4,837 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
Noise-induced behaviors in neural mean field dynamics
The collective behavior of cortical neurons is strongly affected by the
presence of noise at the level of individual cells. In order to study these
phenomena in large-scale assemblies of neurons, we consider networks of
firing-rate neurons with linear intrinsic dynamics and nonlinear coupling,
belonging to a few types of cell populations and receiving noisy currents.
Asymptotic equations as the number of neurons tends to infinity (mean field
equations) are rigorously derived based on a probabilistic approach. These
equations are implicit on the probability distribution of the solutions which
generally makes their direct analysis difficult. However, in our case, the
solutions are Gaussian, and their moments satisfy a closed system of nonlinear
ordinary differential equations (ODEs), which are much easier to study than the
original stochastic network equations, and the statistics of the empirical
process uniformly converge towards the solutions of these ODEs. Based on this
description, we analytically and numerically study the influence of noise on
the collective behaviors, and compare these asymptotic regimes to simulations
of the network. We observe that the mean field equations provide an accurate
description of the solutions of the network equations for network sizes as
small as a few hundreds of neurons. In particular, we observe that the level of
noise in the system qualitatively modifies its collective behavior, producing
for instance synchronized oscillations of the whole network, desynchronization
of oscillating regimes, and stabilization or destabilization of stationary
solutions. These results shed a new light on the role of noise in shaping
collective dynamics of neurons, and gives us clues for understanding similar
phenomena observed in biological networks
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
A chimera state is a spatio-temporal pattern in a network of identical
coupled oscillators in which synchronous and asynchronous oscillation coexist.
This state of broken symmetry, which usually coexists with a stable spatially
symmetric state, has intrigued the nonlinear dynamics community since its
discovery in the early 2000s. Recent experiments have led to increasing
interest in the origin and dynamics of these states. Here we review the history
of research on chimera states and highlight major advances in understanding
their behaviour.Comment: 26 pages, 3 figure
Basins of Attraction for Chimera States
Chimera states---curious symmetry-broken states in systems of identical
coupled oscillators---typically occur only for certain initial conditions. Here
we analyze their basins of attraction in a simple system comprised of two
populations. Using perturbative analysis and numerical simulation we evaluate
asymptotic states and associated destination maps, and demonstrate that basins
form a complex twisting structure in phase space. Understanding the basins'
precise nature may help in the development of control methods to switch between
chimera patterns, with possible technological and neural system applications.Comment: Please see Ancillary files for the 4 supplementary videos including
description (PDF
Synchronization and Noise: A Mechanism for Regularization in Neural Systems
To learn and reason in the presence of uncertainty, the brain must be capable
of imposing some form of regularization. Here we suggest, through theoretical
and computational arguments, that the combination of noise with synchronization
provides a plausible mechanism for regularization in the nervous system. The
functional role of regularization is considered in a general context in which
coupled computational systems receive inputs corrupted by correlated noise.
Noise on the inputs is shown to impose regularization, and when synchronization
upstream induces time-varying correlations across noise variables, the degree
of regularization can be calibrated over time. The proposed mechanism is
explored first in the context of a simple associative learning problem, and
then in the context of a hierarchical sensory coding task. The resulting
qualitative behavior coincides with experimental data from visual cortex.Comment: 32 pages, 7 figures. under revie
Chimera states in pulse coupled neural networks: the influence of dilution and noise
We analyse the possible dynamical states emerging for two symmetrically pulse
coupled populations of leaky integrate-and-fire neurons. In particular, we
observe broken symmetry states in this set-up: namely, breathing chimeras,
where one population is fully synchronized and the other is in a state of
partial synchronization (PS) as well as generalized chimera states, where both
populations are in PS, but with different levels of synchronization. Symmetric
macroscopic states are also present, ranging from quasi-periodic motions, to
collective chaos, from splay states to population anti-phase partial
synchronization. We then investigate the influence disorder, random link
removal or noise, on the dynamics of collective solutions in this model. As a
result, we observe that broken symmetry chimera-like states, with both
populations partially synchronized, persist up to 80 \% of broken links and up
to noise amplitudes 8 \% of threshold-reset distance. Furthermore, the
introduction of disorder on symmetric chaotic state has a constructive effect,
namely to induce the emergence of chimera-like states at intermediate dilution
or noise level.Comment: 15 pages, 7 figure, contribution for the Workshop "Nonlinear Dynamics
in Computational Neuroscience: from Physics and Biology to ICT" held in Turin
(Italy) in September 201
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