Global continuation of nonlinear waves in a ring of neurons

Abstract

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.