Observability and Controllability of Nonlinear Networks: The Role of Symmetry

Observability and controllability are vitally important in networks, but almost all of the present theory was developed for linear networks without symmetries. To advance beyond the study of such generic networks, we quantify observability and controllability in small (3 node) nonlinear neuronal networks as a function of 1) the connection topology and symmetry, and 2) the measured and controlled nodes. We find that typical observability metrics for 3 neuron motifs range over several orders of magnitude, depending upon topology, and for motifs containing symmetry we find that the network observability and controllability decreases. Networks containing dynamical synchrony facilitated by sufficiently strong coupling also cause decreases in observability and controllability, and that under certain connection motifs, an optimal network coupling strength is found for maximum observability and controllability. Additionally, the number of connections into and out of a network node contributes to the observability (controllability) of the full system as measured (controlled) from that node. Our findings demonstrate that such nonlinear networks are partially observable (controllable), and suggest their potential efficacy in reconstructing network dynamics from limited measurement data. Brain networks are characterized by highly overrepresented small motifs, often with retained topological symmetries. Observability and controllability are essential to the design of predictive network observer models and controllers of brain and their use in experimental settings is a subject for future experimental work.