A Semi-Group of Operators as a Model of the Lateral Inhibition Process in the Sensory System

Abstract

When we look at an object in front of us, its image is formed on the retina. But even if the focus is adjusted by the lens of the eye, the image is diffuse because the light coming through is scattered in the cornea, lens and vitreous body. Nevertheless, we recognize the object clearly if the focus is adjusted. This fact means that the diffusion of light is compensated for by our nervous system. In this paper we consider a mathematical model of the process of compensation in the nervous system. Let us define the operator N which maps the object to the diffused image. Then we can denote the object and the image by f and Nf respectively. The diffusion is expressed by the operator N, and the compensation or the sharpening of the image in the nervous system is expressed as a mapping of Nf to f. On the other hand it is well known by physiologists that, when a small spot of light is thrown on the retina, that part of the retina is stimulated and excited but its surroundings are inhibited (Hartline and Ratliff (1957), Ratliff (1965)). This is called 'lateral inhibition'. Figure 1 shows its pattern. It is the result of interaction among neurons in the retina. The lateral inhibition is observed throughout the visual path in the nervous system and widely seen in the sensory system (B6k6sy (1959)). It is considered that the lateral inhibition is closely related to the process of sharpening of the diffused images formed in the distal ends of the sensory systems (i.e. the receptors). The purpose of this paper is to derive the pattern of the lateral inhibition by modelling mathematically the process of sharpening. If this can be done successfully without specific restriction, we can conclude that the necessity of lateral inhibition is proved. If the necessity of a certain biological function is proved mathematically, we can say that mathematical biology has been used successfully. I think such modelling, which proves mathematically the necessity of some biological function, should become one of the main methods of mathematical biology. In this paper the problem is treated under certain