Dynamical Behavior of Neural Networks with Anti-Symmetrical Cyclic Connections

Abstract

We show that a unit-group, which represents a group of contiguous units with the same sign of output, is a dominant component for the dynamical behavior of a neural network with anti-symmetrical cyclic connections for the nearest neighbor connections and global connections. In transient state, it is shown that the unit-group has the dynamics such that the amount n of units which belong to the unit-group increases with time, and that the increasing rate of n decreases with increasing n. The dynamics cause the large difference of the number of limit-cycles between discrete and continuous time models. Additionally, the period of the limit-cycle depends on the size of the unit-groups. This dependency is obtained from computer simulations and two approximation methods. These approximations provide the lower and the upper bounds of the periods which depend on the gain of an activation function. Using these approximations, we also obtain detailed relations between a period and the other network parameters analytically.