Sensitivity to initial conditions in an extended activator-inhibitor model for the formation of patterns

원문 초록

Despite simplicity, the synchronous cellular automaton [D.A. Young, Math. Biosci. 72, 51 (1984)] enables reconstructing basic features of patterns of skin. Our extended model allows studying the formatting of patterns and their temporal evolution also on the favourable and hostile environments. As a result, the impact of different types of an environment is accounted for the dynamics of patterns formation. The process is based on two diffusible morphogens, the short-range activator and the long-range inhibitor, produced by differentiated cells (DCs) represented as black pixels. For a neutral environment, the extended model reduces to the original one. However, even the reduced model is statistically sensitive to a type of the initial distribution of DCs. To compare the impact of the uniform random distribution of DCs (R-system) and the non-uniform distribution in the form of random Gaussian-clusters (G-system), we chose inhibitor as the control parameter. To our surprise, in the neutral environment, for the chosen inhibitor-value that ensures stable final patterns, the average size of final G-populations is lower than in the R-case. In turn, when we consider the favourable environment, the relatively bigger shift toward higher final concentrations of DCs appears in the G. Thus, in the suitably favourable environment, this order can be reversed. Furthermore, the different critical values of the control parameter for the R and the G suggest some dissimilarities in temporal evolution of both systems. In particular, within the proper ranges of the critical values, their oscillatory behaviours are different. The respective temporal evolutions are illustrated by a few examples.