Search

The phase reduction approach has manifested its power in analyzing the rhythmic behaviors for limit cycle oscillators. For coherent oscillation purely induced by noise, e.g., the coherence resonance oscillator, the stochastic dynamics exhibit almost deterministic limit cycle phenomenon which inspires the application of the phase reduction approach to this kind of systems. In this paper, the FitzHugh–Nagumo system in coherence resonance is modeled as a jump process. The phase sensitivity is obtained by applying the phase reduction approach for the hybrid system. A modified direct method is proposed to compare the theoretical results with those of Monte Carlo simulation for the stochastic system, which shows a relatively good agreement for the perturbation not being too small. The phase reduction results of the coherent oscillators are applied to two coupled FitzHugh–Nagumo neurons. It is interesting that the phase difference of the coupled coherent oscillators does not converge as the deterministic oscillators, but forms a distribution where the peaks and valleys correspond to the stable and unstable synchronizations predicted by the phase coupling functions. The idea in this paper could be applied to other coherent oscillators where the stochastic dynamics follow almost deterministic paths.

The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime., 16pages, 4figures

The path from resting to spiking threshold can generate a typical action potential in neurons. For excitable systems with ubiquitous noise, this process can be investigated in the framework of large deviation theory. In this paper, a special position (named as excitability inflection point) of the exit process in the canonical Class 1 excitable system is identified, which is both the extremal position of the dispersion and the momentum. The role of this point is examined by adding a Gaussian white noise and a pulse with a special position into the deterministic system. Surprisingly, by calculating the firing rate, its extremal position can be made closer to the theoretical excitability inflection point with vanishing pulse instead of vanishing noise. The influences of finite noise and pulse are investigated in detail by fixing one of them and changing the other. They showed distinct impacts on the real extremal positions of the firing rate. Finally, some open problems are given.

Noise is ubiquitous and has been verified to play constructive roles in various systems, among which the inverse stochastic resonance (ISR) has aroused much attention in contrast to positive effects such as stochastic resonance. The ISR has been observed in both bistable and monostable systems for which the mechanisms are revealed as noise-induced biased switching and noise-enhanced stability, respectively. In this paper, we investigate the ISR phenomenon in the monostable and bistable Hindmarsh–Rose neurons within a unified framework of large deviation theory. The critical noise strengths for both cases can be obtained by matching the timescales between noise-induced boundary crossing and the limit cycle. Furthermore, different stages of ISR are revealed by the bursting frequency distribution, where the gradual increase of the peak bursting frequency can also be explained within the same framework. The perspective and results in this paper may shed some light on the understanding of the noise-induced complex phenomena in stochastic dynamical systems.

Phase reduction approach is a useful tool to reduce the dimension of limit-cycle oscillators. In this paper, it is applied to the FitzHugh-Nagumo neuron model with large timescale separation. To quantitate the response of the oscillator to the external perturbations, the asymptotic behaviors of the phase sensitivity functions for fast and slow variables are obtained. It is shown that in the relaxation limit, apart from the jump points which are the vertices of the cubic nullcline, the phase is insensitive to the perturbations on the fast variable. By using the phase sensitivity functions, two cases, namely, periodic pulse train perturbation and common noise perturbation, are investigated. Theoretical and numerical results suggest better performance for the synchronization behaviors of the perturbations on the slow variable for large timescale separation.

The exit problem in the framework of the large deviation theory has been a hot topic in the past few decades. The most probable escape path in the weak-noise limit has been clarified by the Freidlin–Wentzell action functional. However, noise in real physical systems cannot be arbitrarily small while noise with finite strength may induce nontrivial phenomena, such as noise-induced shift and noise-induced saddle-point avoidance. Traditional Monte Carlo simulation of noise-induced escape will take exponentially large time as noise approaches zero. The majority of the time is wasted on the uninteresting wandering around the attractors. In this paper, a new method is proposed to decrease the escape simulation time by an exponentially large factor by introducing a series of interfaces and by applying the reinjection on them. This method can be used to calculate the exit location distribution. It is verified by examining two classical examples and is compared with theoretical predictions. The results show that the method performs well for weak noise while may induce certain deviations for large noise. Finally, some possible ways to improve our method are discussed.

In the present paper, the locking phenomenon induced by distance-dependent delay in ring structured neuronal networks is investigated, wherein each neuron is modeled by a FitzHugh-Nagumo neuron. Through increasing the element time delay, the different spatiotemporal patterns are observed. By calculating the interspike interval and its value that is divided by the delay of the nearest neurons, it is found that these patterns are actually the lockings between the period of spiking and the distance-dependent delay of the connected neurons. The lockings could also be revealed by the mean time lag of the neurons and in different connection topologies. Furthermore, the influences of the network size and the coupling strength are investigated, wherein the former seems to play a negligible role on these locking patterns; in contrast, too small coupling strengths will blur the boundaries of different patterns and too large ones may destroy the high ratio locking patterns. Finally, one may predict the locking order which determines the emergence order of the patterns in the networks.

We study the subthreshold and suprathreshold vibrational resonance in the FitzHugh-Nagumo neuron model. For the subthreshold situation, two cases where the stationary states are equilibrium point and limit cycle are considered, where different natures of vibrational resonance are observed via theoretical and numerical methods. Especially when the frequency of the high-frequency driving force is near the so-called canard-resonance frequency, the firing rate can be significantly enhanced at the presence of noise. For the suprathreshold situation, we show that the local maxima of the response amplitude are located at the transition boundaries of different phase-locking patterns. The minimal required forcing amplitudes of high-frequency signal of the firing onset are just multiples of the spiking frequency. Furthermore, phase portraits and time series show that the presence of the global maxima of the response results from not only the suprathreshold but also the subthreshold phase-locking modes. In spite of the distinct characteristics for two stationary states on subthreshold oscillation, the suprathreshold vibrational resonance showed no qualitative difference between the two cases.

We study firing behaviors and the transitions among them in small-world noisy neuronal networks with electrical synapses and information transmission delay. Each neuron is modeled by a two-dimensional Rulkov map neuron. The distance between neurons, which is a main source of the time delay, is taken into consideration. Through spatiotemporal patterns and interspike intervals as well as the interburst intervals, the collective behaviors are revealed. It is found that the networks switch from resting state into intermittent firing state under Gaussian noise excitation. Initially, noise-induced firing behaviors are disturbed by small time delays. Periodic firing behaviors with irregular zigzag patterns emerge with an increase of the delay and become progressively regular after a critical value is exceeded. More interestingly, in accordance with regular patterns, the spiking frequency doubles compared with the former stage for the spiking neuronal network. A growth of frequency persists for a larger delay and a transition to antiphase synchronization is observed. Furthermore, it is proved that these transitions are generic also for the bursting neuronal network and the FitzHugh-Nagumo neuronal network. We show these transitions due to the increase of time delay are robust to the noise strength, coupling strength, network size, and rewiring probability.

In this paper, the collective behaviors for ring structured bursting neuronal networks with electrical couplings and distance-dependent delays are studied. Each neuron is modeled by the Hindmarsh-Rose neuron. Through changing time delays between connected neurons, different spatiotemporal patterns are obtained. These patterns can be explained by calculating the ratios between the bursting period and the delay which exhibit clear locking relations. The holding and the failure of the lockings are investigated via bifurcation analysis. In particular, the bursting death phenomenon is observed for large coupling strengths and small time delays which is in fact the result of the partial amplitude death in the fast subsystem. These results indicate that the collective behaviors of bursting neurons critically depend on the bifurcation structure of individual ones and thus the variety of bifurcation types for bursting neurons may create diverse behaviors in similar neuronal ensembles.