Hindmarsh-Rose Neuron Model Research Articles

Dynamics of excitable cells: spike-adding phenomena in action

We study the dynamics of action potentials of some electrically excitable cells: neurons and cardiac muscle cells. Bursting, following a fast–slow dynamics, is the most characteristic behavior of these dynamical systems, and the number of spikes may change due to spike-adding phenomenon. Using analytical and numerical methods we give, by focusing on the paradigmatic 3D Hindmarsh–Rose neuron model, a review of recent results on the global organization of the parameter space of neuron models with bursting regions occurring between saddle-node and homoclinic bifurcations (fold/hom bursting). We provide a generic overview of the different bursting regimes that appear in the parametric phase space of the model and the bifurcations among them. These techniques are applied in two realistic frameworks: insect movement gait changes and the appearance of Early Afterdepolarizations in cardiac dynamics.

Parsimonious physics-informed random projection neural networks for initial value problems of ODEs and index-1 DAEs.

We present a numerical method based on random projections with Gaussian kernels and physics-informed neural networks for the numerical solution of initial value problems (IVPs) of nonlinear stiff ordinary differential equations (ODEs) and index-1 differential algebraic equations (DAEs), which may also arise from spatial discretization of partial differential equations (PDEs). The internal weights are fixed to ones while the unknown weights between the hidden and output layer are computed with Newton's iterations using the Moore-Penrose pseudo-inverse for low to medium scale and sparse QR decomposition with L 2 regularization for medium- to large-scale systems. Building on previous works on random projections, we also prove its approximation accuracy. To deal with stiffness and sharp gradients, we propose an adaptive step-size scheme and address a continuation method for providing good initial guesses for Newton iterations. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled and the number of basis functions are "parsimoniously" chosen based on bias-variance trade-off decomposition. To assess the performance of the scheme in terms of both numerical approximation accuracy and computational cost, we used eight benchmark problems (three index-1 DAEs problems, and five stiff ODEs problems including the Hindmarsh-Rose neuronal model of chaotic dynamics and the Allen-Cahn phase-field PDE). The efficiency of the scheme was compared against two stiff ODEs/DAEs solvers, namely, ode15s and ode23t solvers of the MATLAB ODE suite as well as against deep learning as implemented in the DeepXDE library for scientific machine learning and physics-informed learning for the solution of the Lotka-Volterra ODEs included in the demos of the library. A software/toolbox in Matlab (that we call RanDiffNet) with demos is also provided.

Hidden firing patterns and memristor initial condition-offset boosting behavior in a memristive Hindmarsh-Rose neuron model

Electromagnetic induction can effectively induce abundant firing patterns in neurons. In modeling a neuron model with the electromagnetic induction effect, an electromagnetic induction current is frequently added to the state equation of membrane potential. To more properly reflect the non-uniform distribution of the ions inside and outside the neuron membrane, an ideal flux-controlled memristor with sinusoidal memductance function and non-linearly modulated input is raised to depict an electromagnetic induction effect on a Hindmarsh–Rose neuron model, and thereby, a three-dimensional (3D) memristive Hindmarsh–Rose (mHR) neuron model is built in this paper. The proposed mHR neuron model possesses no equilibrium point since the involvement of the ideal flux-controlled memristor, which induces the generation of hidden dynamics. Numerical results declare that the mHR neuron model can generate abundant hidden dynamics, i.e., periodic spiking, chaotic spiking, period-doubling bifurcation route, tangent bifurcation, and chaos crisis. These hidden dynamics are much related to the memristor coupling strength and externally applied stimulus. Afterward, the memristor initial condition-offset boosting behavior is revealed. This can trigger the generation of infinite multiple coexisting firing patterns along the memristor variable coordinate. These coexisting firing patterns have identical attractor topology but different locations in the phase plane. Finally, an analog circuit is designed for implementing the mHR neuron model, and PSIM-based circuit simulation is executed. The circuit-simulated results perfectly verify the generation of hidden infinite multiple coexisting initial condition-offset boosting firing patterns in the proposed mHR neuron model.

Noise-driven bursting birhythmicity in the Hindmarsh-Rose neuron model.

The stochastic Hindmarsh-Rose model is studied in the parameter region where two bursting limit cycles of different types coexist. We show that under the influence of noise, transitions between basins of attractions appear, which generates stochastic bursting oscillations of mixed modes. The formation of this new regime is accompanied by anti-coherence and coherence resonances as well as by the transition to chaos. We investigate the probabilistic mechanism of the noise-driven bursting birhythmicity using the stochastic sensitivity functions and confidence domains method.

Multiplierless low‐cost implementation of Hindmarsh–Rose neuron model in case of large‐scale realization

SummaryImplementation of neural networks in case of hardware helps us to understand the different parts of the human brain operation, using artificial intelligence (AI). This paper presents a new model of the Hindmarsh–Rose (HR) Neuron that is based on basic polynomial functions called Nyquist‐look up table‐Hindmarsh–Rose (N‐LUT‐HR) based on an accurate sampling of the original model. The proposed approach is investigated in terms of its digital realization feasibility. According to high matching between the original and proposed terms, it is showed that the new modified model can follow all spiking patterns of primary model with low‐error computations. In hardware case, the proposed and original models are implemented on Xilinx FPGA XC2VP30 chip to validate different aspects of the simulation results. Hardware results demonstrate that our model regenerates the desired patterns in low‐cost and high‐frequency (speed‐up) in comparison with the other similar works. Overall saving in FPGA resources show that this new model is capable of being used in large‐scale networks in case of minimum required resources (FPGA costs). In addition, the analysis of hardware indicates that the new circuits can work in a maximum frequency of 123 MHz with 98.25% saving in FPGA costs (resources utilization of FPGA).

Hybrid Projective Synchronization of Fractional-Order Extended Hindmarsh–Rose Neurons with Hidden Attractors

In view of the diversity of stimulated current that neurons may experience, an extended Hindmarsh–Rose neuron model is proposed and the corresponding fractional-order neuron model, with no equilibrium point, is depicted. Additionally, various hidden attractors of the addressed neuron model are analyzed by changing system parameters and the order of fractional-order neuron system. Furthermore, hybrid projective synchronizations of the proposed neurons are investigated and schemes are obtained by designing suitable controllers according to fractional stability theory. Besides, the validity of the theoretical results is verified through numerical simulations. In short, the research results have potential application in revealing the dynamical behaviors of neuron system and controlling the behaviors of neuron into certain status.

Dynamics and Jacobi stability of the controlled 3D Hindmarsh-Rose neuron model

Dynamics and Jacobi stability of the controlled 3D Hindmarsh-Rose neuron model

Synchronization in functional brain networks of children suffering from ADHD based on Hindmarsh-Rose neuronal model

Attention Deficit Hyperactivity Disorder (ADHD) is one of the common psychological disorders in children, which causes improper recognition of others' emotions. These children often have very poor social interaction and a lack of self-confidence. Brain disorders are typically studied by investigating the functional connectivity of different areas in the brain networks of individuals and measuring synchronization. This study investigates synchronization in the simulated functional brain networks of 22 boys with ADHD and 22 healthy boys using electroencephalogram (EEG) signals during angry, happy, neutral, and sad emotions. In these simulated networks, a Hindmarsh-Rose neuronal model is considered for each network node. The edges' weights connecting these nodes represent the synchronization level between the EEG electrode pairs. By changing the coupling strength between the network nodes, different synchronization patterns in the ADHD and healthy brain networks can be observed in all four types of facial emotions. This study shows a higher synchronization, especially in happiness emotion in the frontal and occipital brain lobes of the ADHD group, responsible for human emotional and visual processing, respectively. Results also indicate the chimera phenomenon in both of the brain networks of ADHD and healthy individuals. However, this phenomenon occurs in the ADHD group at a lower coupling strength. These findings may represent a deficit in the brain's emotional and visual processing centers in the ADHD group, especially during the happy emotional stimuli.

Route to Chaos and Chimera States in a Network of Memristive Hindmarsh-Rose Neurons Model with External Excitation

In this paper we have introduced and investigated the collective behavior of a network of memristive Hindmarsh-Rose (HR) neurons. The proposed model was built considering the memristive autapse of the traditional 2D HR neuron. Using the one-parameter bifurcation diagram and its corresponding maximal Lyapunov exponent graph, we showed that the proposed model was able to exhibit a reverse period doubling route to chaos, phenomenon of interior and exterior crises. Three different configurations of the ring-star network of the memristive HR neuron model, including ring-star, ring, and star, have been considered. The study of those network configurations revealed incoherent, coherent , chimera and cluster state behaviors. Coherent behavior is characterized by synchronization of the neurons of the network, while incoherent behaviors are characterized by the absence of synchronization. Chimera states refer to a differet state where there is a coexistence of synchroniaed and asynchronized nodes of the network. One of the interesting result of the paper is the prevalence of double-well chimera states in both ring and ring-star network and has been first mentioned in the case of memrisitve HR neuron model.

Four-dimensional Hindmarsh–Rose neuron model with hidden firing multistability based on two memristors

Since memristors can be used to describe electromagnetic induction effects, this paper proposes a novel 4D HindMarsh-Rose (HR) neuron model based on two flux-controlled memristors to show complex dynamics of neuronal electrical activity. It has no equilibrium point, revealing hidden dynamical behaviors. The complex dynamics of the system are illustrated by phase portraits, the time sequences, bifurcation diagrams, and Lyapunov exponents spectra. The presented 4D HR neuron model can produce coexisting multiple hidden firing patterns, for instance, periodic spiking, chaotic spiking, transient chaotic spiking, periodic bursting, chaotic bursting, transient chaotic bursting, stochastic bursting, and transient stochastic bursting. Besides, rich nonlinear dynamics, such as anti-monotonicity and initial offset boosting, are also found. Finally, Multisim circuit simulation is performed and the results are in accordance with numerical simulation.

Superextreme spiking oscillations and multistability in a memristor-based Hindmarsh–Rose neuron model

In this paper, we investigate the occurrence of superextreme spiking (SES) oscillations and multistability behavior in a memristor-based Hindmarsh–Rose neuron model. The presence of SES oscillations has been identified as arising due to the occurrence of an interior crisis. As the membrane current I(t), considered as the control parameter is varied, the system transits from bounded chaotic spiking (BCS) oscillations to SES oscillations. These transitions are captured numerically using geometrical representations like time series plots, phase portraits and inter-spikes interval return maps. The characterization of SES from the BCS oscillations is made using statistical tools such as phase shift analysis and probability density distribution function. The multistability nature has been observed using bifurcation analysis and confirmed by the Lyapunov exponents for two different sets of initial conditions. The numerical simulations are substantiated through real-time hardware experiments realized through a nonlinear circuit constructed using an analog model of the memristor.

Numerical and Electrical Simulation of a Hindmarsh-Rose Neuron Model

Numerical and Electrical Simulation of a Hindmarsh-Rose Neuron Model

Application of discrete memristors in logistic map and Hindmarsh–Rose neuron

Continuous memristor (CM) has been widely designed and applied in chaotic and neuromorphic systems. However, discrete memristor (DM) and its application are less studied. This paper reports two kinds of generalized DM model based on sampling discretization of generalized CM model. It is found that the DM models can still retain the characteristics of the CM models. Based on the proposed DM models, the Logistic map (DM-L map) and two-dimensional (2D) discrete Hindmarsh–Rose neuron model (DM-HR model) are constructed. Through multiple numerical measures, it is found that DM-L map has linear fixed points, whose stability are only related to system parameter, and that DM-HR model does not have fixed point thus can generate hidden periodic and chaotic attractors. In particular, the coexistence of point and periodic attractors with different topological structures occur in DM-L map when different initial values are taken, which are rarely reported previously. At the same time, the chaotic region becomes wider when DM is introduced into Logistic map. Therefore, we believe that DM can effectively improve chaotic complexity of Logistic map. However, in addition to hidden complex discharge phenomenon, the coexistence of periodic and chaotic attractors with different positions and shapes are observed in DM-HR model. Finally, the DSP-based hardware platform is constructed to implement these maps, and the experimental results are consistent with the numerical simulation.

Effect of higher-order interactions on synchronization of neuron models with electromagnetic induction

Recent studies have shown that higher-order interactions have a vital role in exploring the collective dynamics of the networks. In particular, the collective behavior of a network of neuron models with many-body interactions has received much attention among researchers in recent times. In this paper, we study the effect of higher-order interactions in the synchronization stability of the network of neuron models, namely Hindmarsh-Rose and Morris-Lecar models, with electromagnetic induction. We consider both two-body and three-body interactions to be diffusive and analyze their effect on the synchronization of the network of neurons. Our analysis shows that higher-order interactions can make the neurons synchrony with the minimal value of first-order coupling strengths in both neuron models. Besides, electromagnetic flux coupling strength also has a significant effect on the synchronization of neurons. In the Hindmarsh-Rose neuron model, the flux coupling demands higher coupling strength in both the first and second-order interactions for the synchronization of neurons. However, the Morris-Lecar neuron model shows a notable distinct effect, where the flux coupling enhances the synchronization of neurons with lesser first and second-order coupling strengths.

Global dynamics analysis of a Filippov Hindmarsh–Rose neuron model

The electromagnetic induction induced by neuron membrane potential plays an important role in regulating its electrical activities, bistability and hidden dynamics. In this paper, a nonsmooth threshold control strategy is proposed, in which the membrane potential is utilized as the threshold to determine the switching function corresponding to the electromagnetic induction intensity and external stimulus current. Consequently, a four-dimensional Filippov-type HR neuron model is established. First, the existence, stability and global bifurcation behaviors of equilibrium points of two subsystems are discussed by using stability theory and numerical simulation. Then, the bistable behavior and evolution modes of subsystems are investigated based on the two-parameter bifurcation analysis. Further, the existence of various equilibrium points and sliding dynamics of the system are analyzed by Filippov convex method and Utkin’s equivalent control method. Finally, the sliding firing modes and multistable features under the control of the threshold are revealed by the method of fast-slow variable dissection. These results will provide useful theoretical support for understanding the hidden dynamic mechanism of neurons and constructing functional neural networks.

Analog/Digital Multiplierless Implementations for Nullcline-Characteristics-Based Piecewise Linear Hindmarsh-Rose Neuron Model

Multipliers are essential in implementing nonlinear neuron models, but they take huge implementation costs. Many multiplierless fitting schemes have been proposed to simplify the implementation of nonlinearities in neuron models. To optimize these schemes, this paper presents a nullcline-characteristics- based piecewise linear (NC-PWL) fitting scheme for multiplierless implementations of Hindmarsh-Rose (HR) neuron model. This NC-PWL fitting scheme uses as few line segments as possible to approximate the critical nonlinearity characteristics of the local nullclines. A NC-PWL HR neuron model that reproduces diverse firing patterns of the original one is successfully established. Using off-the-shelf low-cost components, an analog multiplierless circuit is designed for this fitting model and welded on print circuit board (PCB). Meanwhile, by logical shift method, a digital multiplierless circuit with low resource consumption is developed for this fitting model on field-programmable gate array (FPGA) platform. Experimental results of the analog and digital multiplierless hardware implementations verify the numerical simulations and show the simplicity and feasibility of the presented fitting scheme.

Role of assortativity in predicting burst synchronization using echo state network.

In this study, we use a reservoir computing based echo state network (ESN) to predict the collective burst synchronization of neurons. Specifically, we investigate the ability of ESN in predicting the burst synchronization of an ensemble of Rulkov neurons placed on a scale-free network. We have shown that a limited number of nodal dynamics used as input in the machine can capture the real trend of burst synchronization in this network. Further, we investigate the proper selection of nodal inputs of degree-degree (positive and negative) correlated networks. We show that for a disassortative network, selection of different input nodes based on degree has no significant role in the machine's prediction. However, in the case of assortative network, training the machine with the information (i.e., time series) of low degree nodes gives better results in predicting the burst synchronization. The results are found to be consistent with the investigation carried out with a continuous time Hindmarsh-Rose neuron model. Furthermore, the role of hyperparameters like spectral radius and leaking parameter of ESN on the prediction process has been examined. Finally, we explain the underlying mechanism responsible for observing these differences in the prediction in a degree correlated network.

An image encryption scheme by applying memristive Hindmarsh-Rose neuron model

Neurophysiological studies indicate that there are rich and complex dynamical characteristics in electrical actives between neurons. When neurons are stimulated by appropriate electromagnetic, the dynamical behavior for the nervous system can be changed, and electromagnetic has an important effect for the effectiveness of neurological diseases. To explore the complex dynamic effect of magnetic field in neuron activities, a new memristive HindMarsh-Rose neuron model can be developed by connecting a memristor into a simple 2 dimentional HindMarsh-Rose neuron model. It fond that the new neuron model has rich nonlinear dynamics and generate hidden attractors when changing electromagnetic induction. Neuron model as a nonlinear system, it has properties similar to a chaotic system system. So the memristive HindMarsh-Rose neuron model can be considered as a chaotic system to apply in image encryption algorithm. And an image encryption algorithm based on the memristive HindMarsh-Rose neuron model is proposed in this paper. The security analysis confirmed that enryption algorithm is well.

Topology identification and dynamical pattern recognition for Hindmarsh–Rose neuron model via deterministic learning

Studies have shown that Parkinson’s, epilepsy and other brain deficits are closely related to the ability of neurons to synchronize with their neighbors. Therefore, the neurobiological mechanism and synchronization behavior of neurons has attracted much attention in recent years. In this contribution, it is numerically investigated the complex nonlinear behaviour of the Hindmarsh–Rose neuron system through the time responses, system bifurcation diagram and Lyapunov exponent under different system parameters. The system presents different and complex dynamic behaviors with the variation of parameter. Then, the identification of the nonlinear dynamics and topologies of the Hindmarsh–Rose neural networks under unknown dynamical environment is discussed. By using the deterministic learning algorithm, the unknown dynamics and topologies of the Hindmarsh–Rose system are locally accurately identified. Additionally, the identified system dynamics can be stored and represented in the form of constant neural networks due to the convergence of system parameters. Finally, based on the time-invariant representation of system dynamics, a fast dynamical pattern recognition method via system synchronization is constructed. The achievements of this work will provide more incentives and possibilities for biological experiments and medical treatment as well as other related clinical researches, such as the quantifying and explaining of neurobiological mechanism, early diagnosis, classification and control (treatment) of neurologic diseases, such as Parkinson’s and epilepsy. Simulations are included to verify the effectiveness of the proposed method.

Synchronization of Fractional-Order Hindmarsh-Rose Neurons with Hidden Attractor via Only One Controller

Compared with the integer-order neuron model, the fractional-order neuron model can depict richer electrical activities of the neuron system and becomes a hot topic. To better understand the complex phenomenon of neuron, according to Caputo’s fractional derivative operator, the fractional-order Hindmarsh-Rose neuron model is introduced, and dynamics of it are investigated. Firstly, the hidden attractor of the proposed model is discussed via theoretical analysis and numerical simulation. Secondly, synchronization between fractional-order Hindmarsh-Rose neurons is realized by designing one controller whether the order is the same or different. Simultaneously, the impact of the order on the synchronization speed of considered systems with the same order is explored, and it is found that lower order is beneficial for speeding up synchronization. Theoretical results are confirmed via numerical simulations. Research results contribute to reveal some complex phenomena of neuron systems and control the complex dynamics effectively.