Hindmarsh-Rose Model Research Articles

Inhibitory electromagnetic induction current induces enhancement instead of reduction of neural bursting activities

Nonlinear dynamics play an important role in identifying complex behaviors, dynamical mechanisms, and physiological functions of the neural firing activities. In contrast to the common viewpoint that the inhibitory effect should induce the reduction of neural firing activities, the present paper presents a novel example that inhibitory effect mediated by the electromagnetic induction current can induce the enhancement of bursting activities, which is characterized by the increase of firing frequency and period-adding bifurcations of bursting patterns, i.e. more spikes appearing within a burst. With the fast/slow variable dissection method and bifurcation analysis of the fold/homoclinic bursting pattern of the Hindmarsh–Rose model, the dynamical mechanism of the enhanced bursting activities induced by inhibitory effect is acquired and is different from the one induced by excitatory effect. With increasing the inhibitory effect of induction current, more spikes appear in a burst due to that the burst becomes wider, which is induced by the left shift of the initial phase of the burst, i.e. the fold/saddle-node bifurcation point of equilibrium of the fast subsystem. However, with increasing the excitatory effect of the stimulation, the period of limit cycle/spiking of the fast subsystem becomes short, which leads to that the interspike interval within a burst becomes short and then the burst contains more spikes. Furthermore, the enhancement of bursting activities induced by the inhibitory and excitatory effects is verified in a digital circuit which is designed in the Digital Signal Processor environment of MATLAB system. The result provides a novel example of nonlinear phenomenon contrast to the traditional viewpoint of the inhibitory effect and implies that the inhibitory induction current may be used as a possible measure to modulate the firing frequency and bursting patterns.

Unstable discrete modes in Hindmarsh–Rose neural networks under magnetic flow effect

Abstract The competitive effect between electric and magnetic flux couplings is used, in the context of modulational instability, to describe the collective dynamics in a modified Hindmarsh–Rose neural networks. The multiple-scale expansion is utilized to reduce the system to a nonlinear differential-difference equation, whose plane wave solutions are found to be unstable for some values of parameters. Particular interest is given to the influence of changing both the electric and magnetic coupling strengths, and confirmation of analytical results is given via numerical integration of the generic Hindmarsh–Rose model. The model presents a rich variety of spatiotemporal patterns propagating in the network, as the result of the interplay between nonlinear and dispersive effects. The electromagnetic induction appears to be responsible for regular bursting patterns and synchronous states in the network. With increasing the electric coupling, full synchronization is difficult to realize and irregular spatiotemporal patterns of action potentials are predominant.

Birth and death of spiral waves in a network of Hindmarsh–Rose neurons with exponential magnetic flux and excitable media

Spiral waves are spatiotemporal patterns, observed in complex networks of biological oscillators such as heart and brain. In this paper, at first a modified Hindmarsh–Rose model with external electromagnetic radiation and discontinuous exponential flux coupling is introduced. Then a 2-dimensional network of the proposed model is constructed. The spatiotemporal patterns of the network are investigated in two different cases of periodic and quasi-periodic external magnetic excitations, by varying the external force parameters. As a consequence, the effective factors on the formation and elimination of spiral waves are reported.

Bifurcation analysis and diverse firing activities of a modified excitable neuron model.

Electrical activities of excitable cells produce diverse spiking-bursting patterns. The dynamics of the neuronal responses can be changed due to the variations of ionic concentrations between outside and inside the cell membrane. We investigate such type of spiking-bursting patterns under the effect of an electromagnetic induction on an excitable neuron model. The effect of electromagnetic induction across the membrane potential can be considered to analyze the collective behavior for signal processing. The paper addresses the issue of the electromagnetic flow on a modified Hindmarsh-Rose model (H-R) which preserves biophysical neurocomputational properties of a class of neuron models. The different types of firing activities such as square wave bursting, chattering, fast spiking, periodic spiking, mixed-mode oscillations etc. can be observed using different injected current stimulus. The improved version of the model includes more parameter sets and the multiple electrical activities are exhibited in different parameter regimes. We perform the bifurcation analysis analytically and numerically with respect to the key parameters which reveals the properties of the fast-slow system for neuronal responses. The firing activities can be suppressed/enhanced using the different external stimulus current and by allowing a noise induced current. To study the electrical activities of neural computation, the improved neuron model is suitable for further investigation.

Canards Existence in the Hindmarsh–Rose model

In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only onefastvariable and, on the other hand for four-dimensional singularly perturbed systems with twofastvariables [J.M. Ginoux and J. Llibre,Qual. Theory Dyn. Syst.15(2016) 381–431; J.M. Ginoux and J. Llibre,Qual. Theory Dyn. Syst.15(2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with twofastvariables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability offolded singularities(pseudo singular pointsin this case) of thenormalized slow dynamicsdeduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model.

Electromagnetic induction on a map-based action potential model

Neurons and cardiac cells are known to be susceptible to electromagnetic radiation. Although many mathematical models exist to represent these cells, only recently there was an effort to include the electromagnetic induction on the membrane potential equations. In this paper, we investigate the effects of the induction on the logistic KTz, a computationally efficient map-based action potential model, and compare them to the more widely used Hindmarsh–Rose model. We study the effects of a self-induced current on a single cell and the synchronization of cells coupled through an induction current caused by the magnetic flux of the neighbor. We also study the emergence of aperiodic behaviors and the presence of chaos, as an effect of the inclusion of the induction. Besides, we use a simple network of KTz elements to show that the electromagnetic induction is relevant for the study of pattern formation. Additionally, we report for the first time the presence of cardiac spikes in the Hindmarsh–Rose model. Our results demonstrate the importance of implementing the induction current on different models and we provide a computationally efficient alternative to better understand how the induction acts on neuronal and cardiac cells.

Neuron dynamics variability and anomalous phase synchronization of neural networks.

Anomalous phase synchronization describes a synchronization phenomenon occurring even for the weakly coupled network and characterized by a non-monotonous dependence of the synchronization strength on the coupling strength. Its existence may support a theoretical framework to some neurological diseases, such as Parkinson's and some episodes of seizure behavior generated by epilepsy. Despite the success of controlling or suppressing the anomalous phase synchronization in neural networks applying external perturbations or inducing ambient changes, the origin of the anomalous phase synchronization as well as the mechanisms behind the suppression is not completely known. Here, we consider networks composed of coupled neurons in a small-world topology for two well known neuron models, namely, the Hodgkin-Huxley-like and the Hindmarsh-Rose models, both displaying the anomalous phase synchronization regime. We show that the anomalous phase synchronization may be related to the individual behavior of the coupled neurons; particularly, we identify a strong correlation between the behavior of the inter-bursting-intervals of the neurons, what we call neuron variability, to the ability of the network to depict anomalous phase synchronization. We corroborate the ideas showing that external perturbations or ambient parameter changes that eliminate anomalous phase synchronization and at the same time promote small changes in the individual dynamics of the neurons, such that an increasing individual variability of neurons implies a decrease of anomalous phase synchronization. Finally, we demonstrate that this effect can be quantified using a well known recurrence quantifier, the "determinism." Moreover, the results obtained by the determinism are based on only the mean field potential of the network, turning these measures more suitable to be used in experimental situations.

Elimination of spiral waves in excitable media by magnetic induction

The formation of spiral waves in excitable media is a fascinating example of the beauty of nonlinear dynamics in spatiotemporal systems. Apart from the beauty of the patterns, the subject also has many practical application. For example, the emergence of spiral waves in cardiac tissue can lead to arrhythmias. Cortical spiral waves are also involved in epileptic seizures. Motivated by this, we here study the effects of magnetic induction on the formation of spiral waves in excitable media. An external sinusoidal magnetic induction with different amplitudes and angular frequencies is applied in order to study whether spiral waves could be eliminated. We use a network of coupled neurons as a model for the excitable medium. The four-variable magnetic Hindmarsh–Rose model is used for the local dynamics of each isolated neuron. The distribution of the cell membrane potential over time, affected by magnetic induction, is determined and the results are depicted as snapshots of the 2D network. Our research reveals that the continuance of rotating spiral seeds is impaired by high-amplitude magnetic induction. Moreover, we show that low-frequency induction is not capable of breaking the reorganizing rhythm of the spiral seeds, while much higher frequencies can be too fast to overcome this special rhythm.

Period-adding bifurcation and chaos in a hybrid Hindmarsh–Rose model

Recently, the hybrid neuron models which combine the basic neuron models with impulsive effect(the state reset process) had been proposed, however, the preset value and the reset value of membrane potential were both fixed constants in the known models. In this paper, the Hindmarsh–Rose neuron model with nonlinear reset process is presented where the preset value and the reset value of membrane potential are variable constants. We conduct a qualitative analysis in the vicinity of the equilibrium point or the limit cycle of the proposed system by using the theories of impulsive semi-dynamical systems. Firstly, the more detailed impulsive set and phase set are given, then using the fixed point of Poincaré map, the existences of order-1and order-k(k>=2) period solutions are investigated subsequently. Furthermore, numerical investigations including period-adding bifurcation, multiple attractors coexistence, switch-like behavior are presented to further describe the bifurcation and chaos phenomena. Finally, the obtained results and possible applications of the proposed model are elaborated.

Asymmetry in electrical coupling between neurons alters multistable firing behavior.

The role of asymmetry in electrical synaptic connection between two neuronal oscillators is studied in the Hindmarsh-Rose model. We demonstrate that the asymmetry induces multistability in spiking dynamics of the coupled neuronal oscillators. The coexistence of at least three attractors, one chaotic and two periodic orbits, for certain coupling strengths is demonstrated with time series, phase portraits, bifurcation diagrams, basins of attraction of the coexisting states, Lyapunov exponents, and standard deviations of peak amplitudes and interspike intervals. The experimental results with analog electronic circuits are in good agreement with the results of numerical simulations.

AC-induced coexisting asymmetric bursters in the improved Hindmarsh–Rose model

In this paper, an external alternating current (AC) is injected into the Hindmarsh–Rose (HR) neuron model to imitate the periodic stimulus effect on the membrane potential in the axon of a neuron and then an improved HR model is proposed. The AC equilibrium point and its stability in the proposed model are investigated theoretically, and the AC-induced coexisting behaviors of asymmetric bursters are revealed by MATLAB numerical simulations. Due to the injection of the AC item, the stability distribution of the unique AC equilibrium point in the improved HR model varies between unstable and stable intervals with the periodic evolution of the time, which leads to the emergence of various types of coexisting asymmetric bursters under different initial conditions of the bursting variable, such as hyperchaotic and periodic bursters, chaotic and periodic bursters, quasiperiodic and periodic bursters, two periodic bursters with different periodicities, and so on. Additionally, a simulated circuit model is designed and PSIM circuit simulations are performed to exhibit coexisting behaviors of asymmetric bursters, which effectively confirm the numerically simulated results.

Dynamics of coupled mode solitons in bursting neural networks.

Using an electrically coupled chain of Hindmarsh-Rose neural models, we analytically derived the nonlinearly coupled complex Ginzburg-Landau equations. This is realized by superimposing the lower and upper cutoff modes of wave propagation and by employing the multiple scale expansions in the semidiscrete approximation. We explore the modified Hirota method to analytically obtain the bright-bright pulse soliton solutions of our nonlinearly coupled equations. With these bright solitons as initial conditions of our numerical scheme, and knowing that electrical signals are the basis of information transfer in the nervous system, it is found that prior to collisions at the boundaries of the network, neural information is purely conveyed by bisolitons at lower cutoff mode. After collision, the bisolitons are completely annihilated and neural information is now relayed by the upper cutoff mode via the propagation of plane waves. It is also shown that the linear gain of the system is inextricably linked to the complex physiological mechanisms of ion mobility, since the speeds and spatial profiles of the coupled nerve impulses vary with the gain. A linear stability analysis performed on the coupled system mainly confirms the instability of plane waves in the neural network, with a glaring example of the transition of weak plane waves into a dark soliton and then static kinks. Numerical simulations have confirmed the annihilation phenomenon subsequent to collision in neural systems. They equally showed that the symmetry breaking of the pulse solution of the system leaves in the network static internal modes, sometime referred to as Goldstone modes.

The route to synchrony via drum head mode and mixed oscillatory state in star coupled Hindmarsh–Rose neural network

Network of coupled oscillators exhibit different types of spatiotemporal patterns. We report that as the coupling strength increases the unidirectionally coupled Hindmarsh–Rose neuron star network will synchronize. The condition for synchronization has been evaluated using Lyapunov function method. We also discuss the dynamics of the system in the presence of controllers. The control input generate interesting behaviors which consist of clusters of spatially coherent domains depending on the coupling strength. Drum head mode, mixed oscillatory state, desynchrony, and multi cluster states are formed and cluster reduction takes place before settling to complete synchrony. The evolution of a perfectly synchronized state via drum head mode, mixed oscillatory state, and clusters from a desynchronized state is reported for the first time. The parameter values which lead to stable cluster formation is also discussed. Our results suggest that in the presence of controllers the common oscillator in the star network behaves as a driver and generates the transitions and cluster formation acts as a precursor to complete synchrony in Hindmarsh–Rose model with unidirectional star coupling.

An Efficient and Reconfigurable Synchronous Neuron Model

This brief presents a reconfigurable and efficient 2-D neuron model capable of extending to higher dimensions. The model is applied to the Izhikevich and FitzHugh-Nagumo neuron models as 2-D case studies and to the Hindmarsh-Rose model as a 3-D case study. Hardware synthesis and physical implementations show that the resulting circuits can reproduce neural dynamics with acceptable precision and considerably low hardware overhead compared to previously published piecewise linear models.

Revival of oscillations via common environment

We explore the behavior of groups of Landau-Stuart oscillators, with diffusive mean-field coupling within the group and indirect inter-group coupling mediated by a common medium. Interestingly, we find that oscillations are revived in groups with fixed point dynamics by coupling to an oscillatory group through the common medium. We also investigate groups of Hindmarsh–Rose model neurons and demonstrate the emergence of spiking in indirectly coupled inactive neuronal groups, indicating the vital role of the common medium in restoring oscillations.

Critical transitions and perturbation growth directions.

Critical transitions occur in a variety of dynamical systems. Here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions, and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.

A new approach to optimal control of conductance-based spiking neurons

This paper presents an algorithm for solving the minimum-energy optimal control problem of conductance-based spiking neurons. The basic procedure is (1) to construct a conductance-based spiking neuron oscillator as an affine nonlinear system, (2) to formulate the optimal control problem of the affine nonlinear system as a boundary value problem based on Pontryagin’s maximum principle, and (3) to solve the boundary value problem using the homotopy perturbation method. The construction of the minimum-energy optimal control in the framework of the homotopy perturbation technique is novel and valid for a broad class of nonlinear conductance-based neuron models. The applicability of our method in the FitzHugh–Nagumo and Hindmarsh–Rose models is validated by simulations.

Robustness and versatility of a nonlinear interdependence method for directional coupling detection from spike trains.

The detection of directional couplings between dynamics based on measured spike trains is a crucial problem in the understanding of many different systems. In particular, in neuroscience it is important to assess the connectivity between neurons. One of the approaches that can estimate directional coupling from the analysis of point processes is the nonlinear interdependence measure L. Although its efficacy has already been demonstrated, it still needs to be tested under more challenging and realistic conditions prior to an application to real data. Thus, in this paper we use the Hindmarsh-Rose model system to test the method in the presence of noise and for different spiking regimes. We also examine the influence of different parameters and spike train distances. Our results show that the measure L is versatile and robust to various types of noise, and thus suitable for application to experimental data.

Chimera-like states in a neuronal network model of the cat brain

Neuronal systems have been modelled by complex networks in different description levels. Recently, it has been verified that the networks can simultaneously exhibit one coherent and other incoherent domain, known as chimera states. In this work, we study the existence of chimera-like states in a network considering the connectivity matrix based on the cat cerebral cortex. The cerebral cortex of the cat can be separated in 65 cortical areas organised into the four cognitive regions: visual, auditory, somatosensory-motor and frontolimbic. We consider a network where the local dynamics is given by the Hindmarsh–Rose model. The Hindmarsh–Rose equations are a well known model of the neuronal activity that has been considered to simulate the membrane potential in neuron. Here, we analyse under which conditions chimera-like states are present, as well as the effects induced by intensity of coupling on them. We identify two different kinds of chimera-like states: spiking chimera-like state with desynchronised spikes, and bursting chimera-like state with desynchronised bursts. Moreover, we find that chimera-like states with desynchronised bursts are more robust to neuronal noise than with desynchronised spikes.

Lévy noise improves the electrical activity in a neuron under electromagnetic radiation.

As the fluctuations of the internal bioelectricity of nervous system is various and complex, the external electromagnetic radiation induced by magnet flux on membrane can be described by the non-Gaussian type distribution of Lévy noise. Thus, the electrical activities in an improved Hindmarsh-Rose model excited by the external electromagnetic radiation of Lévy noise are investigated and some interesting modes of the electrical activities are exhibited. The external electromagnetic radiation of Lévy noise leads to the mode transition of the electrical activities and spatial phase, such as from the rest state to the firing state, from the spiking state to the spiking state with more spikes, and from the spiking state to the bursting state. Then the time points of the firing state versus Lévy noise intensity are depicted. The increasing of Lévy noise intensity heightens the neuron firing. Also the stationary probability distribution functions of the membrane potential of the neuron induced by the external electromagnetic radiation of Lévy noise with different intensity, stability index and skewness papremeters are analyzed. Moreover, through the positive largest Lyapunov exponent, the parameter regions of chaotic electrical mode of the neuron induced by the external electromagnetic radiation of Lévy noise distribution are detected.