Heterarchy in biological systems: A logic-based dynamical model of abstract biological network derived from time-state-scale re-entrant form

Heterarchical structure is important for understanding robustness and evolvability in a wide variety of levels of biological systems. Although many studies emphasize the heterarchical nature of biological systems, only a few computational representations of heterarchy have been created thus far. We propose here the time-state-scale re-entrant form to address the self-referential property derived from setting heterarchical structure. In this paper, we apply the time-state-scale re-entrant form to abstract self-referential modeling for a functional manifestation of biological network presented by [Tsuda, I., Tadaki, K., 1997. A logic-based dynamical theory for a genesis of biological threshold. BioSystems 42, 45-64]. The numerical results of this system show different intermittent phase transitions and power-law distribution of time spent in activating functional manifestation. The Hierarchically separated time-scales obtained from spectrum analysis imply that the reactions at different levels simultaneously appear in a dynamical system. The results verify the mutual inter-relationship between heterarchical structure in biological systems and the self-referential property of computational heterarchical systems.