Stationary processes and equilibrium states in non-symmetric neural networks

Abstract

Stationary processes and equilibrium states are discussed in finite separable recurrent neural networks with sequential dynamics and away from saturation. We describe thermal fluctuations of the dynamical order parameters originated as finite size effects of order ${\mathcal{O}}\left( N^{-1/2}\right)$ by means of their corresponding Fokker-Planck equation, and find their time dependent probability distribution. We introduce the concept of extended entropy of fluctuations in order to find a general condition to characterize stationary states in Neural Networks with non symmetric interactions. Divergence and rotational of the probability current in the space of fluctuations are also used to differentiate between stationary and equilibrium states. Besides, algebraic conditions are found to know when stationary states can exist. The results are illustrated by analyzing a neural network with a macroscopic dynamical fixed point but not satisfying detailed balance at microscopic level.