Chaotic Bursting Research Articles

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

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

A memristive chaotic system with rich dynamical behavior and circuit implementation

In this paper, a novel four-dimensional autonomous chaotic system based on memristive diode bridge is presented, and it consists of a simple linear oscillation and a memristor. According to the mathematical model of the system, the equilibrium point is analyzed, and the phase portraits, time-domain sequences, bifurcation diagrams and Lyapunov exponent spectrums are numerically simulated. The rich dynamical behavior of proposed system is investigated. It includes multistability, offset boosting and chaotic bursting. In addition, the circuit simulation and Field-Programmable Gate Array (FPGA) hardware experiment are carried out to verify the feasibility of the system. Then the application of proposed system in chaotic image encryption is presented. By carrying out some security performance analyses, we show that the proposed system has good security performance.

Spontaneous locomotion of phoretic particles in three dimensions

The motion of an autophoretic spherical particle in a simple fluid is analyzed. This motion is powered by a chemical species which is absorbed or emitted by the particle and which diffuses and is advected in the surrounding fluid. The transition from the nonmotile to the motile state occurs if the Péclet number Pe (defined as the ratio of the solute emission rate over the solute diffusion rate) is sufficiently large. We first analyze the axisymmetric case (restricting the particle to a unique direction). In this case, we find that the motion of the particle transits from a motionless to a directed motion at a given critical Pe. Increasing Pe, we find a second critical value where the particle becomes stagnant in a symmetric flow. A further increase of Pe leads to a recovery of motile motion. When Pe is increased even further, the particle shows a periodic motion undergoing a subharmonic cascade before entering chaos. In this regime, the mean-square displacement behaves quadratically with time (a ballistic regime). When the axisymmetry constraint is relaxed, allowing the particle to freely move in three-dimensional space, we find that at a small Pe the particle moves in a straight manner. There exists a critical value where the particle exhibits an oscillatory motion with a meandering trajectory. Increasing Pe further leads to chaotic bursts for some time, before entering fully into chaos via the intermittency scenario at a critical Pe number. In this regime, the particle shows run-and-tumble–like dynamics: The trajectory is then characterized by a ballistic swimming nature at a short time and a diffusive nature at a long time.5 MoreReceived 31 May 2021Accepted 1 March 2022DOI:https://doi.org/10.1103/PhysRevFluids.7.034003©2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLocomotionSwimmingFluid Dynamics

Memristive Rulkov Neuron Model With Magnetic Induction Effects

The magnetic induction effects have been emulated by various continuous memristive models but they have not been successfully described by a discrete memristive model yet. To address this issue, this article first constructs a discrete memristor and then presents a discrete memristive Rulkov (m-Rulkov) neuron model. The bifurcation routes of the m-Rulkov model are declared by detecting the eigenvalue loci. Using numerical measures, we investigate the complex dynamics shown in the m-Rulkov model, including regime transition behaviors, transient chaotic bursting regimes, and hyperchaotic firing behaviors, all of which are closely relied on the memristor parameter. Consequently, the involvement of memristor can be used to simulate the magnetic induction effects in such a discrete neuron model. Besides, we elaborate a hardware platform for implementing the m-Rulkov model and acquire diverse spiking-bursting sequences. These results show that the presented model is viable to better characterize the actual firing activities in biological neurons than the Rulkov model when biophysical memory effect is supplied.

A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling

A memristor is a vital circuit element that can mimic biological synapses. This paper proposes the memristive version of a recently proposed map neuron model based on the phase space. The dynamic of the memristive map model is investigated by using bifurcation and Lyapunov exponents’ diagrams. The results prove that the memristive map can present different behaviors such as spiking, periodic bursting, and chaotic bursting. Then, a ring network is constructed by hybrid electrical and chemical synapses, and the memristive neuron models are used to describe the nodes. The collective behavior of the network is studied. It is observed that chemical coupling plays a crucial role in synchronization. Different kinds of synchronization, such as imperfect synchronization, complete synchronization, solitary state, two-cluster synchronization, chimera, and nonstationary chimera, are identified by varying the coupling strengths.

Firing activities induced by memristive autapse in Fitzhugh–Nagumo neuron with time delay

Due to the natural non-volatility and distinctive plasticity, memristors are considered as ideal devices to mimic biological synapses. In this study, an autapse which is implemented with a locally active memristor, is introduced into Fitzhugh–Nagumo neuron and thus a new neuron model is established. The local stability of the neuron model with and without time delay is analyzed, respectively. Four coexisting firing patterns, including chaotic spiking, periodic spiking, periodic bursting and chaotic bursting, dependent on the memristor initial values, are explored. We find that the neuron has four regular attraction basins and its firing pattern can be regulated by choosing appropriate initial values. The time delay has an important effect on firing activities and the neuron model transits from periodic spiking, to chaotic bursting, and then to chaotic spiking with the increase of the time delay. Furthermore, the influence of the autaptic intensity on firing activities of the neuron is also revealed. In order to verify the complex firing activities, a neuron circuit is constructed and circuit simulations are performed.

Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser.

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known; however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

Abundant Firing Patterns in a Memristive Morris–Lecar Neuron Model

Improving the neuron model and studying its electrical activities according to the real biophysical environment are significant in human cognitive brain activity and neural behavior. The complex transmembrane motion of ions on the neuronal cell membrane can establish time-varying electromagnetic fields and affect the transition firing patterns of neurons. In this paper, a threshold memristor is used to describe the electromagnetic induction and magnetic field effects of neuron cell membrane ion exchange to improve the neuron model, and a memristive Morris–Lecar (mM–L) neuron model is proposed. Numerical simulation confirms that different intensities of electromagnetic fields can produce distinct pattern transitions in electrical activities of the neuron, such as periodic bursting, periodic spiking, chaotic bursting. From the perspective of neuron’s interspike interval (ISI), the ISIs bifurcation in the multiparameter planes, ISIs firing periods, the variance of ISIs and other methods are used to find the trend of the mM–L neuron firing pattern transition. Finally, based on the 4D nonlinear differential equation of the mM–L neuron model, the complete electronic implementation of the model is designed. The output of the designed circuit is consistent with the theoretical prediction, which is extremely useful for studying the dynamics of a single neuron.

Extreme events in globally coupled chaotic maps

Understanding and predicting uncertain things are the central themes of scientific evolution. Human beings revolve around these fears of uncertainties concerning various aspects like a global pandemic, health, finances, to name but a few. Dealing with this unavoidable part of life is far tougher due to the chaotic nature of these unpredictable activities. In the present article, we consider a global network of identical chaotic maps, which splits into two different clusters, despite the interaction between all nodes are uniform. The stability analysis of the spatially homogeneous chaotic solutions provides a critical coupling strength, before which we anticipate such partial synchronization. The distance between these two chaotic synchronized populations often deviates more than eight times of standard deviation from its long-term average. The probability density function of these highly deviated values fits well with the generalized extreme value distribution. Meanwhile, the distribution of recurrence time intervals between extreme events resembles the Weibull distribution. The existing literature helps us to characterize such events as extreme events using the significant height. These extremely high fluctuations are less frequent in terms of their occurrence. We determine numerically a range of coupling strength for these extremely large but recurrent events. On-off intermittency is the responsible mechanism underlying the formation of such extreme events. Besides understanding the generation of such extreme events and their statistical signature, we furnish forecasting these events using the powerful deep learning algorithms of an artificial recurrent neural network. This long short-term memory (LSTM) can offer handy one-step forecasting of these chaotic intermittent bursts. We also ensure the robustness of this forecasting model with two hundred hidden cells in each LSTM layer.

Effect of temperature sensitive ion channels on the single and multilayer network behavior of an excitable media with electromagnetic induction

The dynamical behavior of the neurons directly depends on the transition from resting to spiking states. These transitions show different types of bifurcations and has different spiking periods. The transitions are also affected by the temperature exposure of the ionic channels. To understand such effects, we investigate the Morris-Lecar (ML) neuron model with temperature affected calcium, potassium and leak current channels. The presented ML model is considered with electromagnetic field coupling considering a simple cubic memristor flux relation. Firstly the basic dynamical properties of the ML model is analyzed considering the current temperature as the control parameter. The temperature affected ionic channels in the ML model leads to various types of oscillations from periodic spiking to chaotic bursting. These bifurcation patterns are well discussed with corresponding Lyapunov exponents. To study the wave propagation in the temperature dependent ML model (TDML), we have constructed two different types of network structure. In the first a simple lattice network is considered with the local nodes of the TDML neurons and the temperature effects on the wave propagation is studied individually for the three channels. In the second type of network, we have considered inter coupled three lattice layers of TDML neurons. This discussion is subdivided in to two cases and in the first case the layers are constructed such that the first, second and third layers having temperature affected calcium, potassium and leaky current channels respectively. In the second case we considered only one channel to have temperature effects and the others have no temperature affected ion channels. The wave propagation phenomenon in both the types of network is analyzed considering the current temperature as the control parameter.

Neural Bursting and Synchronization Emulated by Neural Networks and Circuits

Nowadays, research, modeling, simulation and realization of brain-like systems to reproduce brain behaviors have become urgent requirements. In this paper, neural bursting and synchronization are imitated by modeling two neural network models based on the Hopfield neural network (HNN). The first neural network model consists of four neurons, which correspond to realizing neural bursting firings. Theoretical analysis and numerical simulation show that the simple neural network can generate abundant bursting dynamics including multiple periodic bursting firings with different spikes per burst, multiple coexisting bursting firings, as well as multiple chaotic bursting firings with different amplitudes. The second neural network model simulates neural synchronization using a coupling neural network composed of two above small neural networks. The synchronization dynamics of the coupling neural network is theoretically proved based on the Lyapunov stability theory. Extensive simulation results show that the coupling neural network can produce different types of synchronous behaviors dependent on synaptic coupling strength, such as anti-phase bursting synchronization, anti-phase spiking synchronization, and complete bursting synchronization. Finally, two neural network circuits are designed and implemented to show the effectiveness and potential of the constructed neural networks.

Dynamical robustness and firing modes in multilayer memristive neural networks of nonidentical neurons

On the basis of Hindmarsh-Rose neuron model, dynamical robustness and the transition of firing modes of multilayer memristive neural network consisting of nonidentical neurons have been investigated in detail. The dynamic effects of memristive synapses coupling which are either cubic order flux or quadratic flux are detected. Our results suggest that for the cubic order flux-controlled memristors increasing electromagnetic induction parameter and parameter of memristor show the tendency to spoil the dynamical robustness of the multilayer memristive neural network. For quadratic flux-controlled memristors, weak electromagnetic induction can spoil the dynamical robustness while strong electromagnetic induction and parameter of memristor have little influence on dynamical robustness. We found that the ratio of inactive neurons switches the firing patterns of the active neuron among periodic bursting, chaotic bursting and spiking-like. In the case of memristive synapse coupling by cubic order flux-controlled memristors, chaotic bursting alternates with period bursting as the ratio of inactive neurons is increased, and the chaotic bursting occupy a large range of the ratio of inactive neurons. In the case of memristive synapse coupling by quadratic flux-controlled memristors, the main firing pattern of the active neuron is chaotic bursting. The distribution of interspike interval can be broadly classified into two groups: those that are long interspike intervals and those that are short interspike intervals. The dynamics of multilayer memristive neural network is also verified in the circuits built on Multisim. These results could give new mechanism explanation for aging transition by applying electromagnetic field to neural network.

Rhythmicity and firing modes in modular neuronal network under electromagnetic field

Based on a modified Hindmarsh–Rose model with electromagnetic induction being considered, rhythmicity and firing modes in modular neural network consisting of nonidentical neurons have been explored in detail. The global rhythmicity of the system shows a resonancelike behavior with the increase in the ratio of inactive neurons for strong coupling among neurons inside the subnetwork. We found that interaction between different subnetworks and electromagnetic induction parameter show the tendency to weaken dynamical robustness, whereas self-induction parameter could efficiently improve dynamical robustness of modular network. The dynamics of the active neuron on the road to aging transition is detected. The ratio of inactive neurons switches response-firing modes of the active neuron among spiking, periodic bursting and chaotic bursting. Our model system and results can provide new mechanism explanation for electrical activity in biological systems that are prone to suffer damage or deterioration.

Functional brain network dynamics based on the Hindmarsh–Rose model

In order to reveal the dynamics of brain network, we proposed a new research method based on the Hindmarsh–Rose model. In the method, a neural network model was developed by constructing a functional brain network topology based on functional magnetic resonance imaging resting-state data and using Hindmarsh–Rose neurons as nodes in place of the brain regions belonging to the functional brain network. The dynamics of the functional brain network were investigated using the dynamics model. The simulation results showed that the dynamic behaviors of the brain regions in the functional brain network could be divided into three types: stable, chaotic, and periodical bursts. A state space was introduced to analyze the dynamic behavior of the brain regions in the network. We find that increasing excitation and mutual connection strength among brain regions enhanced network communication capabilities in the state space. Both the periodic and stable modes exhibited stronger communication capabilities than the chaotic mode. Despite individual differences in the dynamics of brain regions among subjects, brain regions in the periodic mode were highly consistent and corresponded to key regions of the default mode network in the resting state.

Dynamics of a discontinuous coupled electro-mechanical system oscillator with strong irrational nonlinearities and with two outputs

The dynamics of the nonlinear electromechanical device, consisting of a mechanical part with two outputs and an electrical part which acts as the server is strongly investigated in the present work. The mechanical part consists of two nonlinear oscillators with strong irrational nonlinearities having smooth or discontinuous characteristics, where nonlinearity is just due to the inclination of springs, the geometric configuration, which are both elastically coupled. While the electrical part is the Rayleigh equation. By using the Lagrangian formulation, the model equations are established and used to investigate the equilibrium points and their stabilities. Nest by using the multiple time scales method, the analytical solutions are found both for the case of large amplitude and the weak amplitude, leading to interesting bifurcation sets of the equilibria by varying the control parameters, the inclination angles and driven frequency. Finally, numerical investigations of the exact equation of the system are used to justify the validity of analytical results and to find new phenomena such as chaotic impulses, chaotic bursting and the train of kink signal generations.

Symbolic analysis of bursting dynamical regimes of Rulkov neural networks

Neurons modeled by the Rulkov map display a variety of dynamic regimes that include tonic spikes and chaotic bursting. Here we study an ensemble of bursting neurons coupled with the Watts-Strogatz small-world topology. We characterize the sequences of bursts using the symbolic method of time-series analysis known as ordinal analysis, which detects nonlinear temporal correlations. We show that the probabilities of the different symbols distinguish different dynamical regimes, which depend on the coupling strength and the network topology. These regimes have different spatio-temporal properties that can be visualized with raster plots.

Low dimensional models of dynamo action in rotating magnetoconvection

Abstract Two low-dimensional models for nonlinear dynamo action in Rayleigh-Benard convection in presence of rigid body rotation about vertical axis are constructed for metallic fluids with finite magnetic Prandtl number ( P m ) and small (or zero) thermal Prandtl number ( P r ). Dynamo effect is seen for P m ≥ 0.75 with P r = 0.025 and Taylor number, 0 T a ≤ 1000 . The value of reduced Rayleigh number at the dynamo onset ( r d ) varies with P m , P r and T a . The value of r d decreases with increase in P m and T a separately, if the other two parameters are kept fixed to some small values. However r d increases slightly if P r is increased, the value being minimum for P r = 0 . When 0.75 ≤ P m 4 , dynamo onset appears as intermittent chaotic burst. The intermittent burst changes to continuous chaos with rise in P m . The probability mass of the height of peak in average magnetic energy follows power law when P m is small. For 4 ≤ P m 6 and 600 ≤ T a 1000 dynamo effect starts at the onset of convection as finite oscillation. This effect continues in a small window of r and then disappears. The dynamo action again appears as quasi-periodic wave or chaotic wave for further increase in r .

Dynamic mechanism of multiple bursting patterns in a whole-cell multiscale model with calcium oscillations

The dynamic mechanism of a whole-cell model containing electrical signalling and two-compartment Ca signalling in gonadotrophs is investigated. The transition from spiking to bursting by Hopf bifurcation of the fast subsystem about the slow variable is detected via the suitable parameters. When the timescale of K gating variable is changed, the relaxation oscillation with locally small fluctuation, chaotic bursting and mixed-mode bursting (MMB) are revealed through chaos. In addition, the bifurcation of with regard to is analysed, showing periodic solutions, torus, period doubling solutions and chaos. Finally, hyperpolarizations and torus canard-like behaviours of the full system under a set of specific parameters are elucidated.

Dynamical transitions of the coupled Class I (II) neurons regulated by an astrocyte

Based on the minimal neural network models consisting of a pyramidal (PY) neuron, an interneuron (IN), and an astrocyte (AS), we investigate the regulating effects of astrocyte on dynamical transitions in two coupled neurons of the same type of excitability, either Class I or Class II, which is characterized by saddle-node on invariant circle (SNIC) and Hopf (HB) bifurcation, respectively. It is found that without the regulation of AS, the coupled Class I PY and IN neurons show the regular $$L^s$$ mixed-mode oscillations (MMOs) composed of L large amplitude oscillations (LAOs) and s small amplitude oscillations (SAOs) in one period. Under the AS action, the calcium signals in AS can induce and regulate the mixed-mode low-frequency bursting firings. By contrast, for the coupled Class II PY and IN neurons, when AS is ignored, as the coupling strength increases, the system shows the periodic LAO oscillations, $$L^s$$ mode of oscillations, and the chaotic behaviors. Furthermore, the period-doubling bifurcation is clearly captured. However, the presence of AS makes the PY neuron exhibit the mixed-mode low-frequency chaotic bursting activities. Interestingly, we also discover a new transition route ( $$1^0\rightarrow 1^1\rightarrow 1^2\rightarrow \ldots $$ ) of calcium signals due to the period adding bifurcation of the system, which can shape the firing patterns of the PY neuron. Our results suggest that calcium signals in AS indeed involve in and even shape the coupling dynamics of PY and IN neurons. In particular, AS may exert differential roles in modulating the dynamical properties of Class I and Class II neurons.

Route to hyperbolic hyperchaos in a nonautonomous time-delay system.

We consider a self-oscillator whose excitation parameter is varied. The frequency of the variation is much smaller than the natural frequency of the oscillator so that oscillations in the system are periodically excited and decayed. Also, a time delay is added such that when the oscillations start to grow at a new excitation stage, they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to nonlinearity, the seeding from the past arrives with a doubled phase so that the oscillation phase changes from stage to stage according to the chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period, we found a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as an alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with frequent visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; and (d) transformation of hyperchaos to the hyperbolic form.