Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions

It is shown that a cascade of the neural net n₀ into the net n₁ has oscillatory (convergent) dynamics if the nets n₀ and n₁ have oscillatory (convergent) dynamics and the net n₁ has a special structure. The special structure required of n₁ is that it consists of a finite number of units which can be arranged along a line segment with only nearest neighbor connections between units and where each adjacent pair of units interacts in either a mutally inhibitory manner or in a mutually excitatory manner. These nets, described in Hirsch (1989), have convergent dynamics if given constant input, by a result of Smillie (1984), and have oscillatory dynamics if given oscillatory input, by a result of the author (1990).