Frequency and phase synchronization in stochastic systems

The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.     

Brains swinging in concert: cortical phase synchronization while playing guitar

Background  

Brains interact with the world through actions that are implemented by sensory and motor processes. A substantial part of these interactions consists in synchronized goal-directed actions involving two or more individuals. Hyperscanning techniques for assessing fMRI simultaneously from two individuals have been developed. However, EEG recordings that permit the assessment of synchronized neuronal activities at much higher levels of temporal resolution have not yet been simultaneously assessed in multiple individuals and analyzed in the time-frequency domain. In this study, we simultaneously recorded EEG from the brains of each of eight pairs of guitarists playing a short melody together to explore the extent and the functional significance of synchronized cortical activity in the course of interpersonally coordinated actions.     

Nonparametric estimation of phase variance in auditory-nerve fiber's responses to tonal stimuli.

Statistical estimation of the phase variance from the auditory-nerve fiber's action potential timing data is studied in this paper. A detailed derivation of the sample-based estimation formulas, which deals specifically with the circularity of the phase variable, is given. The development of the estimator is based on nonparametric statistical inference techniques, making no assumptions on the parametric form of the phase distribution (i.e., shape of period histogram). Some desirable properties of the estimator are demonstrated through numerical examples and applications of the estimator in auditory research are discussed.     

Noise-induced phase synchronization and synchronization transitions in chaotic oscillators

Whether common noise can induce complete synchronization in chaotic systems has been a topic of great relevance and long-standing controversy. We first clarify the mechanism of this phenomenon and show that the existence of a significant contraction region, where nearby trajectories converge, plays a decisive role. Second, we demonstrate that, more generally, common noise can induce phase synchronization in nonidentical chaotic systems. Such a noise-induced synchronization and synchronization transitions are of special significance for understanding neuron encoding in neurobiology.     

Neuromagnetic steady-state responses to auditory stimuli

Steady-state magnetic responses to clicks presented at rates between 10 and 70 Hz have been recorded in healthy humans. The responses were highest in amplitude around 40 Hz. This amplitude enhancement is satisfactorily explained by summation of responses evoked by single clicks. The field maps suggest activation of the auditory cortex at all stimulus frequencies. Similar responses were obtained with gated noise bursts and by pauses in a series of clicks. The mean "apparent latency," determined from the phase lag at rates 30-70 Hz, was 54 ms. The physiological relevance of this quantity is shown to be questionable.     

Interaction of cortical evoked potentials to electric and acoustic stimuli

Evoked potentials to a dichotic stimulus composed of either (1) two binaurally presented tone pips or (2) one tone pip and an electrical pulse to the auditory nerve are recorded from the primary auditory cortex of barbiturate anesthetized cats. The composite stimulus is delivered as a time delayed pair where the interstimulus interval (25 ms) is within the relative refractory period of the evoked potential to either stimulus alone. The amplitude of the cortical potential to the trailing stimulus is compared with its single amplitude as the frequency of the trailing tone pip is changed from 250 Hz through 40 kHz. There is an optimal frequency range over which the trailing stimulus is suppressed and this range appears directly related to the current of a preceding electrical pulse. The frequency of maximum suppression shifts according to the position of the electrode in the nerve. In some experiments secondary maxima develop, suggesting stimulus current spread from fibers of one cochlear turn into fibers from another turn.     

Transition from phase to generalized synchronization in time-delay systems

The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.     

Modeling auditory evoked brainstem responses to transient stimuli

A quantitative model is presented that describes the formation of auditory brainstem responses (ABRs) to tone pulses, clicks, and rising chirps as a function of stimulation level. The model computes the convolution of the instantaneous discharge rates using the "humanized" nonlinear auditory-nerve model of Zilany and Bruce [J. Acoust. Soc. Am. 122, 402-417 (2007)] and an empirically derived unitary response function which is assumed to reflect contributions from different cell populations within the auditory brainstem, recorded at a given pair of electrodes on the scalp. It is shown that the model accounts for the decrease of tone-pulse evoked wave-V latency with frequency but underestimates the level dependency of the tone-pulse as well as click-evoked latency values. Furthermore, the model correctly predicts the nonlinear wave-V amplitude behavior in response to the chirp stimulation both as a function of chirp sweeping rate and level. Overall, the results support the hypothesis that the pattern of ABR generation is strongly affected by the nonlinear and dispersive processes in the cochlea.     

Transition from phase synchronization to complete synchronization in mutually coupled nonidentical Nd:YAG lasers

Using mutually coupled nonidentical continuous-wave Nd:YAG lasers, we experimentally confirmed the recently proposed transition route from phase synchronization to complete synchronization. As evidence of this transition we obtained the probability distribution of the intermittent synchronization time near the threshold of the complete synchronization transition.     

Role of multistability in the transition to chaotic phase synchronization

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.     

Detecting phase synchronization in noisy systems

A quantitative method for automatic detection of phase synchronization in noisy experimental bivariate time series is proposed, based on the fact that instantaneous phases of phase-synchronized (sub) systems are mutually dependent in a specific way irrespective of a relation between the original time series. The level of dependence between the instantaneous phases is quantified by a statistical dependence parameter, which also reflects the strength of the systems' phase synchronization. Ranges of the parameter values, for which the detection of the phase synchronization can be considered reliable, are estimated by using the technique of surrogate data. Possible applications of the proposed method are demonstrated by using both numerically generated and real experimental data, namely solutions of two coupled Rössler systems, mammalian cardio-respiratory data, and long-term recordings of surface atmospheric temperature and sunspot numbers.     

Self-organized and driven phase synchronization in coupled maps

We study the phase synchronization and cluster formation in coupled maps on different networks. We identify two different mechanisms of cluster formation: (a) self-organized phase synchronization which leads to clusters with dominant intracluster couplings and (b) driven phase synchronization which leads to clusters with dominant intercluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others. We also discuss the dynamical origin of these two mechanisms for small networks with two and three nodes.     

Three types of transitions to phase synchronization in coupled chaotic oscillators

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.     

The relationship between magnetic and electrophysiological responses to complex tactile stimuli

Background  

Magnetoencephalography (MEG) has become an increasingly popular technique for non-invasively characterizing neuromagnetic field changes in the brain at a high temporal resolution. To examine the reliability of the MEG signal, we compared magnetic and electrophysiological responses to complex natural stimuli from the same animals. We examined changes in neuromagnetic fields, local field potentials (LFP) and multi-unit activity (MUA) in macaque monkey primary somatosensory cortex that were induced by varying the rate of mechanical stimulation. Stimuli were applied to the fingertips with three inter-stimulus intervals (ISIs): 0.33s, 1s and 2s.     

Analytical calculation of the transition to complete phase synchronization in coupled oscillators

Here we present a system of coupled phase oscillators with nearest neighbors coupling, which we study for different boundary conditions. We concentrate at the transition to the total synchronization. We are able to develop exact solutions for the value of the coupling parameter when the system becomes completely synchronized, for the case of periodic boundary conditions as well as for a chain with fixed ends. We compare the results with those calculated numerically.      

Repulsive synchronization in an array of phase oscillators

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.