Temporal Association by Hebbian Connections: the Method of Characteristic System Temporal Association by Hebbian Connections: the Method of Characteristic System

While it is generally accepted that the stability of a static memory pattern corresponds to a certain point attractor in the dynamics of the underlying neural network, when temporal order is introduced and a sequence of memory patterns need to be retrieved successively in continuous time, it becomes less clear what general method should be used to decide whether the transient dynamics is robust or not. In this paper, it is shown that such a general method can be developed if all the connections in the neural network are Hebbian. This method is readily applied to various structures of coupled networks as well as to the standard temporal association model with asymmetric time-delayed connections. The basic idea is to introduce new variables made of memory-overlap projections with alternating signs, and then circumvent the nonlinearity of the sigmoid function by exploiting the dynamical symmetry inherent in the Hebb rule. The result is a self-contained, low-dimensional deterministic system. A powerful feature of this approach is that it can translate questions about the stability of the sequential memory transitions in the original neural network into questions about the stability of the periodic oscillation in the corresponding characteristic system. This correspondence enables direct, quantitative prediction of the behaviors of the original system, as being connrmed by computer simulations on the conjugate networks, the \tri-synaptic loop" networks, and the time-delayed network. In particular, the conjugate networks (consisting of two Hoppeld subnets coupled by asymmetric Hebbian connections) ooer a simple but suucient structure for the storage and retrieval of sequential memory patterns without any additional temporal mechanisms besides the intrinsic dynamics. The same structure can also be used for the recognition of a temporal sequence of sparse memory patterns. Other topics include the storage capacity of the conjugate networks, the exact solution of the limit cycle in the characteristic system, the sequential retrieval at variable speeds, and the problem of equivalence between coupling and time delays.