Experimental observation of coherence resonance in an excitable chemical reaction system

White-noise-induced double coherence resonances in reduced Hodgkin-Huxley neuron model near subcritical Hopf bifurcation

Coherence resonance (CR) describes a counterintuitive phenomenon in which the optimal oscillatory responses in nonlinear systems are shaped by a suitable noise amplitude. This phenomenon has been observed in neural systems. In this research, the generation of double coherence resonances (DCRs) due to white noise is investigated in a three-dimensional reduced Hodgkin-Huxley neuron model with multiple-timescale feature. We show that additive white noise can induce DCRs from the resting state near a subcritical Hopf bifurcation. The appearance of DCRs is related to the changes of the firing pattern aroused by the increases of the noise amplitude. The underlying dynamical mechanisms for the appearance of the DCRs and the changes of the firing pattern are interpreted using the phase space analysis and the dynamics of the stable focus-node near the subcritical Hopf bifurcation. We find that the multiple-timescale dynamics is essential for generating the DCRs and different firing patterns. The results not only present a case in which noise can induce DCRs near a Hopf bifurcation but also provide its dynamical mechanism, which enriches the phenomena in nonlinear dynamics and provides further understanding on the roles of noise in neural systems with multiple-timescale feature.

Coherence resonance in neuronal populations: Mean-field versus network model

The counterintuitive phenomenon of coherence resonance describes a nonmonotonic behavior of the regularity of noise-induced oscillations in the excitable regime, leading to an optimal response in terms of regularity of the excited oscillations for an intermediate noise intensity. We study this phenomenon in populations of FitzHugh-Nagumo (FHN) neurons with different coupling architectures. For networks of FHN systems in an excitable regime, coherence resonance has been previously analyzed numerically. Here we focus on an analytical approach studying the mean-field limits of the globally and locally coupled populations. The mean-field limit refers to an averaged behavior of a complex network as the number of elements goes to infinity. We apply the mean-field approach to the globally coupled FHN network. Further, we derive a mean-field limit approximating the locally coupled FHN network with low noise intensities. We study the effects of the coupling strength and noise intensity on coherence resonance for both the network and the mean-field models. We compare the results of the mean-field and network frameworks and find good agreement in the globally coupled case, where the correspondence between the two approaches is sufficiently good to capture the emergence of coherence resonance, as well as of anticoherence resonance.

Vibrational antiresonance in nonlinear coupled systems

We examine the response of a system of coupled nonlinear oscillators driven by a rapidly varying field, to a low frequency weak periodic excitation of one of the oscillators. The response amplitude of the weak field-driven oscillator at an optimal strength of the rapidly varying field exhibits a strong suppression accompanied by a large negative shift in its oscillation phase. The minimum can be identified as vibrational antiresonance in between the two maxima corresponding to vibrational resonance. This vibrational antiresonance can be observed only in nonlinear coupled systems and not in linearly coupled systems or in a single nonlinear oscillator, under similar physical condition. We discuss the underlying dynamical mechanism, the role of nonlinearity and high frequency in characterizing this counter-resonance effect. Our theoretical analysis is corroborated by detailed numerical simulations.

Pulse-density modulation control of chemical oscillation far from equilibrium in a droplet open-reactor system

The design, construction and control of artificial self-organized systems modelled on dynamical behaviours of living systems are important issues in biologically inspired engineering. Such systems are usually based on complex reaction dynamics far from equilibrium; therefore, the control of non-equilibrium conditions is required. Here we report a droplet open-reactor system, based on droplet fusion and fission, that achieves dynamical control over chemical fluxes into/out of the reactor for chemical reactions far from equilibrium. We mathematically reveal that the control mechanism is formulated as pulse-density modulation control of the fusion–fission timing. We produce the droplet open-reactor system using microfluidic technologies and then perform external control and autonomous feedback control over autocatalytic chemical oscillation reactions far from equilibrium. We believe that this system will be valuable for the dynamical control over self-organized phenomena far from equilibrium in chemical and biomedical studies.

Biological systems typically operate at conditions far from chemical equilibrium. Here, the authors model and develop a microfluidic reactor allowing control over time-variable supply and dissipation of chemicals by droplet fusion and fission, allowing non-equilibrium chemical reactions to be regulated.

Intrinsic noise induced coherence resonance in a glow discharge plasma

Experimental evidence of intrinsic noise induced coherence resonance in a glow discharge plasma is being reported. Initially the system is started at a discharge voltage (DV) where it exhibited fixed point dynamics, and then with the subsequent increase in the DV spikes were excited which were few in number and with further increase of DV the number of spikes as well as their regularity increased. The regularity in the interspike interval of the spikes is estimated using normalized variance. Coherence resonance was determined using normalized variance curve and also corroborated by Hurst exponent and power spectrum plots. We show that the regularity of the excitable spikes in the floating potential fluctuation increases with the increase in the DV, up to a particular value of DV. Using a Wiener filter, we separated the noise component which was observed to increase with DV and hence conjectured that noise can play an important role in the generation of the coherence resonance. From an anharmonic oscillator equation describing ion acoustic oscillations, we have been able to obtain a FitzHugh-Nagumo like model which has been used to understand the excitable dynamics of glow discharge plasma in the presence of noise. The numerical results agree quite well with the experimental results.

Event-triggered feedback in noise-driven phase oscillators

Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics and a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths.

Temperature-driven coherence resonance and stochastic resonance in a thermochemical system

We perform the stochastic analysis of a thermochemical system using a master equation which describes a chemical reaction and includes discrete and continuous temperature jumps. We study the time evolution of the system selecting the temperature of the thermostat as an easily tunable control parameter. Depending on the thermostat temperature, the system can be in an excitable, oscillatory, or stationary regime. Stochastic time series for the system temperature are generated and the distributions of interspike intervals are analyzed in the three dynamical regimes separated by a homoclinic bifurcation and a Hopf bifurcation. Different constructive roles of internal fluctuations are exhibited. A noise-induced transition is observed in the vicinity of the Hopf bifurcation. Coherence resonance and stochastic resonance are found in the oscillatory regime. In a range of thermostat temperatures, a nontrivial behavior of the highly nonlinear system is revealed by the existence of both a minimum and a maximum in the scaled standard deviation of interspike intervals as a function of particle number. This high sensitivity to system size illustrates that controlling dynamics in nanoreactors may remain a difficult task.

Synaptic symmetry increases coherence in a pair of excitable electronic neurons

We study how the synaptic connections in a pair of excitable electronic neurons affect the coherence of their spike trains when the neurons are submitted to noise from independent sources. The coupling is provided by electronic circuits which mimic the dynamics of chemical AMPA synapses. In particular, we show that increasing the strength of an unidirectional synapse leads to a decrease of coherence in the post-synaptic neuron. More interestingly, we show that the decrease of coherence can be reverted if we add a synapse of sufficient strength in the reverse direction. Synaptic symmetry plays an important role in this process and, under the right choice of parameters, increases the network coherence beyond the value achieved at the resonance due to noise alone in uncoupled neurons. We also show that synapses with a longer time scale sharpen the dependency of the coherence on the synaptic symmetry. The results were reproduced by numerical simulations of a pair of synaptically coupled FitzHugh-Nagumo models.

Nonlinear vibrational resonance

We examine the nonlinear response of a bistable system driven by a high-frequency force to a low-frequency weak field. It is shown that the rapidly varying temporal oscillation breaks the spatial symmetry of the centrosymmetric potential. This gives rise to a finite nonzero response at the second harmonic of the low-frequency field, which can be optimized by an appropriate choice of vibrational amplitude of the high-frequency field close to that for the linear response. The potential implications of the nonlinear vibrational resonance are analyzed.

Dynamics of an excitable glow-discharge plasma under external forcing

Glow discharge plasma in the excitable regime shows rich dynamical behavior under external forcing. By perturbing the plasma with a subthreshold sawtooth periodic signal, we obtained small subthreshold oscillations that showed resonance with the perturbation frequency. The resonance phenomenon can be useful to estimate characteristic of an excitable system. However, for suprathreshold perturbation, frequency entrainment was observed. In this case, the system showed harmonic frequency entrainment for the perturbation frequencies greater than the characteristic frequency of the system and the excitable behavior for the perturbation frequencies well below the characteristic frequency. The experiments were performed in a glow-discharge plasma where excitability was achieved at a suitable discharge voltage and gas pressure.

Control of noise-induced coherent behaviors in an array of excitable elements by time-delayed feedback

We investigate feedback-controlled coherent dynamics in a two-dimensional array of excitable elements. We demonstrate that one can freely enhance or reduce the spatiotemporal coherence of noise-induced oscillation, such as coherence resonance and phase synchronization, by controlling both the delay time and the feedback gain. Furthermore, we find that noise-induced oscillations are entrained by the feedback force in a certain range of the delay time. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator model.

Modeling noise-induced resonance in an excitable system: an alternative approach

Recently, it has been observed [Md. Nurujjaman, Phy. Rev. E 80, 015201(R) (2009)] that in an excitable system, one can maintain noise-induced coherency in the coherence resonance by blocking the destructive effect of the noise on the system at higher noise level. This phenomenon of constant coherence resonance (CCR) cannot be explained by the existing way of simulation of the model equations of an excitable system with added noise. In this paper, we have proposed a general model which explains the noise-induced resonance phenomenon CCR as well as coherence resonance (CR) and stochastic resonance (SR). The simulation has been carried out considering the basic mechanism of noise-induced resonance phenomena: noise only perturbs the system control parameter to excite coherent oscillations, taking proper precautions so that the destructive effect of noise does not affect the system. In this approach, the CR has been obtained from the interference between the system output and noise and the SR has been obtained by adding noise and a subthreshold signal. This also explains the observation of the frequency shift of coherent oscillations in the CCR with noise level.

Coherence resonance in a unijunction transistor relaxation oscillator

The phenomenon of coherence resonance is investigated in an unijunction transistor relaxation oscillator and quantified by estimating the normal variance (NV). Depending on the measuring points, two types of NV curves have been obtained. We have observed that the degradations in coherency at higher noise amplitudes in our system is probably the result of direct interference of coherent oscillations and the stochastic perturbation. Degradation of coherency may be minimal if this direct interference of noise and coherent oscillations is eliminated.

Noise-invoked resonances near a homoclinic bifurcation in the glow discharge plasma

Stochastic resonance (SR) and coherence resonance (CR) have been studied experimentally in discharge plasmas close to a homoclinic bifurcation. For the SR phenomenon, it is observed that a superimposed subthreshold periodic signal can be recovered via stochastic modulations of the discharge voltage. Furthermore, it is realized that even in the absence of a subthreshold deterministic signal, the system dynamics can be recovered and optimized using noise. This effect is defined as CR in the literature. In the present experiments, induction of SR and CR is quantified using the absolute mean difference and normalized variance techniques, respectively.

Coherence resonance in a chemical excitable system driven by coloured noise

We investigate how the temporal correlation in excitable systems driven by external noise affects the coherence of the system's response. The coupling to the fluctuating environment is introduced via fluctuations of a bifurcation parameter that controls the local dynamics of the light-sensitive Belousov-Zhabotinsky reaction and of its numerical description, the Oregonator model. Both systems are brought from a highly incoherent regime to a coherent one by an appropriate choice of the correlation time and keeping noise variance constant. This effect has been found both for an Ornstein-Uhlenbeck process and for a dichotomous telegraph signal. In the latter case, we are able to connect the optimal correlation time, for which the system behaviour is most coherent, with a characteristic time scale of the system.

Array-enhanced coherence resonance and phase synchronization in a two-dimensional array of excitable chemical oscillators

We investigate the spatiotemporal dynamics in a two-dimensional array of excitable elements subjected to independent external noise, where elements are prepared by localizing the Belousov-Zhabotinsky reaction in a gel matrix. We experimentally demonstrate that the coherence of noise-induced firings is improved with increasing the array size, i.e., the occurrence of array-enhanced coherence resonance. Furthermore, it is found that synchronization among oscillators which are barely coupled can be achieved via coherence resonance. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator reaction-diffusion model.

Experimental evidence of coherence resonance in a time-delayed bistable system

We report on the experimental observation of coherence resonance in a bistable system with delay. Our system consists of a vertical-cavity surface-emitting laser subject to time-delayed optical feedback simultaneously from a long and from an extremely short external cavity. Coherence resonance is experimentally proven by analysis of the residence time distribution of the polarization mode-hopping regime and of the signal to noise ratio in the power spectrum.

Internal signal transmission in one-way coupled excitable system: noise and coupling effects

We study the spatiotemporal dynamics of a one-way coupled FitzHugh-Nagumo system of twenty neurons, which is subject to external noise at the first neuron. It is shown that noise-induced oscillation (NIO) triggered at the first neuron is propagated along the chain with noise suppression, such that a rather "regular" signal is obtained at the last neuron, which can provide a mechanism for the creation of the informative signal in a neural network. Coherence resonance or coherence biresonance behavior appears in the transmission of NIO at appropriate coupling and the information flow in each neuron can be simultaneously optimized at the optimal value of noise.

Regulating noise-induced spiking using feedback

We report successful manipulation of the noise provoked spiking behavior using delayed feedback control. Experiments were performed in a three electrode electrochemical cell under potentiostatic conditions. The uncontrolled system exhibited noise invoked oscillations whose regularity was quantified using normalized variance (NV). Superimposing delayed feedback, for appropriate values of delay (t), an enhancement in the regularity of the spike sequence was attained. Numerical simulations corroborated experimental observations.

Two distinct mechanisms of coherence in randomly perturbed dynamical systems

We carefully examine two mechanisms--coherence resonance and self-induced stochastic resonance--by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles. We analyze what controls the degree of coherence in these two mechanisms and classify their very different properties. In particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio. In contrast, self-induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both the noise amplitude and the time scale separation ratio.

Canard Explosion and Coherent Biresonance in the Rate Oscillation of CO Oxidation on Platinum Surface

The relationship between canard explosion and coherent biresonance is analyzed by numerically investigating a temporal dynamical model of CO oxidation on Pt surface. Canard explosion, manifesting itself by a dramatic change in the amplitude and period of a periodic orbit within a very narrow interval of a control parameter, is the result of multiple time scales in a dynamical system and is common in excitable systems. Coherent biresonance, namely, two peaks on the signal-to-noise ratio (SNR) curve when varying noise intensity, is a novel phenomenon of coherent resonance which is well-known in excitable systems. When the control parameter is varied from a stable fixed point, crossing the supercritical Hopf bifurcation, one of the peaks that corresponds to relatively larger noise intensity, keeps a constant height and position, while the other becomes higher and moves to lower noise level. When we consider the case in which two control parameters are perturbed by independent noise simultaneously, an interesting picture of one valley between two ridges appears on the 3D surface of SNR.

Fokker-Planck analysis of stochastic coherence in models of an excitable neuron with noise in both fast and slow dynamics

We provide a detailed and quantitative Fokker-Planck analysis of noise-induced periodicity (stochastic coherence, also known as coherence resonance) in both a discrete-time model and a continuous-time model of excitable neurons. In particular, we show that one-dimensional models can explain why the effects of noise added to the fast and slow dynamics of the models are dramatically different. We argue that such effects should occur in any excitable system with two or more distinct time scales and need to be taken into account in experiments investigating stochastic coherence.

Noise-induced spatiotemporal dynamics in a linear array of excitable chemical oscillators

The effects of additive noise on spatiotemporal dynamics are investigated in a one-dimensional array of excitable elements in which the Belousov-Zhabotinsky reaction is localized. At the appropriate separation between adjacent elements, we find that the resonance effect becomes larger for the element being more apart from the first element when only the first element is subjected to the external noise. This phenomenon is a sort of array-enhanced resonance. Furthermore, we find that phase locking between the first element and the other elements is induced via coherence resonance of the first element. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator reaction-diffusion model.

Cooperative dynamics in a network of stochastic elements with delayed feedback

Networks of globally coupled, noise-activated, bistable elements with connection time delays are considered. The dynamics of these systems is studied numerically using a Langevin description and analytically using (1) a Gaussian approximation as well as (2) a dichotomous model. The system demonstrates ordering phase transitions and multistability. That is, for a strong enough feedback it exhibits nontrivial stationary states and oscillatory states whose frequencies depend only on the mean of the time delay distribution function. Other observed dynamical phenomena include coherence resonance and, in the case of nonuniform coupling strengths, amplitude death and chaos. Furthermore, an increase of the stability of the trivial equilibrium with increasing nonuniformity of the time delays is observed.

Noise-induced phase locking in coupled coherence-resonance oscillators

We investigate the effects of additive noise on coupled excitable chemical oscillators, particularly focusing on how oscillatory coupled modes can be induced by noise. We find that phase locking in the weak coupling regime occurs through coherence resonance, although the resulting phase locking modes are apparently similar to those in coupled deterministic oscillators. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator reaction-diffusion model.

Delayed feedback as a means of control of noise-induced motion

Time-delayed feedback is exploited for controlling noise-induced motion in coherence resonance oscillators. Namely, under the proper choice of time delay, one can either increase or decrease the regularity of motion. It is shown that in an excitable system, delayed feedback can stabilize the frequency of oscillations against variation of noise strength. Also, for fixed noise intensity, the phenomenon of entrainment of the basic oscillation period by the delayed feedback occurs. This allows one to steer the time scales of noise-induced motion by changing the time delay.

Dynamics of a stochastic oscillator in an excitable chemical reaction system

Dynamics of an excitable Belousov-Zabotinsky reaction system subject to a subthreshold periodic signal and additive noise is investigated using cation exchange beads loaded with the cationic catalyst. We find two kinds of resonance phenomena with respect to the noise amplitude. At the first resonance, the regular oscillation appears with the period equal to the signal period, which is identified as a conventional stochastic resonance. The second resonance is a great contrast to conventional stochastic resonance, in which the regular oscillation appears in a characteristic fashion like phase locking of deterministic oscillators, depending on the signal period and a noise amplitude. Results are explained in terms of the interplay of periodic forcing and the noise-induced oscillator with the period determined by an intrinsic time scale of the system.