Absolute stability of global pattern formation and parallel memory storage by competitive neural networks

摘要

The process whereby input patterns are transformed and stored by competitive cellular networks is considered. This process arises in such diverse subjects as the short-term storage of visual or language patterns by neural networks, pattern formation due to the firing of morphogenetic gradients in developmental biology, control of choice behavior during macromolecular evolution, and the design of stable context-sensitive parallel processors. In addition to systems capable of approaching one of perhaps infinitely many equilibrium points in response to arbitrary input patterns and initial data, one finds in these subjects a wide variety of other behaviors, notably traveling waves, standing waves, resonance, and chaos. The question of what general dynamical constraints cause global approach to equilibria rather than large amplitude waves is therefore of considerable interest. In another terminology, this is the question of whether global pattern formation occurs. A related question is whether the global pattern formation property persists when system parameters slowly change in an unpredictable fashion due to self-organization (development, learning). This is the question of absolute stability of global pattern formation. It is shown that many model systems which exhibit the absolute stability property can be written in the form ${dx_iover dt} = a_i (x_i)left[b_i(x_i)-sum^n_{k = 1} c_{ik} d_k (x_k)right]$ (1) i = 1, 2, …, n, where the matrix C = ‖cik‖ is symmetric and the system as a whole is competitive. Under these circumstances, this system defines a global Liapunov function. The absolute stability of systems with infinite but totally disconnected sets of equilibrium points can then be studied using the LaSalle invariance principle, the theory of several complex variables, and Sard''s theorem. The symmetry of matrix C is important since competitive systems of the form (1) exist wherein C is arbitraril- close to a symmetric matrix but almost all trajectories persistently oscillate, as in the voting paradox. Slowing down the competitive feedback without violating symmetry, as in the systems ${dx_iover dt} = a_i (x_i)left[b_i(x_i)-sum^n_{k = 1} c_{ik} d_k (y_k)right]$ ${dy_iover dt} = e_i (x_i)[f_i(x_i)-y_i],$ also enables sustained oscillations to occur. Our results thus show that the use of fast symmetric competitive feedback is a robust design constraint for guaranteeing absolute stability of global pattern formation.