Signal Detection and Approximate Adaptation Implies an Approximate Internal Model

The proper function of many biological systems requires that external perturbations be detected, allowing the system to adapt to these environmental changes. It is now well established that this dual detection and adaptation requires that the system have an internal model in the feedback loop. In this paper we relax the requirement that the response of the system adapt perfectly, but instead allow regulation to within a neighborhood of zero. We show that linear systems with the ability to detect input signals and approximately adapt require an approximate model of the input. We illustrate our results by analyzing two well-studied biological systems