Homeostasis in Networks with Multiple Inputs

Homeostasis, also known as adaptation, refers to the ability of a system to counteract persistent external disturbances and tightly control the output of a key observable. Existing studies on homeostasis in network dynamics have mainly focused on perfect adaptation in deterministic single-input single-output networks where the disturbances are scalar and affect the network dynamics via a pre-specified input node. In this paper we provide a full classification of all possible network topologies capable of generating infinitesimal homeostasis in arbitrarily large and complex multiple-input parameter networks. Working in the framework of infinitesimal homeostasis allows us to make no assumption about how the components are interconnected and the functional form of the associated differential equations, apart from being compatible with the network architecture. Remarkably, we show that there are just three distinct mechanisms that generate infinitesimal homeostasis. Each of these three mechanisms generates a rich class of well-defined network topologies - called homeostasis subnetworks. Most importantly, we show that these classes of homeostasis subnetworks provides a topological basis for the classification of homeostasis types: the full set of all possible multiple-input parameter networks can be uniquely decomposed into these special homeostasis subnetworks. We build on previous work that treated the cases of single-input node and multiple-input node, both with a single scalar input parameter. Furthermore, we identify a new phenomenon that occurs in the multiparameter setting, that we call homeostasis mode interaction, in analogy with the well-known characteristic of multiparameter bifurcation theory.