Continuous and discrete models of neural systems in infinite-dimensional abstract spaces

Abstract

Infinite countable or uncountable systems of nonlinear ordinary and partial differential equations, which are discrete and continuous models, respectively, of real-world phenomena and processes were analysed in physics, biology, neuroscience, in studying of neural systems or in pure mathematics. Discrete models and corresponding infinite countable systems have been investigated by numerous authors in Banach sequence spaces ℓ ∞ . But continuous model and corresponding infinite uncountable systems have been investigated by a few authors only. It is interesting that we can attack the above problems by using the same topological fixed-point theory tools as in the countable case. It is the study of infinite uncountable systems of parabolic reaction-diffusion-convection equations that is the subject matter of this paper. Motivation for considering such systems and examples thereof are presented. The paper is an introduction to the study of infinite uncountable systems of parabolic equations.