513 research outputs found
Spike frequency adaptation affects the synchronization properties of networks of cortical oscillators
Oscillations in many regions of the cortex have common temporal characteristics with dominant frequencies centered around the 40 Hz (gamma) frequency range and the 5–10 Hz (theta) frequency range. Experimental results also reveal spatially synchronous oscillations, which are stimulus dependent (Gray&Singer, 1987;Gray, König, Engel, & Singer, 1989; Engel, König, Kreiter, Schillen, & Singer, 1992). This rhythmic activity suggests that the coherence of neural populations is a crucial feature of cortical dynamics (Gray, 1994). Using both simulations and a theoretical coupled oscillator approach, we demonstrate that the spike frequency adaptation seen in many pyramidal cells plays a subtle but important role in the dynamics of cortical networks. Without adaptation, excitatory connections among model pyramidal cells are desynchronizing. However, the slow processes associated with adaptation encourage stable synchronous behavior
Phase Response Curves of Coupled Oscillators
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.Comment: 5 pages 4 figur
Chimera States for Coupled Oscillators
Arrays of identical oscillators can display a remarkable spatiotemporal
pattern in which phase-locked oscillators coexist with drifting ones.
Discovered two years ago, such "chimera states" are believed to be impossible
for locally or globally coupled systems; they are peculiar to the intermediate
case of nonlocal coupling. Here we present an exact solution for this state,
for a ring of phase oscillators coupled by a cosine kernel. We show that the
stable chimera state bifurcates from a spatially modulated drift state, and
dies in a saddle-node bifurcation with an unstable chimera.Comment: 4 pages, 4 figure
Collective synchronization in spatially extended systems of coupled oscillators with random frequencies
We study collective behavior of locally coupled limit-cycle oscillators with
random intrinsic frequencies, spatially extended over -dimensional
hypercubic lattices. Phase synchronization as well as frequency entrainment are
explored analytically in the linear (strong-coupling) regime and numerically in
the nonlinear (weak-coupling) regime. Our analysis shows that the oscillator
phases are always desynchronized up to , which implies the lower critical
dimension for phase synchronization. On the other hand, the
oscillators behave collectively in frequency (phase velocity) even in three
dimensions (), indicating that the lower critical dimension for frequency
entrainment is . Nonlinear effects due to periodic nature of
limit-cycle oscillators are found to become significant in the weak-coupling
regime: So-called {\em runaway oscillators} destroy the synchronized (ordered)
phase and there emerges a fully random (disordered) phase. Critical behavior
near the synchronization transition into the fully random phase is unveiled via
numerical investigation. Collective behavior of globally-coupled oscillators is
also examined and compared with that of locally coupled oscillators.Comment: 18 pages, 18 figure
Dynamically-Coupled Oscillators -- Cooperative Behavior via Dynamical Interaction --
We propose a theoretical framework to study the cooperative behavior of
dynamically coupled oscillators (DCOs) that possess dynamical interactions.
Then, to understand synchronization phenomena in networks of interneurons which
possess inhibitory interactions, we propose a DCO model with dynamics of
interactions that tend to cause 180-degree phase lags. Employing an approach
developed here, we demonstrate that although our model displays synchronization
at high frequencies, it does not exhibit synchronization at low frequencies
because this dynamical interaction does not cause a phase lag sufficiently
large to cancel the effect of the inhibition. We interpret the disappearance of
synchronization in our model with decreasing frequency as describing the
breakdown of synchronization in the interneuron network of the CA1 area below
the critical frequency of 20 Hz.Comment: 10 pages, 3 figure
Time-delayed Spatial Patterns in a Two-dimensional Array of Coupled Oscillators
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.Comment: 5 Figure
Collective modes of coupled phase oscillators with delayed coupling
We study the effects of delayed coupling on timing and pattern formation in
spatially extended systems of dynamic oscillators. Starting from a discrete
lattice of coupled oscillators, we derive a generic continuum theory for
collective modes of long wavelength. We use this approach to study spatial
phase profiles of cellular oscillators in the segmentation clock, a dynamic
patterning system of vertebrate embryos. Collective wave patterns result from
the interplay of coupling delays and moving boundary conditions. We show that
the phase profiles of collective modes depend on coupling delays.Comment: 5 pages, 2 figure
Finite-size and correlation-induced effects in Mean-field Dynamics
The brain's activity is characterized by the interaction of a very large
number of neurons that are strongly affected by noise. However, signals often
arise at macroscopic scales integrating the effect of many neurons into a
reliable pattern of activity. In order to study such large neuronal assemblies,
one is often led to derive mean-field limits summarizing the effect of the
interaction of a large number of neurons into an effective signal. Classical
mean-field approaches consider the evolution of a deterministic variable, the
mean activity, thus neglecting the stochastic nature of neural behavior. In
this article, we build upon two recent approaches that include correlations and
higher order moments in mean-field equations, and study how these stochastic
effects influence the solutions of the mean-field equations, both in the limit
of an infinite number of neurons and for large yet finite networks. We
introduce a new model, the infinite model, which arises from both equations by
a rescaling of the variables and, which is invertible for finite-size networks,
and hence, provides equivalent equations to those previously derived models.
The study of this model allows us to understand qualitative behavior of such
large-scale networks. We show that, though the solutions of the deterministic
mean-field equation constitute uncorrelated solutions of the new mean-field
equations, the stability properties of limit cycles are modified by the
presence of correlations, and additional non-trivial behaviors including
periodic orbits appear when there were none in the mean field. The origin of
all these behaviors is then explored in finite-size networks where interesting
mesoscopic scale effects appear. This study leads us to show that the
infinite-size system appears as a singular limit of the network equations, and
for any finite network, the system will differ from the infinite system
Phase Transitions of an Oscillator Neural Network with a Standard Hebb Learning Rule
Studies have been made on the phase transition phenomena of an oscillator
network model based on a standard Hebb learning rule like the Hopfield model.
The relative phase informations---the in-phase and anti-phase, can be embedded
in the network. By self-consistent signal-to-noise analysis (SCSNA), it was
found that the storage capacity is given by , which is better
than that of Cook's model. However, the retrieval quality is worse. In
addition, an investigation was made into an acceleration effect caused by
asymmetry of the phase dynamics. Finally, it was numerically shown that the
storage capacity can be improved by modifying the shape of the coupling
function.Comment: 10 pages, 6 figure
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