Tae-Wook Ko

Pattern formation in a two-dimensional array of oscillators with phase-shifted coupling

We investigate the dynamics of a two-dimensional array of oscillators with phase-shifted coupling. Each oscillator is allowed to interact with its neighbors within a finite radius. The system exhibits various patterns including squarelike pinwheels, (anti)spirals with phase-randomized cores, and antiferro patterns embedded in (anti)spirals. We consider the symmetry properties of the system to explain the observed behaviors, and estimate the wavelengths of the patterns by linear analysis. Finally, we point out the implications of our work for biological neural networks.

Emergence of chaotic itinerancy in simple ecological systems

Chaotic itinerancy is a universal dynamical concept that describes itinerant motion among many different ordered states through chaotic transition in dynamical systems. Unlike the expectation of the prevalence of chaotic itinerancy in high-dimensional systems, we identify chaotic itinerant behavior from a relatively simple ecological system, which consists only of two coupled consumer-resource pairs. The system exhibits chaotic bursting activity, in which the explosion and the shrinkage of the population alternate indefinitely, while the explosion of one pair co-occurs with the shrinkage of the other pair. We analyze successfully the bursting activity in the framework of chaotic itinerancy, and find that large duration times of bursts tend to cluster in time, allowing the effective burst prognosis. We also investigate the control schemes on the bursting activity, and demonstrate that invoking the competitive rise of the consumer in one pair can even elongate the burst of the other pair rather than shorten it.

Oscillons, kinks and pattern in a model for a periodically forced medium

A dynamic model is devised for the patterns arising in a periodically forced extended medium. This model is based on the assumption of nearest pattern interaction approximation, and contains two control parameters α, β. While β controls the number of stable equilibrium states of the system, α serves as a coupling constant for an interaction of a localized excitation with its neighbors. Despite its extreme simplicity, the model provides a unified perspective of the formation of the patterns such as oscillons, linear and decorated kinks, kink to stripe transitions, skew-varicose and crossroll instabilities, etc.

Phase lead/lag due to degree inhomogeneity in complex oscillator network with application to brain networks

Brain anatomical connectivity is one of the main factors influencing information flow among the brain areas [1] and phase lead/lag relationship between oscillations of brain areas is known to be related to the information flow [2,3]. In this study, we analyze the network effect on the phases of coupled oscillators using Kuramoto model and obtain analytical relationship between phase lead/lag and degrees of network nodes. We also show robustness under various conditions, improving upon the result of ref. [4]. Using the brain anatomical connectivity and the relationship, we can explain the phase distribution across the brain. At first, we investigate the relationship in the oscillator model on a scale-free network of which degree distribution follows a power law [5]. Confirming the result of previous study [4], the phases of higher degree nodes lag the phases of lower degree nodes. Similar behaviors are observed also in random network, where the degree distribution follows a Poisson distribution. Using mean-field approximation, we analytically derive the relationship between phases and node degrees as shown in Figure ​Figure1.1. With various conditions of time delay and coupling strength, we also observe that this phase lead/lag relationship between nodes is robust. Our exact relationship can be well applied to human brain anatomical networks. Figure 1 The phase (black dots) of each oscillator and the calculated phases (red line) from analytic derivation. They match well in locked region and red shaded area represents drifting region. The inset shows the degree for each node.