229 research outputs found
Attractors in fully asymmetric neural networks
The statistical properties of the length of the cycles and of the weights of
the attraction basins in fully asymmetric neural networks (i.e. with completely
uncorrelated synapses) are computed in the framework of the annealed
approximation which we previously introduced for the study of Kauffman
networks. Our results show that this model behaves essentially as a Random Map
possessing a reversal symmetry. Comparison with numerical results suggests that
the approximation could become exact in the infinite size limit.Comment: 23 pages, 6 figures, Latex, to appear on J. Phys.
Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks
Attractors in asymmetric neural networks with deterministic parallel dynamics
were shown to present a "chaotic" regime at symmetry eta < 0.5, where the
average length of the cycles increases exponentially with system size, and an
oscillatory regime at high symmetry, where the typical length of the cycles is
2. We show, both with analytic arguments and numerically, that there is a sharp
transition, at a critical symmetry \e_c=0.33, between a phase where the
typical cycles have length 2 and basins of attraction of vanishing weight and a
phase where the typical cycles are exponentially long with system size, and the
weights of their attraction basins are distributed as in a Random Map with
reversal symmetry. The time-scale after which cycles are reached grows
exponentially with system size , and the exponent vanishes in the symmetric
limit, where . The transition can be related to the dynamics
of the infinite system (where cycles are never reached), using the closing
probabilities as a tool.
We also study the relaxation of the function ,
where is the local field experienced by the neuron . In the symmetric
system, it plays the role of a Ljapunov function which drives the system
towards its minima through steepest descent. This interpretation survives, even
if only on the average, also for small asymmetry. This acts like an effective
temperature: the larger is the asymmetry, the faster is the relaxation of ,
and the higher is the asymptotic value reached. reachs very deep minima in
the fixed points of the dynamics, which are reached with vanishing probability,
and attains a larger value on the typical attractors, which are cycles of
length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge
A "Cellular Neuronal" Approach to Optimization Problems
The Hopfield-Tank (1985) recurrent neural network architecture for the
Traveling Salesman Problem is generalized to a fully interconnected "cellular"
neural network of regular oscillators. Tours are defined by synchronization
patterns, allowing the simultaneous representation of all cyclic permutations
of a given tour. The network converges to local optima some of which correspond
to shortest-distance tours, as can be shown analytically in a stationary phase
approximation. Simulated annealing is required for global optimization, but the
stochastic element might be replaced by chaotic intermittency in a further
generalization of the architecture to a network of chaotic oscillators.Comment: -2nd revised version submitted to Chaos (original version submitted
6/07
Boolean Dynamics with Random Couplings
This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical
Sciences Serie
Global exponential stability of nonautonomous neural network models with unbounded delays
For a nonautonomous class of n-dimensional di erential system with in nite delays, we give
su cient conditions for its global exponential stability, without showing the existence of an
equilibrium point, or a periodic solution, or an almost periodic solution. We apply our main
result to several concrete neural network models, studied in the literature, and a comparison of
results is given. Contrary to usual in the literature about neural networks, the assumption of
bounded coe cients is not need to obtain the global exponential stability. Finally, we present
numerical examples to illustrate the e ectiveness of our results.The paper was supported by the Research Center of Mathematics of University of Minho with the Portuguese Funds from the FCT - âFundação para a CiĂȘncia e a Tecnologiaâ, through the Project UID/MAT/00013/2013. The author thanks the referees for valuable comments.info:eu-repo/semantics/publishedVersio
Global point dissipativity of neural networks with mixed time-varying delays
By employing the Lyapunov method and some inequality techniques, the global point dissipativity is studied for neural networks with both discrete time-varying delays and distributed time-varying delays. Simple sufficient conditions are given for checking the global point dissipativity of neural networks with mixed time-varying delays. The proposed linear matrix inequality approach is computationally efficient as it can be solved numerically using standard commercial software. Illustrated examples are given to show the usefulness of the results in comparison with some existing results. © 2006 American Institute of Physics.published_or_final_versio
Existence of global attractor for a nonautonomous state-dependent delay differential equation of neuronal type
The analysis of the long-term behavior of the mathematical model of a neural
network constitutes a suitable framework to develop new tools for the dynamical
description of nonautonomous state-dependent delay equations (SDDEs).
The concept of global
attractor is given, and some results which establish properties ensuring
its existence and providing a description of its shape, are proved.
Conditions for the exponential stability of the global attractor
are also studied. Some properties
of comparison of solutions constitute a key in
the proof of the main results, introducing methods of monotonicity
in the dynamical analysis of nonautonomous SDDEs.
Numerical simulations of some illustrative models show
the applicability of the theory.Ministerio de EconomĂa y Competitividad / FEDER, MTM2015-66330-PMinisterio de Ciencia, InnovaciĂłn y Universidades, RTI2018-096523-B-I00European Commission, H2020-MSCA-ITN-201
Phase response function for oscillators with strong forcing or coupling
Phase response curve (PRC) is an extremely useful tool for studying the
response of oscillatory systems, e.g. neurons, to sparse or weak stimulation.
Here we develop a framework for studying the response to a series of pulses
which are frequent or/and strong so that the standard PRC fails. We show that
in this case, the phase shift caused by each pulse depends on the history of
several previous pulses. We call the corresponding function which measures this
shift the phase response function (PRF). As a result of the introduction of the
PRF, a variety of oscillatory systems with pulse interaction, such as neural
systems, can be reduced to phase systems. The main assumption of the classical
PRC model, i.e. that the effect of the stimulus vanishes before the next one
arrives, is no longer a restriction in our approach. However, as a result of
the phase reduction, the system acquires memory, which is not just a technical
nuisance but an intrinsic property relevant to strong stimulation. We
illustrate the PRF approach by its application to various systems, such as
Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF
allows predicting the dynamics of forced and coupled oscillators even when the
PRC fails
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