The stable-chaotic transition on cellular automata used to model the immune repertoire

In this paper we study a simplified version of the cellular automata approximation introduced by De Boer, Segel and Perelson to model the immune repertoire. The automaton rule defines an activation window based on the idea of the proliferation function (biphasic dose-response function), which is used to describe the receptor crosslinking involved in the B cell activation. This proliferation function is very sensitive to the activation threshold and activation interval definitions. Here we investigate the influence of these parameters on the automation rule proposed by Stauffer and Weisbuch. Using a fixed window they obtained the stable-“chaotic” transition only for d ≥ 4. We find, contrary to their results, that this transition is always present for d ≥ 2 until a certain critical value of the activation threshold is attained, above which this transition disappears and the system will always evolve towards a stable configuration. The shorter the activation interval the faster the system undergoes to the “chaotic” behaviour. Increasing the activation interval there is a certain critical size from which the system will always exhibit the same behaviour no matter the activation interval size. We also investigate the influence of the initial distribution on the results. Since we defined the relevant parameters of the model, we obtained the phase diagrams exhibiting the regions of stable and “chaotic” behavior. Such diagrams are not easily found in the literature.