526 research outputs found
Model of the early development of thalamo-cortical connections and area patterning via signaling molecules
The mammalian cortex is divided into architectonic and functionally distinct
areas. There is growing experimental evidence that their emergence and
development is controlled by both epigenetic and genetic factors. The latter
were recently implicated as dominating the early cortical area specification.
In this paper, we present a theoretical model that explicitly considers the
genetic factors and that is able to explain several sets of experiments on
cortical area regulation involving transcription factors Emx2 and Pax6, and
fibroblast growth factor FGF8. The model consists of the dynamics of thalamo-
cortical connections modulated by signaling molecules that are regulated
genetically, and by axonal competition for neocortical space. The model can
make predictions and provides a basic mathematical framework for the early
development of the thalamo-cortical connections and area patterning that can be
further refined as more experimental facts become known.Comment: brain, model, neural development, cortical area patterning, signaling
molecule
Formation of antiwaves in gap-junction-coupled chains of neurons
Using network models consisting of gap junction coupled Wang-Buszaki neurons,
we demonstrate that it is possible to obtain not only synchronous activity
between neurons but also a variety of constant phase shifts between 0 and \pi.
We call these phase shifts intermediate stable phaselocked states. These phase
shifts can produce a large variety of wave-like activity patterns in
one-dimensional chains and two-dimensional arrays of neurons, which can be
studied by reducing the system of equations to a phase model. The 2\pi periodic
coupling functions of these models are characterized by prominent higher order
terms in their Fourier expansion, which can be varied by changing model
parameters. We study how the relative contribution of the odd and even terms
affect what solutions are possible, the basin of attraction of those solutions
and their stability. These models may be applicable to the spinal central
pattern generators of the dogfish and also to the developing neocortex of the
neonatal rat
Renewal theory of coupled neuronal pools
A theory is provided to analyze the dynamics of delay-coupled pools of spiking neurons based on stability
analysis of stationary firing. Transitions between stable and unstable regimes can be predicted by bifurcation analysis of the underlying integral dynamics. Close to the bifurcation point the network exhibits slowly changingactivities and allows for slow collective phenomena like continuous attractors
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
Limits and dynamics of stochastic neuronal networks with random heterogeneous delays
Realistic networks display heterogeneous transmission delays. We analyze here
the limits of large stochastic multi-populations networks with stochastic
coupling and random interconnection delays. We show that depending on the
nature of the delays distributions, a quenched or averaged propagation of chaos
takes place in these networks, and that the network equations converge towards
a delayed McKean-Vlasov equation with distributed delays. Our approach is
mostly fitted to neuroscience applications. We instantiate in particular a
classical neuronal model, the Wilson and Cowan system, and show that the
obtained limit equations have Gaussian solutions whose mean and standard
deviation satisfy a closed set of coupled delay differential equations in which
the distribution of delays and the noise levels appear as parameters. This
allows to uncover precisely the effects of noise, delays and coupling on the
dynamics of such heterogeneous networks, in particular their role in the
emergence of synchronized oscillations. We show in several examples that not
only the averaged delay, but also the dispersion, govern the dynamics of such
networks.Comment: Corrected misprint (useless stopping time) in proof of Lemma 1 and
clarified a regularity hypothesis (remark 1
Detecting the temporal structure of intermittent phase locking
This study explores a method to characterize temporal structure of
intermittent phase locking in oscillatory systems. When an oscillatory system
is in a weakly synchronized regime away from a synchronization threshold, it
spends most of the time in parts of its phase space away from synchronization
state. Therefore characteristics of dynamics near this state (such as its
stability properties/Lyapunov exponents or distributions of the durations of
synchronized episodes) do not describe system's dynamics for most of the time.
We consider an approach to characterize the system dynamics in this case, by
exploring the relationship between the phases on each cycle of oscillations. If
some overall level of phase locking is present, one can quantify when and for
how long phase locking is lost, and how the system returns back to the
phase-locked state. We consider several examples to illustrate this approach:
coupled skewed tent maps, which stability can be evaluated analytically,
coupled R\"{o}ssler and Lorenz oscillators, undergoing through different
intermittencies on the way to phase synchronization, and a more complex example
of coupled neurons. We show that the obtained measures can describe the
differences in the dynamics and temporal structure of
synchronization/desynchronization events for the systems with similar overall
level of phase locking and similar stability of synchronized state.Comment: 12 pages, 10 figures. The paper will appear in Phys. Rev.
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
Correlations, fluctuations and stability of a finite-size network of coupled oscillators
The incoherent state of the Kuramoto model of coupled oscillators exhibits
marginal modes in mean field theory. We demonstrate that corrections due to
finite size effects render these modes stable in the subcritical case, i.e.
when the population is not synchronous. This demonstration is facilitated by
the construction of a non-equilibrium statistical field theoretic formulation
of a generic model of coupled oscillators. This theory is consistent with
previous results. In the all-to-all case, the fluctuations in this theory are
due completely to finite size corrections, which can be calculated in an
expansion in 1/N, where N is the number of oscillators. The N -> infinity limit
of this theory is what is traditionally called mean field theory for the
Kuramoto model.Comment: 25 pages (2 column), 12 figures, modifications for resubmissio
Mathematical Analysis and Simulations of the Neural Circuit for Locomotion in Lamprey
We analyze the dynamics of the neural circuit of the lamprey central pattern
generator (CPG). This analysis provides insights 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 behaviour regimes
(characterized by phase and amplitude relationships between oscillators) of
forward/backward swimming, and turning, can be controlled using the neural
connection strengths and external inputs.Comment: 4 pages, accepted for publication in Physical Review Letter
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