Hindmarsh-Rose Model Research Articles

Simple noise-reduction method based on nonlinear forecasting.

Nonparametric detrending or noise reduction methods are often employed to separate trends from noisy time series when no satisfactory models exist to fit the data. However, conventional noise reduction methods depend on subjective choices of smoothing parameters. Here we present a simple multivariate noise reduction method based on available nonlinear forecasting techniques. These are in turn based on state-space reconstruction for which a strong theoretical justification exists for their use in nonparametric forecasting. The noise reduction method presented here is conceptually similar to Schreiber's noise reduction method using state-space reconstruction. However, we show that Schreiber's method has a minor flaw that can be overcome with forecasting. Furthermore, our method contains a simple but nontrivial extension to multivariate time series. We apply the method to multivariate time series generated from the Van der Pol oscillator, the Lorenz equations, the Hindmarsh-Rose model of neuronal spiking activity, and to two other univariate real-world data sets. It is demonstrated that noise reduction heuristics can be objectively optimized with in-sample forecasting errors that correlate well with actual noise reduction errors.

Dynamical Complexities of Non-linear Physical and Biological Systems

The basic equations governing both physical and biological systems are non-linear. Therefore, it is not generally possible to obtain solutions of these systems analytically to decipher the observed complex dynamics. To understand the dynamics of a non-linear system, we use various numerical methods using computer. In this work, we use (a) power spectrum, (b) Poincare-surface of section, (c) correlation dimension, (d) spectral and sample entropy method to understand the complex dynamics observed in simple non-linear systems, for example logistic map, Duffing oscillator, Lorenz system, Hindmarsh–Rose model in neuro-bioscience. We also describe methods to obtain visibility graph from a time series of a dynamical system and further use them to provide a network frame work for any dynamical system.

On integrals, Hamiltonian and metriplectic formulations of polynomial systems in 3D

The first integrals of the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model, and the Oregonator model are derived using the method of Darboux polynomials. It is shown that, the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model can be written in a bi-Hamiltonian/Nambu metriplectic form.

Chaos in Neural Oscillators Induced by Unidirectional Electrical Coupling

We demonstrate that unidirectional electrical coupling between two periodically spiking Hindmarsh-Rose neurons induces bistability in the system. We find that for certain values of intermediate coupling, the slave neuron exhibits coexistence of two attractors. One of them is the periodic orbit similar to the original attractor without coupling, and the other one is a chaotic attractor or a periodic orbit with higher periodicity, depending on the coupling strength. For strong coupling, the slave neuron is monostable at a periodic orbit similar to the attractor of the master neuron. When the master and slave neurons are in a similar attractor they are completely synchronized, whereas being in different states they are in generalized synchronization. We also present the experimental evidence of this behavior with electronic circuits based on the Hindmarsh-Rose model.

Hindmarsh–Rose model: Close and far to the singular limit

Dynamics arising in the Hindmarsh–Rose model are considered from a novel perspective. We study qualitative changes that occur as the time scale of the slow variable increases taking the system far from the slow-fast scenario. We see how the structure of spike-adding still persists far from the singular case but the geometry of the bifurcations changes notably. Particular attention is paid to changes in the shape of the homoclinic bifurcation curves and the disappearance of Inclination-Flip codimension-two points. These transformations seem to be linked to the way in which the spike-adding takes place, the changing from fold/hom to fold/Hopf bursting behavior and also with the way in which the chaotic regions evolve as the time scale of the slow variable increases.

Stochastic Bifurcations and Noise-Induced Chaos in 3D Neuron Model

The stochastically forced three-dimensional Hindmarsh–Rose model of neural activity is considered. We study the effect of random disturbances in parametric zones where the deterministic model exhibits mono- and bistable dynamic regimes with period-adding bifurcations of oscillatory modes. It is shown that in both cases the phenomenon of noise-induced bursting is observed. In the monostable zone, where the only attractor of the system is a stable equilibrium, this effect is connected with a stochastic generation of large-amplitude oscillations due to the high excitability of the model. In a parametric zone of coexisting stable equilibria and limit cycles, bursts appear due to noise-induced transitions between the attractors. For a quantitative analysis of the noise-induced bursting and corresponding stochastic bifurcations, an approach based on the stochastic sensitivity function (SSF) technique is applied. Our estimations of the strength of noise that generates such qualitative changes in stochastic dynamics are in a good agreement with the direct numerical simulation. A relationship of the noise-induced generation of bursts with transitions from order to chaos is discussed.

Estimability and dependency analysis of model parameters based on delay coordinates.

In data-driven system identification, values of parameters and not observed variables of a given model of a dynamical system are estimated from measured time series. We address the question of estimability and redundancy of parameters and variables, that is, whether unique results can be expected for the estimates or whether, for example, different combinations of parameter values would provide the same measured output. This question is answered by analyzing the null space of the linearized delay coordinates map. Examples with zero-dimensional, one-dimensional, and two-dimensional null spaces are presented employing the Hindmarsh-Rose model, the Colpitts oscillator, and the Rössler system.

Chimera states in a Hodgkin-Huxley model of thermally sensitive neurons.

Chimera states occur when identically coupled groups of nonlinear oscillators exhibit radically different dynamics, with one group exhibiting synchronized oscillations and the other desynchronized behavior. This dynamical phenomenon has recently been studied in computational models and demonstrated experimentally in mechanical, optical, and chemical systems. The theoretical basis of these states is currently under active investigation. Chimera behavior is of particular relevance in the context of neural synchronization, given the phenomenon of unihemispheric sleep and the recent observation of asymmetric sleep in human patients with sleep apnea. The similarity of neural chimera states to neural "bump" states, which have been suggested as a model for working memory and visual orientation tuning in the cortex, adds to their interest as objects of study. Chimera states have been demonstrated in the FitzHugh-Nagumo model of excitable cells and in the Hindmarsh-Rose neural model. Here, we demonstrate chimera states and chimera-like behaviors in a Hodgkin-Huxley-type model of thermally sensitive neurons both in a system with Abrams-Strogatz (mean field) coupling and in a system with Kuramoto (distance-dependent) coupling. We map the regions of parameter space for which chimera behavior occurs in each of the two coupling schemes.

Synchronization and Partial Synchronization Experiments with Networks of Time-Delay Coupled Hindmarsh–Rose Neurons

We study experimentally synchronization and partial synchronization in networks of Hindmarsh–Rose model neurons that interact through linear time-delay couplings. Our experimental setup consists of electric circuit board realizations of the Hindmarsh–Rose model neuron and a coupling interface in which the interaction between the circuits is defined. With this experimental setup we test the predictive value of theoretical results about synchronization and partial synchronization in networks.

Multiple modes of electrical activities in a new neuron model under electromagnetic radiation

The three-variable Hindmarsh–Rose model is improved to describe the dynamical behaviors of neuronal activities with electromagnetic induction being considered, and the mode transition of electrical activities in neuron are detected when external electromagnetic radiation is imposed on the neuron. The improved neuron model holds more bifurcation parameters and the mode of electric activities can be selected in larger parameter region. It is found that the electromagnetic radiation can excite quiescent neuron but also can suppress the electrical activities in neuron as well. Particularly, it is important to find that multiple modes of electrical activities can emerge alternatively, and these results are consistent with biological experiments.

Response Ability to External Signal Enhanced by Biological Spatial Configuration in Coupled Hindmarsh-Rose Neural System

The effect of realistic topology configuration of intercellular connections on the response ability in coupled cell system is numerically investigated by using the Hindmarsh-Rose model. For the proper coupling intensity, we set the control parameter to be near the critical value, and the external stimulus is introduced to the first cell in coupled system. It is found that, on one hand, when the cells are coupled with some proper topological structures, the external stimulus could transmit through the system, and shows better response ability and higher sensitivity. On the other hand, the influence of topological configuration on the synchronous ability and selection effect of neural system are also discussed. Our results display that the topology of coupled system may play an important role in the process of signal propagation, which could help us to understand the coordinated performance of cells in tissue.

Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg–Landau equation

We investigate localized wave solutions in a network of Hindmarsh–Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number k → 0) can be governed by the complex fractional Ginzburg–Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindmarsh–Rose models for relatively large network.

Stochastic Generation of Bursting Oscillations in the Three-dimensional Hindmarsh–Rose Model

We study the effect of random disturbances on the dynamics of the three-dimensional Hindmarsh–Rose model of neuronal activity. Due to the strong nonlinearity, even the original deterministic system exhibits diverse and complex dynamic regimes (various types of periodic oscillations, oscillations zones with period doubling and adding, coexistence of several attractors, chaos). In this paper, we consider a parametric zone where a stable equilibrium is the only attractor. We show that even in this zone with simple deterministic dynamics, under the random disturbances, such complex effect as the stochastic generation of bursting oscillations can occur. For a small noise, random states concentrate near the equilibrium. With the increase of the noise intensity, random trajectories can go far from the stable equilibrium, and along with small-amplitude oscillations around the equilibrium, bursts are observed. This phenomenon is analysed using the mathematical methods based on the stochastic sensitivity function technique. An algorithm of estimation of critical values for noise intensity is proposed.

Dynamics of Disordered Network of Coupled Hindmarsh–Rose Neuronal Models

We investigate the effects of disorder on the synchronized state of a network of Hindmarsh–Rose neuronal models. Disorder, introduced as a perturbation of the neuronal parameters, destroys the network activity by wrecking the synchronized state. The dynamics of the synchronized state is analyzed through the Kuramoto order parameter, adapted to the neuronal Hindmarsh–Rose model. We find that the coupling deeply alters the dynamics of the single units, thus demonstrating that coupling not only affects the relative motion of the units, but also the dynamical behavior of each neuron; Thus, synchronization results in a structural change of the dynamics. The Kuramoto order parameter allows to clarify the nature of the transition from perfect phase synchronization to the disordered states, supporting the notion of an abrupt, second order-like, dynamical phase transition. We find that the system is resilient up to a certain disorder threshold, after that the network abruptly collapses to a desynchronized state. The loss of perfect synchronization seems to occur even for vanishingly small values of the disorder, but the degree of synchronization (as measured by the Kuramoto order parameter) gently decreases, and the completely disordered state is never reached.

How single node dynamics enhances synchronization in neural networks with electrical coupling

The stability of the completely synchronous state in neural networks with electrical coupling is analytically investigated applying both the Master Stability Function approach (MSF), developed by Pecora and Carroll (1998), and the Connection Graph Stability method (CGS) proposed by Belykh et al. (2004). The local dynamics is described by Morris–Lecar model for spiking neurons and by Hindmarsh–Rose model in spike, burst, irregular spike and irregular burst regimes. The combined application of both CGS and MSF methods provides an efficient estimate of the synchronization thresholds, namely bounds for the coupling strength ranges in which the synchronous state is stable. In all the considered cases, we observe that high values of coupling strength tend to synchronize the system. Furthermore, we observe a correlation between the single node attractor and the local stability properties given by MSF. The analytical results are compared with numerical simulations on a sample network, with excellent agreement.

Anomalous neuronal responses to fluctuated inputs.

The irregular firing of a cortical neuron is thought to result from a highly fluctuating drive that is generated by the balance of excitatory and inhibitory synaptic inputs. A previous study reported anomalous responses of the Hodgkin-Huxley neuron to the fluctuated inputs where an irregularity of spike trains is inversely proportional to an input irregularity. In the current study, we investigated the origin of these anomalous responses with the Hindmarsh-Rose neuron model, map-based models, and a simple mixture of interspike interval distributions. First, we specified the parameter regions for the bifurcations in the Hindmarsh-Rose model, and we confirmed that the model reproduced the anomalous responses in the dynamics of the saddle-node and subcritical Hopf bifurcations. For both bifurcations, the Hindmarsh-Rose model shows bistability in the resting state and the repetitive firing state, which indicated that the bistability was the origin of the anomalous input-output relationship. Similarly, the map-based model that contained bistability reproduced the anomalous responses, while the model without bistability did not. These results were supported by additional findings that the anomalous responses were reproduced by mimicking the bistable firing with a mixture of two different interspike interval distributions. Decorrelation of spike trains is important for neural information processing. For such spike train decorrelation, irregular firing is key. Our results indicated that irregular firing can emerge from fluctuating drives, even weak ones, under conditions involving bistability. The anomalous responses, therefore, contribute to efficient processing in the brain.

On the analytical solutions of the Hindmarsh–Rose neuronal model

In this article we analytically solve the Hindmarsh–Rose model (Proc R Soc Lond B221:87–102, 1984) by means of a technique developed for strongly nonlinear problems—the step homotopy analysis method. This analytical algorithm, based on a modification of the standard homotopy analysis method, allows us to obtain a one-parameter family of explicit series solutions for the studied neuronal model. The Hindmarsh–Rose system represents a paradigmatic example of models developed to qualitatively reproduce the electrical activity of cell membranes. By using the homotopy solutions, we investigate the dynamical effect of two chosen biologically meaningful bifurcation parameters: the injected current I and the parameter r, representing the ratio of time scales between spiking (fast dynamics) and resting (slow dynamics). The auxiliary parameter involved in the analytical method provides us with an elegant way to ensure convergent series solutions of the neuronal model. Our analytical results are found to be in excellent agreement with the numerical simulations.

Order and chaos in the stochastic Hindmarsh–Rose model of the neuron bursting

We study the stochastic dynamics of the two-dimensional Hindmarsh–Rose model. In the deterministic case, this system demonstrates mono- and bi-stable dynamic regimes. In the parametric zone of the coexisting stable equilibrium and limit cycle, the phenomenon of noise-induced transitions between the attractors is studied. In another parametric region, where the deterministic system has the only stable equilibrium, the stochastic generation of high-amplitude oscillations is also observed. We show that under the random disturbances, the system demonstrates noise-induced bursting: an alternation of small fluctuations near the equilibrium and high-amplitude oscillations. For the quantitative analysis of noise-induced bursting, an approach combining stochastic sensitivity function technique and confidence domains method is suggested. Constructive abilities of this method for the estimation of critical values for noise intensity corresponding to the qualitative changes in stochastic dynamics are demonstrated and confirmed by the good agreement with the direct numerical simulation. An interplay between noise-induced bursting and mutual transformations “order–chaos” is discussed.

Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

In this paper, the effects of different parameters on the dynamic behavior of the nonlinear dynamical system are investigated based on modified Hindmarsh---Rose neural nonlinear dynamical system model. We have calculated and analyzed dynamic characteristics of the model under different parameters by using single parameter bifurcation diagram, time response diagram and two parameter bifurcation diagram. The results show that the period-adding bifurcation (with or without chaos), period-doubling bifurcation and intermittent chaos phenomenon (periodic and intermittent chaotic) can be observed more clearly and directly from the two parameter bifurcation diagram, and the optimal parameters matching interval can also be found easily.

Integral Step Size Makes a Difference to Bifurcations of a Discrete-Time Hindmarsh–Rose Model

A three-dimensional discrete-time Hindmarsh–Rose model obtained by the forward Euler scheme is investigated in this paper. When the integral step size is chosen as a bifurcation parameter, conditions of existence for the fold bifurcation, the flip bifurcation, and the Hopf bifurcation are derived by using the center manifold theorem, bifurcation theory and a criterion of Hopf bifurcation. Numerical simulations including time series, bifurcation diagrams, Lyapunov exponents, phase portraits show the consistence with the analytical analysis. Our research results demonstrate that the integral step size makes a difference corresponding to local and global bifurcations of the three-dimensional discrete-time Hindmarsh–Rose model. These results can supply a solid analytical basis to the study of Hindmarsh–Rose model, and it is necessary to illustrate how much the integral step size is adopted in advance when numerical solutions or approximate solutions of the original continuous-time model is concerned.