A nonequilibrium-potential approach to competition in neural populations

Abstract

Energy landscapes are a highly useful aid for the understanding of dynamical systems, and a particularly valuable tool for their analysis. For a broad class of rate neural- network models of relevance in neuroscience, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained “nonequilibrium potential” (NEP)—the first one obtained for a model of neural circuits—predicts with high accuracy the outcomes of the dynamics in the globally stable cases studied here. Common features of the models in this class are bistability—with implications for working memory and slow neural oscillations—and population bursts, associated with signal detection in neuroscience. Instead, limit cycles are not found for the conditions in which the NEP is defined. Their nonexistence can be proven by resorting to the Bendixson–Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. This NEP constitutes a powerful tool to understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.