Understanding how the multi-layer structure of biological membranes influences neural electrical activity remains an important issue in computational neuroscience and neural circuit modeling. Some dual-membrane neuron models characterize the electrophysiological properties of the cell membrane, but the complete hierarchical structure of biological membranes is not addressed well. Based on the 'sandwich-like' membrane structure as the Davson-Danielli model, this paper constructs a new neuron model consisting of three-layers of membranes, which are described by three capacitors coupled via memristors. An additive capacitor is used to shunt current from the inductive channel and energy level is regulated. The tri-capacitors are connected by two different memristors, and the material property of inner media between capacitive membranes is considered. The proposed neural circuit can maintain signal processing capabilities even when one capacitor breakdowns. The three-layers of membranes exhibit significant inter-layer non-uniform electrical responses under external stimulation, which reveals the possible source mechanism for the negative resting potential of biological cell membranes. Further, energy function is defined to describe the intrinsic relationship between energy distribution and different firing modes, and explores the occurrence conditions for stochastic resonance under noisy excitation. Numerical results show that the proposed model can reproduce multiple firing patterns and reveal the relationship between membrane energy redistribution and firing mode transition. An adaptive control scheme under energy regulation can actively achieve firing mode conversion by adjusting the bifurcation parameter. The proposed three-membrane neuron model not only can reproduce captures possible hierarchical electrical responses in membrane-related circuits of multi-layer membranes and their ion channel encoding effects, but also provides a new theoretical framework for revealing the relationship between membrane structure, energy mechanism, and neural signal regulation, thereby highlighting its potential significance for understanding membrane electrophysiology and for designing artificial neural circuits with multi-modal response capabilities.

Capacitors and inductors respectively serve as the main energy storage components for electric fields and magnetic fields, while the linear resistor R in traditional circuits not only provides a time scale [RC] but also causes thermal power loss. The linear term in the memristor has similar dissipative characteristic of a resistor and Joule heat is consumed. This paper proposes a class of neural circuits that are completely free of using linear resistors, thereby avoiding the problem of thermal dissipation and [LC]1/2 is used as reference benchmark because [RC] is not available as a common reference benchmark (tau=t/RC). Two types of nonlinear neural circuits under external forcing from current and voltage are designed, and the corresponding Hamilton energy function is constructed to characterize their dynamic characteristics. The results show that both models exhibit multiple transitions between periodic states and chaotic states under parameter modulation, with the chaotic state corresponding to the lowest average energy. Besides the dynamical analysis, circuit simulation is presented for showing the response characteristic of outptuts. Moderate noise intensity can induce stochastic resonance, enabling the system to exhibit optimal rhythmicity and energy responses. The energy-driven adaptive control strategy can effectively achieve a stable transition from chaotic state to periodic state. Additionally, the field coupling study shows that an appropriate coupling strength can enable two oscillators to achieve complete synchronization, and a weak random electric field can further enhance the synchronous promotion effect of adaptive control. Our discussions reveal the intrinsic coupling mechanism between parameter modulation, noise effect, and energy control in nonlinear neural circuits, providing new theoretical basis and clues for proposal of new neural circuits with lower or zero-consumption of Joule heat, and then resistor-based electric coupling can be replaced by those non-resistive elements in the coupled neural circuits and networks.

Real cell membranes exhibit a multilayer dielectric architecture, whereas many circuit-derived neuron models still reduce membrane dynamics to a single capacitive variable or, at most, a double-membrane description. As a result, the effects of energy redistribution among the three membrane layers, interlayer dissipation, and structural damage on neural firing remain insufficiently clarified. Motivated by this gap, we develop a tri-capacitor “sandwich-membrane” neural circuit in which the outer, intermediate, and inner membrane layers are represented explicitly and coupled through a linear dissipative medium. Based on the equivalent circuit, a dimensionless neuron model and a Hamilton energy function are derived. Numerical results show that the proposed model responds differently when the external stimulus is applied to different membrane layers, reflecting signal modulation caused by the layered structure and suggesting an intrinsic filtering effect arising from membrane heterogeneity. To assess robustness, staged membrane breakdown is modeled by reduced-order switching, and the remaining circuit is still able to sustain bounded oscillatory encoding after partial damage. In addition, noise-induced coherence resonance is characterized jointly by coefficient of variation (CV) and mean Hamilton energy. These results provide a physically motivated circuit model for studying how layered membranes, dissipation, and robustness jointly shape neuronal signal encoding.

In the neural network, 5 × 5 neurons in the local area are excited by energy injection, which an additive sub-branch circuit injects energy flow into the inductive ion channel, the branch circuit composed of an inductor of the neural circuit by using memristive stimuli within a finite frequency band. A control branch circuit (sub-branch circuit) is built to shunt energy flow from the neural circuit. The memristive current is filtered by the capacitor in the sub-branch circuit, then it is injected into one branch circuit of the neural circuit for continuous energy injection, which creates an energy source in the neural network, and a target wave is induced to occupy the network. It is different from the previous schemes that local periodical forcing is imposed on the membrane potential of each neuron directly, and our scheme emphasizes energy injection into the ion channels for further regulating the membrane potentials indirectly. The control mechanism is that local energy injection into ion channels will increase the energy level of finite neurons, and energy flow is diffused to excite more adjacent neurons accompanying the emergence of a target wave. The pacemaker depends on shunting current from ion channels of local finite neuron, and local energy injection into the ion channels excites finite neurons for further inducing ordered waves. It is also different from the scheme by blocking ion channels of neurons in the local area of the neural network, and our control strategy provides a physical approach and the function of ion channels is understood.

Nonlinear oscillators derived from nonlinear circuits have clear physical significance, and bifurcation analysis provides helpful guidance for further dynamical control and energy regulation. Most mathematical map models have found wide applications in digital signal processing and nonlinear dynamical analysis; however, the assumptions of high order nonlinear terms in the theoretical models lack clear physical interpretations. Euler forward algorithm provides accessible bridge between nonlinear oscillators and maps by applying linear transformation on the variables and intrinsic parameters, and the time step is incorporated as one intrinsic parameter so that the oscillator and its approximate equivalent map model can show similar dynamical characteristic. In particular, energy definition for the obtained map model becomes important when discrete memristor with high order term is introduced, e.g. a cubic term emerges in the map. In this work, a generalized continuous and discrete transformation framework suitable for high order nonlinear maps is proposed. The present approach extends the conventional continuousization theory from quadratic nonlinear systems to cubic and more general high order polynomial maps. By introducing appropriate scaling transformations and time step normalization, discrete high order maps can be approximately transformed into equivalent continuous nonlinear oscillators while preserving their principal dynamical properties, and the corresponding Hamilton energy functions are further constructed. Based on Helmholtz’s theorem, the Hamilton energy functions are derived through the decomposition of the vector field into rotational and gradient components, where the physical oscillator is represented in vector form. Furthermore, a unified transformation form for arbitrary high order polynomial maps is established. The results show that high order nonlinear maps naturally correspond to continuous systems with multi-well potential-energy structures, which may exhibit multistability, and complex oscillatory behaviors. The proposed framework provides an approximate continuous representation for high order discrete maps while preserving their principal dynamical characteristics. The proposed framework not only provides a unified theoretical approach for constructing Hamilton energy functions of high order nonlinear maps, but also offers new insights into the investigation of complex dynamics in memristive systems, nonlinear circuits with multistability, and high order nonlinear systems.

Resistive elements consume Joule heat in the electric circuits, while the capacitors and inductors can save field energy, which is crucial for further energy exchange and maintaining channel currents. Incorporation of specific electric components into the branch circuits of an electric circuit can adjust the energy conversion between capacitive and inductive elements, and continuous energy flow supports continuous oscillations in the electric circuits. In this article, a neural circuit is proposed by connecting a capacitor, inductor and ideal Josephson junction in three paralleled branch circuits, and then two excited neural circuits are coupled via a Josephson junction for maintaining synchronous firing. The energy function is calculated and proved from physical aspect, and coherence resonance is induced under noisy disturbance. The incorporation of Josephson junction just introduces nonlinear modulation on the channel current across the capacitor and inductor, and the neural circuit is excited to present rich firing patterns under external stimulus. The neural circuit is simple but it shows some advantages than most of the known neural circuit because lower Joule heat is consumed in absence of resistor.

In this article, an ion channel diversion (ICD) method is proposed to control the channel current of a neuron model derived from the FitzHugh-Nagumo (FHN) neural circuit. A diversion element (DE) such as capacitor and inductor is intervened into the branch circuit of the FHN circuit, and the channel current across inductor for the inductive ion channel is shunted to the DE for energy regulation and current shunting. That is, a sub-branch circuit is used to shunt current from the inductor of the neural circuit, and a hybrid ion channel is built. Changes in the parameter for the DE (new capacitor or inductor) will modify the shunted current in the sub-branch circuit, and the channel current along inductive channel is changed to regulate the membrane potentials of the neuron. The circuit equations, equivalent theoretical models and Hamilton energy functions are obtained for theoretical analysis, and moderate noise intensity can induce coherence resonance (CR) and stochastic resonance (SR) supporting a maximal value for in this curve vs. noise intensity. Furthermore, a parameter observer is designed to identify the unknown parameter in the theoretical model for further adaptive control in the electrical activities. An adaptive control strategy based on energy regulation is proposed, and the neuron can achieve periodic firing, cluster firing, and chaotic firing by setting a reasonable energy threshold epsilon. The results show that the working state and energy distribution of ion channels can significantly affect the firing patterns and attractor morphology of neurons. The interference in ion channels and shunting current from one branch circuit of neural circuit are effective to control the neural activities in single neuron and collective behaviors in neural networks. The physical significance of this control strategy is that energy control (energy shunting or energy injection) is suitable to modify the energy level and firing modes in the neurons, and then the effect of electromagnetic stimuli on nervous system can be understood from physical aspect.

This study investigates the fundamental mechanisms underlying cross-species, multi-strain transmission in ecosystems from the opinion of group opinion dynamics. A multilayer interaction framework is proposed, incorporating signed-weighted social network dynamics to quantify group-level opinions and dynamically adjust key epidemiological parameters in real time. The analysis reveals that (1) infection pressure alters group opinion thresholds via cognitive-behavioral feedback, while the emerging collective consensus reciprocally regulates transmission intensity, forming a closed-loop feedback mechanism. (2) The topology of the opinion network governs epidemic phase transitions, inducing a bistable regime characterized by either low-risk (opinion cohesion) or high-risk (opinion polarization) states. By identifying critical nodes within the signed social graph, the study transforms group opinion intensity into dynamic warning thresholds, enabling targeted ecological interventions.

A charge-controlled memristor (CCM) is used to couple a FitzHugh-Nagumo (FHN) neural circuit, which composes of an ideal capacitor, inductor, and nonlinear resistor with cubic nonlinear i-v relation, and a memristor-based neural circuit is proposed. A magnetic flux-controlled memristor (MFCM) is utilized to simulate chemical synaptic regulation on two neural circuits coupled via a memristor. Differing from most of the previous works about memristor-based neural circuits, here, a CCM is connected to a voltage source for generating continuous signals within finite frequency band rather than shunting channel current from the signal sources as those capacitive and inductive elements. By applying scale transformation on the physical equations for the coupled neural circuits, a dimensionless theoretical model for coupled memristive neurons is established and the Hamilton energy is derived to discover the relation between energy balance and firing dynamics in neural activities. The influence of synchronicity on the energy distribution is analyzed, and the key role of coupled memristive channel energy in maintaining the system energy balance is revealed as well. Additionally, the Stochastic Resonance (SR) phenomenon is investigated, and it is found that the memristive coupling channel is more sensitive to noisy disturbance. Meanwhile, a parameter observer is designed to identify one of the unknown parameters of the coupled neurons. Finally, an adaptive energy control criterion is proposed to effectively regulate the neuron firing modes, enabling the system to quickly reach a stable state. This research provides a theoretical basis and practical guidance for understanding the dynamic characteristics of memristive neuron systems, the energy regulation mechanism, and its applications in neuromorphic computing and intelligent systems.

Conservative chaotic flows usually originate from dissipative-free dynamical systems. Most of the existing results on conservative chaotic flows appear in ordinary differential systems, while there are relatively few related findings in physical circuits, especially in Josephson junction (JJ) circuits. To this end, this study develops a simple JJ system implemented by a current-driven inductor-capacitor-JJ circuit. This circuit has only one ideal shunted JJ element that achieves sine nonlinearity and is likely to be the simplest structure. Theoretical analyses of the Jacobian matrix trace and generalized energy indicate that the proposed JJ system is volume-conserving but not energy-conserving, and it belongs to a non-Hamiltonian conservative system. In particular, under a set of system parameters, infinitely many chaotic and quasi-periodic flows are disclosed to coexist in heterogeneous and homogeneous forms. Additionally, under another set of system parameters, multi-scroll chaotic flows with scroll growth can also be found. Finally, an equivalent implementation circuit for the JJ system is designed. The numerical findings are verified through circuit simulations and hardware experiments.

Blockers and activators can modify the conductance of the ion channels, and the channel currents are changed to regulate the membrane potential of a neuron for presenting different firing patterns. From dynamical aspect, blocking the ion channel and external stimuli are processed by adding equivalent trans-membrane current on the neuron models, however, the physical processing is not clarified. In this article, a control branch shunts channel current from the inductive channel of the FitzHugh-Nagumo (FHN) neural circuit, and external stimuli can be encoded in the ion channel rather than imposing direct forcing currents on the membrane potential. The neural circuit is controlled by shunting current from the inductor of the branch circuit in the FHN neural circuit, a constant voltage connected with a memristor is used to generate forcing current within finite frequency band rather than sole frequency, which is consistent with realistic external stimuli. The control branch circuit composes a capacitor for filtering the external stimulus and the filtered current will interact with the shunted channel current under energy exchange, as a result, changes of the energy ratio between magnetic field and electric field will modify the capacitor voltage, which corresponds to the membrane potential of the neuron model. Energy function is provided to discern the correlation between firing modes and energy level, and then coherence resonance is induced under noisy disturbance. Our results provide new insights into control of ion channels from physical aspect, that is, external energy injection via the control branch circuit (external sub-branch circuit) into the neural circuit tends to build a hybrid ion channel for regulating the energy level of the neurons. As a result, the ion channel becomes controllable following the injection of energy flow and the channel conductance becomes controllable. Therefore, the external energy is encoded in the ion channel for further adjustment in the energy level of the neuron, and then the energy levels control the neural activities.

Multiple interactions with systems are present in the brain and in many other complex systems. However, how to explore their dynamics is still an open problem. Here, we propose a starlike Hopfield neural network (HNN) with multiple interactions involving neurons, electromagnetic autapse, and synapse. There are two classes of the starlike HNNs activated by introducing memristive electromagnetic autaptic (MEA) current at the central and peripheral nodes, respectively. The starlike MEA-HNN models can exhibit the rapid and slack chaotic bursting, dependent upon the modulation of MEA current at different nodes. The initial values in the improved HNN model can control the alternant occurrence of firing patterns between chaotic bursting and resting. It is found that both electrical synapses and chemical synapses can make homogeneous and heterogeneous coupled networks to achieve synchronization. Under the electrical synapse, the bursting-type synchronization is transitioned as periodic synchronization, suggesting that the original firing pattern has been disrupted. While the coupled model is still the bursting synchronization under the chemical synapse. The obtained results can help understand the complexity of neural networks with different structures and the dynamic mechanism of interaction underlying coupled chaotic bursting oscillators.

Neuromuscular junctions are vital biological interfaces between efferent nerves and muscle fibers, and their emulation remains challenging due to the lack of a clear understanding of the underlying biophysical mechanisms. To avoid complicated electromechanical coupling relationships, we propose an electromechanical model based on the Lagrange-Maxwell equations from an energy-based perspective. The model integrates a Josephson junction-based neuron with a mass-spring-damper module to mimic a biological neuromuscular junction. Numerical simulations and theoretical analysis reveal that the Josephson junction-based neuron exhibits two classes of excitability. The artificial muscle, which incorporates a nonlinear spring, undergoes an interstitial transition between kinetic and static states. Furthermore, the coupling component of the electromechanical model is identified as a meminductor, and the locally active meminductive component significantly influences the modulating vibrational patterns. We also demonstrate the emulation of biological dynamics, such as enhanced artificial muscle activity in response to increased external stimulus current. These findings provide insights into biophysical electromechanical coupling systems and suggest potential applications in bioinspired robotics and neuromorphic devices.

Membrane potential is an observable output, whereas the ion channel pathway governs the internal current partitioning that shapes neural firing modes. Here, we construct a memristive FitzHugh-Nagumo neural circuit in which a tunable diversion branch is connected to the canonical ion channel branch, enabling the regulation of neural firing through the controlled redistribution of channel current. Physical implementation of this strategy is verified by incorporating different electric elements in the sub-branch circuit: (i) a shunting capacitor for differential-type diversion and (ii) a shunting inductor in series with a protective resistor for integral-type diversion. For the two controlled circuits, physical equations and field energy functions are derived, and the corresponding theoretical models and corresponding energy functions are obtained and further checked by the Helmholtz theorem. Numerical analysis shows that capacitive shunting can trigger an abrupt collapse of the inductive energy level in L1 and thereby induce firing-mode transitions, whereas inductive shunting produces markedly weaker modulation over comparable parameter ranges. The model also exhibits noise-induced stochastic resonance, and an adaptive energy-guided regulation law controls the electrical activities effectively. The results suggest that control of firing patterns depends on the physical property of the shunting element and provide a physically interpretable strategy for ion-channel-level regulation in memristive neural circuits.

This paper explores the complex interactions between collective human behavior and the nervous system, where collective behavior is described by diffusion-coupled nonlinear systems, and the nervous system is modeled using the Wilson-Cowan system. Drawing from observed phenomena, we establish the relationship between these two components. Our model accommodates both nondiagonalizable diffusion coupling matrices and non-diagonalizable neural connectivity matrices. Initially, we derive local stability conditions for the synchronization of behavioral groups influenced by the nervous system. We then determine the synchronization conditions for the nervous system that depend on the synchronization of collective behavior. A particular focus is given to the scenario where the rows of neuronal connection matrices are uniformly normalized, allowing us to translate the analysis of synchronization in neural systems with arbitrary connection structures into numerical algorithms. Moreover, we found that modifying the connection weights and structure among neurons can influence the synchronization of the neural system, thereby affecting the synchronization of collective human behavior. Lastly, we present a numerical example to validate our theoretical findings.

Electromagnetic induction has important impact on the propagation and diffusion of intracellular ions in cardiac tissue. Memristive current and magnetic flux can be introduced into the simple cardiac model for estimating the effect of electromagnetic induction during ions diffusion. In this study, a two-variable cardiac tissue model is improved to include a new variable for magnetic flux, and a higher order memristive term is used to describe the induction current. Spiral waves can be induced and developed in the memristive cardiac model, and a statistical function named as synchronization factor is defined to predict the synchronization degree in the cardiac model. Spatial excitation is imposed on local and whole area of the media, and breakup of spiral wave is detected by taming the forcing intensity and noisy disturbance. Selection of high order form in the memristive current emphasizes the field effect and Joule heat is ignored during continuous diffusion of ions and charges in the ion channels and cells. In particular, noisy excitation on the magnetic flux variable estimates the regulation from external electromagnetic field and the membrane potential is controlled synchronously because the memristive current is dependent on the value of magnetic flux variable. This scheme can be used to control and suppress spiral waves for preventing possible fibrillation in heart.

Existing neural network models predominantly rely on pairwise interactions, often neglecting the higher-order interactions inherent in real-world neural systems. Even among studies that do incorporate higher-order structures, the dynamical mechanisms by which hyperedge overlap governs stability and catastrophe behavior remain poorly understood. To bridge this critical gap, this paper proposes a generalized star-topology neural network framework that explicitly incorporates higher-order interactions via hypergraphs, with a specific focus on the dynamical consequences of hyperedge overlap. By distinguishing between low-overlap and high-overlap topological configurations, we rigorously analyze the local stability and the existence of Hopf bifurcations through the derivation of characteristic equations and critical time delay thresholds. Theoretical analysis and extensive numerical simulations reveal a fundamental structural insight: compared to the low-overlap counterpart, the high-overlap configuration functions as a structural stabilizer. It significantly expands the stability domain by elevating bifurcation thresholds and effectively suppresses the amplitude of post-bifurcation limit cycles. Furthermore, scalability and robustness analyses demonstrate that these stabilizing effects persist across varying network scales and provide superior resilience against stochastic perturbations. By contrasting hypergraph dynamics with traditional pairwise frameworks, this work provides a novel mathematical perspective on how higher-order topological redundancy fundamentally shapes and stabilizes the collective dynamics of complex neural networks.

Two electromechanical arms are combined to control the movements of a brush for plotting different spatial patterns and attractors. The brush is attached to the end of one arm of the electromechanical devices, and two electromechanical arms are forced to move along perpendicular directions in a plane. A hyperchaotic Lorenz system is used as the signal source. Its two output variables are encoded and used as driving currents in the coils of the robot arms’ electromechanical devices, and changes to those currents modify the movements of the brush. A scale transformation is applied to the physical equations and physical energy to obtain a dimensionless theoretical model and an energy function. Field energy from the signal sources is shunted to the electromechanical devices. The moving arms generate motional electromotive forces and provide feedback to the driving circuits (signal sources); that is, the coils of the robot arms are considered as load circuits (additive branch circuits) of the driving circuits. The trajectory of the brush seldom produces complete patterns and attractors like those in the driving system, so two controllers are introduced to assist in adaptive drawing. The Hamilton energy function H1 of the multi-wing Lorenz system (MWLS) is provided, and the energy is characteristic of the driving system. An energy-based adaptive control law is proposed that dynamically adjusts the system parameters based on the MWLS’s energy distribution, thereby significantly improving flexibility. Bifurcation parameters, control intensity, and adaptive control parameters affect the dynamic characteristics of a signal source and the coupled robot arms. The energy-guided adaptive control scheme enables switching during drawing by a mechanical arm, approaching the desired graphic, and providing greater control freedom. Our results provide theoretical support for two-dimensional drawing by mechanical arms.

The output voltages for the capacitive elements of a neural circuit model can be mapped into dimensionless capacitive variables, which present firing patterns similar to the membrane potentials detected in biological neurons. The inclusion of a memcapacitor also enables consideration of membrane deformation effects, enhancing the model's capacity to simulate neuronal behavior across varying physiological and environmental conditions. In this study, a capacitor and a memcapacitor are connected through a linear resistor in parallel with other electric components in different branch circuits composed of an inductor and a nonlinear resistor. The electrical activities in a neuron with a double-layer membrane and two capacitive variables are discussed in detail after converting the nonlinear equations for the neural circuit into a theoretical neuron model. A dimensionless neuron model and its corresponding energy function are derived. The field energy function for the neural circuit is converted into an equivalent Hamilton energy function and further validated via the Helmholtz theorem. Furthermore, the average value of energy serves as an indicator for predicting stochastic resonance, as supported by analyzing the distribution of the coefficient of variation. The neuronal firing patterns are shown to be energy-dependent. An adaptive control strategy is proposed to regulate mode transitions in electrical activities of the neuron. An analog equivalent circuit is constructed to experimentally verify the numerical results, thereby supporting the reliability of the proposed neuron model.

Biological neurons and muscle cells often generate thermal energy during their discharge process. The occurrence of electrical activity, such as action potentials, is typically accompanied by a measurable release of heat. In this study, a thermosensitive neuron model is derived from a memristor-coupled neural circuit, which integrates capacitor, inductor, memristor, and thermistor, and thermal effect and interaction with memristive regulation are discussed. The model can reproduce typical firing behaviors including spiking, periodic oscillations, and bursting, and it reveals the emergence of chaotic discharges induced by hidden attractors. From the perspective of energy analysis, both the Hamilton energy and thermal energy are adopted as quantitative metrics, and then the correlation between energy level and firing modes in electrical activities is explained. The results show that chaotic discharges are associated with the lowest average Hamilton energy yet the highest consumption of thermal energy, while periodic discharges exhibit an opposite behavior. Furthermore, by tuning some thermal parameters (e.g., B ' and lambda), environment-related factors (e.g., k2), and the external stimulation frequency omega, desired control over different discharge patterns can be achieved. Additive Gaussian white noise is also independently introduced into each circuit branch to explore stochastic resonance. The findings demonstrate that noise intensity significantly influences both energy levels and rhythmic behavior. This work provides a comprehensive theoretical framework for understanding thermally coupled neural dynamics and offers a novel approach for designing energy-efficient and highly tunable neuromorphic systems.