Hindmarsh-Rose Neurons Research Articles

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.

Coexistence of Attractors and Its Control with Selection of a Desired Attractor in a Model of Extended Hindmarsh–Rose Neuron with Nonlinear Smooth Fitting Function: Microcontroller Implementation

Coexistence of Attractors and Its Control with Selection of a Desired Attractor in a Model of Extended Hindmarsh–Rose Neuron with Nonlinear Smooth Fitting Function: Microcontroller Implementation

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.

Coexistence of infinitely many patterns and their control in heterogeneous coupled neurons through a multistable memristive synapse.

The phenomenon of hidden heterogeneous extreme multistability is rarely reported in coupled neurons. This phenomenon is investigated in this contribution using a model of a 2D FitzHugh-Nagumo neuron coupled with a 3D Hindmarsh-Rose neuron through a multistable memristive synapse. The investigation of the equilibria revealed that the coupled neuron model is equilibrium free and, thus, displays a hidden dynamics. Some traditional nonlinear analysis tools are used to demonstrate that the heterogeneous neuron system is able to exhibit the coexistence of an infinite number of electrical activities involving both periodic and chaotic patterns. Of particular interest, a noninvasive control method is applied to suppress all the periodic coexisting activities, while preserving only the desired chaotic one. Finally, an electronic circuit of the coupled neurons is designed in the PSpice environment and used to further support some results of the theoretical investigations.

Hidden coexisting firing patterns and bubble-like bifurcation in HR neuron model under electromagnetic induction

When ions enter or leave cell membrane, action potentials are generated. The fluctuations or changes in action potential of neurons result in an extra electromagnetic field according to the Faraday’s law of induction. As a result, the electromagnetic field will contribute to generation of membrane potential as a feedback effect, which should not be ignored in neuron modeling. In this paper, a three-dimensional improved Hindmarsh–Rose neuron model considering the influence of electromagnetic induction is proposed, in which magnetic flux is used to describe the effect of electromagnetic induction that contributes to generation of membrane potential as a feedback memristive current. The improved neuron model has no equilibrium point and can present different types of hidden coexisting firing patterns. Besides, the neuron model can show bubble-like bifurcation modes by varying system parameters, which has not been reported in other Hindmarsh–Rose neuron models. Extensive numerical simulations are carried out and bifurcation analyses indicate that the proposed neuron model has a wide variety of hidden coexisting firing patterns and bifurcation modes. Finally, equivalent circuit is designed and performed in Pspice, and the simulation results are consistent well with the numerical ones.

Spiral waves and their characterization through spatioperiod and spatioenergy under distinct excitable media

The external periodic stimulus is typically used to activate each cell in an excitable media which in turn triggers the stable pulse or target waves in the network. In this work, the spatiotemporal pattern in excitable media is investigated by applying a periodic stimulus to a specific cell site. To do so, three distinct arrays of single-layer networks, namely the modified Hindmarsh-Rose neuron, the hybrid neuron, and the Morris-Lecar neuron models are considered. The existence of spiral waves and their characteristics are studied with periodic excitation. We considered the modified Hindmarsh-Rose neuron model, hybrid neuron, and Morris-Lecar neuron model for the investigation. The results obtained are highlighting the existence of disordered broken spiral seeds at low coupling values and the formation of spiral waves as coupling strength increased to higher levels. Importantly, the characteristics of the nodes in the network are manifested using the spatioperiod and spatioenergy. Furthermore, we discover that the core of the spiral has low excitability and energy when compared to the other nodes in the spiral. Notably, the spatioenergy clearly shows the actual position of the spiral core. Finally, we confirmed the spatiotemporal patterns and their characteristics by extending the network to the two and three-layer Morris-Lecar neural systems.

Predefined-time synchronization of 5D Hindmarsh–Rose neuron networks via backstepping design and application in secure communication

In this paper, the fast synchronization problem of 5D Hindmarsh–Rose neuron networks is studied. Firstly, the global predefined-time stability of a class of nonlinear dynamical systems is investigated under the complete beta function. Then an active controller via backstepping design is proposed to achieve predefined-time synchronization of two 5D Hindmarsh–Rose neuron networks in which the synchronization time of each state variable of the master-slave 5D Hindmarsh–Rose neuron networks is different and can be defined in advance, respectively. To show the applicability of the obtained theoretical results, the designed predefined-time backstepping controller is applied to secure communication to realize asynchronous communication of multiple different messages. Three numerical simulations are provided to validate the theoretical results.

Energy analysis of bursting Hindmarsh-Rose neurons with time-delayed coupling

Mathematical modeling is an important tool to study the role of delay in neural systems and to evaluate its effects on the signaling activity of coupled neurons. Models for delayed neurons are often used to represent the dynamics of real neurons, but rarely to assess the energy required to maintain these dynamics. In this work, we address these questions from an energy perspective by considering a pair of Hindmarsh-Rose burst neurons coupled by reciprocal time-delayed coupling with electrical and chemical synapses. We examine the average energy consumption required to maintain cooperative behavior and quantify the contribution of synapses to total energy consumption. We show that unlike electrical coupling, where the time delay appears to reduce the instantaneous average relative weight of the synaptic contribution, in chemical coupling this average synaptic contribution appears to be much higher in delayed coupling than in instantaneous coupling, except at certain values of coupling strength where the instantaneous synaptic contribution is more important.

Firing activities in a fractional-order Hindmarsh–Rose neuron with multistable memristor as autapse

Considering the fact that memristors have the characteristics similar to biological synapses, a fractional-order multistable memristor is proposed in this paper. It is verified that the fractional-order memristor has multiple local active regions and multiple stable hysteresis loops, and the influence of fractional-order on its nonvolatility is also revealed. Then by considering the fractional-order memristor as an autapse of Hindmarsh–Rose (HR) neuron model, a fractional-order memristive neuron model is developed. The effects of the initial value, external excitation current, coupling strength and fractional-order on the firing behavior are discussed by time series, phase diagram, Lyapunov exponent and inter spike interval (ISI) bifurcation diagram. Three coexisting firing patterns, including irregular asymptotically periodic (A-periodic) bursting, A-periodic bursting and chaotic bursting, dependent on the memristor initial values, are observed. It is also revealed that the fractional-order can not only induce the transition of firing patterns, but also change the firing frequency of the neuron. Finally, a neuron circuit with variable fractional-order is designed to verify the numerical simulations.

Coexisting multiple firing behaviors of fractional-order memristor-coupled HR neuron considering synaptic crosstalk and its ARM-based implementation

The crosstalk between synapses has a far-reaching impact on the specificity of synaptic communication in human brain. Therefore, the influence of synaptic crosstalk on the dynamic behaviors of neural network cannot be ignored. In this paper, we investigate the dynamics of fractional-order (FO) memristor-coupled Hindmarsh-Rose (HR) neuron model considering synaptic crosstalk. Firstly, the equilibrium points and stability are investigated via qualitative analysis, from which it can be obtained that the number and stability of equilibrium points have diversity for different crosstalk strength parameters. Secondly, the global coexistence of multiple firing patterns is reflected by phase diagrams, Lyapunov exponent spectrums and bifurcation diagrams. The local attraction basins reveal the multi-stability phenomenon related to the initial value. The parameter related firing behaviors are described by the two-parameter bifurcation diagram. In particular, the model shows the offset boosting control method for a single variable. Then, spectral entropy (SE) and C0 complexity chaos diagram are used to observe the change of system complexity when two parameters change simultaneously. Finally, the digital implementations based on ARM are given to verify the consistency with the numerical simulation results.

Coherence-resonance chimeras in coupled HR neurons with alpha-stable Lévy noise

Abstract In this paper, we have investigated the collective dynamical behaviors of a network of identical Hindmarsh–Rose neurons that are coupled under small-world schemes upon the addition of α-stable Lévy noise. According to the firing patterns of each neuron, we distinguish the neuronal network into spike state, burst state and spike-burst state coexistence of the neuron with both a spike firing pattern and a burst firing pattern. Moreover, the strength of the burst is proposed to identify the firing states of the system. Furthermore, an interesting phenomenon is observed that the system presents coherence resonance in time and chimera states in space, namely coherence-resonance chimeras (CRC). In addition, we show the influences of α-stable Lévy noise (noise intensity and stable parameter) and the small-world network (the rewiring probability) on the spike-burst state and CRC. We find that the stable parameter and noise intensity of the α-stable noise play a crucial role in determining the CRC and spike-burst state of the system.

Stability analysis of the Hindmarsh–Rose neuron under electromagnetic induction

We consider the Hindmarsh–Rose neuron model modified by taking into account the effect of electromagnetic induction on membrane potential. We study the impact of the magnetic flux on the neuron dynamics, through the analysis of the stability of fixed points. Increasing magnetic flux reduces the number of equilibrium points and favors their stability. Therefore, electromagnetic induction tends to regularize chaotic regimes and to affect regular and quasi-regular ones by reducing the number of spikes or even destroying the oscillations.

Collective dynamics of neural network with distance dependent field coupling

In this work, we analyse the collective dynamics of Hindmarsh-Rose neurons with distance dependent memristor and mean field coupling, realised by power law exponent. The memristor characterises the electromagnetic field induced by the magnetic flux. The mean field represents the global effect in the neurons which are usually influenced by enormous number of neighbours. The uniform global mean field drives the memristive neural network from Amplitude Death to synchronised bursting state. Desynchronised, Mixed Oscillatory, imperfect and complete synchronised states are attained by the network due to the interplay between memristor and uniform mean field coupling. The synchronisation scenario has been analysed in distance dependent memristive and mean field network, by varying the values of power law exponent. The distance dependent mean field coupling induces chimera and multichimera states in the system. The strength of incoherence and discontinuity measure are calculated to distinguish incoherent, chimera, multichimera and coherent states. Coherent states are unattainable for high values of power law exponent. The existence of chimera states in system with long range interactions are verified by parameter space. This work may give better understanding about the different aspects of synchronisation involved in neural networks with distance dependent coupling and proves that the non local synaptic coupling is not a prerequisite to realise chimera states.

Nonlinear dynamics, chaos and control of the Hindmarsh-Rose neuron model

Mathematics has changed over time to comprise interdisciplinary fields of research, and considering this, biomathematics has arisen as an interface study. In this work, we analyze the dynamical behavior of the Hindmarsh-Rose (HR) neuron model, which describes the neuronal bursting in a single neuron. A stability study through the Lyapynov exponents method is proposed and evidence of a chaotic dynamics is presented. Therefore, a control design based on the State-Dependent Ricatti Equation (SDRE) is proposed aiming to reduce the oscillation of the system to a desired orbit. The results show that the controller is efficient and robust as a method for preventing epileptic seizures.

Effects of memristive autapse under field coupling on transition of collective dynamics in neural hypernetworks

Electrical and chemical synapses play important role in neural system. Recently, a neural hypernetwork by taking a unidirectional chemical synapse and bidirectional electrical synapse has been proposed. Electromagnetic radiation is also important to dynamical response and mode transition in firing activities of neurons. In this paper, we first investigate the effect of memristive autapse on transformation of firing activity modes on an individual Hindmarsh-Rose (HR) neuron. Different periodic firing activity patterns and chaotic firing activity modes are observed, and mode transition between these firing patterns can be induced by adjusting the value of delay time of memristive autapse. Then, we consider a HR neural hypernetwork with memristive autapse under field coupling. Collective behavior is studied numerically, and different collective dynamics are observed. Chaotic states, chimera states, imperfect traveling chimera states, traveling chimera states, synchronous coherent states are found when coupling parameters and delay time of memristive autapse are varied. These findings indicate that memristive autapse under field coupling plays a crucial role in neural networks.

Numerical study of laser synapse connecting Hindmarsh–Rose neurons

Numerical study of laser synapse connecting Hindmarsh–Rose neurons

Synchronization in Hindmarsh–Rose neurons subject to higher-order interactions

Higher-order interactions might play a significant role in the collective dynamics of the brain. With this motivation, we here consider a simplicial complex of neurons, in particular, studying the effects of pairwise and three-body interactions on the emergence of synchronization. We assume pairwise interactions to be mediated through electrical synapses, while for second-order interactions, we separately study diffusive coupling and nonlinear chemical coupling. For all the considered cases, we derive the necessary conditions for synchronization by means of linear stability analysis, and we compute the synchronization errors numerically. Our research shows that the second-order interactions, even if of weak strength, can lead to synchronization under significantly lower first-order coupling strengths. Moreover, the overall synchronization cost is reduced due to the introduction of three-body interactions if compared to pairwise interactions.