Information spectra and optimal background states for dynamical networks

We consider the notion of stimulus representation over dynamic networks, wherein the network states encode information about the identify of an afferent input (i.e. stimulus). Our goal is to understand how the structure and temporal dynamics of networks support information processing. In particular, we conduct a theoretical study to reveal how the background or ‘default’ state of a network with linear dynamics allows it to best promote discrimination over a continuum of stimuli. Our principal contribution is the derivation of a matrix whose spectrum (eigenvalues) quantify the extent to which the state of a network encodes its inputs. This measure, based on the notion of a Fisher linear discriminant, is relativistic in the sense that it provides an information value quantifying the ‘knowablility’ of an input based on its projection onto the background state. We subsequently optimize the background state and highlight its relationship to underlying state noise covariance. This result demonstrates how the best idle state of a network may be informed by its structure and dynamics. Further, we relate the proposed information spectrum to the controllabilty gramian matrix, establishing a link between fundamental control-theoretic network analysis and information processing.