7,337 research outputs found
Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions
In this manuscript we analyze the collective behavior of mean-field limits of
large-scale, spatially extended stochastic neuronal networks with delays.
Rigorously, the asymptotic regime of such systems is characterized by a very
intricate stochastic delayed integro-differential McKean-Vlasov equation that
remain impenetrable, leaving the stochastic collective dynamics of such
networks poorly understood. In order to study these macroscopic dynamics, we
analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics
and sigmoidal interactions. In that case, we prove that the solution of the
mean-field equation is Gaussian, hence characterized by its two first moments,
and that these two quantities satisfy a set of coupled delayed
integro-differential equations. These equations are similar to usual neural
field equations, and incorporate noise levels as a parameter, allowing analysis
of noise-induced transitions. We identify through bifurcation analysis several
qualitative transitions due to noise in the mean-field limit. In particular,
stabilization of spatially homogeneous solutions, synchronized oscillations,
bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from
static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow
further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence
in space of Brownian motion
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
Hopf Bifurcation and Chaos in a Single Inertial Neuron Model with Time Delay
A delayed differential equation modelling a single neuron with inertial term
is considered in this paper. Hopf bifurcation is studied by using the normal
form theory of retarded functional differential equations. When adopting a
nonmonotonic activation function, chaotic behavior is observed. Phase plots,
waveform plots, and power spectra are presented to confirm the chaoticity.Comment: 12 pages, 7 figure
Hopf Bifurcation and Chaos in Tabu Learning Neuron Models
In this paper, we consider the nonlinear dynamical behaviors of some tabu
leaning neuron models. We first consider a tabu learning single neuron model.
By choosing the memory decay rate as a bifurcation parameter, we prove that
Hopf bifurcation occurs in the neuron. The stability of the bifurcating
periodic solutions and the direction of the Hopf bifurcation are determined by
applying the normal form theory. We give a numerical example to verify the
theoretical analysis. Then, we demonstrate the chaotic behavior in such a
neuron with sinusoidal external input, via computer simulations. Finally, we
study the chaotic behaviors in tabu learning two-neuron models, with linear and
quadratic proximity functions respectively.Comment: 14 pages, 13 figures, Accepted by International Journal of
Bifurcation and Chao
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
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A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays
This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.In this Letter, the analysis problem for the existence and stability of periodic solutions is investigated for a class of general discrete-time recurrent neural networks with time-varying delays. For the neural networks under study, a generalized activation function is considered, and the traditional assumptions on the boundedness, monotony and differentiability of the activation functions are removed. By employing the latest free-weighting matrix method, an appropriate Lyapunov–Krasovskii functional is constructed and several sufficient conditions are established to ensure the existence, uniqueness, and globally exponential stability of the periodic solution for the addressed neural network. The conditions are dependent on both the lower bound and upper bound of the time-varying time delays. Furthermore, the conditions are expressed in terms of the linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. Two simulation examples are given to show the effectiveness and less conservatism of the proposed criteria.This work was supported in part by the National Natural Science Foundation of China under Grant 50608072, an International Joint Project sponsored by the Royal Society of the UK and the National Natural Science Foundation of China, and the Alexander von Humboldt Foundation of Germany
Geometric Analysis of Synchronization in Neuronal Networks with Global Inhibition and Coupling Delays
We study synaptically coupled neuronal networks to identify the role of
coupling delays in network's synchronized behaviors. We consider a network of
excitable, relaxation oscillator neurons where two distinct populations, one
excitatory and one inhibitory, are coupled and interact with each other. The
excitatory population is uncoupled, while the inhibitory population is tightly
coupled. A geometric singular perturbation analysis yields existence and
stability conditions for synchronization states under different firing patterns
between the two populations, along with formulas for the periods of such
synchronous solutions. Our results demonstrate that the presence of coupling
delays in the network promotes synchronization. Numerical simulations are
conducted to supplement and validate analytical results. We show the results
carry over to a model for spindle sleep rhythms in thalamocortical networks,
one of the biological systems which motivated our study. The analysis helps to
explain how coupling delays in either excitatory or inhibitory synapses
contribute to producing synchronized rhythms.Comment: 43 pages, 12 figure
Reverberating activity in a neural network with distributed signal transmission delays
It is known that an identical delay in all transmission lines can destabilize
macroscopic stationarity of a neural network, causing oscillation or chaos. We
analyze the collective dynamics of a network whose intra-transmission delays
are distributed in time. Here, a neuron is modeled as a discrete-time threshold
element that responds in an all-or-nothing manner to a linear sum of signals
that arrive after delays assigned to individual transmission lines. Even though
transmission delays are distributed in time, a whole network exhibits a single
collective oscillation with a period close to the average transmission delay.
The collective oscillation can not only be a simple alternation of the
consecutive firing and resting, but also nontrivially sequenced series of
firing and resting, reverberating in a certain period of time. Moreover, the
system dynamics can be made quasiperiodic or chaotic by changing the
distribution of delays.Comment: 8pages, 9figure
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