Parallel dynamics of the fully connected Blume–Emery–Griffiths neural network

Abstract

The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied at zero temperature using a probabilistic approach. A recursive scheme is found determining the complete time evolution of the order parameters, taking into account all feedback correlations. It is based upon the evolution of the distribution of the local field, the structure of which is determined in detail. As an illustrative example explicit analytic formula are given for the first few time steps of the dynamics. Furthermore, equilibrium fixed-point equations are derived and compared with the thermodynamic approach. The analytic results find excellent confirmation in extensive numerical simulations.Recently, an optimal Hamiltonian has been derived in the statistical mechanics approach to Qstate neural networks starting from the concept of mutual information [1], [2]. Optimal means that the best retrieval properties are guaranteed including, e.g., the largest retrieval overlap, loading capacity, basin of attraction, convergence time. For Q = 3 this Hamiltonian resembles the classical Blume-Emery-Griffiths (BEG) Hamiltonian in the sense that it contains both a bilinear and biquadratic term in the spins [3]. For a fully connected architecture it has been shown, using a thermodynamic replica approach that the maximal loading capacity for the BEG network is indeed bigger than the one for other three-state networks existing in the literature (Ising, Potts...) [2].

The Blume–Emery–Griffiths neural network: dynamics for arbitrary temperature

“…In [1] we discussed the SNA for the BEG model at temperature T = 0. In order to generalize this approach to finite temperatures we introduce auxiliary thermal fields [5] in order to express the stochastic dynamics within the gain function formulation of the deterministic dynamics.…”

“…In order to obtain the recursion relations for the local fields we can then follow the derivation of the zero temperature case [1] (see Section 3) step by step. We do not repeat this calculation here but write down the final results…”

“…Stationary equations for non-zero temperature can, in principle, be obtained from the zero temperature expressions obtained in [1] by introducing auxiliary thermal fields and averaging over them. However, extreme care has to be taken concerning these thermal fluctuations.…”

“…Recently, we have studied the Blume-Emery-Griffiths (BEG) neural network dynamics at zero temperature using the probabilistic signal-to-noise analysis (SNA) [1]. The interest of such a study arises from the fact that the BEG-model optimises the mutual information content for three-state networks [2], [3].…”

Dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model in an oscillating field: the effective-field theory based on the Glauber-type stochastic dynamics

“…Erdem et al [20] used Onsager's theory of irreversible thermodynamics to investigate the relaxation times near the critical and multicritical points. Boll e et al [21] studied the parallel dynamics of the fully connected BEG neural network model at zero temperature using a probabilistic approach and determined the complete time evolution of the order parameters. Keskin and co-workers [22À24] studied the dynamic phase transition (DPT) and presented the dynamic phase diagrams of the BEG model within the MFT based on Glauber-type stochastic dynamics (DMFT) in detail.…”