Hindmarsh-Rose Neuron Model Research Articles

A novel no-equilibrium HR neuron model with hidden homogeneous extreme multistability

• A novel no-equilibrium HR neuron model with memristive electromagnetic induction is proposed. • This memristive HR neuron model can exhibit complicated memristor initial offset boosting dynamics. • The interesting phenomenon of hidden homogeneous extreme multistability is found. • Hardware experiments are performed to verify homogeneous coexisting hidden attractors. • A pseudo-random number generator with excellent randomness is designed based on memristor initial-controlled chaotic sequences. In this paper, a novel no-equilibrium Hindmarsh-Rose (HR) neuron model with memristive electromagnetic induction is proposed. This memristive HR neuron model exhibits complex memristor initial offset boosting dynamics, from which infinitely many coexisting hidden attractors sharing the same shape but with different positions can be generated, therefore breeding the interesting phenomenon of hidden homogeneous extreme multistability. The complicated dynamical behaviors are detailedly investigated via bifurcation diagrams, Lyapunov exponents , time series, attraction basins and spectral entropy (SE) complexity. Moreover, PSIM circuit simulations and DSP hardware experiments are carried out to demonstrate the theoretical analyses and numerical simulations. Finally, a pseudorandom number generator is also designed by using the memristor initial-controlled chaotic sequences extracted from the memristive HR neuron model. The performance analysis results show that these chaotic sequences can yield pseudorandom numbers with excellent randomness, which are more suitable for chaos-based engineering applications.

Synchronization for fractional-order extended Hindmarsh-Rose neuronal models with magneto-acoustical stimulation input

We investigate the generalized projective synchronization (GPS) problem of fractional-order extended Hindmarsh-Rose (FOEHR) neuronal models with magneto-acoustical stimulation input. The improved neuronal model has advantages in depicting the biological characteristics of neurons and therefore exhibits complex firing behaviors. In addition, we consider the nonlinearity and uncertain parameters of the neuronal model as well as the unknown external disturbances, which make the synchronization control of the master-slave neuron system more difficult. For the synchronous firing rhythms of neurons, a neural network (NN) sliding mode algorithm for the FOEHR neuron system is derived by the Lyapunov approach. We use a radial basis NN to approximate the unknown nonlinear dynamics of the error system, and the adaptive parameters are robust to the approximation errors, model uncertainties and unknown external disturbances. Under the proposed control scheme, the master and slave neuron systems can achieve GPS in a finite amount of time and realize resilience for the uncertain parameters and the external disturbances. The simulation results demonstrate that the membrane potentials of the slave neuron synchronize with those of the master neuron in proportion and that the underlying synchronization errors converge towards an arbitrarily small neighborhood of zero.

Multi-scroll hidden attractor in memristive HR neuron model under electromagnetic radiation and its applications.

This paper aims to propose a novel no-equilibrium Hindmarsh-Rose (HR) neuron model with memristive electromagnetic radiation effect. Compared with other memristor-based HR neuron models, the uniqueness of this memristive HR neuron model is that it can generate multi-scroll hidden attractors with sophisticated topological structures and the parity of the scrolls can be controlled conveniently with changing the internal parameters of the memristor. In particular, the number of scrolls of the multi-scroll hidden attractors is also associated with the intensity of external electromagnetic radiation stimuli. The complex dynamics is numerically studied through phase portraits, bifurcation diagrams, Lyapunov exponents, and a two-parameter diagram. Furthermore, hardware circuit experiments are carried out to demonstrate theoretical analyses and numerical simulations. From the perspective of engineering application, a pseudo-random number generator is designed. Besides, an image encryption application and security analysis are also performed. The obtained results show that the memristive HR neuron model possesses excellent randomness and high security, which is suitable for chaos-based real-world applications.

Dynamic expression of a HR neuron model under an electric field

The movement of large amounts of ions (e.g., potassium, sodium and calcium) in the nervous system triggers time-varying electromagnetic fields that further regulate the firing activity of neurons. Accordingly, the discharge states of a modified Hindmarsh–Rose (HR) neuron model under an electric field are studied by numerical simulation. By using the Matcont software package and its programming, the global basins of attraction for the model are analyzed, and it is found that the model has a coexistence oscillation pattern and hidden discharge behavior caused by subcritical Hopf bifurcation. Furthermore, the model’s unstable branches are effectively controlled based on the Washout controller and eliminating the hidden discharge states. Interestingly, by analyzing the two-parametric bifurcation analysis, we also find that the model generally has a comb-shaped chaotic structure and a periodic-adding bifurcation pattern. Additionally, considering that the electric field is inevitably disturbed periodically, the discharge states of this model are more complex and have abundant coexisting oscillation modes. The research results will provide a useful reference for understanding the complex dynamic characteristics of neurons under an electric field.

Hopf bifurcation of the fractional-order Hindmarsh–Rose neuron model with time-delay

The Hopf bifurcation problem of the fractional-order Hindmarsh–Rose neuron model with time-delay is investigated. The effects of time-delay to the bifurcation behavior of system is considered by linearizing the system at the equilibria method. With the bifurcation theory, the condition for the existence of Hopf bifurcation is obtained when considering time-delay τ as variable parameter. As illustrated by the numerical example, the proposed theoretical results work effectively and accurately.

Synchronization of Fractional Order Neurons in Presence of Noise.

The firing rate of some biological neurons such as neocortical pyramidal neurons is consistent with fractional order derivative, and the fractional-order neuron models depict the firing rate of neurons more accurately than other integer order neuron models do. For this reason, first, the dynamical characteristics of fractional order Hindmarsh Rose (HR) neuron are investigated, here and then a two coupled neuronal system based on Hindmarsh Rose neuron is presented. The results show several differences in the dynamical cha.racteristics of integer order and fractional order Hindmarsh Rose neuron model. The integer order model shows only one type of firing characteristics when the parameter of the model remained the same. The fractional-order model depicts several dynamical behaviors even for the same parameters as the order of the fractional operator is varied with the same parameter values. The firing frequency increases as the order of the fractional operator decreases. The fractional-order is therefore key in determining the firing characteristics of biological neuron models. A linearized model of HR neuron is also given for hardware resource minimizations and to implement this neuronal network on a large scale. A synchronized system of two fractional-order fractional Hindmarsh-Rose (HR) neurons in the presence of noise is also presented. The dynamical characteristics of the modified coupled neuron are determined by the parameters of the neuron model and the coupling function. The robustness of the network in the presence of noise is verified by both amplitude and phase synchronization techniques. A simplification of the coupling function is also presented to reduce the hardware cost. The synchronization results show that the model can produce the desired behavior with acceptable error.

Dynamic response and control of neuros based on electromagnetic field theory

The time-varying electric fields generated by continuously pumping and transmitting calcium, potassium and sodium ions in cells not only affect the discharge activity of neurons, but also induce time-varying magnetic fields to further regulate the fluctuation of ions. According to the Maxwell's electromagnetic field theory, time-varying electric field and magnetic field can stimulate each other in the electrophysiological environment inside and outside the cells to produce electromagnetic field. In order to explore the discharge rhythm transition of neurons under the influence of electromagnetic fields, a five-dimensional (5D) HR neuron model (EMFN model for short) is established by introducing magnetic flux variable and electric field variable into a three-dimensional (3D) Hindmarsh-Rose (HR) neuron model. Firstly, the equilibrium distribution and global bifurcation properties of EMFN model are analyzed by Matcont software, then the existence of subcritical Hopf bifurcation, hidden discharge, coexistence of periodic discharge and resting state are found and analyzed. Secondly, by using the tools of two-parameter and one-parameter bifurcation, ISI bifurcation and the maximum Lyapunov exponent for numerical simulation, the period-adding bifurcation with and without chaos, mixed mode discharge and coexistence mode discharge in the EMFN model are analyzed in detail. At the same time, the transition law of discharge rhythm with the influence of electric field and magnetic field intensity is revealed. Finally, the Washout controller is used to convert the subcritical Hopf bifurcation into supercritical Hopf bifurcation, so the topological structure of EMFN model near the bifurcation point is changed for eliminating the hidden discharge. The research results of this paper confirm that the novel neuron model has rich discharge rhythm, which will affect the information transmission and coding, and provide some ideas for improving the neuron models, revealing the influence of electromagnetic field on biological nervous system, and exploring the pathogenic mechanism of some neurological diseases.

Evaluating the effectiveness of several synchronization control methods applying to the electrically and the chemically coupled hindmarsh-rose neurons

This study focuses on the synchronization control between the coupled neurons. The achievements of several synchronization control methods have been checked by evaluating the effects of the synaptic coupling weight alteration on the synchronization. Here, a neural ensemble has been constructed by utilizing the Hindmarsh Rose (HR) Neuron Model. The HR neurons have been linked to each other with the bidirectional coupling. The synchrony or the asynchrony states between these coupled neurons have been observed by using the standard deviation results. Here, firstly, the electrically and the chemically coupled HR neurons have been handled without using any control method, separately and the effects of the synaptic coupling weight alteration on the synchronic firing have been assessed by considering the features of the coupling types. Then, while the electrically coupled HR neurons are generally preferred in the available synchronization control studies; the Lyapunov, the back-stepping, and the feedback synchronization control methods have been adapted to both the electrically and the chemically coupled HR neurons. Thus, a remarkable contribution has been provided to the limited number of studies, which are about the synchronization control of the chemically coupled HR neurons. Also, the synchronization control between the electrically or the chemically coupled HR neurons has been provided by the back-stepping method for the first time. Finally, the differences between the membrane potentials of the coupled neurons have been calculated by utilizing an alternative error function. Since this function calculates the amplitude and the phase errors, separately; the effectiveness of these methods can be evaluated correctly in terms of the performing the minimum differences between the neural dynamics.

DNA encoding for RGB image encryption with memristor based neuron model and chaos phenomenon

In this paper, a deoxyribonucleic acid (DNA) encoding algorithm with memristor based chaotic system is proposed. The study includes four different topics together which are neuron model, circuit design with memristor, chaos theory and DNA encoding based cryptology. Operational Transconductance Amplifier (OTA) based memristive Hindmarsh-Rose neuron model using three-dimensional differential equations is employed in chaotic circuit structure. Cryptographic keys obtained from suggested three-dimensional continuous time chaotic system are utilized in order to encrypt and decrypt the image. Diffusion and confusion methods are essential for cryptology. Circular shift algorithm is used as a diffusion method and logical XOR operation is preferred as a confusion method on image. Security analyses such as histogram, correlation coefficient of two adjacent pixels, information entropy, differential attack, the secret key sensitivity and the initial condition sensitivity analyses have been performed to show the performance of the encryption scheme. Analyses results reveal that the proposed encryption is robust against statistical and differential attacks and this chaotic system also exhibits the behaviour of sensitive dependence on the secret key and the initial condition. The chaotic behaviour of the suggested system has been given performing numerical analyses in MATLAB environment and doing the analog circuit analyses with LTspice simulation software.

Complex bifurcation analysis and synchronization optimal control for Hindmarsh-Rose neuron model under magnetic flow effect.

In this contribution, the complex behaviour of the Hindmarsh-Rose neuron model under magnetic flow effect (mHR) is investigated in terms of bifurcation diagrams, Lyapunov exponent plots and time series when varying only the electromagnetic induction strength. Some exciting phenomena are found including, for instance, various firings patterns by applying appropriate magnetic strength and Hopf-fold bursting through fast-slow bifurcation. In addition to this, the interesting phenomenon of Hopf bifurcation is examined in the model. Thus, we prove that Hopf bifurcation occurs in this memristor-based HR neuron model when an appropriately chosen magnetic flux varies and reaches its critical value. Furthermore, one of the main results of this work was the optimal control approach to realize the synchronization of two mHR. The main advantage of the proposed optimal master-slave synchronization from a control point of view is that, in the practical application, the electrical activities (quiescent, bursting, spiking, period and chaos states) of a neuron can be regulated by a pacemaker (master) associated with biological neuron (slave) to treat some diseases such as epilepsy. A suitable electronic circuit is designed and used for the investigations. PSpice based simulation results confirm that the electrical activities and synchronization between coupled neurons can be modulated by electromagnetic flux.

Wave propagation and spiral wave formation in a Hindmarsh–Rose neuron model with fractional-order threshold memristor synaps

In this paper, a modified Hindmarsh–Rose neuron model is presented, which has a fractional-order threshold magnetic flux. The dynamics of the model is investigated by bifurcation diagrams and Lyapunov exponents in two cases of presence and absence of the external electromagnetic induction. Then the emergence of the spiral waves in the network of the proposed model is studied. To find the effects of different factors on the formation and destruction of spiral waves, the external current, the coupling strength and the external stimuli amplitude are varied. It is observed that all of these parameters have significant impacts on the spiral waves. Furthermore, the external electromagnetic induction influences the existence of spiral waves in specific external current values.

Homoclinic organization in the Hindmarsh-Rose model: A three parameter study.

Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book," gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme.

Bifurcations of Negative Responses to Positive Feedback Current Mediated by Memristor in a Neuron Model with Bursting Patterns

In contrast to traditional viewpoint that positive feedback current always enhances neural firing activities, in the present paper, we identify that the excitatory feedback current mediated by memristor can induce negative responses of bursting patterns, which can be well interpreted with bifurcations. For the Hindmarsh–Rose neuron model without memristor, the period-adding bifurcations of bursting patterns and increase of firing frequency can be induced by increasing the excitatory effect of the background current. After introducing a memristor to simulate the biological synapse or electromagnetic induction effect, inverse period-adding or complex bifurcations of bursting patterns are induced by the excitatory feedback current mediated by the memristor. The number of spikes per burst becomes smaller and the firing frequency becomes lower when increasing the positive feedback gain. Such negative responses of bursting patterns to the positive feedback current are demonstrated in a circuit designed with Digital Signal Processor systems of the MatLab software. Furthermore, the underlying bifurcation mechanism of the negative responses to the positive feedback is acquired with fast–slow variable dissection method. With increasing feedback gain, the initial phase of the burst, which corresponds to a saddle-node bifurcation point of the fast subsystem, delays, while the termination phase of the burst, which corresponds to a saddle-homoclinic bifurcation point, remains unchanged. Therefore, the burst becomes narrower with increasing feedback gain, which leads to decrease in the number of spikes within a burst and decrease in firing frequency. The results present a paradoxical nonlinear phenomenon and the dynamical mechanism, which is helpful for understanding the functions of memristor and roles of the electromagnetic induction current.

Adaptive Synchronization of Fractional-Order Coupled Neurons Under Electromagnetic Radiation

In this paper, we investigate the dynamical characteristics of four-variable fractional-order Hindmarsh–Rose neuronal model under electromagnetic radiation. The numerical results show that the improved model exhibits more complex dynamical behavior with more bifurcation parameters. Meanwhile, based on the fractional-order Lyapunov stability theory, we propose two adaptive control methods with a single controller to realize chaotic synchronization between two coupled neurons. Finally, numerical simulations show the feasibility and effectiveness of the presented method.

Route to hyperchaos and chimera states in a network of modified Hindmarsh-Rose neuron model with electromagnetic flux and external excitation

Analyzing the chaos and bursting phenomenon of neurons has been of interest in the past decade. In this paper, we discuss an extended Hindmarsh-Rose neuron model by taking into consideration the slowly interacting cell phenomenon due to the calcium ions. In the extended model, we consider the effect of an external forcing current, and the electromagnetic coupling between the magnetic flux and the membrane potential of the neuron. We analyze the modified neuron model in the presence of periodic and quasi-periodic excitations. A more complex chaotic behavior (hyperchaos) is identified in the neuron model. The results also demonstrate the multistable nature, which was not explored earlier. To discuss the dynamical behavior of the modified neuron in a network, we construct a ring network of neurons and capture the spatiotemporal patterns of the neuron in the network, in the presence of different excitations.

Various firing activities and finite-time synchronization of an improved Hindmarsh-Rose neuron model under electric field effect.

Nowadays, it is important to realize systems that can model the electrical activity of neurons taking into account almost all the properties of the intracellular and extracellular environment in which they are located. It is in this sense that we propose in this paper, the improved model of Hindmarsh-Rose (HR) which takes into account the fluctuation of the membrane potential created by the variation of the ion concentration in the cell. Considering the effect of the electric field that is produced on the dynamic behavior of neurons, the essential properties of the model such as equilibrium point and its stability, bifurcation diagrams, Lyapunov spectrum, frequency spectra, time series of the membrane potential and phase portraits are thoroughly investigated. We thus prove that Hopf bifurcation occurs in this system when the parameters are chosen appropriately. We also observe that by varying specific parameters of the electric field, the model presents a very rich and striking event, namely hysteresis phenomenon, which justifies the coexistence of multiple attractors. Besides, by applying a suitable sinusoidal excitation current, we prove that the neuron under electric field effect can present several important electrical activities including quiescent, spiking, bursting and even chaos. We propose the improved HR model under electric field effect (mHR) to study the finite-time synchronization between two neurons when performing synapse coupling across the membrane potential and the electric field coupling. As a result, we find that the synchronization between the two neurons is weakly influenced by the variation of the intensity of the electric field coupling while it is strongly impacted when the intensity of the synapse coupling is modified. From these results, it is obvious that the electric field can be another effective bridge connection to encourage the exchange and coding of the signal. Using the finite-time synchronization algorithm, we theoretically quantify the synchronization time between these neurons. Finally, Pspice simulations are presented to show the feasibility of the proposed model as well as that of the developed synchronization strategy.

Adaptive Fuzzy Control for the Generalized Projective Synchronization of Fractional-Order Extended Hindmarsh-Rose Neurons

We investigate the generalized projective synchronization (GPS) control of fractional-order extended Hindmarsh-Rose (FOEHR) neuronal models with transcranial magneto-acoustical stimulation (TMAS) input. This improved neuronal model has advantages in describing the complex firing characteristics of neurons stimulated by alternating current. In this study, a master-slave neuron system consisting of two FOEHR neuronal models is assumed to be subject to uncertain model parameters and unknown external disturbances. To quantify the GPS error, we design a new error variable based on the properties of the fractional-order derivative and construct a related GPS error system. Fuzzy logic systems are introduced to approximate the unknown nonlinear dynamics of the error system. To ensure the synchronous firing rhythms of the master-slave neuron system, an adaptive fuzzy control algorithm is proposed under the Lyapunov approach, in which the adaptive parameters are robust to the estimation errors. By choosing the appropriate design parameters, the proposed control scheme enables the master-slave neuron system to achieve GPS in a finite amount of time and to be resilient to uncertain parameters and unknown disturbances. The simulation results demonstrate that after the designed control inputs are implemented, the states of the slave neuron synchronize with those of the master neuron in specified proportions, and the corresponding synchronization error converges towards an arbitrarily small neighborhood of zero.

FPGA Realization of Fractional Order Neuron

In this paper fractional Hindmarsh Rose (HR) neuron, which mimics several behaviors of a real biological neuron is implemented on field programmable gate array (FPGA). The results show several differences in the dynamic characteristics of integer and fractional order Hindmarsh Rose neuron models. The integer order model shows only one type of firing characteristics when the parameters of model remains same. The fractional order model depicts several dynamical behaviors even for the same parameters as the order of the fractional operator is varied. The firing frequency increases when the order of the fractional operator decreases. The fractional order is therefore key in determining the firing characteristics of biological neurons. To implement this neuron model first the digital realization of different fractional operator approximations are obtained, then the fractional integrator is used to obtain the low power and low cost hardware realization of fractional HR neuron. The fractional neuron model has been implemented on a low voltage and low power circuit and then compared with its integer counter part. The hardware is used to demonstrate the different dynamical behaviors of fractional HR neuron for different type of approximations obtained for fractional operator in this paper. A coupled network of fractional order HR neurons is also implemented. The results also show that synchronization between neurons increases as long as coupling factor keeps on increasing.

Synchronization of Hindmarsh Rose Neurons

Modeling and implementation of biological neurons are key to the fundamental understanding of neural network architectures in the brain and its cognitive behavior. Synchronization of neuronal models play a significant role in neural signal processing as it is very difficult to identify the actual interaction between neurons in living brain. Therefore, the synchronization study of these neuronal architectures has received extensive attention from researchers. Higher biological accuracy of these neuronal units demands more computational overhead and requires more hardware resources for implementation. This paper presents a two coupled hardware implementation of Hindmarsh Rose neuron model which is mathematically simpler model and yet mimics several behaviors of a real biological neuron. These neurons are synchronized using an exponential function. The coupled system shows several behaviors depending upon the parameters of HR model and coupling function. An approximation of coupling function is also provided to reduce the hardware cost. Both simulations and a low cost hardware implementations of exponential synaptic coupling function and its approximation are carried out for comparison. Hardware implementation on field programmable gate array (FPGA) of approximated coupling function shows that the coupled network produces different dynamical behaviors with acceptable error. Hardware implementation shows that the approximated coupling function has significantly lower implementation cost. A spiking neural network based on HR neuron is also shown as a practical application of this coupled HR neural networks. The spiking network successfully encodes and decodes a time varying input.

ESTUDO DO COMPORTAMENTO DINÂMICO DO MODELO NEURONAL DE HINDMARSH-ROSE

Based on the Hindmarsh-Rose (RH) neuronal model for nerve impulse transmission, this paper aims to study the properties and dynamic behavior of the non-linear chaotic system that describes neuronal bursting in a single neuron. On the part of bioengineering, there is great motivation in the study of the HR model because it is well representative of the biological neuron, being able to simulate several behaviors of a real neuron, among them periodic, aperiodic and chaotic behavior. The literature suggests that the chaotic behaviorrepresents in the human being the epileptic or convulsive state. Through computer simulations, considering the system parameters, it was analyzed that the stability is highly sensitive to the initial conditions and producing oscillations, more so, when the oscillation increases the random behavior tends to increase making the system unpredictable.