Scale-Limited Activating Sets and Multiperiodicity for Threshold-Linear Networks on Time Scales

The existing results for multiperiodicity of threshold-linear networks (TLNs) are scale-free on time evolution and hence exhibit some restrictions. Due to the nature of the scale-limited activating set, it is interesting to study the dynamical properties of neurons on time scales. In this paper we analyze and obtain results concerning nondivergence, attractivity, and multiperiodic dynamics of TLNs on time scales. Using the notion of exponential functions on time scales, we obtain results for scale-limited type criteria for boundedness and global attractivity of TLNs. Moreover, by constructing simple algebraic inequalities over scale-limited activating sets, we achieve results regarding multiperiodicity of TLNs. This will show that each scale-limited activating set depends on scale-synchronous self-excitation, and the existence of inactive neurons will slow down convergence of TLNs. At the end of the paper, we perform computer simulations to illustrate the obtained new theories.