C. J. Pérez Vicente

Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions

Abstract The mean field Kuramoto model describing the synchronization of a population of phase oscillators with a bimodal frequency distribution is analyzed (by the method of multiple scales) near regions in its phase diagram corresponding to synchronization to phases with a time-periodic order parameter. The richest behavior is found near the tricritical point where the incoherent, stationarily synchronized, “traveling wave” and “standing wave” phases coexist. The behavior near the tricritical point can be extrapolated to the rest of the phase diagram. Direct Brownian simulation of the model confirms our findings.

Optimised network for sparsely coded patterns

The performance of attractor neural networks storing sparsely coded patterns has been shown to be greatly improved on shifting from the -1, +1 representation of neural states to the 0, 1 representation. Here the authors show that when this shift is considered as a special case of the transformation of the dynamical variables which depends on a continuous parameter, the value of the parameter can be chosen to improve the performance of the network even further for every value of the bias in the patterns.

The target switch algorithm: a constructive learning procedure for feed-forward neural networks

We propose an efficient procedure for constructing and training a feedforward neural network. The network can perform binary classification for binary or analogue input data. We show that the procedure can also be used to construct feedforward neural networks with binary-valued weights. Neural networks with binary-valued weights are potentially straightforward to implement using microelectronic or optical devices and they can also exhibit good generalization.

Exactly Solvable Phase Oscillator Models with Synchronization Dynamics

Populations of phase oscillators interacting globally through a general coupling function fsxd have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, fsxd › sgnsxd, have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved. [S0031-9007(98)07451-1] PACS numbers: 05.45. + b, 87.10. + e

Glassy Synchronization in a Population of Coupled Oscillators

A phase model for a population of oscillators with random excitatory and inhibitory mean-field coupling and subject to external white noise random forces is proposed and studied. In the thermodynamic limit different stable phases for the oscillator population may be found: (i) an incoherent state where all possible values of an oscillator phase are equally probable, (ii) a synchronized state where the population has a nonzero collective phase; (iii) a glassy phase where the global synchronization is zero but the oscillators are in phase with the random disorder; and (iv) a mixed phase where the oscillators are partially synchronized and partially in phase with the disorder. These predictions are based upon bifurcation analysis of the reduced equation valid at the thermodynamic limit and confirmed by Brownian simulation.

Finitely connected vector spin systems with random matrix interactions

We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Finitely connected spin models, although still of a mean-field nature, can be regarded as a convenient level of description in between fully connected and finite-dimensional ones. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d = 2 (finitely connected XY spins with random chiral interactions) and for d = 3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

On the short-time dynamics of networks of Hebbian coupled oscillators

We have analysed the collective behaviour of an assembly of coupled oscillators interacting through Hebbian synaptic weights designed to store information about the phase of each element. Our study focuses on the dynamic properties of the population showing the existence of a relevant time-scale, useful for the processing of information.

Phase Locking in a Network of Neural Oscillators

We present an analytical approach that allows to compute the long-time behaviour of networks with oscillatory behaviour. We show that phase locking is a mechanism to store information in the system. This technique is an interesting alternative to conventional methods of analysis of associative memories.

EXACT SOLUTIONS AND DYNAMICS OF GLOBALLY COUPLED OSCILLATORS

We analyze mean-field models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the Kuramoto–Sakaguchi model. Their probability densities satisfy local partial differential equations similar to the porous medium, Burgers and extended Burgers systems depending on the degree of singularity of the coupling. We show that porous medium oscillators (the most singularly coupled) do not synchronize and that (transient) synchronization is possible only at zero temperature for Burgers oscillators. The extended Burgers oscillators have a nonlocal coupling first introduced by Daido and they may synchronize at any temperature. Exact expressions for their synchronized phases and for Daidos order function are given in terms of elliptic functions.

Phase diagram of a planar XY model with random field

We have studied a infinite-range planar XY model in the presence of a random field. We have analyzed the dynamics of the system through a Fokker-Planck formalism, focussing our attention in the long time behaviour. This technique allows to compute exactly the phase diagram of the model.