A learning scheme for dynamic neural networks: equilibrium manifold and connective stability

Stability of the Hopfield neural network N:x˙_i=-x _i+Σ_j=1ⁿe_ijw_ij g_j(x_j)+I_i, i=1, 2, …, n y _i=g_i(x_i) with a learning rule L:e˙ _ij=μh_ij(e, y, y*), i, j=1, 2, …, n is analyzed using the concept of equilibrium manifold. We show that the concept provides an ideal setting for the formulation of a learning rule L that adaptively teaches network N to acquire y* as one of its asymptotically stable equilibria. Connective stability of a moving equilibrium in the composite two-time-scale system N&L, is established for bounded interconnection weights e_ijw_ij and a sufficiently small learning rate μ. The conditions for connective stability, which are derived using the M-matrices and the concept of vector Lyapunov functions, are suitable for studying the design trade-offs between the bounds on the nominal weights w_ij , the shape of the sigmoid-function g_i(x_i), the learning rate μ, and the size of the stability region containing y*