93,380 research outputs found
Chaos and Asymptotical Stability in Discrete-time Neural Networks
This paper aims to theoretically prove by applying Marotto's Theorem that
both transiently chaotic neural networks (TCNN) and discrete-time recurrent
neural networks (DRNN) have chaotic structure. A significant property of TCNN
and DRNN is that they have only one fixed point, when absolute values of the
self-feedback connection weights in TCNN and the difference time in DRNN are
sufficiently large. We show that this unique fixed point can actually evolve
into a snap-back repeller which generates chaotic structure, if several
conditions are satisfied. On the other hand, by using the Lyapunov functions,
we also derive sufficient conditions on asymptotical stability for symmetrical
versions of both TCNN and DRNN, under which TCNN and DRNN asymptotically
converge to a fixed point. Furthermore, generic bifurcations are also
considered in this paper. Since both of TCNN and DRNN are not special but
simple and general, the obtained theoretical results hold for a wide class of
discrete-time neural networks. To demonstrate the theoretical results of this
paper better, several numerical simulations are provided as illustrating
examples.Comment: This paper will be published in Physica D. Figures should be
requested to the first autho
Non-Euclidean Contraction Analysis of Continuous-Time Neural Networks
Critical questions in dynamical neuroscience and machine learning are related
to the study of continuous-time neural networks and their stability,
robustness, and computational efficiency. These properties can be
simultaneously established via a contraction analysis.
This paper develops a comprehensive non-Euclidean contraction theory for
continuous-time neural networks. First, for non-Euclidean
logarithmic norms, we establish quasiconvexity with
respect to positive diagonal weights and closed-form worst-case expressions
over certain matrix polytopes. Second, for locally Lipschitz maps (e.g.,
arising as activation functions), we show that their one-sided Lipschitz
constant equals the essential supremum of the logarithmic norm of their
Jacobian. Third and final, we apply these general results to classes of
continuous-time neural networks, including Hopfield, firing rate, Persidskii,
Lur'e and other models. For each model, we compute the optimal contraction rate
and corresponding weighted non-Euclidean norm via a linear program or, in some
special cases, via a Hurwitz condition on the Metzler majorant of the synaptic
matrix. Our non-Euclidean analysis establishes also absolute, connective, and
total contraction properties
Cluster synchronization in an ensemble of neurons interacting through chemical synapses
In networks of periodically firing spiking neurons that are interconnected
with chemical synapses, we analyze cluster state, where an ensemble of neurons
are subdivided into a few clusters, in each of which neurons exhibit perfect
synchronization. To clarify stability of cluster state, we decompose linear
stability of the solution into two types of stabilities: stability of mean
state and stabilities of clusters. Computing Floquet matrices for these
stabilities, we clarify the total stability of cluster state for any types of
neurons and any strength of interactions even if the size of networks is
infinitely large. First, we apply this stability analysis to investigating
synchronization in the large ensemble of integrate-and-fire (IF) neurons. In
one-cluster state we find the change of stability of a cluster, which
elucidates that in-phase synchronization of IF neurons occurs with only
inhibitory synapses. Then, we investigate entrainment of two clusters of IF
neurons with different excitability. IF neurons with fast decaying synapses
show the low entrainment capability, which is explained by a pitchfork
bifurcation appearing in two-cluster state with change of synapse decay time
constant. Second, we analyze one-cluster state of Hodgkin-Huxley (HH) neurons
and discuss the difference in synchronization properties between IF neurons and
HH neurons.Comment: Notation for Jacobi matrix is changed. Accepted for publication in
Phys. Rev.
Sensitivity and stability: A signal propagation sweet spot in a sheet of recurrent centre crossing neurons
In this paper we demonstrate that signal propagation across a laminar sheet of recurrent neurons is maximised when two conditions are met. First, neurons must be in the so-called centre crossing configuration. Second, the network’s topology and weights must be such that the network comprises strongly coupled nodes, yet lies within the weakly coupled regime. We develop tools from linear stability analysis with which to describe this regime, and use them to examine the apparent tension between the sensitivity and instability of centre crossing networks
Non-Euclidean Contractivity of Recurrent Neural Networks
Critical questions in dynamical neuroscience and machine learning are related to the study of recurrent neural networks and their stability, robustness, and computational efficiency. These properties can be simultaneously established via a contraction analysis.This paper develops a comprehensive contraction theory for recurrent neural networks. First, for non-Euclidean ℓ 1 /ℓ ∞ logarithmic norms, we establish quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes. Second, for locally Lipschitz maps (e.g., arising as activation functions), we show that their one-sided Lipschitz constant equals the essential supremum of the logarithmic norm of their Jacobian. Third and final, we apply these general results to classes of recurrent neural circuits, including Hopfield, firing rate, Persidskii, Lur’e and other models. For each model, we compute the optimal contraction rate and corresponding weighted non-Euclidean norm via a linear program or, in some special cases, via a Hurwitz condition on the Metzler majorant of the synaptic matrix. Our non-Euclidean analysis establishes also absolute, connective, and total contraction properties
Consensus analysis of multiagent networks via aggregated and pinning approaches
This is the post-print version of of the Article - Copyright @ 2011 IEEEIn this paper, the consensus problem of multiagent nonlinear directed networks (MNDNs) is discussed in the case that a MNDN does not have a spanning tree to reach the consensus of all nodes. By using the Lie algebra theory, a linear node-and-node pinning method is proposed to achieve a consensus of a MNDN for all nonlinear functions satisfying a given set of conditions. Based on some optimal algorithms, large-size networks are aggregated to small-size ones. Then, by applying the principle minor theory to the small-size networks, a sufficient condition is given to reduce the number of controlled nodes. Finally, simulation results are given to illustrate the effectiveness of the developed criteria.This work was jointly supported by CityU under a research grant (7002355) and GRF funding (CityU 101109)
Global Exponential Stability of Delayed Periodic Dynamical Systems
In this paper, we discuss delayed periodic dynamical systems, compare
capability of criteria of global exponential stability in terms of various
() norms. A general approach to investigate global
exponential stability in terms of various () norms is
given. Sufficient conditions ensuring global exponential stability are given,
too. Comparisons of various stability criteria are given. More importantly, it
is pointed out that sufficient conditions in terms of norm are enough
and easy to implement in practice
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