In recent years, discrete neuron and discrete neural network models have played an important role in the development of neural dynamics. This paper reviews the theoretical advantages of well-known discrete neuron models, some existing discretized continuous neuron models, and discrete neural networks in simulating complex neural dynamics. It places particular emphasis on the importance of memristors in the composition of neural networks, especially their unique memory and nonlinear characteristics. The integration of memristors into discrete neural networks, including Hopfield networks and their fractional-order variants, cellular neural networks and discrete neuron models has enabled the study and construction of various neural models with memory. These models exhibit complex dynamic behaviors, including superchaotic attractors, hidden attractors, multistability, and synchronization transitions. Furthermore, the present paper undertakes an analysis of more complex dynamical properties, including synchronization, speckle patterns, and chimera states in discrete coupled neural networks. This research provides new theoretical foundations and potential applications in the fields of brain-inspired computing, artificial intelligence, image encryption, and biological modeling.

Edge deployment solutions based on convolutional neural networks (CNNs) have garnered significant attention because of their potential applications. However, traditional CNNs rely on pooling to reduce the feature size, leading to substantial information loss and reduced network robustness. Herein, we propose a more robust adaptive pooling network (APN) method implemented using memristor technology. Our method introduces an improved pooling layer that reduces input features to an arbitrary scale without compromising their importance. Different coupling coefficients of the pooling layer are stored as conductance values in arrays. We validate the proposed APN on generic datasets, demonstrating significant performance improvements over previously reported CNN architectures. Additionally, we evaluate the APN on a CAPTCHA recognition task with perturbations to assess network robustness. The results show that the APN achieves 92.6% accuracy in 4-digit CAPTCHA recognition and exhibits higher robustness. This brief presents a highly robust and novel scheme for edge computing using memristor technology.

Neural synchronization is associated with various brain disorders, making it essential to investigate the intrinsic factors that influence the synchronization of coupled neural networks. In this paper, we propose a minimal architecture as a prototype, consisting of two bi-neuron Hopfield neural networks (HNNs) coupled via a memristor. This coupling elevates the original two bi-neuron HNNs into a five-dimensional system, featuring an unstable line equilibrium set and rich dynamics absent in the uncoupled case. Our results show that varying the coupling strength and the initial state of the memristor can induce periodic, chaotic, hyperchaotic, and quasi-periodic oscillations, as well as initial-offset-regulated multistability. We derive sufficient conditions for achieving exponential synchronization and identify multiple synchronous regimes with transitions that strongly depend on the initial states. Field-programmable gate array (FPGA) implementation confirms the predicted dynamics and synchronization in real time, demonstrating that the memristive coupler enables complex dynamics and controllable synchronization in the most compact Hopfield architecture, with implications for the study of neuromorphic circuits and synchronization.

Considering the impact of electromagnetic induction on neurons, this paper presents a three-dimensional (3D) memristor Hindmarsh-Rose (HR) neuron model. This model exhibits diverse hidden chaotic dynamics due to the absence of equilibrium points, including bifurcation phenomena, coexisting attractors, transient chaos, state transitions, and offset-boosting control. Since equilibrium points are absent in this model, all observed dynamics are classified as hidden behaviors. The complex dynamics of this neuron model are illustrated through bifurcation diagrams, Lyapunov diagrams, time series plots, and phase portraits. Furthermore, an equivalent circuit for the memristor HR neuron is constructed, and the accuracy of numerical simulations is confirmed via circuit simulation results.

The functionality of the biological brain is closely related to the dynamic behavior generated by synapses in its complex neural system. The self-connection synapse, as a critical form of feedback synapse in Hopfield neurons, plays an essential role in understanding the dynamic behavior of the brain. Synaptic memristors can bring neural network models closer to the complexity of the brain’s neural networks. Inspired by this, this study incorporates the nonlinear memory characteristics of synapses into the Hopfield neural network (HNN) by replacing a single self-synapse in a four-dimensional HNN model with a novel cosine memristor model, aiming to more realistically reproduce the dynamical behavior of biological neurons in artificial systems. By performing a dynamical analysis of the system using numerical methods, we find that the model exhibits infinitely many equilibrium points and can induce the formation of rare transient attractors, as well as an arbitrary number of multi-scroll attractors. Additionally, the model demonstrates complex coexisting attractor dynamics, including transient chaos, periodicity, decaying periodicity, and coexisting chaos. Furthermore, the feasibility of the proposed HNN model is verified using a field-programmable gate array (FPGA). Finally, an electronic codebook (ECB)-mode block cipher encryption algorithm is proposed for image encryption. The encryption performance is evaluated, with an information entropy value of 7.9993, demonstrating the excellent randomness of the system-generated numbers.

Conventional open-loop deep brain stimulation (DBS) systems with fixed parameters fail to accommodate inter-individual pathological differences in Parkinson’s disease (PD) management while potentially inducing adverse effects and causing excessive energy consumption. In this paper, we present an adaptive closed-loop framework integrating a Yogi-optimized proportional-integral-derivative neural network (Yogi-PIDNN) controller. The Yogi-augmented gradient adaptation mechanism accelerates the convergence of general PIDNN controllers in high-dimensional nonlinear control systems while reducing control energy usage. In addition, a system identification method establishes input-output dynamics for pre-training stimulation waveforms, bypassing real-time parameter-tuning constraints and thereby enhancing closed-loop adaptability. Finally, a theoretical analysis based on Lyapunov stability criteria establishes a sufficient condition for closed-loop stability within the identified model. Computational validations demonstrate that our approach restores thalamic relay reliability while reducing energy consumption by (81.0 ±0.7)% across multi-frequency tests. This study advances adaptive neuromodulation by synergizing data-driven pre-training with stability-guaranteed real-time control, offering a novel framework for energy-efficient and personalized Parkinson’s therapy.

This study investigates the bifurcation dynamics underlying rhythmic transitions in a biophysical hippocampal-cortical neural network model. We specifically focus on the membrane potential dynamics of excitatory neurons in the hippocampal CA3 region and examine how strong coupling parameters modulate memory consolidation processes. Employing bifurcation analysis, we systematically characterize the model’s complex dynamical behaviors. Subsequently, a characteristic waveform recognition algorithm enables precise feature extraction and automated detection of hippocampal sharp-wave ripples (SWRs). Our results demonstrate that neuronal rhythms exhibit a propensity for abrupt transitions near bifurcation points, facilitating the emergence of SWRs. Critically, temporal rhythmic analysis reveals that the occurrence of a bifurcation is not always sufficient for SWR formation. By integrating one-parameter bifurcation analysis with extremum analysis, we demonstrate that large-amplitude membrane potential oscillations near bifurcation points are highly conducive to SWR generation. This research elucidates the mechanistic link between changes in neuronal self-connection parameters and the evolution of rhythmic characteristics, providing deeper insights into the role of dynamical behavior in memory consolidation.

The nonlinear memory characteristics of memristors resemble those of biological synapses and ion channels. Therefore, memristors serve as ideal components for constructing artificial neurons. This paper presents a sound-sensitive neuron circuit featuring a memristor-based hybrid ion channel, designed to simulate the dynamic response mechanisms of biological auditory neurons to acoustic signals. In this neural circuit, a piezoelectric ceramic element captures external sound signals, while the hybrid ion channel is formed by connecting a charge-controlled memristor in series with an inductor. The circuit realizes selective encoding of sound frequency and amplitude and investigates the influence of external electric fields on neuronal ion-channel dynamics. In the dynamic analysis, bifurcation diagrams and Lyapunov exponents are employed to reveal the rich nonlinear behaviors, such as chaotic oscillations and periodic oscillations, exhibited by the circuit during the acoustic-electric conversion process, and the validity of the circuit model is experimentally verified. Simulation results show that by adjusting the threshold of the ratio between electric-field energy and magnetic-field energy, the firing modes and parameters of neurons can be adaptively regulated. Moreover, the model exhibits stochastic resonance in noisy environments. This research provides a theoretical foundation for the development of new bionic auditory sensing hardware and opens a new path for the bio-inspired design of memristor-ion-channel hybrid sy

We propose a simplified version of the classic two-dimensional Hindmarsh-Rose neuron (2DHR), resulting in a new 2DHR that exhibits novel chaotic phenomena. Its dynamic characteristics are analyzed through bifurcation diagrams, Lyapunov exponent spectra, equilibrium points, and phase diagrams. Based on this system, a corresponding circuit is designed and circuit simulations are carried out, yielding results consistent with the numerical simulations. To explore practical applications of chaotic systems, 2DHR is employed to improve the solution of the flexible job-shop scheduling problem with dynamic events. The research results demonstrate that applying 2DHR can significantly enhance the convergence rate of the optimization algorithm and improve the quality of the scheduling solution.

Spiral waves, as a typical self-organized structure with chiral characteristics, are widely found in excitable media such as cardiac tissues, chemical reactions, and neural networks. Based on the FitzHugh-Nagumo model, we investigated the mechanisms underlying the effects of direct current electric fields (DCEF), alternating current electric fields (ACEF), and polarized electric fields (PEF) on the interaction and annihilation processes of counter-rotating spiral waves. We found that in a direct current electric field, the drift direction of the spiral wave is determined jointly by its chirality and the electric field direction, which allows selective attraction or repulsion. In an alternating current electric field, the annihilation behavior of spiral waves can be influenced by the phase and intensity of the electric field, where a specific range of parameters induces resonance drift and eventual annihilation. On the other hand, the polarized electric field exhibits a more complex modulation capability on spiral waves: the trajectory and annihilation efficiency of spiral waves can be regulated by both the intensity and phase of the polarized electric field. These results reveal the potential feasibility of regulating multichiral spiral waves through multiple electric fields, providing theoretical insight for the control of spiral waves in relevant systems.