Activation process in excitable systems with multiple noise sources: One and two interacting units.

We consider the coaction of two distinct noise sources on the activation process of a single excitable unit and two interacting excitable units, which are mathematically described by the Fitzhugh-Nagumo equations. We determine the most probable activation paths around which the corresponding stochastic trajectories are clustered. The key point lies in introducing appropriate boundary conditions that are relevant for a class II excitable unit, which can be immediately generalized also to scenarios involving two coupled units. We analyze the effects of the two noise sources on the statistical features of the activation process, in particular demonstrating how these are modified due to the linear or nonlinear form of interactions. Universal properties of the activation process are qualitatively discussed in the light of a stochastic bifurcation that underlies the transition from a stochastically stable fixed point to continuous oscillations.

Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units.

We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfillment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfillment of the quasi-independence approximation and certain forms of synchronization between the individual units.

Triggered dynamics in a model of different fault creep regimes.

The study is focused on the effect of transient external force induced by a passing seismic wave on fault motion in different creep regimes. Displacement along the fault is represented by the movement of a spring-block model, whereby the uniform and oscillatory motion correspond to the fault dynamics in post-seismic and inter-seismic creep regime, respectively. The effect of the external force is introduced as a change of block acceleration in the form of a sine wave scaled by an exponential pulse. Model dynamics is examined for variable parameters of the induced acceleration changes in reference to periodic oscillations of the unperturbed system above the supercritical Hopf bifurcation curve. The analysis indicates the occurrence of weak irregular oscillations if external force acts in the post-seismic creep regime. When fault motion is exposed to external force in the inter-seismic creep regime, one finds the transition to quasiperiodic- or chaos-like motion, which we attribute to the precursory creep regime and seismic motion, respectively. If the triggered acceleration changes are of longer duration, a reverse transition from inter-seismic to post-seismic creep regime is detected on a larger time scale.

Predictions of experimentally observed stochastic ground vibrations induced by blasting.

In the present paper, we investigate the blast induced ground motion recorded at the limestone quarry "Suva Vrela" near Kosjeric, which is located in the western part of Serbia. We examine the recorded signals by means of surrogate data methods and a determinism test, in order to determine whether the recorded ground velocity is stochastic or deterministic in nature. Longitudinal, transversal and the vertical ground motion component are analyzed at three monitoring points that are located at different distances from the blasting source. The analysis reveals that the recordings belong to a class of stationary linear stochastic processes with Gaussian inputs, which could be distorted by a monotonic, instantaneous, time-independent nonlinear function. Low determinism factors obtained with the determinism test further confirm the stochastic nature of the recordings. Guided by the outcome of time series analysis, we propose an improved prediction model for the peak particle velocity based on a neural network. We show that, while conventional predictors fail to provide acceptable prediction accuracy, the neural network model with four main blast parameters as input, namely total charge, maximum charge per delay, distance from the blasting source to the measuring point, and hole depth, delivers significantly more accurate predictions that may be applicable on site. We also perform a sensitivity analysis, which reveals that the distance from the blasting source has the strongest influence on the final value of the peak particle velocity. This is in full agreement with previous observations and theory, thus additionally validating our methodology and main conclusions.

Mean-field approximation of two coupled populations of excitable units.

The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of N stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the interensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset, and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical interpopulation couplings.

Cluster synchronization of spiking induced by noise and interaction delays in homogenous neuronal ensembles.

Properties of spontaneously formed clusters of synchronous dynamics in a structureless network of noisy excitable neurons connected via delayed diffusive couplings are studied in detail. Several tools have been applied to characterize the synchronization clusters and to study their dependence on the neuronal and the synaptic parameters. Qualitative explanation of the cluster formation is discussed. The interplay between the noise, the interaction time-delay and the excitable character of the neuronal dynamics is shown to be necessary and sufficient for the occurrence of the synchronization clusters. We have found the two-cluster partitions where neurons are firmly bound to their subsets, as well as the three-cluster ones, which are dynamical by nature. The former turn out to be stable under small disparity of the intrinsic neuronal parameters and the heterogeneity in the synaptic connectivity patterns.

Spontaneous formation of synchronization clusters in homogenous neuronal ensembles induced by noise and interaction delays.

The spontaneous formation of clusters of synchronized spiking in a structureless ensemble of equal stochastically perturbed excitable neurons with delayed coupling is demonstrated for the first time. The effect is a consequence of a subtle interplay between interaction delays, noise, and the excitable character of a single neuron. The dependence of the cluster properties on the time lag, noise intensity, and the synaptic strength is investigated.

Stability, bifurcations, and dynamics of global variables of a system of bursting neurons.

An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.

Influence of interaction delays on noise-induced coherence in excitable systems.

Influence of the interaction time delay on the noise-induced system size resonance in a system of all-to-all electrically coupled FitzHugh-Nagumo excitable neurons is studied. It is observed that small time lags decrease and that large time lags increase the coherence of spiking. Bifurcations of the system's stationary state are used to explain the observed nonmonotonic dependence of coherence on the time lag.

Synchronization of bursting neurons with delayed chemical synapses.

The synchronization of bursting Hindmarsh-Rose neurons coupled by a time-delayed fast threshold modulation synapse was studied. It is shown that there is a domain of the coupling parameter and nonzero time-lag values such that the bursting neurons are exactly synchronized. Furthermore, and contrary to the case of electrical synapses, such synchronous bursting is stochastically stable.

Influence of noise on dynamics of coupled bursters.

The influence of white noise on the dynamics of a delayed electrically coupled pair of Hidmarsh-Rose bursting neurons is studied. In particular a simple method to predict the intensity of noise that can destabilize the quiescent state is proposed and compared with numerical computations. Furthermore, it is demonstrated that quite small noise completely destroys the exact synchronization of bursting dynamics, and a qualitative explanation of this effect is discussed.

Synchronization of noisy delayed feedback systems with delayed coupling.

Synchronization of delayed coupled and stochastically perturbed systems with delayed nonlinear feedback is studied, using as an example circular chains of three and four delayed coupled Ikeda oscillators. It is proved that in the case of multiplicative noise the exact synchronization in the mean occurs for sufficiently large coupling, and an analytic estimate of the sufficient coupling is given. The sufficiency condition is compared with numerical computations, and typical effects of noise on the exact and some generalized types of synchronization are illustrated.