Stochastic resonance in coupled nonlinear dynamic elements

Theoretical description of logical stochastic resonance and its enhancement: Fast Fourier transform filtering method

Over the past decade, dynamic schemes have been proposed for the use of bistable systems in the design of logic devices. A bistable system in a noisy background can operate as a reliable logic gate in a moderate noise level, which is called a logical stochastic resonance (LSR). In this paper, we theoretically explore the emergence of LSR in general bistable systems and identify the dynamical mechanisms of LSR. The timescale relationship between the measured time and the mean first-pass time of two-state transitions is a key condition in determining whether the system is reliable. Furthermore, we demonstrate that the stability of the logic operation can be significantly improved by choosing the appropriate filtering method. Low-pass filtered noise-driven systems are more stable than Gaussian white noise. However, band-pass and high-pass filtered noise are harmful to the stability of the system due to the filtering of low-frequency components. Our theoretical and numerical simulation results offer perspectives for the development of logic devices.

Effects of chaotic activity and time delay on signal transmission in FitzHugh-Nagumo neuronal system

The influences of chaotic activity and time delay on the transmission of the sub-threshold signal (STS) in a single FitzHugh-Nagumo neuron and coupled neuronal networks are studied. It is found that a moderate chaotic activity level can enhance the system's detection and transmission of STS. This phenomenon is known as chaotic resonance (CR). In a single neuron, the large amplitude and small period of the STS have a positive effect on the CR phenomenon. In the coupled neuronal network, however, the signal transmission performance of chemical synapses is better than that of electrical synapses. The time delay can determine the trend of the system response, and the multiple chaotic resonances phenomenon is observed upon fine-tuning the time delay length. Both sub-harmonic chaotic resonance and chaotic anti-resonance appear when the STS period and time delay are locked. In chained networks, the signal transmission performance between electrical synapses attenuates continuously. Conversely, the performance between chemical synapses reaches a steady state.

Phase-noise-induced resonance in arrays of coupled excitable neural models

Recently, it is observed that, in a single neural model, phase noise (time-varying signal phase) arising from an external stimulating signal can induce regular spiking activities even if the signal is subthreshold. In addition, it is also uncovered that there exists an optimal phase noise intensity at which the spiking rhythm coincides with the frequency of the subthreshold signal, resulting in a phase-noise-induced resonance phenomenon. However, neurons usually do not work alone, but are connected in the form of arrays or blocks. Therefore, we study the spiking activity induced by phase noise in arrays of globally and locally coupled excitable neural models. We find that there also exists an optimal phase noise intensity for generating large neural response and such an optimal value is significantly decreased compared to an isolated single neuron case, which means the detectability in response to the subthreshold signal of neurons is sharply improved because of the coupling. In addition, we reveal two new resonance behaviors in the neuron ensemble with the presence of phase noise: there exist optimal values of both coupling strength and system size, where the coupled neurons generate regular spikes under subthreshold stimulations, which are called as coupling strength and system size resonance, respectively. Finally, the dependence of phase-noise-induced resonance on signal frequency is also examined.

Cooperative dynamics in coupled noisy dynamical systems near a critical point: The dc superconducting quantum interference device as a case study

Dynamical systems that operate near the onset of coupling-induced oscillations can exhibit enhanced sensitivity to external perturbations under suitable operating parameters. This cooperative behavior and the attendant enhancement in the system response (quantified here via a signal-to-noise ratio at the fundamental of the coupling-induced oscillation frequency) are investigated in this work. As a prototype, we study an array of dc superconducting quantum interference device (SQUID) rings locally coupled, unidirectionally as well as bidirectionally, in a ring configuration; it is well known that each individual SQUID can be biased through a saddle-node bifurcation to oscillatory behavior. We show that biasing the array near the bifurcation point of coupling-induced oscillations can lead to a significant performance enhancement.

Potential energy landscape and finite-state models of array-enhanced stochastic resonance

Noise and coupling can optimize the response of arrays of nonlinear elements to periodic signals. We analyze such array-enhanced stochastic resonance (AESR) using finite-state transition rate models. We simply derive the transition rate matrices from the underlying potential energy function of the corresponding Langevin problem. Our implementation exploits Floquet theory and provides useful theoretical and numerical tools. Our framework both facilitates analysis and elucidates the mechanism of AESR. In particular, we show how sublinear coupling diminishes AESR, but superlinear coupling enhances it.

Global spatiotemporal order and induced stochastic resonance due to a locally applied signal

We study the phenomenon of spatiotemporal stochastic resonance (STSR) in a chain of diffusively coupled bistable oscillators. In particular, we examine the situation in which the global STSR response is controlled by a locally applied signal and reveal a wave-front propagation. In order to deepen the understanding of the system dynamics, we introduce, on the time scale of STSR, the study of the effective statistical renormalization of a generic lattice system. Using this technique we provide a criterion for STSR, and predict and observe numerically a bifurcationlike behavior that reflects the difference between the most probable value of the local quasiequilibrium density and its mean value. Our results, tested with a chain of nonlinear oscillators, appear to possess some universal qualities and may stimulate a deeper search for more generic phenomena.

Noisy FitzHugh-Nagumo model: from single elements to globally coupled networks

We study the noisy FitzHugh-Nagumo model, representative of the dynamics of excitable neural elements, and derive a Fokker-Planck equation for both a single element and for a network of globally coupled elements. We introduce an efficient way to numerically solve this Fokker-Planck equation, especially for large noise levels. We show that, contrary to the single element, the network can undergo a Hopf bifurcation as the coupling strength is increased. Furthermore, we show that an external sinusoidal driving force leads to a classical resonance when its frequency matches the underlying system frequency. This resonance is also investigated analytically by exploiting the different time scales in the problem.

Stochastic resonance on two-dimensional arrays of bistable oscillators in a nonlinear optical system

We describe an experimental realization of stochastic resonance in two-dimensional arrays of coupled nonlinear oscillators. The experiment is implemented using an optoelectronic system composed of a liquid crystal light valve in a feedback loop with external, spatially variable noise being added through a liquid crystal display. The behavior of the system differs from previously studied uniform arrays, showing a high signal-to-noise ratio at the output for a broad range of input noise. We show that this behavior is qualitatively the same as that exhibited by computer models where the nonlinear elements of the array have a distribution of biases applied to their switching thresholds.

Cooperative dynamics in a class of coupled two-dimensional oscillators

We study a system of globally coupled two-dimensional nonlinear oscillators [using the two-junction superconducting quantum interference device (SQUID) as a prototype for a single element] each of which can undergo a saddle-node bifurcation characterized by the disappearance of the stable minima in its potential energy function. This transition from fixed point solutions to spontaneous oscillations is controlled by external bias parameters, including the coupling coefficient. For the deterministic case, an extension of a center-manifold reduction, carried out earlier for the single oscillator, yields an oscillation frequency that depends on the coupling; the frequency decreases with coupling strength and/or the number of oscillators. In the presence of noise, a mean-field description leads to a nonlinear Fokker-Planck equation for the system which is investigated for experimentally realistic noise levels. Furthermore, we apply a weak external time-sinusoidal probe signal to each oscillator and use the resulting (classical) resonance to determine the underlying frequency of the noisy system. This leads to an explanation of earlier experimental results as well as the possibility of designing a more sensitive SQUID-based detection system.

Characterization of the stretched-exponential trap-time distributions in one-dimensional coupled map lattices

Stretched-exponential distributions and relaxation responses are encountered in a wide range of physical systems such as glasses, polymers, and spin glasses. As found recently, this type of behavior occurs also for the distribution function of a certain trap time in a number of coupled dynamical systems. We analyze a one-dimensional mathematical model of coupled chaotic oscillators that reproduces an experimental setup of coupled diode resonators and identify the necessary ingredients for stretched-exponential distributions.

Asymmetric kinks: stabilization by entropic forces

Asymmetric kinks bridging two adjacent potential valleys of equal depth but different curvature are unstable against phonon modes. When coupled to a heat bath, a kink-bearing string tends to cross over into the shallower valley; kinks are thus predicted to drift in the appropriate direction with velocity proportional to the temperature, in close agreement with numerical simulation. When contrasted by a mechanical bias, these entropic forces give rise to a rich phenomenology that includes configurational phase transitions, double-kink dissociation, and noise-directed signal transmission.

Theoretical analysis of array-enhanced stochastic resonance in the diffusively coupled FitzHugh-Nagumo equation

The array-enhanced stochastic resonance (AESR) in the diffusively coupled FitzHugh-Nagumo equation is investigated. The two properties of AESR, namely, the scaling of the optimal noise intensity and the enhancement of the maximum value of the correlation coefficient as a function of the coupling strength, are analyzed theoretically. By transforming the dynamics of N elements into that of the mean and the deviation from it, it is found that AESR is caused by the correlation between them. A low-dimensional model that reproduces the above properties is constructed.

Thermal resonance in signal transmission

We use temperature tuning to control signal propagation in simple one-dimensional arrays of masses connected by hard anharmonic springs and with no local potentials. In our numerical model a sustained signal is applied at one site of a chain immersed in a thermal environment and the signal-to-noise ratio is measured at each oscillator. We show that raising the temperature can lead to enhanced signal propagation along the chain, resulting in thermal resonance effects akin to the resonance observed in arrays of bistable systems.

Monostable array-enhanced stochastic resonance

We present a simple nonlinear system that exhibits multiple distinct stochastic resonances. By adjusting the noise and coupling of an array of underdamped, monostable oscillators, we modify the array's natural frequencies so that the spectral response of a typical oscillator in an array of N oscillators exhibits N-1 different stochastic resonances. Such families of resonances may elucidate and facilitate a variety of noise-mediated cooperative phenomena, such as noise-enhanced propagation, in a broad class of similar nonlinear systems.

Stochastic resonance and noise-enhanced order with spatiotemporal periodic signal

Stochastic resonance is investigated in a chain of coupled threshold elements driven by independent noises and a plane traveling wave. Both stochastic resonance in an individual element embedded in the chain, characterized by a maximum of the signal-to-noise ratio for nonzero noise intensity, and stochastic resonance with spatiotemporal signal, characterized by a maximum of the spatiotemporal input-output correlation function, are observed. For a wide range of wavelengths of the plane wave an optimum value of coupling exists for which both kinds of stochastic resonance are most pronounced, i.e., the phenomenon of array enhanced stochastic resonance is observed. For large wavelengths the enhancement of stochastic resonance coincides with a maximum of spatiotemporal synchronization among elements with the same phase of the periodic signal at inputs. This synchronization is a manifestation of spatiotemporal order induced in the system by the cooperative influence of noise and periodic signal.

Nonlinear stochastic resonance: the saga of anomalous output-input gain

We reconsider stochastic resonance (SR) for an overdamped bistable dynamics driven by a harmonic force and Gaussian noise from the viewpoint of the gain behavior, i.e., the signal-to-noise ratio (SNR) at the output divided by that at the input. The primary issue addressed in this work is whether a gain exceeding unity can occur for this archetypal SR model, for subthreshold signals that are beyond the regime of validity of linear response theory: in contrast to nondynamical threshold systems, we find that the nonlinear gain in this conventional SR system exceeds unity only for suprathreshold signals, where SR for the spectral amplification and/or the SNR no longer occurs. Moreover, the gain assumes, at weak to moderate noise strengths, rather small (minimal) values for near-threshold signal amplitudes. The SNR gain generically exhibits a distinctive nonmonotonic behavior versus both the signal amplitude at fixed noise intensity and the noise intensity at fixed signal amplitude. We also test the validity of linear response theory; this approximation is strongly violated for weak noise. At strong noise, however, its validity regime extends well into the large driving regime above threshold. The prominent role of physically realistic noise color is studied for exponentially correlated Gaussian noise of constant intensity scaling and also for constant variance scaling; the latter produces a characteristic, resonancelike gain behavior. The gain for this typical SR setup is further contrasted with the gain behavior for a "soft" potential model.

Stochastic resonance in noisy maps as dynamical threshold-crossing systems

Interplay of noise and periodic modulation of system parameters for the logistic map in the region after the first bifurcation and for the kicked spin model with Ising anisotropy and damping is considered. For both maps two distinct symmetric states are present that correspond to different phases of the period-2 orbit of the logistic map and to disjoint attractors of the spin map. The periodic force modulates the transition probabilities from any state to the opposite one symmetrically. It follows that the maps behave as threshold-crossing systems with internal dynamics, and stochastic resonance (maximum of the signal-to-noise ratio in the signal reflecting the occurrence of jumps between the symmetric states) in both models is observed. Numerical simulations agree qualitatively with analytic results based on the adiabatic theory.