55 research outputs found
Transitions in active rotator systems: invariant hyperbolic manifold approach
Our main focus is on a general class of active rotators with mean field
interactions, that is globally coupled large families of dynamical systems on
the unit circle with non-trivial stochastic dynamics. Each isolated system is a
diffusion process on a circle, with drift -delta V', where V' is a periodic
function and delta is an intensity parameter. It is well known that the
interacting dynamics is accurately described, in the limit of infinitely many
interacting components, by a Fokker-Planck PDE and the model reduces for
delta=0 to a particular case of the Kuramoto synchronization model, for which
one can show the existence of a stable normally hyperbolic manifold of
stationary solutions for the corresponding Fokker-Planck equation (we are
interested in the case in which this manifold is non-trivial, that happens when
the interaction is sufficiently strong, that is in the synchronized regime of
the Kuramoto model). We use the robustness of normally hyperbolic structures to
infer qualitative and quantitative results on the |delta|< delta0 cases, with
delta0 a suitable threshold: as a matter of fact, we obtain an accurate
description of the dynamics on the invariant manifold for delta=0 and we link
it explicitly to the potential V . This approach allows to have a complete
description of the phase diagram of the active rotators model, at least for
|delta|< delta0, thus identifying for which values of the parameters (notably,
noise intensity and/or coupling strength) the system exhibits periodic pulse
waves or stabilizes at a quiescent resting state. Moreover, some of our results
are very explicit and this brings a new insight into the combined effect of
active rotator dynamics, noise and interaction. The links with the literature
on specific systems, notably neuronal models, are discussed in detail.Comment: 29 pages, 4 figures. Version 2: some changes in introduction, added
reference
The heterogeneous gas with singular interaction: Generalized circular law and heterogeneous renormalized energy
We introduce and analyze dimensional Coulomb gases with random charge
distribution and general external confining potential. We show that these gases
satisfy a large deviations principle. The analysis of the minima of the rate
function (which is the leading term of the energy) reveals that at equilibrium,
the particle distribution is a generalized circular law (i.e. with spherical
support but non-necessarily uniform distribution). In the classical
electrostatic external potential, there are infinitely many minimizers of the
rate function. The most likely macroscopic configuration is a disordered
distribution in which particles are uniformly distributed (for , the
circular law), and charges are independent of the positions of the particles.
General charge-dependent confining potentials unfold this degenerate situation:
in contrast, the particle density is not uniform, and particles spontaneously
organize according to their charge. In that picture the classical electrostatic
potential appears as a transition at which order is lost. Sub-leading terms of
the energy are derived: we show that these are related to an operator,
generalizing the Coulomb renormalized energy, which incorporates the
heterogeneous nature of the charges. This heterogeneous renormalized energy
informs us about the microscopic arrangements of the particles, which are
non-standard, strongly depending on the charges, and include progressive and
irregular lattices.Comment: 26 pages, 10 figure
Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation
International audienceMotivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge.In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two "extreme" cases.Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20
Dynamics of a structured neuron population
We study the dynamics of assemblies of interacting neurons. For large fully connected networks,the dynamics of the system can be described by a partial differential equation reminiscent of age-structure models used in mathematical ecology, where the "age" of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network. We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behavior of solutions; both for small and large J, we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing
The real Ginibre ensemble with real eigenvalues
We consider the ensemble of Real Ginibre matrices with a positive fraction
of real eigenvalues. We demonstrate a large deviations principle for
the joint eigenvalue density of such matrices and we introduce a two phase
log-gas whose stationary distribution coincides with the spectral measure of
the ensemble. Using these tools we provide an asymptotic expansion for the
probability that an Ginibre matrix has real eigenvalues and we characterize the spectral measures of these
matrices.Comment: 19 pages, 3 figure
Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators
We study the dynamics of the large N limit of the Kuramoto model of coupled
phase oscillators, subject to white noise. We introduce the notion of shadow
inertial manifold and we prove their existence for this model, supporting the
fact that the long term dynamics of this model is finite dimensional. Following
this, we prove that the global attractor of this model takes one of two forms.
When coupling strength is below a critical value, the global attractor is a
single equilibrium point corresponding to an incoherent state. Conversely, when
coupling strength is beyond this critical value, the global attractor is a
two-dimensional disk composed of radial trajectories connecting a saddle
equilibrium (the incoherent state) to an invariant closed curve of locally
stable equilibria (partially synchronized state). Our analysis hinges, on the
one hand, upon sharp existence and uniqueness results and their consequence for
the existence of a global attractor, and, on the other hand, on the study of
the dynamics in the vicinity of the incoherent and synchronized equilibria. We
prove in particular non-linear stability of each synchronized equilibrium, and
normal hyperbolicity of the set of such equilibria. We explore mathematically
and numerically several properties of the global attractor, in particular we
discuss the limit of this attractor as noise intensity decreases to zero.Comment: revised version, 28 pages, 4 figure
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