Self organization and evolution in mathematical models

Abstract

Through several examples we illustrate to non mathematicians how biological situations can be translated in mathematical terms. Even simple mathematical models can be very efficient to clarify a complex biological situation and the same basic mathematical procedures can be found in different biological frameworks. In particular stability analysis under small fluctuations of non linear models possessing different time scales is fundamental, especially bifurcation properties. An important issue is also to choose between deterministic and stochastic models. We illustrate these points through classical examples coming from the formation of patterns (morphogenesis), vision (visual hallucinations), onset of periodic oscillations in the neuronal activity due to random noise and a mathematically related example of stochastic resonance. Our presentation in the present paper is primarily directed towards non mathematicians. Through several examples we would like to show how biological situations can be translated in mathematical terms. Mathematical models have at least two advantages: they help exhibiting universal properties commun to a priori very different situations, and also they enable identifying the relevant parameters among the available collection of parameters describing a system which may be quite large. We want to stress that even simple mathematical models can be very efficient to clarify a complex biological situation. In particular the stability analysis under small fluctuations of non linear models possessing different time scales is fundamental, especially their bifurcation properties. An important issue is also to choose between deterministic and stochastic models. We will illustrate these points through classical examples coming from the formation of patterns (morphogenesis), vision (visual hallucinations), onset of periodic oscillations in the neuronal activity due to random noise and a mathematically related example of stochastic resonance. Our main reference for deterministic models is Murray's book (9) on mathematical modelling in biology. 2. MORPHOGENESIS The mathematical modeling of morphogenesis was iniated by Turing who created the notion of diffusion driven instability. The formation of patterns is a nonlinear phenomenon. However a good indication of the possible patterns is provided by the analysis of the linear stability. In particular the linear analysis gives the range