Langevin approach with rescaled noise for stochastic channel dynamics in Hodgkin–Huxley neurons

The Langevin approach has been applied to model the random open and closing dynamics of ion channels. It has long been known that the gate-based Langevin approach is not sufficiently accurate to reproduce the statistics of stochastic channel dynamics in Hodgkin–Huxley neurons. Here, we introduce a modified gate-based Langevin approach with rescaled noise strength to simulate stochastic channel dynamics. The rescaled independent gate and identical gate Langevin approaches improve the statistical results for the mean membrane voltage, inter-spike interval, and spike amplitude.     

Effects of channel noise on synchronization transitions in delayed scale-free network of stochastic Hodgkin–Huxley neurons

We numerically study the effect of the channel noise on the spiking synchronization of a scale-free Hodgkin–Huxley neuron network with time delays. It is found that the time delay can induce synchronization transitions at an intermediate and proper channel noise intensity, and the synchronization transitions become strongest when the channel noise intensity is optimal. The neurons can also exhibit synchronization transitions as the channel noise intensity is varied, and this phenomenon is enhanced at around the time delays that can induce the synchronization transitions. It is also found that the synchronization transitions induced by the channel noise are dependent on the coupling strength and the network average degree, and there is an optimal coupling strength or network average degree with which the synchronization transitions become strongest. These results show that by inducing synchronization transitions, the channel noise has a big regulation effect on the synchronization of the neuronal network. These findings could find potential implications for the information transmission in neural systems.     

Coherence resonance and synchronization of Hindmarsh-Rose neurons with noise

Noise effects on coherence resonance and synchronization of Hindmarsh-Rose (HR) neuron model are studied. The coherence resonance of a single HR neuron with Gaussian white noise added to the membrane potential is investigated in situations before, near and after the Hopf bifurcation, separately, with the external direct current as a bifurcation parameter. It is shown that even though there is no coupling between neurons, uncoupled identical HR neurons driven by a common noise can achieve complete synchronization when the noise intensity is higher than a critical value. Furthermore, noise also enhances complete synchronization of weakly coupled neurons. It is concluded that synchronization in bursting neurons is easier to be induced than in spiking ones, and coupling enhances the sensitivity of synchronization of neurons to noise stimulus.     

Response of Hodgkin–Huxley stochastic bursting neuron to single-pulse stimulus

We investigate Hodgkin–Huxley neuron model with external Gaussian noise in the range of parameters where it exhibits bistability of silent and firing states, and noise-induced bursts occur. We study the response of the system to brief single pulse of current. When noise amplitude increases, the delay time between the stimulus and the first spike decreases substantially even for subthreshold stimulus. The mean number of spikes in a post-stimulus burst has a maximum in a certain range of noise amplitudes. Therefore, we found that Hodgkin–Huxley neuron in the stochastic bursting regime has more improved sensitivity to single-pulse stimulus than in the silent one.     

Langevin approach for stochastic Hodgkin–Huxley dynamics with discretization of channel open fraction

The random opening and closing of ion channels establishes channel noise, which can be approximated and included into stochastic differential equations (Langevin approach). The Langevin approach is often incorporated to model stochastic ion channel dynamics for systems with a large number of channels. Here, we introduce a discretization procedure of a channel-based Langevin approach to simulate the stochastic channel dynamics with small and intermediate numbers of channels. We show that our Langevin approach with discrete channel open fractions can give a good approximation of the original Markov dynamics even for only 10 K⁺$10{\text{K}}^{+}$ channels. We suggest that the better approximation by the discretized Langevin approach originates from the improved representation of events that trigger action potentials.     

Collective firing regularity of a scale-free Hodgkin–Huxley neuronal network in response to a subthreshold signal

We consider a scale-free network of stochastic HH neurons driven by a subthreshold periodic stimulus and investigate how the collective spiking regularity or the collective temporal coherence changes with the stimulus frequency, the intrinsic noise (or the cell size), the network average degree and the coupling strength. We show that the best temporal coherence is obtained for a certain level of the intrinsic noise when the frequencies of the external stimulus and the subthreshold oscillations of the network elements match. We also find that the collective regularity exhibits a resonance-like behavior depending on both the coupling strength and the network average degree at the optimal values of the stimulus frequency and the cell size, indicating that the best temporal coherence also requires an optimal coupling strength and an optimal average degree of the connectivity.     

Effects of temperature and electromagnetic induction on action potential of Hodgkin–Huxley model

Based on an improved the Hodgkin–Huxley (HH) neuron model which is driven by the electromagnetic induction, the effects of temperature and electromagnetic induction on the action potential of neuron are investigated by numerical computations. It is very interesting that, under the fixed condition of electromagnetic induction, there is a region for the electrical activity of neuron in the external current and temperature parameters plane, the region of electrical firing is similar to the Arnold’ tongue-like structure, and the Arnold’ tongue originates from the nonlinear variation of temperature with the increasing of threshold external current. The effects of temperature and electromagnetic induction on neuronic electrical activity are respectively discussed by using numerical simulations. Our results provide new insights into the roles of temperature in the improved HH neuron model, the existence of Arnold’ tongue-like structure might give some insights for the treatment of neurological diseases such as the epilepsia.     

Response of a Duffing—Rayleigh system with a fractional derivative under Gaussian white noise excitation

In this paper,we consider the response analysis of a Duffing-Rayleigh system with fractional derivative under Gaussian white noise excitation.A stochastic averaging procedure for this system is developed by using the generalized harmonic functions.First,the system state is approximated by a diffusive Markov process.Then,the stationary probability densities are derived from the averaged Ito stochastic differential equation of the system.The accuracy of the analytical results is validated by the results from the Monte Carlo simulation of the original system.Moreover,the effects of different system parameters and noise intensity on the response of the system are also discussed.     

Restoration of Decentered Gaussian Beam with the Combination of a Lens and a Gaussian Aperture

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Response probability density functions of Duffing–Van der Pol vibro-impact system under correlated Gaussian white noise excitations

This paper aims to investigate the stationary probability density functions (PDFs) of a Duffing–Van der Pol vibro-impact system excited by correlated Gaussian white noise. With the help of non-smooth transformation, the stationary PDFs are formulated analytically by the stochastic averaging of energy envelope. The analytical results are verified by numerical simulation results. Stochastic bifurcations for different parameters are considered, and several special PDF forms are observed in this paper. The first form is the shape of the PDF of total energy can be similar to a crater, which has a minimum and a maximum. The second one is the shape of the joint PDF with three peaks, that is to say, the section of joint PDF has three maximum and two minimum. In addition, the influence of the degree of the correlation of Gaussian white noises is explored.     

Specifics of the stochastic ionization of a Rydberg collision complex with Förster resonance

The physics of the formation of dynamic nonlinear resonances in an isolated Rydberg collisional complex is described. The development of the stochastic instability of Rydberg electron trajectories due to charge exchange in the complex is considered. The realization of the resonance in external statistic magnetic and electric fields is predicted to be accompanied by a significant narrowing of areas of stochastic motion with a concurrent decrease in the rates of the ionization of real quasimolecular systems proceeding through the migration over the Rydberg crowding of quantum states.     

Studies of phase return map and symbolic dynamics in a periodically driven Hodgkin–Huxley neuron

How neuronal spike trains encode external information is a hot topic in neurodynamics studies.In this paper,we investigate the dynamical states of the Hodgkin–Huxley neuron under periodic forcing.Depending on the parameters of the stimulus,the neuron exhibits periodic,quasiperiodic and chaotic spike trains.In order to analyze these spike trains quantitatively,we use the phase return map to describe the dynamical behavior on a one-dimensional(1D)map.According to the monotonicity or discontinuous point of the 1D map,the spike trains are transformed into symbolic sequences by implementing a coarse-grained algorithm—symbolic dynamics.Based on the ordering rules of symbolic dynamics,the parameters of the external stimulus can be measured in high resolution with finite length symbolic sequences.A reasonable explanation for why the nervous system can discriminate or cognize the small change of the external signals in a short time is also presented.     

Steady-state analysis subject to a coloured and a white additive of a bistable system multiplicative noise noise with coloured cross-correlated noises

The steady-state properties of a bistable system are investigated when both the multiplicative noise and the coupling between additive and multiplicative noises are coloured with different values of noise correlation times T1 and T2. After introducing a dimensionless parameter R（R = α/D, D is the intensity of the multiplicative noise and a is the intensity of the additive noise）, and performing the numerical computations, we find the following points： （1） For the case of R 〉 1, A （the intensity of correlation between additive and multiplicative noises）, T1 and T2 can induce the stationary probability distribution （SPD） transition from bimodal to unimodal in structure, but for the cases of R _〈 1, the bimodal structure is preserved; （2） a can also induce the SPD transition from bimodal to unimodal in structure; （3） the bimodal structure of the SPD exhibits a symmetrical structure as D increases.     

The analysis of stochastic resonance and bearing fault detection based on linear coupled bistable system under lévy noise

In this paper, the stochastic resonance (SR) phenomenon of the linear coupled bistable system induced by Lévy noise is analyzed. Meanwhile, the characteristics of Lévy noise is also analyzed according to its probability density functions (PDFs) of different stability index α, symmetry parameter β, scale parameter σ and location index μ. The mean of signal-noise ratio increase (MSNRI) is regarded as an index to measure the SR phenomenon. Then, the rules for MSNRI affected by noise intensity D are explored under different charastic indexes of Lévy noise, system parameters a, b, c and coupling coefficient r. The results are beneficial to the numerical simulation of single-frequency and multi-frequency weak signals detection based on single bistable system and linear coupled system respectively. It is found that the performance of the proposed system is better than single bistable system and results of bearing fault detection could also verify the conclusion.     

Analysis of the photon–photon resonance influence on the direct modulation bandwidth of dual-longitudinal-mode distributed feedback lasers

The paper explores the possibilities to extend the direct modulation bandwidth in dual-longitudinal-mode distributed feedback lasers by exploiting the photon–photon resonance induced by the interaction of the two modes in the laser cavity. The effects on the direct amplitude modulation and on the direct modulation of the difference frequency between the two modes are analyzed using simulation and experimental results. When the photon–photon resonance, which occurs at the difference frequency between the two modes, is properly placed at a higher frequency than the carrier-photon resonance, the small-signal amplitude modulation (AM) bandwidth of the laser can be significantly increased. However, both simulations and experiments point out that a high small-signal AM bandwidth does not lead to a high large-signal AM bandwidth if the small-signal modulation response has significant variations across the modulation bandwidth. The paper shows that a high large-signal AM bandwidth is obtained when the two modes are significantly unbalanced, whereas a high-bandwidth difference frequency modulation can be best detected when the two modes are balanced and the DC bias is properly chosen.     

Painleve Analysis and Symmetries of the Hirota–Satsuma Equation

Abstract

The singular manifold expansion of Weiss, Tabor and Carnevale [1] has been successfully applied to integrable ordinary and partial differential equations. They yield information such as Lax pairs, Bäcklund transformations, symmetries, recursion operators, pole dynamics, and special solutions. On the other hand, several recent developments have made the application of group theory to the solution of the differential equations more powerful then ever. More recently, Gibbon et. al. [2] revealed interrelations between the Painlevè property and Hirota’s bilinear method. And W. Strampp [3] hase shown that symmetries and recursion operators for an integrable nonlinear partial differential equation can be obtained from the Painlevè expansion. In this paper, it has been shown that the Hirota–Satsuma equation passes the Painlevé test given by Weiss et al. for nonlinear partial differential equations. Furthermore, the data obtained by the truncation technique is used to obtain the symmetries, recursion operators, some analytical solutions of the Hirota–Satsuma equation.     

Cooperative dynamics of coupled motor–cargoes system with stochastic interactions in the crowded environment

In this study, we investigate the cooperative transport behaviors of coupled motor–cargoes system, in which multiple passive cargoes stochastically interact with one active Brownian motor. The environment with stochastic interactions is characterized by the concentration (reflecting the cargo’s number in unit volume) and switching rate (reflecting the interacting stability between motor and cargoes), based on which the stationary multiple-state process can be employed to describe the fluctuating-cargo state in the coupled system. By analyzing the average probability current of decoupled system in the thermodynamic limit, we effectively study the possibility of cooperative transport through stochastic cargoes to behave rich dynamical behaviors, including the directed current, current reversal, stochastic resonance (SR) and stochastic inhibition (SI), inverse SR and SI, even without the effect of external driving force. Based on numerical results, we systematically discuss the transport dependence on various parameters, including the cargo concentration in the crowded environment, cargo capacity of the motor, driving amplitude of external periodic force, and medium temperature. Obviously, the sensitivity of transport process to parameter changes can be used by the environment to regulate its cargo traffic, which also provides latent support for manipulating the transport performance and optimizing the coupled structure in artificial nano-machines.