Inverse stochastic resonance in Hodgkin–Huxley neural system driven by Gaussian and non-Gaussian colored noises

Inverse stochastic resonance (ISR) is the phenomenon of the response of neuron to noise, which is opposite to the conventional stochastic resonance. In this paper, the ISR phenomena induced by Gaussian and non-Gaussian colored noises are studied in the cases of single Hodgkin–Huxley (HH) neuron and HH neural network, respectively. It is found that the mean firing rate of electrical activities depends on the Gaussian or non-Gaussian colored noises which can induce the phenomenon of ISR. The ISR phenomenon induced by Gaussian colored noise is most obvious under the conditions of low external current, low reciprocal correlation rate and low noise level. The ISR in neural network is more pronounced and lasts longer than the duration of a single neuron. However, the ISR phenomenon induced by non-Gaussian colored noise is apparent under low noise correlation time or low departure from Gaussian noise, and the ISR phenomena show different duration ranges under different parameter values. Furthermore, the transition of mean firing rate is more gradual, the ISR lasts longer, and the ISR phenomenon is more pronounced under the non-Gaussian colored noise. The ISR is a common phenomenon in neurodynamics; our results might provide novel insights into the ISR phenomena observed in biological experiments.

     

Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

The phenomenon of stochastic bifurcation driven by the correlated non-Gaussian colored noise and the Gaussian white noise is investigated by the qualitative changes of steady states with the most probable phase portraits. To arrive at the Markovian approximation of the original non-Markovian stochastic process and derive the general approximate Fokker-Planck equation (FPE), we deal with the non-Gaussian colored noise and then adopt the uni¯ed colored noise approximation (UCNA). Subsequently, the theoretical equation concerning the most probable steady states is obtained by the maximum of the stationary probability density function (SPDF). The parameter of the uncorrelated additive noise intensity does enter the governing equation as a non-Markovian effect, which is in contrast to that of the uncorrelated Gaussian white noise case, where the parameter is absent from the governing equation, i.e., the most probable steady states are mainly controlled by the uncorrelated multiplicative noise. Additionally, in comparison with the deterministic counterpart, some peculiar bifurcation behaviors with regard to the most probable steady states induced by the correlation time of non-Gaussian colored noise, the noise intensity, and the non-Gaussian noise deviation parameter are discussed. Moreover, the symmetry of the stochastic bifurcation diagrams is destroyed when the correlation between noises is concerned. Furthermore, the feasibility and accuracy of the analytical predictions are verified compared with those of the Monte Carlo (MC) simulations of the original system.

     

Stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises

The stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises is investigated by using the theory of signal-to-noise (SNR) in the adiabatic limit. The analytic expression of the SNR is obtained for arbitrary signal amplitude without being restricted to small amplitudes. The SNR is a nonmonotonic function of intensities of multiplicative and additive noises and the noise intensity ratio R=D/Q, so stochastic resonance exhibits in the bias monostable system. We investigate the effect of any system parameter (such as D,Q,R,r) on the SNR. It is shown that the SNR is a nonmonotonic function of the static asymmetry r, also; the SNR is decreased when |r| is increased. Moreover, the SNR is increased when the noise intensity ratio R=D/Q is increased.     

Stochastic resonance phenomenon in an underdamped bistable system driven by weak asymmetric dichotomous noise

Stochastic resonance in an underdamped bistable system subjected to a weak asymmetric dichotomous noise is investigated numerically. Dichotomous noise is a non-Gaussian color noise and more complex than Gaussian white noise, whose waiting time complies with the exponential distribution. Utilizing an efficiently numerical algorithm, we acquire the asymmetric dichotomous noise accurately. Then the system responses and the averaged power spectrum as the signatures of the stochastic resonance are calculated by the fourth-order Runge?CKutta algorithm. The effects of the noise strength, the forcing frequency, and the asymmetry of dichotomous noise on the system responses and the effects of the forcing frequency on the averaged power spectrum are discussed, respectively. It is found that the increasing of the noise strength or the forcing frequency could strengthen the passage between the stable points of the system, and the system responses also display the asymmetry for the asymmetric dichotomous noise, which has not been discovered in other investigated results. Additionally, the averaged power spectrum exhibits the sharp peaks, which indicates the occurrence of stochastic resonance, and we also discover two critical forcing frequencies: one denoting the transformation of the peaks and another for the optimum on stochastic resonance.     

Stochastic resonance in a harmonic oscillator with damping trichotomous noise

In this paper, the phenomenon of stochastic resonance (SR) in a prototype fluctuating damping harmonic oscillator with trichotomous Markovian noise is investigated. The exact expression of output amplitude gain has been calculated using the well-known Shapiro–Loginov formula. The phenomenon of SR has been found in a broad sense—that is, the non-monotonic behavior of output amplitude gain as a function of noise parameters. Then the influences of noise amplitude, noise switching rate, and noise flatness on the output amplitude gain have also been discussed. Finally, the reverse resonance phenomenon has been presented.     

Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure: influence of noise

This paper presents an analysis of the effects of noise and precision on a simplified model of the clarinet driven by a variable control parameter. When the control parameter is varied, the clarinet model undergoes a dynamic bifurcation. A consequence of this is the phenomenon of bifurcation delay: the bifurcation point is shifted from the static oscillation threshold to a higher value, called dynamic oscillation threshold. In a previous work Bergeot et al. in Nonlinear Dyn. doi:10.1007/s11071-013-0806-y, (2013), the dynamic oscillation threshold is obtained analytically. In the present article, the sensitivity of the dynamic threshold on precision is analyzed as a stochastic variable introduced in the model. A new theoretical expression is given for the dynamic thresholds in presence of the stochastic variable, providing a fair prediction of the thresholds found in finite-precision simulations. These dynamic thresholds are found to depend on the increase rate and are independent on the initial value of the parameter, both in simulations and in theory.     

Flow-excited acoustic resonance of two side-by-side cylinders in cross-flow

The aeroacoustic response of two side-by-side circular cylinders in cross-flow is investigated experimentally. In order to investigate the effect of the gap between the cylinders on the acoustic resonance mechanism, six spacing ratios between the cylinders, in the range of T/D=1.25–3, have been investigated, where D is the diameter of the cylinders and T the centre-to-centre distance between them. Special attention is given to the intermediate spacing ratio range, which exhibits bistable flow regimes in the absence of resonance. During the tests, the acoustic cross-modes of the duct housing the cylinders are self-excited. For the intermediate spacing ratios, T/D=1.25, 1.35, 1.46 and 1.75, two distinct vortex-shedding frequencies at the off-resonance conditions are observed. These are associated with the wide and narrow wakes of the cylinders, as described in the literature. In this case, acoustic resonances occur at a Strouhal number, which is between those observed before the onset of resonance. The acoustic resonance synchronizes vortex shedding in the two wakes and thereby eliminates the bistable flow phenomenon. For large spacing ratios, T/D=2.5 and 3, vortex shedding occurs at a single Strouhal number at which the acoustic resonance is excited.     

Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state

Van der Pol–Duffing oscillator, which can be used a model for many dynamical system, has been widely concerned. However, most of the systems by scholars are either stable steady states or limit cycles. Here, the self-sustained oscillator with the coexistence of steady state and limit cycles, which is famous for describing the flutter of airfoils with large span ratio in low-speed wind tunnels, is treated in this paper. Using the energy balance method, the deterministic bifurcation of the tristable system with time-delay feedback is investigated. The presence of time-delay feedback expands the bifurcation range of the parameters, making the bifurcation phenomenon more abundant. In addition, according to the stationary probability density function obtained by the stochastic averaging method, stochastic bifurcation of the system with time-delay feedback and noise is explored theoretically. The numerical results confirm the correctness of the theoretical analysis. Transition between the unimodal structure, the bimodal structure and the trimodal structure is found. Many rich bifurcations are available by adjusting the time-delay and noise intensity, which may be conductive to achieve the desired phenomenon in the real-world application.

     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

Existence of Bubbling Solutions for Chern–Simons Model on a Torus

We study the existence of bubbling solutions for the the following Chern–Simons–Higgs equation: $$\Delta u +\frac1{\varepsilon^2} {\rm e}^u(1-{\rm e}^u) = 4\pi \sum_{i=1}^{2k}\delta_{p_i},\quad \text{in}\,\Omega,$$ where Ω is a torus. If k = 1, for any critical point q of the associated sum of the Green functions, we introduce a quantity D(q) (see (1.11) below). We show that for any non-degenerate critical point q with D(q) < 0, the above problem has a solution u _ε satisfying that ε → 0, u _ε blows up at q. The calculations in this paper also show that, if a sequence of solutions u _ε blows up at q as ε → 0, then q must be a critical point of the associated sum of the Green functions, and ${D(q) \leqq 0}$ . So, the condition D(q) < 0 is almost necessary to obtain our result. We also construct solutions with k bubbles for ${k \geqq 2}$ .     

Numerical and experimental investigation of flow-acoustic resonance of side-by-side cylinders in a duct

The phenomenon of flow-excited acoustic resonance is a design concern in many engineering applications, especially when wakes of bluff bodies are encountered in ducts, piping systems, heat exchangers, and other confined systems. In this paper, the case of self-excited acoustic resonance of two side-by-side cylinders in a duct with cross-flow is investigated both numerically and experimentally for a single spacing ratio of T/D=2.5, where D is the diameter of the cylinders and T is the centre-to-centre distance between them. The numerical investigation is performed using a finite-volume method at a Reynolds number of 3.0×10⁴ to simulate the unsteady flow field, which is then coupled with an imposed resonant sound field of the first acoustic cross-mode of the duct calculated through the use of Finite Element Analysis (FEA). The experimental investigation has been performed using phase-locked Particle Image Velocimetry (PIV) of the flow field during the occurrence of a self-excited acoustic resonance condition in the duct. The results of both methods reveal that the flow-excited acoustic resonance produces a strong oscillatory flow pattern in the cylinder wakes, with strong in-phase vortex shedding being synchronized by the acoustic resonance. The distribution and strength of the aeroacoustic sources and sinks within the flow field have been computed by means of Howe׳s theory of aerodynamic sound for both the experimental and numerical cases, with the results of the two methods comparing favourably, showing comparable trends in the oscillating flow fields, and very similar trends in the distribution of net acoustic power.     

Fluid flow in charged nanotubes

The dynamics of fluid flow through nanochannels is different from those in macroscopic systems. By using the molecular dynamics simulations, we investigate the influence of surface polarity of nanotube on the transport properties of the water fluid. The nanotube used here resembles the carbon nanotube, but carries charges of q on some atoms; overall, the nanotube is charge-neutral. Our simulation results show that water flux decreases sharply with the increasing of q for q < 1.6 e; however, the water flux for shells far away from nanotube wall increases slightly when q > 1.6 e. The mechanism behind the interesting phenomenon is discussed. Our findings may have implications for development of nano-fluidic devices and for understanding the movement of confined fluid inside the hydrophilic nanochannel.     

FLOW VISUALIZATION AROUND A CIRCULAR CYLINDER NEAR TO A PLANE WALL

Flow visualization, particle image velocimetry and hot-film anemometry have been employed to study the fluid flow around a circular cylinder near to a plane wall for Reynolds numbers, based on cylinder diameter, between 1200 and 4960. The effect of changing the gap between the cylinder and the wall, G, from G=0 (cylinder touching the wall) to G/D=2, was investigated. It is shown that the flow may be characterized by four distinct regions. (a) For very small gaps, G/D≤0·125, the gap flow is suppressed or extremely weak, and separation of the boundary layer occurs both upstream and downstream of the cylinder. Although there is no regular vortex shedding, there is a periodicity associated with the outer shear-layer. (b) In the “small gap ratio” region, 0·125<G/D<0·5, the flow is very similar to that for very small gaps, except that there is now a pronounced pairing between the inner shear-layer shed from the cylinder and the wall boundary layer. (c) Intermediate gap ratios, 0·5<G/D<0·75, are characterized by the onset of vortex shedding from the cylinder. (d) For the fourth region, characterized by the largest gap ratios considered, G/D>1·0, there is no separation of the wall boundary layer, either upstream or downstream of the cylinder.     

Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary

Two-degree-of-freedom vortex-induced vibrations (VIV) of a circular cylinder close to a plane boundary are investigated numerically. The Reynolds-Averaged Navier-Stokes (RANS) equations are solved using the Arbitrary Lagrangian Eulerian (ALE) scheme with a k-ω turbulence model closure. The numerical model is validated against experimental data of VIV of a cylinder in uniform flow and VIV of a cylinder close to a plane boundary at low mass ratios. The numerical results of the vibration mode, vibration amplitude and frequency agree well with the experimental data. VIV of a circular cylinder close to a plane boundary is simulated with a mass ratio of 2.6 and gap ratios of e/D=0.002 and 0.3 (gap ratio is defined as the ratio of gap between the cylinder and the bed (e) to cylinder diameter (D)). Simulations are carried out for reduced velocities ranging from 1 to 15 and Reynolds numbers ranging from 1000 to 15 000. It is found that vortex-induced vibrations occur even if the initial gap ratio is as small as e/D=0.002, although reported research indicated that vortex shedding behind a fixed circular cylinder is suppressed at small gap ratios (e/D<0.3 or 0.2). It was also found that vibration amplitudes are dependant on the bouncing back coefficient when the cylinder hits the plane boundary. Three vortex shedding modes are identified according to the numerical results: (i) single-vortex mode where the vortices are only shed from the top of the cylinder; (ii) vortex-shedding-after-bounce-back mode; (iii) vortex-shedding-before-bounce-back mode. It was found that the vortex shedding mode depends on the reduced velocity.     

Hyperbolic stress equations for compressible slabs

A constitutive law, chosen to model isotropic compressible slabs, is specialized for plane deformation and then substituted into the equations of motion. The resulting system is quasi-linear hyperbolic. Solutions depend on in situ measurements of a deformation-rate parameter, D. The characteristics of the hyperbolic system span a distance 2DH, where H is the penetration of the characteristics into the slab.     

Bifurcations in a birhythmic biological system with time-delayed noise

We study the effects of recycled noise on the dynamics of a birhythmic biological system. This noise is generated by the superposition of a primary Gaussian white noise source with a second component (its replicas delayed of time τ). We find that under the influence of this kind of noise, the dynamics of the birhythmic biological system can be well characterized through the concept of stochastic bifurcation, consisting in a qualitative change of the stationary probability distribution. Analytical results are obtained following the quasiharmonic assumption through the Langevin and Fokker–Planck equations. Comparing the analytical and numerical results, we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. We also find that the increase of noise intensity leads to coherence resonance.     

On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance

A bistable dynamical system with the Duffing potential, fractional damping, and random excitation has been modelled. To excite the system, we used a stochastic force defined by Wiener random process of Gaussian distribution. As expected, stochastic resonance appeared for sufficiently high noise intensity. We estimated the critical value of the noise level as a function of derivative order \(q\) . For smaller order \(q\) , damping enhancement was reported.     

Vortex shedding and acoustic resonance of single and tandem finned cylinders

The effect of fins on vortex shedding and acoustic resonance is investigated for isolated and two tandem cylinders exposed to cross-flow in a rectangular duct. Three spacing ratios between the tandem cylinders (S/D_e=1.5, 2 and 3) are tested for a Reynolds number range from 1.6×10⁴ to 1.1×10⁵. Measurements of sound pressure as well as mean and fluctuating velocities are performed for bare and finned cylinders with three different fin densities. The effect of fins on the sound pressure generated before the onset of acoustic resonance as well as during the pre-coincidence and coincidence resonance is found to be rather complex and depends on the spacing ratio between cylinders, the fin density and the nature of the flow-sound interaction mechanism.For isolated cylinders, the fins reduce the strength of vortex shedding only slightly, but strongly attenuate the radiated sound before and during the occurrence of acoustic resonance. This suggests that the influence of the fins on correlation length is stronger than on velocity fluctuations. In contrast to isolated cylinders, the fins in the tandem cylinder case enhance the vortex shedding process at off-resonant conditions, except for the large spacing case which exhibits a reversed effect at high Reynolds numbers. Regarding the acoustic resonance of the tandem cylinders, the fins promote the onset of the coincidence resonance, but increasing the fin density drastically weakens the intensity of this resonance. The fins are also found to suppress the pre-coincidence resonance for the tandem cylinders with small spacing ratios (S/D_e=1.5, 2 and 2), but for the largest spacing case (S/D_e=3), they are found to have minor effects on the sound pressure and the lock-in range of the pre-coincidence resonance.     

Experimental research on the thermal performance of converging slot holes with different divergence angles

Thermal performances of two kinds of converging slot-hole (console) with different divergence angles have been measured using transient liquid crystal measurement technique which can process the nonuniform initial wall temperature. Four momentum ratios are tested. Consoles with different divergence angles produce different cooling effectiveness distributions in the upstream region. However, the cooling effectiveness distributions of the two consoles are similar in the downstream. The laterally averaged cooling effectiveness results show that the differences between the two consoles are very small and the best momentum ratio for both consoles’ cooling effectiveness distribution is around two. With the momentum ratio increasing, the normalized heat transfer coefficient h/h₀ of both consoles increases, but the h/h₀ value of small divergence case is higher and becomes progressively higher than that of large divergence case. Moreover, the effect of the couple vortices on the heat transfer coefficient distributions is more significant for the large divergence case. Both consoles provide the surface a certain degree of thermal protection, especially in the upstream region. The distributions of heat flux ratio q/q₀ are similar with those of η because the influence of η on q/q₀ is much larger than that of h/h₀ on q/q₀.     

含噪双稳杜芬振子矩方程的分岔与随机共振

研究了含噪声的双稳杜芬振子矩方程的分岔与随机共振的关系，并根据它们的关系, 从另 一个角度揭示了随机共振发生的机制. 首先在It?方程的基础上，导出了双稳杜芬振子在白噪声和弱周期信号作用下的矩方程，其次以噪声强度 为分岔参数分析了矩方程的分岔特性，再次分析了矩方程的分岔与双稳杜芬振子随机共振 之间的关系，最后根据该对应关系从另一种观点提出了双稳杜芬振子随机共振的机制，该 机制是由于以噪声强度为分岔参数的矩方程发生了分岔，而分岔使得原系统响应均值的能量分布发生了转移，使能 量向频率等于输入信号频率的分量处集中，使得弱信号得到了放大，随机共振发生了.