We investigate coupled identical phase oscillators with scale-free distribution of coupling strength. It is shown that partially locked states can occur due to the inhomogeneity in coupling and some properties of the coupling function. Various quantities of the partially locked states are computed through a self-consistency argument and the values show good agreement with simulation results.

We use weakly coupled oscillator theory to study the effects of delays on coupled systems of neuronal oscillators. We explore, first, simple pairs with constant delays and then examine the role of distributed delays as would occur in systems with dendritic branches or in networks where there is a distance-dependent conductance delay. In the latter, we use mean field theory to show the emergence of travelling waves and the loss of synchronization. Next, we consider phase models with stronger coupling and delays in the state variables. We show that they have a richer dynamics but one that is still similar to the weakly coupled case.

Many real oscillators are coupled to other oscillators and the coupling can affect the response of the oscillators to stimuli. We investigate phase response curves (PRCs) of coupled oscillators. The PRCs for two weakly coupled phase-locked oscillators are analytically obtained in terms of the PRC for uncoupled oscillators and the coupling function of the system. Through simulation and analytic methods, the PRCs for globally coupled oscillators are also discussed.PACS numbers: 05.45. Xt, 87.19.La Many systems in physics, chemistry and biology are modeled as interacting nonlinear oscillators [1,2,3,4,5,6]. One of the easiest ways to characterize an oscillator is its phase response curve (PRC) [3,4,5,6,7]. The PRC is defined as the steady phase shift of an oscillation relative to the unperturbed oscillation as a function of the timing of perturbation to the oscillator. It provides a useful information for understanding the oscillator's behavior when the oscillator is subjected to external stimuli or signals from other oscillators.In most of previous studies, the PRC is obtained when the oscillator is isolated from other oscillators [3,4,5,7]. However, many oscillators in real systems are coupled to others when they are under the influence of external stimuli, and the coupling can affect the response of the oscillators. To better understand the dynamics of oscillators such as the response of neuronal population to signals from other brain region [5] or to controlling stimulations [6], it is necessary to study how the coupling changes the PRCs. This study can also give insights into the phase response of a giant oscillator (for example, circadian rhythm generators [3]) composed of many individual oscillators [8]. In this letter, we study the PRC of coupled oscillators using the average phase of the system and the relative phases between the oscillators comprising the system. The PRC is shown to depend on the PRC of the isolated oscillator, the nature of the coupling, and the relative phases between the oscillators. For some cases, the PRCs are analytically obtained. Our approach differs from that of Ref. [8] in that we analytically approximate the PRC while they require the numerical evaluation of the adjoint of a certain linear operator.If coupling between a network of oscillators is sufficiently "weak", the possibly high-dimensional system can be reduced to a network of coupled phase models [2,4,5]. In the following we exploit this fact and restrict our analysis to coupled phase models. Consider, first, two weakly coupled phase-locked oscillators subjected to a common * Electronic address: taewook.ko@gmail.com † Electronic address: bard@math.pitt.edu perturbation characterized by their individual PRC: (2) where θ i (t) is the phase of oscillator i at time t, ω i is the natural frequency of the oscillator i and K(≥ 0) is the coupling strength. H(θ) is the coupling function obtained by the phase reduction [2,4,5]. Aδ(t − t 1 ) denotes a Dirac delta impulse with amplitude A at time t 1 which is sufficiently large s...

We investigated the effect of time delays on phase configurations in a set of two-dimensional coupled phase oscillators. Each oscillator is allowed to interact with its neighbors located within a finite radius, which serves as a control parameter in this study. It is found that distance-dependent time-delays induce various patterns including traveling rolls, square-like and rhombus-like patterns, spirals, and targets. We analyzed the stability boundaries of the emerging patterns and briefly pointed out the possible empirical implications of such time-delayed patterns.

We investigate the effects of axonal time delay when the neuronal oscillators are coupled by sparse and random connections. Using phase-reduced models with general coupling functions, we show that a small fraction of connections with time delay can destabilize synchronous states and induce near-regular wave states. An order parameter is introduced to characterize those states. We analyze the systems using mean-field-type approximation.