This chapter presents detailed descriptive information about functions and differential equations central to the book. It serves as a foundational guide, introducing critical theorems and lemmas, which are instrumental for exploring neural networks in the subsequent chapters. It begins with discussing the basic properties and types of functions encountered in this work, emphasizing their roles in modeling dynamic systems. Key concepts such as Poisson stability and alpha unpredictability are introduced, laying the groundwork for more advanced discussions. Compartmental functions are considered, which integrate periodicity, quasi-periodicity, and almost periodicity with alpha unpredictability or Poisson stability. A significant portion of the chapter is dedicated to discussing the existence of alpha unpredictable and Poisson stable solutions of systems, encompassing ordinary differential equations, impulsive systems, and differential equations with piecewise constant arguments subjected to irregular perturbations. Theoretical frameworks and methods for analyzing these solutions are presented, emphasizing the role of B-topology in demonstrating the presence of alpha unpredictable solutions within the space of discontinuous functions. All theoretical concepts are validated throughout the chapter by practical examples with simulations. These illustrations clarify the theoretical results and demonstrate their applicability, providing a foundation for the advanced topics on neural networks discussed in the remaining chapters.