237 research outputs found
Spatiotemporal coding of inputs for a system of globally coupled phase oscillators
Copyright © 2008 The American Physical SocietyWe investigate the spatiotemporal coding of low amplitude inputs to a simple system of globally coupled phase oscillators with coupling function g(ϕ)=−sin(ϕ+α)+rsin(2ϕ+β) that has robust heteroclinic cycles (slow switching between cluster states). The inputs correspond to detuning of the oscillators. It was recently noted that globally coupled phase oscillators can encode their frequencies in the form of spatiotemporal codes of a sequence of cluster states [P. Ashwin, G. Orosz, J. Wordsworth, and S. Townley, SIAM J. Appl. Dyn. Syst. 6, 728 (2007)]. Concentrating on the case of N=5 oscillators we show in detail how the spatiotemporal coding can be used to resolve all of the information that relates the individual inputs to each other, providing that a long enough time series is considered. We investigate robustness to the addition of noise and find a remarkable stability, especially of the temporal coding, to the addition of noise even for noise of a comparable magnitude to the inputs
Higher order approximation of isochrons
Phase reduction is a commonly used techinque for analyzing stable
oscillators, particularly in studies concerning synchronization and phase lock
of a network of oscillators. In a widely used numerical approach for obtaining
phase reduction of a single oscillator, one needs to obtain the gradient of the
phase function, which essentially provides a linear approximation of isochrons.
In this paper, we extend the method for obtaining partial derivatives of the
phase function to arbitrary order, providing higher order approximations of
isochrons. In particular, our method in order 2 can be applied to the study of
dynamics of a stable oscillator subjected to stochastic perturbations, a topic
that will be discussed in a future paper. We use the Stuart-Landau oscillator
to illustrate the method in order 2
Universal behavior in populations composed of excitable and self-oscillatory elements
We study the robustness of self-sustained oscillatory activity in a globally
coupled ensemble of excitable and oscillatory units. The critical balance to
achieve collective self-sustained oscillations is analytically established. We
also report a universal scaling function for the ensemble's mean frequency. Our
results extend the framework of the `Aging Transition' [Phys. Rev. Lett. 93,
104101 (2004)] including a broad class of dynamical systems potentially
relevant in biology.Comment: 4 pages; Changed titl
Collective dynamical response of coupled oscillators with any network structure
We formulate a reduction theory that describes the response of an oscillator
network as a whole to external forcing applied nonuniformly to its constituent
oscillators. The phase description of multiple oscillator networks coupled
weakly is also developed. General formulae for the collective phase sensitivity
and the effective phase coupling between the oscillator networks are found. Our
theory is applicable to a wide variety of oscillator networks undergoing
frequency synchronization. Any network structure can systematically be treated.
A few examples are given to illustrate our theory.Comment: 4 pages, 2 figure
Asynchronous response of coupled pacemaker neurons
We study a network model of two conductance-based pacemaker neurons of
differing natural frequency, coupled with either mutual excitation or
inhibition, and receiving shared random inhibitory synaptic input. The networks
may phase-lock spike-to-spike for strong mutual coupling. But the shared input
can desynchronize the locked spike-pairs by selectively eliminating the lagging
spike or modulating its timing with respect to the leading spike depending on
their separation time window. Such loss of synchrony is also found in a large
network of sparsely coupled heterogeneous spiking neurons receiving shared
input.Comment: 11 pages, 4 figures. To appear in Phys. Rev. Let
Strong Effects of Network Architecture in the Entrainment of Coupled Oscillator Systems
Entrainment of randomly coupled oscillator networks by periodic external
forcing applied to a subset of elements is numerically and analytically
investigated. For a large class of interaction functions, we find that the
entrainment window with a tongue shape becomes exponentially narrow for
networks with higher hierarchical organization. However, the entrainment is
significantly facilitated if the networks are directionally biased, i.e.,
closer to the feedforward networks. Furthermore, we show that the networks with
high entrainment ability can be constructed by evolutionary optimization
processes. The neural network structure of the master clock of the circadian
rhythm in mammals is discussed from the viewpoint of our results.Comment: 15 pages, 11 figures, RevTe
Gauge Theory for the Rate Equations: Electrodynamics on a Network
Systems of coupled rate equations are ubiquitous in many areas of science,
for example in the description of electronic transport through quantum dots and
molecules. They can be understood as a continuity equation expressing the
conservation of probability. It is shown that this conservation law can be
implemented by constructing a gauge theory akin to classical electrodynamics on
the network of possible states described by the rate equations. The properties
of this gauge theory are analyzed. It turns out that the network is maximally
connected with respect to the electromagnetic fields even if the allowed
transitions form a sparse network. It is found that the numbers of degrees of
freedom of the electric and magnetic fields are equal. The results shed light
on the structure of classical abelian gauge theory beyond the particular
motivation in terms of rate equations.Comment: 4 pages, 2 figures included, v2: minor revision, as publishe
Reaction-diffusion equations with spatially distributed hysteresis
The paper deals with reaction-diffusion equations involving a hysteretic
discontinuity in the source term, which is defined at each spatial point. In
particular, such problems describe chemical reactions and biological processes
in which diffusive and nondiffusive substances interact according to hysteresis
law. We find sufficient conditions that guarantee the existence and uniqueness
of solutions as well as their continuous dependence on initial data.Comment: 30 pages, 14 figure
- …