Cooperation and Competition in Central Nervous System Development: A Unifying Approach

This paper presents a family of differential equations which describe in a concise way various phenomena of ontogenetic organisation in the Central Nervous System. We consider four different types of “pattern formation”: (i) development of stimulus selectivity in individual neurons, (ii) spatial organisation of selectivity in the nervous tissue, (iii) development of topographic mappings, (iv) development of spatially periodic mappings. In the last two decades, many mathematical models have been proposed with the common goal of explaining how such global, or “macroscopic”, regularities can arise from local, or “microscopic”, interactions. This work is essentially an effort towards unification: in providing a simple evolution scheme, the asymptotic behaviour of which is sufficiently rich to account for the main patterns of organisation, our aim is to make explicit what the minimal ingredients are. These ingredients, or principles, present in virtually all the models proposed so far, are cooperation and competition. It is our belief that analogous principles are responsible for many self-organisation phenomena in various domains. Thus, the model that is presented here, in providing some suggestions as to how the elementary bricks should be put together, could help to design a framework for a simple theory of self-organisation.