Phase Oscillator Model Research Articles

Mathematical modeling of frogs’ calling behavior and its possible application to artificial life and robotics

This paper theoretically and qualitatively describes the calling behavior of the Japanese tree frog Hyla japonica with a simple model of phase oscillators. Experimental analysis showed that while an isolated single male frog called nearly periodically, two interacting male frogs called periodically but alternately, with little overlap. We model these phenomena as a system of coupled phase oscillators, where each isolated oscillator behaves periodically as a model of the calling of a single frog, and two coupled oscillators show antiphase synchronization, reflecting the alternately calling behavior of two interacting frogs. Then, we extend the model to a system of three coupled oscillators virtually corresponding to three interacting male frogs, and analyse the nonlinear dynamics and the bifurcation. We also discuss the biological meaning of the calling behavior and its possible application to artificial life and robotics.

On the derivation and tuning of phase oscillator models for lamprey central pattern generators

Using phase response curves and averaging theory, we derive phase oscillator models for the lamprey central pattern generator from two biophysically-based segmental models. The first one relies on network dynamics within a segment to produce the rhythm, while the second contains bursting cells. We study intersegmental coordination and show that the former class of models shows more robust behavior over the animal's range of swimming frequencies. The network-based model can also easily produce approximately constant phase lags along the spinal cord, as observed experimentally. Precise control of phase lags in the network-based model is obtained by varying the relative strengths of its six different connection types with distance in a phase model with separate coupling functions for each connection type. The phase model also describes the effect of randomized connections, accurately predicting how quickly random network-based models approach the determinisitic model as the number of connections increases.

Phase transitions in the Kuramoto model

We consider the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal function with the plateau section in the middle representing the maximum and symmetric tails falling off predominantly as |omega-omega0|m, m>0, in the vicinity of the flat region. It is found that the phase transition is of first order as long as there is a finite flat region and that in the vicinity of the critical coupling the following scaling law holds r-rc proportional, variant(K-Kc)2/(2m+3), where r is the order parameter and K is the coupling strength of the interacting oscillators.

Delayed feedback control of synchronization in locally coupled neuronal networks

We present a novel, particularly robust technique for effective desynchronization of neuronal populations in the presence of noise. Delayed feedback signals are administered in a spatially coordinated way via four stimulation sites using different delays for each stimulation site, respectively. The technique is numerically tested in a phase oscillator model and in a physiologically realistic model. We propose our methods as novel, particularly mild and effective stimulation protocols for the therapy of patients suffering from Parkinson's disease, essential tremor or epilepsy.

Mathematical analysis and simulations of the neural circuit for locomotion in lampreys.

We analyze the dynamics of the neural circuit of the lamprey central pattern generator. This analysis provides insight into how neural interactions form oscillators and enable spontaneous oscillations in a network of damped oscillators, which were not apparent in previous simulations or abstract phase oscillator models. We also show how the different behavior regimes (characterized by phase and amplitude relationships between oscillators) of forward or backward swimming, and turning, can be controlled using the neural connection strengths and external inputs.

Phase Model Analysis of the Long-Range Excitation in the Hippocampal CA1 Model

nite conduction delay time needs to be analyzed. This work attempts such analysis, utilizing the reduced phase oscillator model. It is shown that the long-range gamma rhythm remains unstable, regardless of the presence of the excitatory connection. However, the beta rhythm is stable over a broad range of conduction time delay, which cannot apparently be tolerated by the long-range gamma rhythm. These synchronization features are consistent with experimental observations which imply that gamma rhythms are used for local computations, whereas beta rhythms are used for higher level interactions involving more distant structures.

Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling.

The phenomenon of slow switching in populations of globally coupled oscillators is discussed. This characteristic collective dynamics, which was first discovered in a particular class of the phase oscillator model, is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population. We argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom. We also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators. Although a heteroclinic loop might seem to arise only exceptionally, we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony. We argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle-node bifurcations. Slow switching arises when a system with a heteroclinic loop is weakly perturbed. This will be demonstrated with a vector model by applying weak noises. Other types of weak symmetry-breaking perturbations can also cause slow switching.

Selective visual attention in a neurocomputational model of phase oscillators.

In order to understand the dynamic property of covert selective visual attention, which is different from the proposed mechanism of the spotlight metaphor, a two-layered network of phase oscillators was developed. The first layer is related to the hippocampus and controls attention focus formation. The second layer is related to the visual cortex, and each cortical oscillator in it simulates an assembly of cells coding for a particular stimulus in the sense of feature binding. Selective visual attention is interpreted as the result of the emergent synchronization of hippocampus oscillators and a part of cortical oscillators. Numerical experiments are presented to illustrate attention focus formation and attention shifting from one set of stimuli to another. From a neurocomputational point of view, our results demonstrate that attention is an emergent property of the dynamical cell assemblies responding to the whole visual field.

Exactly Solvable Phase Oscillator Models with Synchronization Dynamics

Populations of phase oscillators interacting globally through a general coupling function $f(x)$ have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, $f(x)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{sgn}(x)$, have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved.

Synchronization in an oscillator neural network model with time-delayed coupling

We study collective synchronization in a neural network model of phase oscillators with time-delayed coupling. The network contains a central oscillator and peripheral oscillators which interact via the central oscillator. The introduction of delays in connections between elements of the network significantly changes the network dynamics. It is shown that, under certain conditions for the delays and coupling strengths, the network has a multitude of synchronization frequencies in contrast to the case without delays where it has, at most, one frequency. The coexistence of different synchronous states with their own basins of attraction is the characteristic property of the network with time-delayed coupling. The criteria for the existence of synchronous states and their stability are derived. Analysis of the asymptotic behaviour of the network under increasing connection strengths (K) and delays (τ) has shown that the number of synchronous regimes in the network is increasing, and even when tau is a small ...

Desynchronization in a Population of Globally Coupled Identical Oscillators

Desynchronization in a coupled system of oscillation and deformation is studied with a mean-field type phase oscillator model. The asynchronous state is analyzed with a self-consistent equation

Stabilized generation of three phase sinewaves

The paper deals with the derivation of a three phase oscillator model. The model is unusual in embracing a third order three-dimensional dynamics. Another feature of the model is in enabling an undistorted generation of precise sinewaves. The oscillator is in this latter respect similar to the well known two phase precise quadrature oscillators, which are also characterized in possessing a method of amplitude stabilization which does not distort the oscillator waveforms. The main effort of the present work has been, therefore, in extending the fundamental principles of the second order quadrature oscillator system from a two dimensional space into a three dimensional space. The paper concludes in suggesting possible applications of the present model.