Hindmarsh-Rose Neuron Model Research Articles

Memristors-coupled neuron models with multiple firing patterns and homogeneous and heterogeneous multistability

Abstract Memristors are extensively used to estimate the external electromagnetic stimulation and synapses for neurons. In this paper, two distinct scenarios, i.e., an ideal memristor serves as external electromagnetic stimulation and a locally active memristor serves as a synapse, are formulated to investigate the impact of a memristor on a two-dimensional Hindmarsh–Rose neuron model. Numerical simulations show that the neuronal models in different scenarios have multiple burst firing patterns. The introduction of the memristor makes the neuronal model exhibit complex dynamical behaviors. Finally, the simulation circuit and DSP hardware implementation results validate the physical mechanism, as well as the reliability of the biological neuron model.

Fuzzy-Based Sampled-Data Synchronization of the Hindmarsh–Rose Neuronal Model

Fuzzy-Based Sampled-Data Synchronization of the Hindmarsh–Rose Neuronal Model

A Hindmarsh–Rose neuron model with electromagnetic radiation control for the mechanical optimization design

This paper proposes a Hindmarsh–Rose(HR) neuron model controlled by electromagnetic radiation to solve the mechanical optimization design problem. A novel memristor for simulating electromagnetic radiation is established to control the dynamical behavior of HR neuron models. By analyzing the HR model and using the numerical analysis methods of phase portrait, bifurcation diagram, Lyapunov exponent and spectral entropy (SE) complexity, the HR model is shown to have a hidden attractor and rich chaotic dynamics. Then, a chaotic optimization algorithm based on HR neuron model is designed to solve the mechanical optimization design of tension/compression coil springs and pressure vessels. The chaotic sequence of this algorithm which has high-quality pseudo-random number, has stronger global search ability. Simulation of two mechanical optimization design problems in comparison with some well-known optimization algorithms demonstrates the superiority of the HR in the solution quality.

Effects of external stimuli on the dynamics of deterministic and stochastic Hindmarsh–Rose neuron models

Effects of external stimuli on the dynamics of deterministic and stochastic Hindmarsh–Rose neuron models

Constructing Memristive Hindmarsh–Rose Neuron with Countless Coexisting Firings

Memristor synapses have been widely introduced into neuronal models to investigate the effects of external magnetic fields. However, there is a relative lack of research on the external-induced electric fields in neurons. In this paper, a 4D-memristive Hindmarsh–Rose neuron model is constructed by introducing a memristor and an electric field variable, which can generate complex neural firing. Notably, numerical simulations reveal that the initial conditions of the memristor can induce different firing patterns, exhibiting a unique fractal structure in the basin of attractions. Remarkably, the offset parameters of the internal variables of the neuron can be canceled out so that the offset boosting of the variables can be achieved according to the initial values, giving rise to an uncountably many hidden attractors with homogeneous multistability. This model provides the first example of generating uncountably many attractors in a memristive neuron model without relying on trigonometric functions, significantly advancing our understanding of neuronal dynamics. Finally, a digital circuit is designed and implemented on the RISC-V platform to verify the numerical simulation and theoretical analysis. The findings of this study have a certain implication for the development of advanced neuromorphic computing systems and the understanding of complex neuronal behaviors in the presence of external electric fields.

A chaotic memristive Hindmarsh-Rose neuron with hybrid offset boosting

In recent years, memristor has been widely introduced into neuronal models to simulate magnetically induced currents. However, there are relatively few studies on electric field stimulation of neuronal models. This paper presents a 4D memristive Hindmarsh-Rose (HR) neuron model that introduces a sinusoidal function memristor and electric field variables, which can produce heterogeneous multistability and homogeneous multistability. More interestingly, six modes of offset boosting can be triggered in the neuronal model by varying the electric field coupling coefficients, the first three of which are parameter-dependent two-dimensional offset boosting modes, and the second three are initial-value-dependent offset-boosting modes. Notably, homogeneous multistability can also combined with the offset-boosting, and then iterating cubic infinitely many attractors. This model presents a model where multiple offset boosting is freely controllable for the first time. Finally, the theoretical analysis is verified by digital circuits on a RISC-V platform.

Multiple firing patterns, energy conversion and hardware implementation within Hindmarsh-Rose-improved neuron model

The transmission of information between neurons is accomplished in living organisms through synapses. The memristor is an electronic component that simulates the tunability of the strength of biological synaptic connections in artificial neural networks. This article constructs a novel type of locally active memristor and verifies by nonlinear theoretical analysis, locally active analysis and circuit simulation. The designed memristor is simulated as a biological autapse of Hindmarsh-Rose(HR) neuron to obtain the improved HR neuron model of memristive autapse, and the Hamilton energy is obtained according to Helmholtz theorem. By varying the external forcing current and the memristive autapse strength, this article analyses the changes of the Hamilton energy and explores its self-excited and hidden firing behavior. The analog circuit simulation and digital circuit implementation of the HR model confirm the consistency between the mathematical model and the actual behavior, which can advance the field of neuroscience and artificial intelligence.

Noise-induced extreme events in integer and fractional-order memristive Hindmarsh–Rose neuron models: a comprehensive study

Noise-induced extreme events in integer and fractional-order memristive Hindmarsh–Rose neuron models: a comprehensive study

Periodic firing evolution of a Hindmarsh–Rose neuron model and FPGA circuit implementation

Periodic firing evolution of a Hindmarsh–Rose neuron model and FPGA circuit implementation

Efficient digital design of the nonlinear behavior of Hindmarsh–Rose neuron model in large-scale neural population

Spiking networks, as the third generation of neural networks, are of great interest today due to their low power consumption in cognitive processes. This important characteristic has caused the hardware implementation techniques of spiking networks in the form of neuromorphic systems attract a lot of attention. For the first time, the focus is on the digital implementation based on CORDIC approximation of the Hindmarsh–Rose (HR) neuron so that the hardware implementation cost is lower than previous studies. If the digital design of a neuron is done efficient, the possibility of implementing a population of neurons is provided for the feasibility of low-consumption implementation of high-level cognitive processes in hardware, which is considered in this paper through edge detector, noise removal and image magnification spiking networks based on the proposed CORDIC_HR model. While using less hardware resources, the proposed HR neuron model follows the behavior of the original neuron model in the time domain with much less error than previous study. Also, the complex nonlinear behavior of the original and the proposed model of HR neuron through the bifurcation diagram, phase space and nullcline space analysis under different system parameters was investigated and the good follow-up of the proposed model was confirmed from the original model. In addition to the fact that the individual behavior of the original and the proposed neurons is the same, the functional and behavioral performance of the randomly connected neuronal population of original and proposed neuron model is equal. In general, the main contribution of the paper is in presenting an efficient hardware model, which consumes less hardware resources, follows the behavior of the original model with high accuracy, and has an acceptable performance in image processing applications such as noise removal and edge detection.

Effects of high-order interactions on synchronization of a fractional-order neural system

In this study, effects of high-order interactions on synchronization of the fractional-order Hindmarsh–Rose neuron models have been examined deeply. Three different network situations in which first-order coupling, high-order couplings and first-plus second-order couplings included in the neuron models, have been considered, respectively. In order to find the optimal values of the first- and high-order coupling parameters by minimizing the cost function resulted from pairwise and triple interactions, the particle swarm optimization algorithm is employed. It has been deduced from the numerical simulation results that the first-plus second-order couplings induce the synchronization with both reduced first-order coupling strength and total cost compared to the first-order coupled case solely. When the only first-order coupled case is compared with the only second-order coupled case, it is determined that the neural network with only second-order couplings involved could achieve synchronization with lower coupling strength and, as a natural result, lower cost. On the other hand, solely second- and first-plus second-order coupled networks give very similar results each other. Therefore, high-order interactions have a positive effect on the synchronization. Additionally, increasing the network size decreases the values of the both first- and high-order coupling strengths to reach synchronization. However, in this case, total cost should be kept in the mind. Decreasing the fractional order parameter causes slower synchronization due to the decreased frequency of the neural response. On the other hand, more synchronous network is possible with increasing the fractional order parameter. Thus, the neural network with higher fractional order as well as high-order coupled is a good candidate in terms of the neural synchronization.

Stability and synchronization in neural network with delayed synaptic connections.

In this paper, we investigate the stability of the synchronous state in a complex network using the master stability function technique. We use the extended Hindmarsh-Rose neuronal model including time delayed electrical, chemical, and hybrid couplings. We find the corresponding master stability equation that describes the whole dynamics for each coupling mode. From the maximum Lyapunov exponent, we deduce the stability state for each coupling mode. We observe that for electrical coupling, there exists a mixing between stable and unstable states. For a good setting of some system parameters, the position and the size of unstable areas can be modified. For chemical coupling, we observe difficulties in having a stable area in the complex plane. For hybrid coupling, we observe a stable behavior in the whole system compared to the case where these couplings are considered separately. The obtained results for each coupling mode help to analyze the stability state of some network topologies by using the corresponding eigenvalues. We observe that using electrical coupling can involve a full or partial stability of the system. In the case of chemical coupling, unstable states are observed whereas in the case of hybrid interactions a full stability of the network is obtained. Temporal analysis of the global synchronization is also done for each coupling mode, and the results show that when the network is stable, the synchronization is globally observed, while in the case when it is unstable, its nodes are not globally synchronized.

Reservoir computing with higher-order interactive coupled pendulums.

The reservoir computing approach utilizes a time series of measurements as input to a high-dimensional dynamical system known as a reservoir. However, the approach relies on sampling a random matrix to define its underlying reservoir layer, which leads to numerous hyperparameters that need to be optimized. Here, we propose a nonlocally coupled pendulum model with higher-order interactions as a novel reservoir, which requires no random underlying matrices and fewer hyperparameters. We use Bayesian optimization to explore the hyperparameter space within a minimal number of iterations and train the coupled pendulums model to reproduce the chaotic attractors, which simplifies complicated hyperparameter optimization. We illustrate the effectiveness of our technique with the Lorenz system and the Hindmarsh-Rose neuronal model, and we calculate the Pearson correlation coefficients between time series and the Hausdorff metrics in the phase space. We demonstrate the contribution of higher-order interactions by analyzing the interaction between different reservoir configurations and prediction performance, as well as computations of the largest Lyapunov exponents. The chimera state is found as the most effective dynamical regime for prediction. The findings, where we present a new reservoir structure, offer potential applications in the design of high-performance modeling of dynamics in physical systems.

Coexisting Firing Patterns in an Improved Memristive Hindmarsh-Rose Neuron Model with Multi-Frequency Alternating Current Injection.

With the development of memristor theory, the application of memristor in the field of the nervous system has achieved remarkable results and has bright development prospects. Flux-controlled memristor can be used to describe the magnetic induction effect of the neuron. Based on the Hindmarsh-Rose (HR) neuron model, a new HR neuron model is proposed by introducing a flux-controlled memristor and a multi-frequency excitation with high-low frequency current superimposed. Various firing patterns under single and multiple stimuli are investigated. The model can exhibit different coexisting firing patterns. In addition, when the memristor coupling strength changes, the multiple stability of the model is eliminated, which is a rare phenomenon. Moreover, an analog circuit is built to verify the numerical simulation results.

FPGA implementation of memristive Hindmarsh–Rose neuron model: Low cost and high-performing through hybrid approximation

Recent memristive neuron models have improved the ability of researchers to describe the operation of biological neurons. We employ FPGA implementation for a memristive neuron model, FPGA implementation provides an improved hardware system for mimicking brain behavior, and provides a better understanding of neurological phenomena. In this manuscript, a memristive Hindmarsh–Rose neuron model taking into account the influence of electromagnetic radiation and two multiplierless approximations of this model are presented. These approximations are piecewise linear (PWL) model and hybrid model, which mimic the behaviors of the original model. In the hybrid model, power 2-based approximation and a novel approximation are used simultaneously. The proposed models were subjected to error analysis, revealing that the PWL model exhibited mean NRMSEs of 3.56% and 2.3% for bursting and spiking modes, respectively. On the other hand, the hybrid model displayed mean NRMSEs of 1.92% and 1.34% for bursting and spiking modes. According to the FPGA implementation results, the PWL model yields the overall resources saving of 75.82% and 47.8% speed-up. But in the hybrid model, by optimizing the approximation functions, in addition to obtaining higher accuracy, better results are provided with a 67.3% improvement in the maximum frequency and the 78.64% saving in hardware resources.

Switching dynamics analysis and synchronous control of a non-smooth memristive Hindmarsh–Rose neuron model

Switching dynamics analysis and synchronous control of a non-smooth memristive Hindmarsh–Rose neuron model

Predefined-time smooth stability analysis of nonlinear chaotic systems with applications in the PMSM system and Hindmarsh-Rose neuron model

This article investigates the predefined-time stabilization of nonlinear chaotic systems with applications in the permanent magnet synchronous motor (PMSM) system and Hindmarsh-Rose neuron model. Distinguished from the traditional predefined-time control methods, this investigation develops the smooth control protocols, in which the discontinuous absolute value and signum functions are not used anymore, so that the unfavorable chattering phenomenon can be avoided effectively. By the Lyapunov stability analysis, the sufficient condition is derived to achieve the predefined-time stable for nonlinear chaotic systems, in which the upper-bound time estimation (TE) of arriving at the stable state is explicit in contrast to the traditional finite-/fixed-time convergence. Specifically, the analytical results are successfully applied into stabilizing the PMSM system and Hindmarsh-Rose neuron model within the predefined-time. Finally, the numerical simulations for stabilizing the chaotic PMSM system and Hindmarsh-Rose neuron model verify the effectiveness and advantages of theoretical analysis.

Learning-based sliding mode synchronization for fractional-order Hindmarsh-Rose neuronal models with deterministic learning.

In recent years, extensive research has been conducted on the synchronous behavior of neural networks. It is found that the synchronization ability of neurons is related to the performance of signal reception and transmission between neurons, which in turn affects the function of the organism. However, most of the existing synchronization methods are faced with two difficulties, one is the structural parameter dependency, which limits the promotion and application of synchronous methods in practical problems. The other is the limited adaptability, that is, even when faced with the same control tasks, for most of the existing control methods, the control parameters still need to be retrained. To this end, the present study investigates the synchronization problem of the fractional-order HindmarshRose (FOHR) neuronal models in unknown dynamic environment. Inspired by the human experience of knowledge acquiring, memorizing, and application, a learning-based sliding mode control algorithm is proposed by using the deterministic learning (DL) mechanism. Firstly, the unknown dynamics of the FOHR system under unknown dynamic environment is locally accurately identified and stored in the form of constant weight neural networks through deterministic learning without dependency of the system parameters. Then, based on the identified and stored system dynamics, the model-based and relearning-based sliding mode controller are designed for similar as well as new synchronization tasks, respectively. The synchronization process can be started quickly by recalling the empirical dynamics of neurons. Therefore, fast synchronization effect is achieved by reducing the online computing time. In addition, because of the convergence of the identification and synchronization process, the control experience can be constantly replenished and stored for reutilization, so as to improve the synchronization speed and accuracy continuously. The thought of this article will also bring inspiration to the related research in other fields.

Synchronization in memristive HR neurons with hidden coexisting firing and lower energy under electrical and magnetic coupling

Employing memristors to mimic biological synapses and to describe electromagnetic induction effects attracts much attention in current researches. In this paper, a memristive Hindmarsh–Rose neuron model is proposed by using an electromagnetic inductive current involving quadratic memconductance to replace the adaptive current in the classical 3-D HR model. The memristive neuron shows hidden firing and coexisting state. The energy function of memristive neuron is deduced by the generalized Hamiltonian formalism. In particular, two memristive neurons with irregular firing reach phase synchronization under the electrical coupling, while achieve complete synchronization under the magnetic coupling. The coupled neurons operate with economy energy in the region of whether complete synchronization or phase synchronization. Corresponding results are further demonstrated by designing digital circuit. These theoretical investigations can better choose appropriate model to explore dynamics of larger-scale neuronal network.

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Similar to period-doubling bifurcation of fixed points, periodic orbits, it has been found since 1980's that a corresponding doubling bifurcation can also be found in the case of quasiperiodic orbits. Doubling bifurcations of quasiperiodic orbits has an important consequence on the dynamics of the system under consideration. Recently, it has been shown that subsequent doublings of quasiperiodic closed invariant curves lead to the formation of Shilnikov attractors. In this contribution, we illustrate for the first time in a discrete neuron system, the phenomenon of doubling of closed invariant curves. We also show the presence of mode-locked orbits and the geometry of one-dimensional unstable manifolds associated with them resulting in the formation of a resonant closed invariant curve. Moreover, we illustrate the phenomenon of crisis and multistability in the system.