Graph-theoretic description of the interplay between non-linearity and connectivity in biological systems

The purpose of this article is to stress the implications that the consider ation of nonlinearity has upon the extension and strength of connectivity, if this is understood as a characterization of the degree of interrelation between parts of the system. This objective is reached within the QP formal ism for non-linear ODEs. The formalism is developed in a graph-theoretic se tting, with the help of which the connectionist aspect of non-linearity bec omes apparent. Topology-preserving transformations involve an exchange betw een the degree of non-linearity and the strengths of interactions, thus ass embling systems of apparently different nature into classes of equivalence. We argue that, if we have in mind a classification of systems according to behavior, these classes of equivalence should be given their proper singul arity. We characterize globally the connectivity of a class with an index, although we point out during the discussion that the mathematical conceptio n of the complex idea of connectivity is still incomplete. (C) 1999 Elsevie r Science Inc. All rights reserved.