7,634 research outputs found
Nonlinear Oscillations and Bifurcations in Silicon Photonic Microresonators
Silicon microdisks are optical resonators that can exhibit surprising
nonlinear behavior. We present a new analysis of the dynamics of these
resonators, elucidating the mathematical origin of spontaneous oscillations and
deriving predictions for observed phenomena such as a frequency comb spectrum
with MHz-scale repetition rate. We test predictions through laboratory
experiment and numerical simulation.Comment: Main text: 5 pages, 6 figures. Supplemental material: 12 pages, 8
figure
Understanding Epileptiform After-Discharges as Rhythmic Oscillatory Transients
Electro-cortical activity in patients with epilepsy may show abnormal
rhythmic transients in response to stimulation. Even when using the same
stimulation parameters in the same patient, wide variability in the duration of
transient response has been reported. These transients have long been
considered important for the mapping of the excitability levels in the
epileptic brain but their dynamic mechanism is still not well understood.
To understand the occurrence of abnormal transients dynamically, we use a
thalamo-cortical neural population model of epileptic spike-wave activity and
study the interaction between slow and fast subsystems.
In a reduced version of the thalamo-cortical model, slow wave oscillations
arise from a fold of cycles (FoC) bifurcation. This marks the onset of a region
of bistability between a high amplitude oscillatory rhythm and the background
state. In vicinity of the bistability in parameter space, the model has
excitable dynamics, showing prolonged rhythmic transients in response to
suprathreshold pulse stimulation. We analyse the state space geometry of the
bistable and excitable states, and find that the rhythmic transient arises when
the impending FoC bifurcation deforms the state space and creates an area of
locally reduced attraction to the fixed point. This area essentially allows
trajectories to dwell there before escaping to the stable steady state, thus
creating rhythmic transients. In the full thalamo-cortical model, we find a
similar FoC bifurcation structure.
Based on the analysis, we propose an explanation of why stimulation induced
epileptiform activity may vary between trials, and predict how the variability
could be related to ongoing oscillatory background activity.Comment: http://journal.frontiersin.org/article/10.3389/fncom.2017.00025/ful
Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations
This work continues the analysis of complex dynamics in a class of
bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal
voltage dynamics with adaptation and spike emission. We show that these models
can generically display a form of mixed-mode oscillations (MMOs), which are
trajectories featuring an alternation of small oscillations with spikes or
bursts (multiple consecutive spikes). The mechanism by which these are
generated relies fundamentally on the hybrid structure of the flow: invariant
manifolds of the continuous dynamics govern small oscillations, while discrete
resets govern the emission of spikes or bursts, contrasting with classical MMO
mechanisms in ordinary differential equations involving more than three
dimensions and generally relying on a timescale separation. The decomposition
of mechanisms reveals the geometrical origin of MMOs, allowing a relatively
simple classification of points on the reset manifold associated to specific
numbers of small oscillations. We show that the MMO pattern can be described
through the study of orbits of a discrete adaptation map, which is singular as
it features discrete discontinuities with unbounded left- and
right-derivatives. We study orbits of the map via rotation theory for
discontinuous circle maps and elucidate in detail complex behaviors arising in
the case where MMOs display at most one small oscillation between each
consecutive pair of spikes
Data-Driven Diagnostics of Mechanism and Source of Sustained Oscillations
Sustained oscillations observed in power systems can damage equipment,
degrade the power quality and increase the risks of cascading blackouts. There
are several mechanisms that can give rise to oscillations, each requiring
different countermeasure to suppress or eliminate the oscillation. This work
develops mathematical framework for analysis of sustained oscillations and
identifies statistical signatures of each mechanism, based on which a novel
oscillation diagnosis method is developed via real-time processing of phasor
measurement units (PMUs) data. Case studies show that the proposed method can
accurately identify the exact mechanism for sustained oscillation, and
meanwhile provide insightful information to locate the oscillation sources.Comment: The paper has been accepted by IEEE Transactions on Power System
Formation of Pillars at the Boundaries between H II Regions and Molecular Clouds
We investigate numerically the hydrodynamic instability of an ionization
front (IF) accelerating into a molecular cloud, with imposed initial
perturbations of different amplitudes. When the initial amplitude is small, the
imposed perturbation is completely stabilized and does not grow. When the
initial perturbation amplitude is large enough, roughly the ratio of the
initial amplitude to wavelength is greater than 0.02, portions of the IF
temporarily separate from the molecular cloud surface, locally decreasing the
ablation pressure. This causes the appearance of a large, warm HI region and
triggers nonlinear dynamics of the IF. The local difference of the ablation
pressure and acceleration enhances the appearance and growth of a multimode
perturbation. The stabilization usually seen at the IF in the linear regimes
does not work due to the mismatch of the modes of the perturbations at the
cloud surface and in density in HII region above the cloud surface. Molecular
pillars are observed in the late stages of the large amplitude perturbation
case. The velocity gradient in the pillars is in reasonably good agreement with
that observed in the Eagle Nebula. The initial perturbation is imposed in three
different ways: in density, in incident photon number flux, and in the surface
shape. All cases show both stabilization for a small initial perturbation and
large growth of the second harmonic by increasing amplitude of the initial
perturbation above a critical value.Comment: 21 pages, 8 figures, accepted for publication in ApJ. high resolution
figures available upon reques
Synchronization of electrically coupled resonate-and-fire neurons
Electrical coupling between neurons is broadly present across brain areas and
is typically assumed to synchronize network activity. However, intrinsic
properties of the coupled cells can complicate this simple picture. Many cell
types with strong electrical coupling have been shown to exhibit resonant
properties, and the subthreshold fluctuations arising from resonance are
transmitted through electrical synapses in addition to action potentials. Using
the theory of weakly coupled oscillators, we explore the effect of both
subthreshold and spike-mediated coupling on synchrony in small networks of
electrically coupled resonate-and-fire neurons, a hybrid neuron model with
linear subthreshold dynamics and discrete post-spike reset. We calculate the
phase response curve using an extension of the adjoint method that accounts for
the discontinuity in the dynamics. We find that both spikes and resonant
subthreshold fluctuations can jointly promote synchronization. The subthreshold
contribution is strongest when the voltage exhibits a significant post-spike
elevation in voltage, or plateau. Additionally, we show that the geometry of
trajectories approaching the spiking threshold causes a "reset-induced shear"
effect that can oppose synchrony in the presence of network asymmetry, despite
having no effect on the phase-locking of symmetrically coupled pairs
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