Doubly periodic traveling waves in cellular neural networks with polynomial reactions

In a study (Szekely, 1965) [1] of the locomotion of salamanders, it is observed that a ‘doubly periodic traveling wave solution’ of a logical neural network can be used to explain a dynamic pattern of movements. We show here that nonlinear and nonlogical artificial neural network can also be built by means of reaction diffusion principles and existence or nonexistence of doubly periodic traveling waves can be guaranteed by adjusting parameters built into these networks. Our derivations for existence are based on implicit function theorems and the invariance properties of our model; while nonexistence is based on boundedness properties of the polynomial reaction term. We also give illustrative examples as well as comments on the differences between present results with those obtained for linear models studied earlier in Cheng and Lin (2009) [2] and [3].