Input Novelty as a Control Metric for Time Varying Linear Systems

TL;DR

This paper proposes a novel input control metric based on input novelty for time-varying linear systems, with applications to neural networks, providing analytical solutions and insights into controllability.

Contribution

It introduces a new input novelty metric for controllability analysis, including analytical solutions and a framework for complex linear systems and neural networks.

Findings

Analytical conditions for existence and uniqueness of minimally novel inputs.

Closed-form solution for continuous-time systems.

Convex optimization approach for discrete-time systems.

Abstract

This paper introduces a framework for quantitative characterization of the controllability of time-varying linear systems (or networks) in terms of input novelty. The motivation for such an approach comes from the study of biophysical sensory networks in the brain, wherein responsiveness to both energy and salience (or novelty) are 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 the past input history. Then, by constraining input energy, we define a non-convex optimal control problem to obtain the minimally novel input that effects a given state transfer. We provide analytical conditions for existence and uniqueness in continuous-time, as well as an explicit closed-form expression for the solution. In discrete time, we show that a relaxed convex optimization formulation provides the global optimal solution of the original non-convex problem. Finally, we show how the minimum novelty control can be used as a metric to study control properties of large scale recurrent neuronal networks and other complex linear systems. In particular, we highlight unique aspects of a system's controllability that are captured through the novelty-based metric. The result suggests that a multifaceted approach, combining energy-based analysis with other specific metrics, may be useful for obtaining more complete controllability characterizations.