Network Of Relaxation Oscillators Research Articles

Robust universal approach to identify travelling chimeras and synchronized clusters in spiking networks

We propose a robust universal approach to identify multiple network dynamical states, including stationary and travelling chimera states based on an adaptive coherence measure. Our approach allows automatic disambiguation of synchronized clusters, travelling waves, chimera states, and asynchronous regimes. In addition, our method can determine the number of clusters in the case of cluster synchronization. We further couple our approach with a new speed calculation method for travelling chimeras. We validate our approach by an example of a ring network of type II Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections where we identify a rich repertoire of coherent and wave states. We propose that the method is robust for the networks of phase oscillators and extends to a general class of relaxation oscillator networks.

Abnormal route to aging transition in a network of coupled oscillators.

In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.

Cloning of Chimera States in a Large Short-term Coupled Multiplex Network of Relaxation Oscillators

A new phenomenon of the chimera states cloning in a large two-layer multiplex network with short-term couplings has been discovered and studied. For certain values of strength and time of multiplex interaction, in the initially disordered layer, a state of chimera is formed with the same characteristics (the same average frequency and amplitude distributions in coherent and incoherent parts, as well as an identical phase distribution in coherent part), as in the chimera which was set in the other layer. The mechanism of the chimera states cloning is examined. It is shown that the cloning is not related with synchronization, but arises from the competition of oscillations in pairs of oscillators from different layers.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity.

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

Susceptibility of switching between in-phase and anti-phase patterns in the network of relaxation oscillators

Neural networks composed of two or four cells with combined, electrical and inhibitory, synapses and realized for various network topologies were examined. The aim of this study was to determine a set of phases of oscillatory cycle in which different patterns of activity, characteristic for such networks, can be switched under an external stimulus. In particular, we studied susceptibility of switching between in-phase (IP) and anti-phase (AP) patterns (and vice versa). Our results demonstrate that windows of switching between patterns are similar for networks with electrical and mixed synapses and, in general, relatively independent of the network topology. The only effect of the network topology is an increase of the robustness of the AP pattern in networks of ring-like connectivity. The switching window width and thereby the robustness of the transitions between patterns decreases with the increase of the electrical coupling strength.

An oscillatory correlation model of auditory streaming

We present a neurocomputational model for auditory streaming, which is a prominent phenomenon of auditory scene analysis. The proposed model represents auditory scene analysis by oscillatory correlation, where a perceptual stream corresponds to a synchronized assembly of neural oscillators and different streams correspond to desynchronized oscillator assemblies. The underlying neural architecture is a two-dimensional network of relaxation oscillators with lateral excitation and global inhibition, where one dimension represents time and another dimension frequency. By employing dynamic connections along the frequency dimension and a random element in global inhibition, the proposed model produces a temporal coherence boundary and a fissure boundary that closely match those from the psychophysical data of auditory streaming. Several issues are discussed, including how to represent physical time and how to relate shifting synchronization to auditory attention.

Synchronization in relaxation oscillator networks with conduction delays.

We study locally coupled networks of relaxation oscillators with excitatory connections and conduction delays and propose a mechanism for achieving zero phase-lag synchrony. Our mechanism is based on the observation that different rates of motion along different nullclines of the system can lead to synchrony in the presence of conduction delays. We analyze the system of two coupled oscillators and derive phase compression rates. This analysis indicates how to choose nullclines for individual relaxation oscillators in order to induce rapid synchrony. The numerical simulations demonstrate that our analytical results extend to locally coupled networks with conduction delays and that these networks can attain rapid synchrony with appropriately chosen nullclines and initial conditions. The robustness of the proposed mechanism is verified with respect to different nullclines, variations in parameter values, and initial conditions.

Texture segmentation using Gaussian-Markov random fields and neural oscillator networks

We propose an image segmentation method based on texture analysis. Our method is composed of two parts. The first part determines a novel set of texture features derived from a Gaussian-Markov random fields (GMRF) model. Unlike a GMRF-based approach, our method does not employ model parameters as features or require the extraction of features for a fixed set of texture types a priori. The second part is a 2D array of locally excitatory globally inhibitory oscillator networks (LEGION). After being filtered for noise suppression, features are used to determine the local couplings in the network. When LEGION runs, the oscillators corresponding to the same texture tend to synchronize, whereas different texture regions tend to correspond to distinct phases. In simulations, a large system of differential equations is solved for the first time using a recently proposed method for integrating relaxation oscillator networks. We provide results on real texture images to demonstrate the performance of our method.

Fast numerical integration of relaxation oscillator networks based on singular limit solutions

Abstract-Relaxation oscillations exhibiting more than one time scale arise naturally from many physical systems. When relaxation oscillators are coupled in a way that resembles chemical synapses, we propose a fast method to numerically integrate such networks. The numerical technique, called the singular limit method, is derived from analysis of relaxation oscillations in the singular limit. In such limit, system evolution gives rise to time instants at which fast dynamics takes place and intervals between them during which slow dynamics takes place. A full description of the method is given for a locally excitatory globally inhibitory oscillator network (LEGION), where fast dynamics, characterized by jumping which leads to dramatic phase shifts, is captured in this method by iterative operation and slow dynamics is entirely solved. The singular limit method is evaluated by computer experiments, and it produces remarkable speedup compared to other methods of integrating these systems. The speedup makes it possible to simulate large-scale oscillator networks.

Relaxation oscillators with time delay coupling

We study networks of relaxation oscillators coupled with time delays synapses. A pair of oscillators is analyzed and shown to attain loosely synchronous solutions for a wide range of initial conditions and time delays. Simulations of one- and two-dimensional oscillator networks indicate that locally coupled oscillators are also loosely synchronous. Desynchronous solutions are possible when system parameters are varied. To characterize lossely synchronous networks, we introduce a measure of synchrony, the maximum time difference between any two oscillators. In locally excitatory globally inhibitory oscillator networks with time delays, we find that desynchronous solutions for different groups of oscillators are maintained, and the number of groups that can be segregated is related to the maximum time difference within each group. To examine the maximum time difference, we display its histograms for oscillator networks in one and two dimensions. Also, a range of initial conditions is given so that the maximum time difference is contained as the system evolves.

Primitive auditory segregation based on oscillatory correlation

Auditory scene analysis is critical for complex auditory processing. We study auditory segregation from the neural network perspective, and develop a framework for primitive auditory scene analysis. The architecture is a laterally coupled two-dimensional network of relaxation oscillators with a global inhibitor. One dimension represents time and another one represents frequency. We show that this architecture, plus systematic delay lines, can in real time group auditory features into a stream by phase synchrony and segregate different streams by desynchronization. The network demonstrates a set of psychological phenomena regarding primitive auditory scene analysis, including dependency on frequency proximity and the rate of presentation, sequential capturing, and competition among different perceptual organizations. We offer a neurocomputational theory—shifting synchronization theory—for explaining how auditory segregation might be achieved in the brain, and the psychological phenomenon of stream segregation. Possible extensions of the model are discussed.

Primitive Auditory Segregation Based on Oscillatory Correlation

Auditory scene analysis is critical for complex auditory processing. We study auditory segregation from the neural network perspective, and develop a framework for primitive auditory scene analysis. The architecture is a laterally coupled two‐dimensional network of relaxation oscillators with a global inhibitor. One dimension represents time and another one represents frequency. We show that this architecture, plus systematic delay lines, can in real time group auditory features into a stream by phase synchrony and segregate different streams by desynchronization. The network demonstrates a set of psychological phenomena regarding primitive auditory scene analysis, including dependency on frequency proximity and the rate of presentation, sequential capturing, and competition among different perceptual organizations. We offer a neurocomputational theory—shifting synchronization theory—for explaining how auditory segregation might be achieved in the brain, and the psychological phenomenon of stream segregation. Possible extensions of the model are discussed.