515 research outputs found
Propagation of chaos in neural fields
We consider the problem of the limit of bio-inspired spatially extended
neuronal networks including an infinite number of neuronal types (space
locations), with space-dependent propagation delays modeling neural fields. The
propagation of chaos property is proved in this setting under mild assumptions
on the neuronal dynamics, valid for most models used in neuroscience, in a
mesoscopic limit, the neural-field limit, in which we can resolve the quite
fine structure of the neuron's activity in space and where averaging effects
occur. The mean-field equations obtained are of a new type: they take the form
of well-posed infinite-dimensional delayed integro-differential equations with
a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We
also show how these intricate equations can be used in practice to uncover
mathematically the precise mesoscopic dynamics of the neural field in a
particular model where the mean-field equations exactly reduce to deterministic
nonlinear delayed integro-differential equations. These results have several
theoretical implications in neuroscience we review in the discussion.Comment: Updated to correct an erroneous suggestion of extension of the
results in Appendix B, and to clarify some measurability questions in the
proof of Theorem
Kac's chaos and Kac's program
In this note I present the main results about the quantitative and
qualitative propagation of chaos for the Boltzmann-Kac system obtained in
collaboration with C. Mouhot in \cite{MMinvent} which gives a possible answer
to some questions formulated by Kac in \cite{Kac1956}. We also present some
related recent results about Kac's chaos and Kac's program obtained in
\cite{MMWchaos,HaurayMischler,KleberSphere} by K. Carrapatoso, M. Hauray, C.
Mouhot, B. Wennberg and myself
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
Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D
We consider a particle system in 1D, interacting via repulsive or attractive
Coulomb forces. We prove the trajectorial propagation of molecular chaos
towards a nonlinear SDE associated to the Vlasov-Poisson-Fokker-Planck
equation. We obtain a quantitative estimate of convergence in expectation, with
an optimal convergence rate of order . We also prove some exponential
concentration inequalities of the associated empirical measures. A key argument
is a weak-strong stability estimate on the (nonlinear) VPFP equation, that we
are able to adapt for the particle system in some sense.Comment: 30 p, with respect to v1: some typos corrected and a more precise
theorem of propagation of chao
Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons
We derive the mean-field equations arising as the limit of a network of
interacting spiking neurons, as the number of neurons goes to infinity. The
neurons belong to a fixed number of populations and are represented either by
the Hodgkin-Huxley model or by one of its simplified version, the
Fitzhugh-Nagumo model. The synapses between neurons are either electrical or
chemical. The network is assumed to be fully connected. The maximum
conductances vary randomly. Under the condition that all neurons initial
conditions are drawn independently from the same law that depends only on the
population they belong to, we prove that a propagation of chaos phenomenon
takes places, namely that in the mean-field limit, any finite number of neurons
become independent and, within each population, have the same probability
distribution. This probability distribution is solution of a set of implicit
equations, either nonlinear stochastic differential equations resembling the
McKean-Vlasov equations, or non-local partial differential equations resembling
the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of
these equations, i.e. the existence and uniqueness of a solution. We also show
the results of some preliminary numerical experiments that indicate that the
mean-field equations are a good representation of the mean activity of a finite
size network, even for modest sizes. These experiment also indicate that the
McKean-Vlasov-Fokker- Planck equations may be a good way to understand the
mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure
Limits and dynamics of randomly connected neuronal networks
Networks of the brain are composed of a very large number of neurons
connected through a random graph and interacting after random delays that both
depend on the anatomical distance between cells. In order to comprehend the
role of these random architectures on the dynamics of such networks, we analyze
the mesoscopic and macroscopic limits of networks with random correlated
connectivity weights and delays. We address both averaged and quenched limits,
and show propagation of chaos and convergence to a complex integral
McKean-Vlasov equations with distributed delays. We then instantiate a
completely solvable model illustrating the role of such random architectures in
the emerging macroscopic activity. We particularly focus on the role of
connectivity levels in the emergence of periodic solutions
Collective fluctuation by pseudo-Casimir-invariants
In this study, we propose a universal scenario explaining the
fluctuation, including pink noises, in Hamiltonian dynamical systems with many
degrees of freedom under long-range interaction. In the thermodynamic limit,
the dynamics of such systems can be described by the Vlasov equation, which has
an infinite number of Casimir invariants. In a finite system, they become
pseudoinvariants, which yield quasistationary states. The dynamics then exhibit
slow motion over them, up to the timescale where the pseudo-Casimir-invariants
are effective. Such long-time correlation leads to fluctuations of
collective variables, as is confirmed by direct numerical simulations. The
universality of this collective fluctuation is demonstrated by taking a
variety of Hamiltonians and changing the range of interaction and number of
particles.Comment: 13 pages, 12 figure
Contributions of plasma physics to chaos and nonlinear dynamics
This topical review focusses on the contributions of plasma physics to chaos
and nonlinear dynamics bringing new methods which are or can be used in other
scientific domains. It starts with the development of the theory of Hamiltonian
chaos, and then deals with order or quasi order, for instance adiabatic and
soliton theories. It ends with a shorter account of dissipative and high
dimensional Hamiltonian dynamics, and of quantum chaos. Most of these
contributions are a spin-off of the research on thermonuclear fusion by
magnetic confinement, which started in the fifties. Their presentation is both
exhaustive and compact. [15 April 2016
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