implications for the study of neural dynamics. Finally, the correctness of the proposed model is verified through hardware experiments. This study provides valuable guidance for neural dynamics researches and the application of memristors.

Nowadays, it is important to realize systems that can model the electrical activity of neurons taking into account almost all the properties of the intracellular and extracellular environment in which they are located. It is in this sense that we propose in this paper, the improved model of Hindmarsh–Rose (HR) which takes into account the fluctuation of the membrane potential created by the variation of the ion concentration in the cell. Considering the effect of the electric field that is produced on the dynamic behavior of neurons, the essential properties of the model such as equilibrium point and its stability, bifurcation diagrams, Lyapunov spectrum, frequency spectra, time series of the membrane potential and phase portraits are thoroughly investigated. We thus prove that Hopf bifurcation occurs in this system when the parameters are chosen appropriately. We also observe that by varying specific parameters of the electric field, the model presents a very rich and striking event, namely hysteresis phenomenon, which justifies the coexistence of multiple attractors. Besides, by applying a suitable sinusoidal excitation current, we prove that the neuron under electric field effect can present several important electrical activities including quiescent, spiking, bursting and even chaos. We propose the improved HR model under electric field effect (mHR) to study the finite-time synchronization between two neurons when performing synapse coupling across the membrane potential and the electric field coupling. As a result, we find that the synchronization between the two neurons is weakly influenced by the variation of the intensity of the electric field coupling while it is strongly impacted when the intensity of the synapse coupling is modified. From these results, it is obvious that the electric field can be another effective bridge connection to encourage the exchange and coding of the signal. Using the finite-time synchronization algorithm, we theoretically quantify the synchronization time between these neurons. Finally, Pspice simulations are presented to show the feasibility of the proposed model as well as that of the developed synchronization strategy.

The human brain is an extremely sophisticated system. Several neural models have been proposed to mimic and understand brain function. Most of them incorporate memristors to simulate autapse or self-coupling, electromagnetic radiation and the synaptic weight of the neuron. This article introduces and studies the dynamics of a Hopfield neural network (HNN) consisting of four neurons, where one of the synaptic weights of the neuron is replaced by a memristor. Theoretical aspects such as dissipation, the requirements for the existence of attractors, symmetry, equilibrium states and stability are studied. Numerical investigations of the model reveal that it develops very rich and diverse behaviors such as chaos, hyperchaos and transient chaos. These results obtained numerically are further supported by the results obtained from an electronic circuit of the system, constructed and simulated in PSpice. Both approaches show good agreement. In light of the findings from the numerical and experimental studies, it appears that the 4D Hopfield neural network under consideration in this work is more complex than its original version, which did not include a memristor.

In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one’s amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate electronic simulator is proposed for the investigations of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results yields a very good agreement.

This paper addresses the problem of optimization of the synchronization of a chaotic modified Rayleigh system. We first introduce a four-dimensional autonomous chaotic system which is obtained by the modification of a two-dimensional Rayleigh system. Some basic dynamical properties and behaviors of this system are investigated. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the proposed system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. Furthermore, we propose an optimal robust adaptive feedback which accomplishes the synchronization of two modified Rayleigh systems using the controllability functions method. The advantage of the proposed scheme is that it takes into account the energy wasted by feedback coupling and the closed loop performance on synchronization. Also, a finite horizon is explicitly computed such that the chaos synchronization is achieved at an established time. Numerical simulations are presented to verify the effectiveness of the proposed synchronization strategy. Pspice analog circuit implementation of the complete master–slave controller system is also presented to show the feasibility of the proposed scheme.

This paper deals with the analog circuit implementation and synchronization of a model consisting of a van der Pol oscillator coupled to a Duffing oscillator. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of the systems solutions (i.e., elastic coupling). The primary motivation of our investigations lays in the fact that coupled attractors of different types might serve as a good model for real systems in nature (e.g., electromechanical, physical, biological, or economic systems). The stability of fixed points is examined. The bifurcation structures of the system are analyzed with particular emphasis on the effects of nonlinearity. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results shows a very good agreement. By exploiting recent results on adaptive control theory, a controller is designed that enables both synchronization of two unidirectionally coupled systems and the estimation of unknown parameters of the drive system.

This chapter contains sections titled: Introduction The Signal Processing Aim Representative Testing Performance Metrics Statistical Validation Conclusions References

In this paper, a new four-dimensional dissipative chaotic system which can produce multiple asymmetric attractors is designed and its dynamical behaviors are analyzed. The basin of attraction reveals the asymmetric multistability of the system. In addition, it is very interesting to observe different types of asymmetric coexisting attractors as the bifurcation parameters change. The spectral entropy complexity chaotic diagrams are used to observe the changes in the sequence complexity when the two bifurcation parameters change simultaneously. Moreover, the difference of the system complexity between the two different initial values is analyzed. In order to facilitate engineering applications, the offset boosting control is introduced to the state variable, and the numerical simulation shows that the offset boosting control scheme can flexibly change the polarity of the chaotic signal. Finally, an analog circuit and a digital circuit were designed to verify the new chaotic system. The new research results will enrich the theoretical basis of multistability, offset boosting control and circuit implementation of chaos.

It is believed that local activation is the origin of all complexities, and the locally active memristive synaptic neural network can generate complex chaotic dynamic behaviors, such as hyperchaotic, multi-scroll, multi-stability and hidden dynamical behaviors. However, there are few studies on the simultaneous occurrence of multiple complex dynamic behaviors in neural networks. No chaotic system of multi-scroll hyperchaotic hidden attractors based on neural network has been found yet. To solve the problem, in this paper, we propose a new locally active memristive Hopfield neural network (HNN) model based on a multi-segment function, which is affected by electromagnetic radiation and external current. The multi-scroll hyperchaotic hidden attractors are found in the memristive HNN for the first time. Theoretical analysis and numerical simulation results show that the memristive HNN model has no equilibrium point, and the number of multi-scroll attractors is controlled by the state equation parameters of the memristive synapse. In addition, the structures and number of scrolls are also affected by electromagnetic radiation and external current. At the same time, under the appropriate parameter conditions, by modifying the initial value of the system, the memristive HNN has a controllable number of coexisting attractors, showing extreme multi-stability. Finally, a memristive HNN analog circuit is designed. The hardware experiment results reproduce the multi-scroll dynamics phenomenon, which verifies the correctness of the theoretical analysis and numerical simulation.