Activity homogeneity: a measure for comparing time discretization and state quantization in ODE simulation

In this work, we introduce the concept of activity homogeneity in the solutions of ordinary differential equations (ODEs) and characterize it through a metric called Homogeneity Factor . This indicator quantifies the degree of similarity in the temporal evolution of the system state variables. We show that this measure can be related to the convenience of using classic numerical integration schemes or quantization-based methods such as Quantized State System (QSS) algorithms. The developed notions, which are also extended to systems exhibiting discontinuities, provide a theoretical argument that corroborates observations from previous works, indicating that QSS methods offer advantages when activity is heterogeneous, systems are sparse, and/or frequent discontinuities occur. The concepts are applied to two case studies: an advection-diffusion-reaction model and a spiking neural network. Theoretical predictions are compared with empirical results obtained through simulations using different numerical integration methods, confirming that the proposed metric consistently identifies the integration strategy that is more computationally efficient in practice.