Observability limits for networked oscillators

Inspired by the neuro-scientific problem of predicting brain dynamics from electroencephalography (EEG) measurements of the brain’s electrical activity, this paper presents limitations on the observability of networked oscillators sensed with quantised measurements. The problem of predicting highly complex brain dynamics sensed with relatively limited amounts of measurement is abstracted to a study of observability in a network of oscillators. It is argued that a low-dimensional quantised measurement is in fact, by itself, an exceptionally poor observer for a large-scale oscillator network, even for the case of a completely connected graph. The main rational is based on (i) an information-theoretic argument based on ideas of entropy in measure preserving maps, (ii) a linear deterministic observability argument, and (iii) a linear stochastic approach using Kalman filtering. For prediction of brain network activity, the findings indicate that the classic EEG signal is just not precise enough to be able to provide reliable prediction and tracking in a clinical setting in view of the complexity of underlying neural dynamics.