A Nonequilibrium-Potential Approach to Competition in Neural Populations - Deza, Roberto R.; Deza, Ignacio; Martínez, Nataniel; Mejías, Jorge F.; Wio, Horacio S. (Frontiers in Physics, 2019)
Energy landscapes are a useful aid for the understanding of dynamical systems, and a valuable tool for their analysis. For a broad class of rate models of neural networks, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained "nonequilibrium potential" (NEP) 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 burst", also relevant in neuroscience. Instead, limit cycles are not found. Their nonexistence can be proven by resort to the Bendixson-Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. Hopefully, this NEP will help understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.