Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model

Neural networks contain, very often, asymmetric bonds. The interactions $J_{\mathrm{ij}}$ and $J_{\mathrm{ji}}$ between the ith and the jth neurons are not identical. In this paper we study the Langevin dynamics of fully connected spin systems whose interaction matrix contains a random antisymmetric part. The symmetric part consists of independent random bonds whose mean is either zero or ferromagnetic. We also consider a more general class of systems such as the asymmetric Hopfield model and other neural-network models. Within the framework of mean-field theory, the spin fluctuations are viewed as local, thermally averaged, time-dependent magnetic moments. These moments are induced by excess (i.e., nonthermal) internal noise which, in the presence of asymmetry, is time dependent and does not vanish even in the high-temperature phase. The mean-field equations are solved using a simplified, spherical model, in which the spins are linear variables except for a global constraint on the total level of their fluctuations. Random asymmetry of arbitrary strength destroys spin-glass freezing. Ferromagnetic phases, as well as ‘‘retrieval’’ states in neural networks, are affected only slightly by weak random asymmetry, in agreement with the conclusions of Hertz et al.