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

Neural networks contain, very often, asymmetric bonds. The interactions Jij and Jji 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. The dynamical behavior of a system with weak asymmetry is studied in some detail. In the spin-glass case at low temperatures, when the strength of the asymmetry decreases, the internal excess noise does not vanish but slows down with a characteristic correlation time τ∝k^-6. The parameter k denotes the relative strength of the antisymmetric components of the bonds. The system behaves as a frozen symmetric spin glass on time scales t≪τ and as a paramagnet on scales t≫τ. The thermal fluctuations decay with a characteristic time τT∝k^-4. The spherical model exhibits a completely frozen spin-glass state at zero temperature. As T→0, fluctuations exhibit a critical slowing down with time τ∝T^-1 for all values of k>0. This T=0 spin-glass transition is probably an artifact of the spherical model and is not expected to exist in nonlinear systems. The relevance of the results to the performance of neural networks is discussed.