Coupled Hindmarsh-Rose Neurons Research Articles

An experimental set-up design for synchronization and control of coupled Hindmarsh–Rose neurons with Markov-jump dynamics: A case study on finite-time sliding-mode synchronization

This paper introduces a real-time experimental set-up design to realize the robust synchronization of chaotic systems with Markov-Jump behavior, where the control law is designed based on a finite-time sliding-mode controller. First, master and slave chaotic systems are designed with the dynamics of biological Hindmarsh–Rose (HR) neurons using electronic circuit elements. In order to create the Markov-Jump behavior, the printed circuit board is adjusted to add or remove the resistors so that it is possible to obtain the time-varying dynamic of a chaotic system in the synchronization process. In the realization of chaotic neurons, Matlab and Multisim environments are used for the simulations, and Proteus software is utilized for the design of PCB layout. The Arduino microcontroller is used for signal processing and closed-loop control, where the states of chaotic neurons are recorded and control signals are produced and applied to the slave chaotic system to be synchronized. The real-time data corresponding to the behavior of the circuits was recorded, and Lyapunov exponents were calculated to check whether the neuron circuits are chaotic or not. Second, a sliding-mode controller (SMC) was designed to guarantee a finite-time convergence of synchronization error where its parameters are optimized using a recently developed efficient optimization method, namely the adolescent-identity search algorithm. Robust finite-time synchronization results with sliding-mode control were recorded in simulation and real-time experiments. Finally, the difficulties encountered in general and unachievable results related to experimental system design and finite-time synchronization are discussed for future studies.

Collective dynamics of a coupled Hindmarsh–Rose neurons with locally active memristor

A Locally active memristors can mimic neural synapses, resulting in rich neuro-morphological dynamics in biological neurons. To illustrate the impact of a local active memristive synapse, we consider coupled Hindmarsh–Rose (HR) neurons. Firstly, the dynamical transitions of the proposed system are investigated using bifurcation analysis and Lyapunov exponents, and we find that the transition between periodic and chaotic states depends on the input currents and memristive coupling strength. By performing the two-parameter analysis, the existence of periodic and chaotic regions is revealed. The collective behavior is then examined by expanding the network to include memristive coupled HR neurons under different network connectivities. We show that the system achieves synchronization behavior for all network connectivities, including regular, random, and small-world, when the strength of the memristive coupling is increased.

Chemically coupled Hindmarsh–Rose neurons with cross interactions between membrane potential and magnetic flux

In this work, we have analysed the synchronous dynamics and pattern formation in Hindmarsh–Rose neurons with cross interactions between membrane potential and magnetic flux, in the chemical mode. The self, mixed and cross interactions are realised by varying coupling phase. The magnetic flux induces plateau bursting and amplitude death in the network. The self chemical coupling induces synchrony, whereas, the cross coupling is incapable of it. However, the cross coupling acts along with self coupling to form mixed coupling and induces synchrony in the system. The stability of the synchronous state has been studied by master stability approach. The parameter space reveals the bifurcation point at which cross coupling overrides self coupling effects. The synchronising ability of interactions are justified in a network of neurons as well. The statistical factor of synchronisation quantifies the amount of synchrony in the network in different interaction modes. The combined effect of non local interactions and mixed coupling of variables initiates the emergence of chimera and multichimera states. However, in cross-coupled systems, only incoherent states are present. The existence of chimera and multichimera states are confirmed by calculating the strength of incoherence and discontinuity measure. The analysis of spatiotemporal patterns reveals the presence of travelling chimeras within the network. The Hamilton energy function indicate that a greater amount of energy is required to sustain coherent neurons at higher potential. This work may enhance the understanding of chimera states and improve its applicability to real-world systems.

A synaptic plasticity induced change in synaptic intensity variation and neurodynamic transition during awakening-sleep cycle

It has been found experimentally that learning during wakefulness leads to a net enhancement of synaptic strength, accompanied by the neural dynamical transition from tonic to bursting firing, while the net synaptic strength decreases to a baseline level during sleep, accompanied by the transition from bursting to tonic firing. In this paper, we establish a model of synaptic plasticity, which can realize synaptic strength changes and neural dynamical transitions in wakefulness-sleep cycle by using the coupled Hindmarsh-Rose neurons. Through numerical simulation and theoretical analysis, it is further found that the average synaptic weight of the neural network can reach a stable value during either prolonged wakefulness or prolonged sleep, which depends on the ratio of some specific parameters in the model. Particularly, the synaptic weight exhibits a stable log-normal distribution observed in a real neural system, when the average synaptic weight reaches a stable value. Moreover, the fluctuation of this weight distribution is positively correlated with the fluctuation of noise in the synaptic plasticity model. The provided model of the synaptic plasticity and its dynamics results can provide a theoretical reference for studying the physiological mechanism of synaptic plasticity and neuronal firings during the wakefulness-sleep cycle, and they are expected to have potential applications in the development of therapeutic interventions for sleep disorders.

Transition of Chimera States and Synchronization in Two-Layer Networks of Coupled Hindmarsh–Rose Neurons

The present work focuses on the existence of chimera states in two-layer networks of locally, nonlocally, and globally coupled Hindmarsh–Rose neurons. We show that chimera states occur in all three coupled neuronal networks by changing synaptic coupling strength and the number of coupled neighbors, and traveling chimera state exists in networks by using local and nonlocal couplings. Particularly, we find that in the nonlocally coupled network, there exist regions of coherent state, which can produce a wave such that the period becomes longer with the increase of the number of coupled neighbors. Interestingly, we observe a new chimera state with the coexistence of interlayer synchronous and asynchronous chimera states, which we named as the interlayer semi-synchronous chimera state. In addition, the results indicate that the interaction between layers in neuronal networks can induce different types of chimera states and firing patterns, which could be useful to control neural firing patterns and deepen the understanding of neuronal evolution where coherent and incoherent dynamics coexist.

Synaptic changes modulate spontaneous transitions between tonic and bursting neural activities in coupled Hindmarsh-Rose neurons.

Experimentally, certain cells in the brain exhibit a spike-burst activity with burst synchronization at transition to and during sleep (or drowsiness), while they demonstrate a desynchronized tonic activity in the waking state. We herein investigated the neural activities and their transitions by using a model of coupled Hindmarsh-Rose neurons in an Erdős-Rényi random network. By tuning synaptic strength, spontaneous transitions between tonic and bursting neural activities can be realized. With excitatory chemical synapses or electrical synapses, slow-wave activity (SWA) similar to that observed during sleep can appear, as a result of synchronized bursting activities. SWA cannot appear in a network that is dominated by inhibitory chemical synapses, because neurons exhibit desynchronized bursting activities. Moreover, we found that the critical synaptic strength related to the transitions of neural activities depends only on the network average degree (i.e., the average number of signals that all the neurons receive). We demonstrated, both numerically and analytically, that the critical synaptic strength and the network average degree obey a power-law relation with an exponent of -1. Our study provides a possible dynamical network mechanism of the transitions between tonic and bursting neural activities for the wakefulness-sleep cycle, and of the SWA during sleep. Further interesting and challenging investigations are briefly discussed as well.

Controlling phase synchrony in the mean field coupled Hindmarsh–Rose neurons

The phase synchronization in a network of mean field coupled Hindmarsh–Rose neurons and the control of phase synchrony by an external input has been analyzed in this work. The analysis of interspike interval, with varying coupling strength, reveals the dynamical change induced in each neuron in the network. The bursting phase lines depict that mean field coupling induces phase synchrony in excitatory mode and desynchrony in inhibitory mode. The coefficient of variability, in spatial and temporal domain, signifies the deviations in firing times of neurons, in a collective manner. The Kuramoto order parameter quantifies the intermittent and complete phase synchrony, induced by excitatory mean field coupling. The capability of external input, in the form of spikes, to control the intermittent and complete phase synchrony has been analyzed. The coefficient of variability and Kuramoto order parameter has been studied by varying the amplitude, pulse width and frequency of the input. The studies have shown that high-frequency spike input, with optimum amplitude and pulse width, has high desynchronizing ability, which is substantiated by the parameter space analysis. The control of synchrony in the network of neurons may find application in rectifying neural disorders.

Effects of electric field on multiple vibrational resonances in Hindmarsh-Rose neuronal systems

• The effect of electric field plays an important role in the electrical activities of neuronal systems. • Vibration resonance is widely used in weak signal detection and propagation. • The electric field weakens the multiple vibrational resonance of a single • Hindmarsh-Rose (HR) ne4uron, but enhances the multiple vibrational resonance of the coupled HR neuron system. • Multiple vibrational resonances implies the matching of multiple frequencies giving rise to an enhanced response. • Anti-resonance has certain practical significance in reducing some unexpected vibration in the field of aerospace engineering and machinery The effects of electric field on vibrational resonance in single Hindmarsh-Rose (HR) neuron and coupled HR neurons system are investigated by using Fourier coefficient, respectively. It is found that the multiple vibrational resonances (MVR) can be observed in a single HR neuron model no matter the electric field is considered or not, and the electric field weakens the MVR. When bidirectional coupling between two HR neurons is considered, the occurrence of MVR can also be detected, it is very interesting to observe that the electric field can enhance the MVR. The higher the frequency of the low-frequency signal is, the less the number of resonance peaks of the system response to the low-frequency signal will be. Moreover, the local anti-resonance is also observed when appropriate parameters are selected. The effects of coupling strength and other system parameters on Fourier coefficient are also illustrated here. The systems manifesting MVR have better capacity for detecting and propagating signals.

Collective behavior in a two-layer neuronal network with time-varying chemical connections that are controlled by a Petri net.

In this paper, we propose and study a two-layer network composed of a Petri net in the first layer and a ring of coupled Hindmarsh-Rose neurons in the second layer. Petri nets are appropriate platforms not only for describing sequential processes but also for modeling information circulation in complex systems. Networks of neurons, on the other hand, are commonly used to study synchronization and other forms of collective behavior. Thus, merging both frameworks into a single model promises fascinating new insights into neuronal collective behavior that is subject to changes in network connectivity. In our case, the Petri net in the first layer manages the existence of excitatory and inhibitory links among the neurons in the second layer, thereby making the chemical connections time-varying. We focus on the emergence of different types of collective behavior in the model, such as synchronization, chimeras, and solitary states, by considering different inhibitory and excitatory tokens in the Petri net. We find that the existence of only inhibitory or excitatory tokens disturbs the synchronization of electrically coupled neurons and leads toward chimera and solitary states.

Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor.

A model is introduced by coupling two three-dimensional Hindmarsh-Rose models with the help of a nonsmooth memristor. The firing patterns dependent on the external forcing current are explored, which undergo a process from adding-period to chaos. The stability of equilibrium points of the considered model is investigated via qualitative analysis, from which it can be gained that the model has diversity in the number and stability of equilibrium points for different coupling coefficients. The coexistence of multiple firing patterns relative to initial values is revealed, which means that the referred model can appear various firing patterns with the change of the initial value. Multiple firing patterns of the addressed neuron model induced by different scales are uncovered, which suggests that the discussed model has a multiscale effect for the nonzero initial value.

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.

Synchronization and multistability in the coupled neurons with propagation and processing delays

The dynamics of spiking and bursting activities have attracted interest from scientists due to their significance in the brain and neural systems. In this work, the effects of processing delay, specifically the time delay in coupling, on the dynamical behaviors of the coupled Hindmarsh–Rose neurons with resting state are investigated. The results reveal that the processing delay can induce rich spiking patterns, such as anti-phase synchronous spiking, in-phase synchronous periodic bursting, anti-phase synchronous periodic bursting and anti-phase synchronous chaotic bursting, and maintained the bistability and tristability of different spiking patterns. This emphasizes that the processing delay-induced spiking patterns might not be observed in the diffusively coupled neurons with resting state with and without propagation delay. The results can help deepen the knowledge on neural synchronization and information processing in the neuronal network and brain.

Synchronization of boundary coupled Hindmarsh–Rose neuron network

In this work, we present a new mathematical model of a boundary coupled neuron network described by the partly diffusive Hindmarsh–Rose equations. We prove the global absorbing property of the solution semiflow and then the main result on the asymptotic synchronization of this neuron network at a uniform exponential rate provided that the boundary coupling strength and the stimulating signal exceed a quantified threshold in terms of the parameters.

Synchronization Patterns in Modular Neuronal Networks: A Case Study of C. elegans

We investigate synchronization patterns and chimera-like states in the modular multilayer topology of the connectome of Caenorhabditis elegans. In the special case of a designed network with two layers, one with electrical intra-community links and one with chemical inter-community links, chimera-like states are known to exist. Aiming at a more biological approach based on the actual connectivity data, we consider a network consisting of two synaptic (electrical and chemical) and one extrasynaptic (wireless) layers. Analyzing the structure and properties of this layered network using Multilayer-Louvain community detection, we identify modules whose nodes are more strongly coupled with each other than with the rest of the network. Based on this topology, we study the dynamics of coupled Hindmarsh-Rose neurons. Emerging synchronization patterns are quantified using the pairwise Euclidean distances between the values of all oscillators, locally within each community and globally across the network. We find a tendency of the wireless coupling to moderate the average coherence of the system: for stronger wireless coupling, the levels of synchronization decrease both locally and globally, and chimera-like states are not favored. By introducing an alternative method to define meaningful communities based on the dynamical correlations of the nodes, we obtain a structure that is dominated by two large communities. This promotes the emergence of chimera-like states and allows to relate the dynamics of the corresponding neurons to biological neuronal functions such as motor activities.

Graph-theoretic approach to synchronization of fractional-order coupled systems with time-varying delays via periodically intermittent control

Abstract This paper deals with synchronization problem of fractional-order coupled systems (FOCSs) with time-varying delays via periodically intermittent control. Here, nonlinear coupling, time-varying internal delay and time-varying coupling delay are considered when modeling, which makes our model more general in comparison with the most existing fractional-order models. It is the first time that periodically intermittent control is applied to synchronizing FOCSs with time-varying delays. Combining Lyapunov method with graph-theoretic approach, some synchronization criteria are obtained. Moreover, the synchronization criteria we derive depend on the fractional order α, control gain, control rate and control period. Besides, the synchronization issues of fractional-order coupled chaotic systems with time-varying delays and fractional-order coupled Hindmarsh–Rose neuron systems with time-varying delays are also investigated as applications of our theoretical results, and relevant sufficient conditions are derived. Finally, numerical simulations with two examples are provided in order to demonstrate the effectiveness of the theoretical results and the feasibility of control strategy.

Nonlinear optimal control for the synchronization of biological neurons under time-delays.

The article proposes a nonlinear optimal control method for synchronization of neurons that exhibit nonlinear dynamics and are subject to time-delays. The model of the Hindmarsh-Rose (HR) neurons is used as a case study. The dynamic model of the coupled HR neurons undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure relies on Taylor series expansion of the model and on computation of the associated Jacobian matrices. For the approximately linearized model of the coupled HR neurons an H-infinity controller is designed. For the selection of the controller's feedback gain an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The stability properties of the control loop are proven through Lyapunov analysis. First, it is shown that the H-infinity tracking performance criterion is satisfied. Moreover, it is proven that the control loop is globally asymptotically stable.

Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction

The membrane potential difference between two adjacent neurons can induce an electromagnetic induction current that behaves like a memristor synapse effect using to couple these two neurons. Based on two-dimensional (2D) Hindmarsh–Rose (HR) neuron and non-ideal threshold memristor, this paper presents a five-dimensional (5D) neuron model of two adjacent neurons coupled by memristive electromagnetic induction. With the 5D memristor-coupled neuron model, the equilibrium states and their stabilities are investigated by qualitative analyses, and the coexisting phenomena of multiple firing patterns and initials-depending bifurcation routes along with extreme events are then uncovered by numerical simulations. Due to complex stabilities of the seven equilibrium states, the attraction basin of one neuron in the 5D memristor-coupled neuron model is not only related to the coupling strength of memristor synapse but also associated with another neuron initials, leading to that the coexisting multiple firing patterns are emerged from different regions in the attraction basin and the bifurcation routes are closely dependent to the initials. Furthermore, circuit syntheses using the off-the-shelf components and breadboard experiments for the proposed 5D memristor-coupled neuron model are executed so that the coexisting multiple firing patterns and extreme events are then conformed perfectly, which have not yet been previously reported in the coupled HR neuron model.

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.

Synchronization of “light-sensitive” Hindmarsh–Rose neurons

The suprachiasmatic nucleus is a network of synchronized neurons whose electrical activity follows a 24 h cycle. The synchronization phenomenon (among these neurons) is not completely understood. In this work we study, via experiments and numerical simulations, the phenomenon in which the synchronization threshold changes under the influence of an external (bifurcation) parameter in coupled Hindmarsh–Rose neurons. This parameter “shapes” the activity of the individual neurons the same way as some neurons in the brain react to light. We corroborate this experimental finding with numerical simulations by quantifying the amount of synchronization using Pearson’s correlation coefficient. In order to address the local stability problem of the synchronous state, Floquet theory is applied in the case where the dynamic systems show continuous periodic solutions. These results show how the sufficient coupling strength for synchronization between these neurons is affected by an external cue (e.g. light).

Chaotic synchronization of time-delay coupled Hindmarsh–Rose neurons via nonlinear control

Chaotic synchronization of two time-delay coupled Hindmarsh–Rose neurons via nonlinear control is investigated in this paper. Both the intrinsic slow current delay in a single Hindmarsh–Rose neuron and the coupling delay between the two neurons are considered. When there is no control, chaotic synchronization occurs for a limited range of the coupling strength and the time-delay values. To obtain complete chaotic synchronization irrespective of the time-delay or the coupling strength, we propose two nonlinear control schemes. The first uses adaptive control for chaotic synchronization of two electrically coupled delayed Hindmarsh–Rose neuron models. The second derives the sufficient conditions to ensure a complete synchronization between master and slave models through appropriate Lyapunov–Krasovskii functionals and the linear matrix inequality technique. Numerical simulations are carried out to show the effectiveness of the proposed methods.