Sensitivity of linear systems to input orientation and novelty

This paper introduces a framework for quantitative characterization of the sensitivity of time-varying linear systems (or networks) in terms of input orientation. The motivation for such an approach comes from the study of biophysical sensory networks in the brain, wherein responsiveness to both energy and salience (in terms of input orientation and novelty) is presumably critical for mediating behavior and function. Here, we use an inner product to define the angular separation of the current input with respect to past inputs. Then, by constraining input energy, we define an optimal control problem to obtain the minimally novel input-the one that has the smallest relative angle-that effects a given state transfer. We provide analytical conditions for existence and uniqueness for the solution in both continuous and discrete-time. A closed-form expression for the minimally novel input is derived and from this solution, a fundamental relationship between control energy and input orientation sensitivity is highlighted. We provide an example that demonstrates the utility of the developed sensitivity analysis. (C) 2018 Elsevier Ltd. All rights reserved.